sournal qf economic theory 8, 361-388...

SOURNAL QF ECONOMIC THEORY 8, 361-388 (1974)

Risk Aversion with Many ~Q~~~~~it~~s*

RICHARD E. KIHLSTROM AND LEONARD J. IRMAI’J

State University of New York at Stony Brook

Received November 6. 1952

The introduction of the risk aversion measure V(X) = -(u”(x)/u’(x)), corresponding to a utility of wealth function U, by Grrow [I] and has made it possible to study questions involving the effect of risk aversion on economic behavior under uncertainty. One of the first results of this type to appear in the literature concerns the portfolio selection behavior of an investor when there is one risky and one non-risky asset. Arrow shows in [l] that if the individual whose utility function is u becomes more risk averse as his wealth rises; i.e., if r(x) is an increasing function of x; then the amount invested in the risky asset decreases as wealth increases. Pratt states essentially the same result in a different way. He considers two investors with utility functions u1 and zk2 and shows that investor 1 always invests less in the risky asset than investor 2 if y1 = -(Z&L,‘) is always larger than r2 = -(ul/u,‘>.

The major drawback of the Arrow-Pratt approach is that its applica- bility is limited to situations in which utility is a function of .one argument. There are, however, problems in which utility is a function of two or more commodities where it is of interest to study the effect of attitudes towards risk on economic behavior. For example, in the situation where a consumer must decide what part of his income to save and what part to consume, utility is a function of two commodities, consumption now and consumption later. When the rate of return on savings is uncertain, it is important to know how the consumption savings choice is affected by risk aversion, even though the Arrow-Pratt approach is not appiicable.

The purpose of this paper is to extend the Arrow-Pratt concept of risk aversion to utility functions of n variables. The basis for the extension is provided by Pratt’s Theorem 1, which demonstrat.es that u1 is more risk averse than u2 if and only if u1 is an increasing concave transformation of U$ . Pn view of this result, it is natural to designate the ~-dimensional

* Research supported under NSF grants GS-3046 and 68-3228.

361 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

362 KIHLSTROM AND MIRMAN

utility function ~8 to be more risk averse than the utility function 9 if k is an increasing concave transformation of u2.1 While this approach is initially appealing it is not immediately obvious that the economic arguments used to justify the Arrow-Pratt approach can be extended to n-dimensions. In one dimension, the basic justification for using Y(X) as a measure of risk aversion is its relