S I M O N G R A N T

A S T R O N G ( R O S S ) C H A R A C T E R I Z A T I O N OF

M U L T I V A R I A T E R I S K A V E R S I O N

ABSTRACT. Using the 'addition of uncorrelated noise' as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with the same ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers with distinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981) global condition for strongly more risk averse.

Keywords: multivariate risk, strongly more risk averse.

I N T R O D U C T I O N

Ross (1981) provided a characterization of the notion of comparative risk aversion by permitting agents to face all possible removals of risks from distributions (partial or, but not necessarily, complete), and comparing the unconditional premiums that these agents would be willing to pay for the removal of such risks. Machina and Neilson (1987) pointed out that such premiums may also be taken to be (nonnegative) random variables. This was a valuable insight since in many economic situations not only are particular risks insurable while others are not, the effective (or real) premium required for the removal of these insurable risks may also be random. For example a premium specified in nominal terms when there is uncertainty about prices constitutes a random (albeit nonnegative) payment in terms of the foregone consumption opportunities that it represents.

Thus risk attitudes may be captured in the following broad sense: let F be a distribution, and G the resulting distribution after 'some risk is removed' from F. Let H be a third distribution which is dominated by G in the sense of first order stochastic dominance. (The 'difference' between G and H captures the random premium to be 'paid' for the 'removal' of risk.) The comparison of risk aversion can then be based upon the preferences between pairs of distributions such as H and F. That is, if one agent prefers It to F, thereby revealing a willingness to pay the premium for the removal of risk, then a more risk averse agent should also prefer H to F.

This paper seeks to generalize the characterization achieved by the above

Theory and Decision 38: 131-152, 1995. © 1995 Kluwer Academic Publishers. Printed in the Netherlands.

132 SIMON GRANT

comparison of attitudes towards risk in the case where the outcome space is multidimensional or, to be precise, a finite-dimensional vector space. The motivation for this task stems from the observation that many economic problems are specified in an explicitly multivariate framework. For example, the consumer in neoclassical economics is identified by a preference relation defined over the space of consumption bundles. Even where the consumption commodities that are purchased in the same period are aggregated into a single good, the individual's labor/leisure choice or consumption/saving choice still renders the analysis multivariate in nature. 1

There are two parts to this task: first, one must capture the notion of 'risk removal', or equivalently, characterize when F is riskier than G, and secondly, one must construct an appropriate notion of first-order stochastic dominance for this multidimensional case.

Approaches for both these parts exist already in the literature. For a multi- variate definition of 'increasing risk', I adopt Fishburn and Vickson's (1978) notion of the 'addition of uncorrelated noise' as the natural multivariate extension to Rothschild and Stiglitz's (1970) definition of increasing risk for univariate distributions. The second part addresses the issue of the class of (possibly stochastic) premiums to be considered. In the univariate case, the premiums are clearly payments of 'money'. In the multidimensional case, however, there are at least two options: (1) consider all premiums (vector payments) that lower an agent's utility; and (2) consider only those premiums that are nonnegative vectors. If we view outcomes as consumption bundles of 'goods' and thus assume that utility functions are increasing with respect to the vector ordering, then the premiums given by (1) are a superset of the premiums given by (2). The partial ordering that compares the risk aversion of agents, derived from (1), will consequently be 'more partial' than the ordering from (2).

After some preliminaries and explanation of notation used in the paper in Section 1, Section 2 considers premiums given by cases (1) and (2) and obtains for each a direct extension of Ross's characterization of 'strongly more risk averse'. Case (1) yields the additional necessary condition of Kihlstrom and Mirman (1974), that to compare the attitudes to risk between two individuals, it must be the case that their ordinal preferences over the outcome space are the same. If the somewhat more restricted class (2) of premiums is considered, then there is scope for some difference in ordinal 'riskless' preferences. Section 3 demonstrates that such differences may not be arbitrary and thus the ordinal properties of the commodity preferences may still confound attempts to compare risk attitudes. In addition to this negative result for global comparisons, however, Section 3 also presents conditions sufficient for local characterizations of strongly more risk averse comparisons and demonstrates how comparisons may be possible between pairs of time-separable expected utility maximizers. All proofs appear in the Appendix, except where noted.

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 133

1. NOTATION

Let X be the set of outcomes. I assume X = I-[i=1 Xi, where for each i, X i is a closed interval of the real line. Such an outcome space, which I shall refer to as a (multidimensional) commodity space, is special in that it is endowed v?ith a natural although only partial ordering (except for the case where l = 1). Outcomes will thus be viewed as commodity bundles.

The choice set is ~q(X), the set of all multivariate cumulative distribution functions (CDFs), F over X. That is, F is a nondecreasing function (with respect to the natural partial order) from X to [0, 1] where F(x) is interpreted as the probability that the lottery represented by F assigns to all commodity bundles less than or equal to x. The degenerate distribution function that assigns a probability mass of one to a single commodity bundle x E X is denoted by S x.

I assume that an agent's preferences over distributions, denoted ~ , satisfy the hypotheses of expected utility theory and are strongly monotonic. That is, there exists a strictly increasing yon Neumann-Morgenstern (vN-M) utility index, u :X--~R, such that for all F, G ~ ( X ) , F ~ G iff ~xcxU(X) dF(x)>1 ~ e x u(x) dG(x) .2

2. MULTIVARIATE RISK AND RISK AVERSION

In order to be able to label preferences or behavior as risk averse, we first need to define what it means for one lottery to be 'more risky' than another and, secondly, what is to be traded in return for removing or reducing risk.

The standard definition of 'more risky' used extensively in the literature for univariate lotteries (i.e. money lotteries) is that of Rothschild and Stiglitz (1970). This has been generalized by Fishburn and Vickson (1978) for multivariate lotteries by defining increasing risk as corresponding to the notion that one distribution can be viewed as being obtained from the other by the addition of uncorrelated noise.

DEFINITION (Multivariate More Risky). For all F, G E 5f(X)F is multivariate more risky (MVMR) than G if and only if there exist random variables X, Y and Z, where

(i) G (resp. F) is the CDF of X (resp. Y), and (ii) Y = X + Z with E[Z]X = x] = 0 for all x E range(X) (where the first equality means equal in distribution).

The attractive feature of this generalization is that, in an analogous fashion to the Rothschild and Stiglitz definition, the class of expected utility maximizers averse to such multivariate increases in risk are exactly those with concave von Neumann-Morgenstern utility indices for commodity bundles.

134 SIMON GRANT

PROPOSITION 2.1 (Fishburn and Vickson (1978) Theorem 2.10 p. 95). The following statements are equivalent for all F, G @ 5f(X):

(A) F MVMR t3

(B) f u(x) dF(x)<~ f u(x) dG(x) for all concave u(. ) x X

2.1. A Kihlstrom-Mirman [KM] Style Multivariate Characterization

One can define a notion of first order stochastic dominance using the complete ordering over the multidimensional commodity space induced by an agent's preferences for degenerate lotteries. In this context one distribution would be considered to first order stochastically dominate another if, roughly speaking, it places more probability weight on commodity bundles in higher upper contour sets. Let u:X--->~ be a utility function over commodity bundles, and for all F E 5g(X) let F(m; u) --- S{~: u(~)~.,} dF(x).

DEFINITION. F first order stochastically dominates (with respect to preferences u) (FSD,)G iff F(m; u) ~ G(m; u) for all real m and strict inequality holding for some m'.

PROPOSITION 2.2. The following three statements are equivalent for all F, G E ~(X):

(A)

(B)

F FSD. G

X x

for all v which are monotonic increasing transformations of u.

(c) There ex&t random variables X, Y where:

(i) F (resp. G) is the CDF of X (resp. Y) , and

(ii) Pr[u(X) - u(Y) >i 0] = 1 and with

Pr[u(X) - u(Y) > 0 I X = x] = 1 )¢br some x E range(X).

We are now in a position to present a first possible generalization of the Ross-Machina-Nielson notion of strongly more risk averse, for multivariate distributions that is analogous to KM's extension of the Arrow-Prat t more risk averse comparison.

DEFINITION (KM). Agent i is strongly more risk averse (in the KM sense)

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 135

than agent j, if and only if, for all F, G, H E Z f ( X ) s.t. F MVMR G and G F S D . j H ,

H ~ J F ~ H ~ iF

We can view the transformation of F into H as first removing some risk to create G, and then taking away a first order risk premium. From Propositions 2.1 and 2.2 we can find three random variables X, Y and Z, such that E [ Z I X = x] -- 0 for all x E range(X), Pr[u(X) - u ( Y ) >! 0] = 1 and that F, G and II are the CDFs of X + Z, X and Y, respectively. Viewing X as random base wealth, Z as an uncorrelated additional risk, and X - Y as a random premium, the definition says that whenever j is willing to pay the premium X - Y in order to avoid the

multivariate noise Z, then so is i. I contend that this definition is in the spirit of KM's approach since the payment

of the premium's X - Y represents a first order stochastic worsening with respect to j 's preferences for commodities. In their paper, they allowed the constant premium, that was paid in return for the complete removal of risk, to be in any direction that represented a fall in utility for j.

With this definition we can obtain the following multivariate characterization of strongly more risk averse.

P R O P O S I T I O N 2.3. Let i and j be two expected utility maximizers with v N - M utility indices u i and u j, respectively, i is strongly more risk averse (in the K M

sense) than j, i f and only if:

(a) u i and u j represent the same ordinal preferences for commodit ies , and (b) u i ( x ) = Au~(x) + v(x) fo r some A > 0 and v( . ) that is nonincreasing and

concave in x.

Proof . The proof of the necessity of (a) comes directly from KM's demonstra- tion of how different ordinal preferences for commodity bundles may confound comparisons of agents' attitudes towards risk. Consider two risk averse expected utility maximizers, i and j, whose indifference curves intersect at outcome x. Select two bundles y, z such that ~yl + ½z = x. By the continuity of i's and j 's preferences for lotteries there exists an a E (0, 1) and outcome x' such that •x >/(D( 1 - a)t~x + a ( ½6y -F l ¢~z) > i(J)6 x, and the indifference curves in outcome space for i and j cross at x'. We can then select outcomes w i and w; such that ui(x ') > uJ(w j) while ui(w j) > ui(x '); and, uJ(w i) > uJ(x ') while ui(x ') > ui(wi) .

Set F = (1 - a)6 x + a(½~y + ½6z), G = 6 x, H j = 6wJ and H i = 6w i . By construction, F M V M R G and G FSD, jH j. Hence, as H i > ~F and F > itlJ, i is not strongly more risk averse (in the KM) sense than j. Similarly, as G FSDuJtt i, H i > iF and F > JH i, j is not strongly more risk averse than i.

136 SIMON GRANT

For the remainder of the proof of the proposition see the Appendix.

2.2. An Ambarish-Kaltberg Style Multivariate Characterization

One curious property on the KM definition is that, for l > 1, no strictly risk averse agent is more risk averse than any risk neutral or risk loving agent. 3'4 Part of the motivation for Ambarish and Kallberg [AK] (1987) in their analysis of multi- variate risk aversion was to allow such a ranking. Their approach was to restrict the premiums that could be exchanged in return for reducing or removing risk to be in directions that agreed with the partial ordering of the outcome space. Their paper focused exclusively on small risks and deterministic premiums but we can extend it to cover large risks and random (but nonnegative) premiums by first defining the following multivariate version of first order stochastic dominance that depends only on the natural although partial ordering of the commodity space.

DEFINITION (Comprehensive Sets). A subset S of X is comprehensive if for all x, y E X, x E S and y <~ x implies that y E S .

DEFINITION (Multivariate First Order Stochastic Dominance). For all F, G E ~(X) , F multivariately first order stochastically dominates (MVFSD) G if and only if ~x~s dF(x) ~< Sxcs dG(x) for every open comprehensive S c X and F # G.

The definition accords with the intuition that first order stochastic dominance in a multivariate context corresponds to the notion that the dominant distribution can be obtained from the dominated one by shifting probability weight onto bigger (with respect to the natural partial ordering) commodity bundles. Hence, as Levhari et al. (1975) showed, any expected utility maximizer with a strictly increasing vN-M utility index for commodity bundles will agree with the partial ordering of MVFSD. Moreover, as Fishburn and Vickson (1978) proved, the dominated distribution can be viewed as obtained from the other by adding a random variable that only takes on negative values.

PROPOSITION 2.4. The following statements are equivalent for all F, G E 5f(X):

(A) V MVFSD G

(B) f f u(x)dG(x) X X

for all u(. ) strictly increasing (non-decreasing) in x.

(C) There exist random variables X, Y where:

(i) F (resp. G) is the CDF of X (resp. Y ) , and

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 137

( i i ) P r [ X - Y t> 0] = 1 and with Pr[X - Y > 0 I X = x ] = 1

for some x E range(X).

We can now amend the definition of Subsection 2.1 to only allow premia that agree with the natural and partial ordering of the commodity space.

D E F I N I T I O N (AK). Agent i is more strongly risk averse (in the AK sense) than agent j, iff for all F, G, H ~ 5f(X) s.t. F MVMR G and G MVFSD It

H ~ J F ~ H 2 ~F

With this definition, the next proposition shows the relation that must hold between two individuals' functional representations of their preferences over lotteries in order for one to be deemed more risk averse than the other is an immediate generalization of Ross's characterization.

Of particular note is that (as AK showed for the case of deterministic premiums and small risks) we do not necessarily require the individuals to have the same ordinal preferences. Moreover (unlike AK), the result requires neither the concavity nor the differentiability of the agents' utility functions. And necessity of the Ross characterization is demonstrated for all pairs of utility functions, not simply additively separable ones as is demonstrated in AK.

P R O P O S I T I O N 2.5. Let i and j be two expected utility maximizers with v N - M utility indices u i and u j, respectively, i is strongly more risk averse (in the A K sense) than j, if and only if:

(*) u'(x) = AuJ(x) + v(x)

for some A > 0 and v(. ) that is nonincreasing and concave in x.

To see the intuition for the sufficiency of (*) note that from Propositions 2.1 and 2.4 we can find random variables X, Y, and Z, such that E[Z I X = x] = 0 for all x E range(X), Pr[X - Y i> 0] = 1 and that F, G, and I-I are the CDFs of X + Z, X and Y, respectively. Again viewing X as base wealth, Z as an uncorrelated but insurable risk added to base wealth, and X - Y as a random first order risk premium, H 2 iF corresponds to j being willing to pay the premium X - Y in order to avoid the risk Z. That is,

eu (6) - e U J ( r i )

EUJ( )-EUJ(r) ~<1.

If i's and j 's utility indices for commodities satisfy (*) then;

138 SIMON GRANT

(i) E U i ( G ) - EUi(F)>t)t[EUJ(G)- EUJ(F)] as v(-) is concave and F MVMR G, and

(ii) E U i ( G ) - E U ~ ( H ) ~ A [ E U J ( G ) - EUJ(H)] as v(. ) is nonincreasing and G MVFSD H.

Thus

EU~(G) - EUi(H)

EU~(G) - EU~(F)

EUJ(C) - :UJ(H) EUJ(G) - EUJ(F)

That is, the ratio of the loss of utility for paying the premium over the loss from experiencing the risk is less for i, and so i will pay such a premium to avoid a risk whenever j will.

As an immediate corollary of Proposition 2.5 we have that a risk averse agent is indeed strongly more risk averse (in the AK) sense than a risk neutral or risk loving agent. If u i is concave, u j is convex (or linear) and both functions are continuously differentiable on X then v(x) = ui(x) - ,~uJ(x) is concave and nonin- creasing for sufficiently large h > 0. 5

Proposition 2.5 extension of the Ross characterization of the 'strongly more risk averse than' relation to multivariate risks (that is, condition (*)) bears a strong resemblance to Demers and Demers's (1991, p. 15) Theorem 3, condition (b). Much of their discussion is conducted in terms of price and income risks, where the uis are indirect utility functions and the premia are (possibly random) nonnegative amounts of money. They also point out that indirect utility functions that satisfy their conditions need not represent the same ordinal preferences. One could recouch the analysis in this paper in terms of indirect utility functions where the first commodity is (money) income. In order for the utility functions to be strictly monotonic, the other commodities could be identified as either negative prices or inverse prices (with appropriate redefinition of the domain of the utility function). Partial insurance would thus be available for some joint income and price (or inverse price) risk. While a (possibly random) vector premium would correspond to a payment of nonnegative quantities of income and/or higher average (either arithmetic or geometric) prices. Thus Proposition 2.5 generalizes Demers and Demers' result to premiums that can have price as well as income components, which might be relevant for price stabilization policy applications.

In addition to condition (*) Demers and Demers' Theorem 3, condition (b) also requires a fairly complicated expression formed from second-order deriva- tives to hold for all pairs of vector outcomes. This condition is redundant if the two utility functions being compared are both concave. But as they note, indirect utility functions are not concave in both income and prices. Proposition 2.5 does not need this condition to hold (even for noncave utility functions) because of a subtle difference between their definition of 'more risk averse' and the one

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 139

employed in this paper. In my definition, for agent i to be more risk averse than agent j, i must be willing to pay a non negative (possibly) random premium to avoid a partial (insurable) risk whenever j is. Demers and Demers (following Machina and Nielsen, 1987) in addition require that, whenever j is willing to receive a nonnegative (possibly) random premium to forego a partial risk, then so is i. That is, not only do Demers and Demers require i to be more risk averse than j whenever j wishes to avoid a risk, but they also require i to be less risk loving than j whenever j wishes to face a risk. The two relations, although closely related, are logically distinct. If comparisons of risk aversion are used in insurance contexts to compare willingness to pay insurance premia, then it seems reasonable to restrict attention to the class of risks that the less risk averse person wishes to avoid and not require any relationship between the two agents with regard to their willingness to pay for desirable gambles. Interestingly, the weaker notion used in this paper has in Proposition 2.5 a considerably more elegant characteriza- tion that eschews the need even for the utility functions to be differentiable.

3. HOW MAY PREFERENCES DIFFER ORDINALLY?

A naive interpretation of Proposition 2.5 is that it is straightforward to find for a given v N - M utility index u i, expected utility maximizers who are strongly more risk averse in this AK sense, for example u i = u j - a.x. where a is a nonnegative vector, or u i = u j - x r W W r x where W is a square matrix with nonnegative elements.

A more interesting issue is what pairs of ordinal preferences over commodities could conceivably be ranked in terms of this relation of strongly more risk averse. That is, given two preference relations over the commodity space, does there exist two utility representations of those commodity preferences, u i and u j, such that u i

is strongly more risk averse than u J? To narrow our focus let us consider for the moment only those ordinal preferences with a utility representation that is twice continuously differentiable and concave. For these types of preferences a corollary of Proposition 2.5 is a Ross style condition for strong risk aversion in the 'small'.

For a twice continuously differentiable vN-M utility index, u, let D~u(x) (resp. D~xu(x ) correspond to the gradient (resp. Hessian) of u evaluated at commodity bundle x.

C O R O L L A R Y 3.1. Let u i and u j be two twice cont inuously differentiabIe and

concave v N - M utility indices, i is strongly more risk averse (in the A K sense) than

j , i f and only i f

- z r D ~ u i ( x ) z -ZrDxxUJ(X)Z (**) i D~u ( y ) . h DxuJ(y ) .h

140 SIMON G R A N T

for all z, for all h >1 O, h ¢: 0 and for all x, y E X.

We see in particular that a necessary condition for i to be more strongly risk averse than j is for (**) to hold for y = x .

- z T DxxUi(X)Z --z r D~xuJ(x)z

i DxUJ(x).h , (3.1) DxU(X).h /> Vz , Vh~>0 h~a0 .

This necessary condition has more chance of holding if we take an increasing concave transformation of u i, say u* = go u i where g ' > 0 and g" < 0. Since,

-zrDxxu*(x)z --zTDx~ui(x)z g"(ui(x))(D~ui(x).z) 2

(3.2) D~u*(x).h - D, ui(x).h g'(u~(x))(D, ui(x).h)

Note that a concave transformation of u ~ does not help for directions orthogonal to the gradient of preferences at x, i.e. z such that Dxui(x).z = O. Hence, if, for two v N - M strictly increasing concave utility functions, u i and u j, (3.1) does not hold for some x' = y ' , z ' and h ' >/0 then (3.1) will not hold with these parameters for any other utility representations of the pair of ordinal preferences described by u i and u j. In essence, the difference in the ordinal preferences are confounding any comparison of relative aversion to risk.

A stark example of how differences in ordinal preferences may prevent risk aversion comparisons is provided by a pair of two-good Cobb-Douglas utility functions. In certain circumstances, however, some of the differences in the ordinal preferences may aid strong risk aversion comparisons. To see this consider now two two-good constant elasticity of substitution (CES) utility functions,

u i ( x l , x2 )= l l~(ax~l -n)+(1-a)x~21-") ) and

To see if the necessary condition (3.1) can hold for x = (~, x') take z ' = ( [ 1 - i ~ a]~", - a ~ ) , as it is orthogonal to Dxu (x, 2).

For h' = ([1 - h], h),

_ _ z t T i D x, u ( x, x-)z' n(1 - a)a.~ (2"-~) = and

Dxui(£, x").h' (1 - h)a + h(1 - a)

__Zr T J ~ Dx, U (x, x')z' 7[(1 - a)2b + a2(1 - b)12 (2"-1)

D x#( L x-).h' (1 - h ) a + h(X - b)

Hence, if b ~< a,

R O S S C H A R A C T E R I Z A T I O N O F M U L T I V A R I A T E R I S K A V E R S I O N 141

rT i ~ ] T j N f-z min " i--------'~ - - - 7 ' "- (h) L D,u (x,x').h / DxU~(Y,x').h'

(1 - a)b "/ [1 - a)2b + a2(1 - b)l '

Thus for (3.1) to hold at (E, £) for a direction z that is orthogonal to D=ui(E, ~ , we require:

(3.3) _~/> (1 - a ) [ ( 1 - a)b] + a [ a ( 1 - b ) l

y ( 1 - a)b

The RHS of (3.3) is at least 1 since a ( 1 - b)i> ( 1 - a)b. So, in particular, (3.3) cannot hold if , /=2 / and b < a . A pair of Cobb-Douglas utility functions corresponds to this case, since it can be thought of as the limit obtained when one allows 3' to go to zero. On the other hand if 7/is sufficiently greater than 3" then (3.3) obtains. Thus there exists an agent i ' whose v N - M utility function is a sufficiently concave transformation of d ( - ), such that the necessary condition for i ' to be strongly more risk averse than j is satisfied at (E, E). Actually, if (3.1) can be shown to hold with strict inequality for some commodity bundle x, then that is both necessary and sufficient for i' to be strongly more risk averse than ] for lotteries whose support lies on an open ball around x.

P R O P O S I T I O N 3.2. Let u i and u j be twice continuously differentiabIe and concave and let S "-1 = {a E N" I ]]all = 1}, that is the surface of the n-dimensional unit sphere. I f for some x E X

T i - z D~xu (x)z -zrDxxu](x)z

i D~uJ(x).h , (3.2 ') Dxu(x)'h > VZ, h E S n-1 h>~O

then there exists 6 > 0 such that ui(y)= A u J ( y ) + v ( y ) where ) t > 0 and v(. ) is nonincreasing and concave for all y such that I l y - x]l < a.

We can interpret -zrDxxU(X)Z/DxU(X).h as the analog Ar row-Pra t t measure of local risk aversion for local risks in the direction z and local premiums in the direction h. Thus from Propositions 3.2 and 2.5 we can conclude that if for all possible directions of local risks and all positive directions for premiums around a commodity bundle x, i is locally strictly more risk averse (in an Ar row-Pra t t sense) than j, then i is locally strongly more risk averse (in an AK sense) than j.

For the CES example, if (3.3) holds with strict inequality, then there exists a concave transformation of u i such that (3.1) also holds with strict inequality for x = (E, ~ and thus is locally around (E, ~ strongly more risk averse than u j.

142 S I M O N G R A N T

Actually we can go further and show by simple algebraic manipulations that given b ~< a (resp b > a) if

y > b((i-----~(1 - b)a (resp. m-(i C b-J ) ( 1 - a)b

(3.1') holds for 0 7, ~ and thus i itself is locally around (2, ~ strongly more risk averse than j.7

This result is particularly interesting if we view the commodity bundle x as a consumption stream of a single (aggregate) consumption good through time. It is well known that if we require preferences for consumption streams to be both homothetic and additively separable across time then the vN-M utility function must take the CES form

1 (t=~ 1 . ( 1 - ~ ) ~ 8 U ( X ) = ~ _ _ ~ _ at.~ t ] .

With this interpretation for our two good example, (1-a) /a [ resp . ( 1 - b ) / b ] corresponds to i's (resp. j's) discount factor and ~/(resp. Y) both the within period coefficient of relative risk aversion and the reciprocal of the elasticity of substitution. Roughly speaking, if i's within period coefficient of relative risk aversion is sufficiently greater than j's (i.e.

~_ { ( 1 - a ) / a ( 1 - b ) / b ) - ~ m a x ~ ( 1 b)/b' (1 a - ~ !

then around any constant consumption stream (Y, ~ , i is locally strongly more risk averse than ].

Notice that if we employ a KM style definition of 'more risk averse than' and so require that the two agents being compared have the same ordering over degenerate outcomes then for comparative risk aversion in expected utility consumption we require one utility function to be a concave transformation of the other. As Epstein and Zin (1989) have pointed out, this may result in an implausible dependence of present risk attitudes on the past. Hence Chew and Epstein (1990) and Epstein (1988) consider nonexpected utility preferences that relax the inversely proportional relationship between an agent's elasticity of intertemporal substitution and the within period degree of risk aversion enabling lottery preferences to differ while inducing the same ordering over degenerate outcomes.

The AK definition of the 'more risk averse than' relation, on the other hand allows for comparisons of risk aversion between additively separable and homothetic expected utility maximizers. And the interpretation of what it takes for one agent to be locally (strongly) more risk averse than another is quite

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 143

intuitive. For one good the discount factor is trivially one and hence for constant relative risk aversion preferences, whether one individual is locally strongly more risk averse than the other depends on whether its coefficient of relative risk aversion is larger. With two goods that is no longer sufficient if preferences differ because of different discount factors. However, a sufficiently greater coefficient of within period relative risk aversion can swamp the effect of the different discount factors once again allowing a comparison of risk aversion.

In many respects this is an appealing result. Expected utility is still the dominant paradigm in the economics of uncertainty, and homothetic, additively separable utility functions are standard in intertemporal representative consumer models. It is, however, important to acknowledge that whether it is possible to find a pair of vN-M utility functions which can be compared in terms of risk aversion and that represent a given pair of ordinal preferences over commodities may critically depend on the relative shape of those ordinal preferences. Thus the ordinal properties of the commodity preferences may still confound comparisons of aversion towards risk.

APPENDIX

Proof of Proposition 2.2. By construction:

(A) if and only if (A') F(u) FSD G(u); (B) if and only if (B') fm m' dV(m'; u) >fm m' dG(m'; u) ; (C) if and only if there exists X, distributed F(u) and Y, distributed G(u) such

that Pr[X, - Y, t> 0] = 1.

But (A') ¢:~ (B') (respectively, (A') ¢:> (C')) is a restatement of Fishburn and Vickson's (1978) Theorem 2.1 p. 65 (resp, Theorem 2.5 p. 89).

Proof of Proposition 2.3. In the text I showed the necessity of (a) (i.e. that u ~ and u i represent the same ordinal preferences), it remains to show that given (a) condition (b) is necessary and sufficient.

LEMMA A.1. I f u is strictly monotonic in x and G FSD u H then there exists I-I' such that H'(u)= It(u) and G MVFSD H'.

Proof. G FSD, H implies G(u) FSD H(u). Thus by Fishburn and Vickson's (1978) Theorem 2.5 (p. 89) there exist random variables Xu, Yu from the state space [0, 1] (with Lebesque measure) into [_u, t/] (where u =min u(x) and l /=

x ~ X

max u(x)) such that X, (resp. Y~) is distributed G(u) (resp. It(u)) and for all x C X

w E [0, 1]X~(o~) i> Yu(w) with strict inequality for some w E [0, 1]. From probabili- ty theory we can construct a measure space (fF, ~ , P) (where J2' = [0, 1] x 5 ~,

144 SIMON GRANT

is the Borel o--algebra for ~2' and P is a probability measure over ~2') and a random variable X from this measure space into X such that for all o) E [0, 1], (i) u(X(w, s)) = Xu(~o ) for all s E 5g, and (ii) X(o)) is distributed G I ~xEx lu(x)=x~(o~)(~)~. It follows from this construction that X is distributed G. Now implicitly define a function a :1~--->[0, 1] by u(a(w,s)X(o,s))= Y,(~o). Such a function is welt defined since u is strictly monotonic in x. Finally, define the random variable Y:F~---~X by Y(o),s)=a(o),s)X(o~,s). Let H' be the distribution of Y. By the construction of Y, G MVFSD H' and H'(u) = H(u). •

With Lemma A.1, condition (b) is now equivalent to (*) of Proposition 2.5 so proof proceeds analogously.

Proof of Proposition 2.4. The equivalence of (B) and (C) is a generalization of Fishburn and Vickson's (1978) Theorem 2.5. ( A ) O ( B ) is a restatement of Levhari 's et aI.'s (1975) Theorem 1.

Proof of Proposition 2.5. Sufficiency. Assume (*) and that H ~ iF. EUi(H) -

EU'(F)

= ~ u'(x) d [ n - F ] (x ) x

= A[EUJ(H) - EUJ(F)] + [ v(x) d[H - V](x) X

= A[EUJ(H) - EUJ(F)] + f o(x) diG - F](x) X

+ f [-v(x)] d[C - HI(x) -~ 0 x

(adding and subtracting fx v(x)dG(x)).

The first term is nonnegative since H ~ ~F. Since F MVMR G and v is concave in x, it follows from Proposition 2.1 that the second term is nonnegative. Finally, as G MVFSD H and - v is nondecreasing in x, by Proposition 2.4 we can conclude that the third term is also nonnegative. Thus H ;~ iF.

Necessity. The method of proof is to assume that (*) does not hold and to show that this implies that we can construct a triple of lotteries for which j is willing to pay a first order stochastic premium in order to avoid a risk but i is not. As I have not assumed that the local utility functions are differentiable in x, it is first useful to provide the following topological definitions of concavity and weak monotonici- ty.

R O S S C H A R A C T E R I Z A T I O N OF M U L T I V A R I A T E R I S K A V E R S I O N 145

D E F I N I T I O N (Local concavity), u is locally concave for outcome x, if there exists an open neighborhood of x, nbd(x), such that for all x0, x 1 E nbd(x0) and

for all a @ (0, 1), u(ax o + (1 - a)x l ) >t otU(Xo) + (1 - a)u(x~).

Clearly a concave function is locally concave for all x, and if a function is not concave there exists an outcome x ' for which it is not locally concave.

D E F I N I T I O N (Local nonincreasing), u is locally nonincreasing at outcome x, if there exists an open neighborhood of x, nbd(x), such that for all x+ , x_ E nbd(x), with x+ >~ x_ , x+ # x_ , u(x+) <-u(x_).

Again it is clear that if a function is nonincreasing in x, then it is locally nonincreasing for all x, and that if it is not nonincreasing then there exists an outcome x' for which it is not locally nonincreasing.

Denote C" C X as the set of outcomes for which u is locally concave, and N" C_ X as the set of outcomes for which u is locally nonincreasing.

There are two cases to consider. The first is where both agents are risk averse everywhere, i.e. C "~= C "J= X for all H E ~ ( X ) and the second is where C"~C_ C "~ C_ X for some H ~ 5f(X). I show below that C "i C_ C "~ immediately implies that i is not more strongly risk averse than j.

Case I: u ~ and u j are concave in x. If (*) does not hold then for all h > 0v(. ,h) --- u~( • ) - huJ( • ) is either not locally nonincreasing for some x and /o r

not locally concave for some x. That is, for all h > 0 , X'JVv(~)#0 and /o r X~C v(~) # 0 . Define the following correspondences, II, ~ : R + - ~ 2 x, by II(A)--- X'kN o(A) and qb(A) - X~C o(a). That is, H(A) is the set of outcomes for which v(A) is not locally nonincreasing, and q~(A) the set of outcomes for which v(A) is not locally concave. Note II(0) = X, (-'/~ II(A) = 0 and VA 2 > A~H(A2) C_ H(A~) and

• (0) = ~1, U x qb(A) = X and VA 2 > AltO(A2) ~ ~(A1). Let A= = inf{A : FI(A) = 0} and A. = sup{A:q~(A)= 0}. Such A= and A. exist with suitable bounded con- ditions on u i and u j.

L E M M A A.2. A,~ E inf{A : H(A) = 0) and A® ~ sup{A :@(A) = 0}. Proof. Consider sequence A, converging to A,~ from above. By construction,

for each A,, II(A,) = 0. Suppose A ,~ in f{A : II(A) = 0}, then there is an open set on which v(. ,A,~) is strictly increasing, but if n is large enough v(. ,A,) - v(. ,A,~) is arbitrarily small and thus v( - ,A, ) is strictly increasing on that open set. A contradiction.

Similarly for h . ~ sup{h : ~ ( h ) = 0}. •

Subcase 1.1 A,~ <~ A~,. For 2, such that h,~ <~ 2~< A. , II()~) = 0 and qb(/~) = 0 and thus v(. ,A) is nonincreasing and concave in x and so (*). A contradiction.

146 S I M O N G R A N T

Subcase 1.2 A= > A¢. For 2, such that A= t> , ~ A. , take y ~ II(2) and y+ , y_ E nbd(y) s.t. y+ ~>y_, y+ # y _ and

(A-l) v(y+, 2)>v(y_, 2)

and take z E dp(2), and z0, zz E nbd(z) s.t. z~ = ~z 0 + (1 - &)za, and

(A-Z) v(z~, 2) < dV(Zo, 2) + (1 - d ) v ( z l , 2)

Let F =/356z0 +/3(1 - ~)6z~ + (1 -/3)6y+

G =/3~z~ + (1 -/3)~y+

n =/3~z~ + (1 - / 3 ) ( 1 - ~ ) ¢ + (1 - / 3 ) ~ y _ .

As F MVMR G and G MVFSD H we know G > ]F and G > ill. So assume/3 and Y set so that F :~ H and F - J H . Thus E U i ( H ) - EUi(F)

= 2[EuJ(I-I) - EUJ(~)] + [ v(x, 2) d i G - r](x) X

+ [ [-v(x, 2)] d6; - n ) ( x ) X

=/3[vfza, 2) - aOfZo, 2) - (1 - 8)ofz~, 2)]

- (1 - / 3 ) y [ v ( y + , 2) - v (y , 2)1

The first term is negative by inequality (A-2) and the second term is negative by inequality (A-l) , hence F > ill, so i is not more strongly multivariate risk averse than j.

Case II: ui( . ) and u]( - ) are not necessarily concave in x. The problem that may arise with the proof above when u i and u j are not concave is that constructing an F which is MVMR than G does not ensure that G > IF, so I cannot assume that /3 and y can be set so that F # H and F > ]H. Hence the construction of F wilt have to involve 'adding noise' in a region where u j is risk averse (i.e. locally concave). I first show that wherever j is locally concave then so must i be.

L E M M A A.3. I f C "j is not a subset o f C "~ then i is not strongly more risk averse (in the Ambarish-Kallberg sense) than j.

Proof. If C "~ is not a subset of C"', then we can choose an outcome x j C C "j N X~C ~' and an associated nbd(x j) over which u j is concave. Thus there exists x 0 , x 1 E nbd(x ]) and ~ E (0, 1) such that:

ROSS C H A R A C T E R I Z A T I O N OF M U L T I V A R I A T E RISK A V E R S I O N 147

( A - 3 ) ui(flXo + (1 - ~ / ) x l ) < ~ui(Xo) + (1 - ~)ui(xl), and

(A-4) uJ(dxo + (1 - d ) x l ) I> duJ(x0) + (1 - ~)uJ(xl).

Le t F = £~8xo + (1 - ~ ) 6 ~ , G = 6Sxo+(~-s)~ and t I = -/6~0+(~ ~)Xl q- (1 - 7 ) 6 y , w h e r e y_ <~ fix + (1 - c/)x 1 , y ¢ (/x 0 + (1 - {/)x I . Fo r y sufficiently close to 1 we h a v e I I ~ iF. H o w e v e r since F > ~G and t3 ~ i I I for any y E (0, 1] i is no t s t rongly

m o r e risk averse than j.

C O R O L L A R Y A.3.1 . C"J C "~ (and thus X'kC"' CX'kC ~j) is a necessary condition for i to be strongly more risk averse (in the Ambarish-Kallberg sense) than j.

W h e n C "j ~ X (and pe rhaps also C ui ~ X) we still have I I (0 ) = X and (-~ ~ H(A) =

0 but now ~ ( 0 ) = X~C"' and U ~ ~ ( h ) D C "j. H o w e v e r , for the co r r e spondence q~c(h)--- XkCV(A) D C uj, q~c(0) = 0 , for all A2>A 1 (I)(A2) D (I)(A1) and U A °Pc(A) =

C "j. Le t t ing A= = inf{h : I I ( h ) = 0} and A,c = sup{h : ~ c ( h ) = 0} again sui table b o u n d e d condi t ions on u s and u j ensure their existence and a similar a r g u m e n t to

the one p re sen t ed in the p r o o f of L e m m a A.2 shows tha t bo th these ' l imits ' are

con ta ined in the respect ive sets by which they are defined. Subcase H.1. h T > h®c. This subcase is ana logous to Subcase 1.2 above and an

equ iva len t cons t ruc t ion of a triple of dis tr ibut ions can be e m p l o y e d to show tha t i

is not s t rongly m o r e risk averse than j. Subcase H.2. A T ~< A~c. ( T h a t i s , for s o m e As v(A) is non- increas ing and locally

concave for all o u t c o m e s in C"~.) If for some h E [A~, A.~], @(h) = 0 then (*) is

satisfied for this A, so a s sume O ( A ) ~ 0 for all such As. T h e m e t h o d now is to p roceed much as we did in Subcase 1.2 above , but to

have an ex t ra m e a n preserv ing sp read for the m o r e r isky dis t r ibut ion H in a reg ion whe re j is risk averse . By manipu la t ing the relat ive weights on the two

m e a n prese rv ing spreads it will be possible to m a k e j indifferent b e t w e e n the m o r e risk), d is t r ibut ion F and the first o rde r s tochast ical ly d o m i n a t e d dis t r ibut ion H .

Cons ide r v ( . ,h +) --- u~( - ) - h+uJ( • ) for s o m e h + E [A~, A,~].

C h o o s e an x E C "i for which u j is no t l inear in x. As v( - ,h +) is non- increas ing

and u J( • ) is locally concave a round x, we can find x0, x~ E nbd(x) such that:

1 1 (A-5) O < v ( Y , A + ) - - ~ v ( x o , h + ) - - ~ v ( x l , A + ) = C , and

1 (A-6) 0 < u i ( x ~ - ~ u ( x 0 ) - uJ(x i )=J,

where 2 = i x 0 + i x 1 . N o w choose a z ~ q~(h+). As v( . ,A ~ ) and u J( • ) are not locally concave a round

z, we can find z0, z I E nbd(z) and c~ ~ (0, 1) such that:

148

(A-7)

( A - S )

(A-9)

SIMON GRANT

0 < - - [ v ( z a , a +) -- gev(z o , A +) -- (1 -- G ) v ( z l , A+)] = B ,

0 < - [ u J ( z a ) - 6,uJ(Zo) - (1 - a ) u J ( Z l ) ] ~-- E ,

where za = Gz o + (1 - ~ )z I and for which

B J = C E .

and

Finally, as v ( . ,k +) is nonincreasing in x while u J( • ) is strictly increasing in x we can choose y + , y_ E X , such that y+ ~>y_, y+ # y _ ,

(A-10) 0 < - [ v ( y + , A + ) - v ( y _ , A + ) l = A ,

(A-11) O < u J ( y + ) - u i ( y _ ) = D , and for which

(A-12) D C + A J = B D + A E .

[(A-5) plus A + t imes (A-6) and (A-7) plus A + t imes (A-8) gives

1 1 (A-11) 0 < ui(y) - -~ ui(xo) - -~ u i (xa) = C + A +J

(A-12) 0 < - [ u i ( z a ) - Gui(Zo) - (1 - G)u~(zl)] = B + A + E ,

tha t is, u i is locally concave at x but not locally concave at z.] Le t

1 1 F = e(1 - ¢)a, + ~¢ 28x0 + e~b -~ axl

+ 13(i - .)az~ + 13~Gazo + 13. ( i - ~)az2,

G = e6~ + fi6~. + ( 1 - / 8 - e)ay+, and

H = e6,. + tiara + (1 - fl - e)(1 - y)ay+ + (1 - 13 - e )yay_.

No te for all e, 13, y, ¢ , v F M V M R G and G M V F S D H. Hence it only remains to show that there exists (e, 13, y, ~b, v) E (0, 1) s with e + 13 < 1 such that:

(A-13) E U J ( H ) - E U J ( F ) = 0 , and

(A-14) EU~(I - I ) - e U * ( V ) < 0 .

Subtracting and adding EUi (G) to the LHS of (A-13) we obtain:

i /. 1 (A-15) - ( 1 - 13 - e ) y [ u J ( y + ) - u i (y_ ) ] + e ¢ [ u J ( 2 ) - "~ u (Xo) - 2 u j ( x l - ]

- 1 3 " [ u J ( z ~ ) - ~ " J ( Z o ) - (1 - ~ ) u J ( z , ) ] = 0

Substi tut ing f rom (A-6), (A-8) and ( A - t l ) and rearranging we get

ROSS CHARACTERIZATION OF MULTIVARIATE RISK AVERSION 149

~3[rE - yD] + yD (1 - f i ) yD + fivE (A-16) e = [¢J + yD] - [¢J + ~/DI

Substituting A+u]( • ) + v(" ,A +) for ui( • ) in the LHS of (A-t4) and subtracting and adding fx v(x', A +) dG(x') we get:

(A-17) A +[EUJ(H) - EUJ(F)I - (1 - / 3 - e)y[v(y+, ;t +) - v ( y _ , A+)]

1 1 + ~[v(~, ~+) --~V(Xo, ~+) --~v(xl, ~+)l

- / 3 v [ v ( z s , A +) - SV(Zo, A +) - (1 - ~)v(z 1 , A +)1 < 0.

Substituting from (A-5), (A-7), (A-10) and (A-13); and rearranging (A-17) simplifies to:

(A-18) e < f l[yA + vB] - TA [OC - yA] (provided [4~C - yA] > O)

Combining (A-16) and (A-18) and solving for 13 leads to:

y¢[DC + A J] (A-19) /3 >

y ¢ [ D C + A J] + ~u[BJ - CE] + wy[BD + AE] "

(A-20) /3 > ~b +----~

Combining (A-16) and (A-20);

(A-21) e +/3 - [~bJ + TD] > vyD + ~o2J + 4)yD + ~TD "

Hence if v/y < D / E and y/~b < C/A, (A-16), (A-18), (A-20) and (A-21) show that it is possible to find e > 0 and /3 > 0 such that e +/3 < 1 and E U i ( F ) = EUi (H) yet EU~(H)< EU~(F). That is, the triple of distributions F, G and H demonstrate that i is not strongly more risk averse than ].

Proof o f Corollary 3.1. Differentiating (*), ui(x)= Aui(x)+ v(x), twice we derive:

(A-22)

(A-23)

D~ui(y).h = AD~uJ(y).h + D~v(y).h

<~ AD~uJ(y).h (for all h t> 0, h ¢ 0 )

- zrDxxui (x )z = A[-zrDxxu](x)z] - zrDxxv(x)z

>~A[-zTDx~ui(x)z] (i>0) (for all z)

( * *) follows immediately.

Proof o f Proposition 3.2.

L E M M A A.4. For u i and u j twice continuously differentiable and concave

150 SIMON GRANT

T i - z DxxU (x)z -zrD~uJ(x)z (A-24) Dxu~(x).h > D~uJ(x).h Vz, h E S" ~, h ~ 0

if and only if there exists h x > 0 , such that for ti(hx, y)=-u~(y)- A~uJ(y), D~6( A~, x )< 0 and D~x~(A~, x) is negative definite.

m~, inf{h > 0 : D ~ 0 ( h , x ) ~<0}. Again with u ~ and u j suitably Proof. Let h~ = bounded, such an infimum exists as u i and u j are assumed to be strictly increasing in x. An analogous argument to the one used in the proof of Lemma A.2 can show that the infimum is in the set. Differentiating the definition of 6(. ,. ) twice with respect to x and pre- and post-multiplying by a vector z E S "-1, we have

(A-25) r . . . . . in T i z u~xvta ~ , x ) z = z Dx~U(X)z-'mi" T,, j, , a x Z IJxxU (X)Z

i ~ m i n As D~u (x) > 0 and D~uJ(x) > 0 it follows that Dxv(h~ , x) is not strictly less than n - 1 ~ ~ ^ ra in 0 and so we can find a vector /~ES , 0, such that Dxv(h ~ , x ) . h = 0 (i.e.

D~u(x).h+hmin~i ~ ~ L~UJ''tx).h).~ Thus dividing (A-25) by Dxui(x).h~ (>0) we obtain: T ^ m i n Z D~u (x)z zTD~uJ(x)z z D~v(A~ , x ) z _ T i

(A-26) D~u (x).h ~ - OxU (x).h Dxu'(x).h < 0

(by (A-24)). ^ m i n m i n Hence D~y(h , , x) is negative definite. Note if we take 1, + e, for some e > 0

sufficiently small, , min ~ min D~v(h~ +e ,x ) is still negative definite but D~v(A x + e, x) < O. So set hx = h m~n + e for some e > 0 sufficiently small. •

From Lemma A.4 by construction Dxt)(hx,x ) < 0 and Dxx~(hx,x ) is negative definite. That is,

max D x 6 ( h x , x ) . h = - e , where e > 0 ; and h E s n - l , h>-O)

max ZrDxxO(hx,x).z = - y , where 7 > 0. ZES n-1 )

By the uniform continuity of 6(hx,x).h (resp. zrO(h~,x)z) in (x, h) (resp. in (x, z)) there exists 6 > 0 such that:

ID~6(A~, y).h' - Dx6(h~, x).h[ < e ; and

Iz'rD~x6(h~, y).z'--zTD~x6(hx, X).Z[ < y for all y E X ,

all h', z ' ~ S n-I, s.t. I l y - x l l < ~ , I I h ' - h f l < ~ , and t l z ' - z l l < ~ . Thus for all y s u c h t h a t IIY - xl l <

Dx6(h x, y).h < 0 and zrD~xg(A~, y).z < 0Vh, z E S n 1, h >t 0 . •

ACKNOWLEDGEMENTS

I would like to thank Jerry Green and Andreu Mas-Colell for their guidance and suggestions on this work. I am also very grateful to Atsushi Kajii for his assistance

R O S S C H A R A C T E R I Z A T I O N O F M U L T I V A R I A T E R I S K A V E R S I O N 151

with some of the proofs, and to two anonymous referees for detailed suggestions on how to improve the exposition. Naturally, I remain solely responsible for any errors or omissions.

N O T E S

For specific applications of multivariate risk analysis to such economic examples see Chew and Epstein (1990), Demers and Demers (1991), Diamond and Stiglitz (1974), Epstein and Zin (1989) and Karni (1982). z A t considerable cost in terms of notational and expositional complexity the analysis of this paper can be readily extended to preferences that are 'smooth ' in Machina's (1982) sense and hence are representable by a real-valued preference functional V that is smooth with respect to the topology of weak convergence on Le(X). The interested reader is referred to Karni (1989) which is a 'generalized expected utility' extension of Kihlstrom and Mirman (1974). 3 Notice that a multivariate risk neutral agent has linear utility function which implies that his or her indifference map are parallel hyperplanes in X. A risk loving agent has a convex utility function. 4 I thank an anonymous referee for drawing this point to my attention. 5 I thank an anonymous referee for suggesting this corollary. 6 If b ~> a (4.3) would instead read

• t ( 1 - b ) [ ( 1 - b)a] + b [ b ( 1 - a ) ]

3' a(1 - b)

7 As u i and u j are increasing and concave, (3.1') is equivalent to:

T i m

min z Dx~u (x)z >max Dxui(x)'h r j <z~ - z D~u (x)z <h> D y ( x ) . h

F o r x = ( Y , x - ) , a n d b ~ < a

L H S = rain r / [ a ( 1 - z ) + ( 1 - a ) z ] ~ - ("+1~ _ ~ / (1 - a ) ;~(~_,~ (ze[0,11) y [ b ( 1 - z) + (I - b) z l£ (,+1) 3'(1 b)

and

[a(l - h) + (1 - a)h]'2 -~ a £(~ ~ . R H S = max = - -

(h~10,11~ [b(1 - h) + (1 - b)h]~-" b

8 For constant elasticity of substitution of 1, u(x) is the logarithm of the Cobb-Douglas utility function (i.e. f,r_ 1 a, log(x,)).

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152 SIMON GRANT

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and Extension', Econometrica, 55, 1139-1149. Paroush, J.: 1975, 'Risk Premium with Many Commodities', J. Econ. Theory, 11, 283-286. Pratt, J.: 1964, 'Risk Aversion in the Large and the Small', Econometrica, 32, 122-136. Ross, S.: 1981, 'Some Stronger Measures of Risk Aversion in the Small and the Large', Econometrica,

49, 62t-638. Rothblum, U. G.: 1975, 'Multivariate Constant Risk Posture', J. Econ. Theory, 10, 309-332, Rothschild, M. and Stiglitz, J.: 1970, 'Increasing Risk I: A Definition, J. Econ. Theory, 2,225-243.

E c o n o m i c s Programme , R S S S ,

The Austral ian Nat ional University,

A C T 0200, Austral ia.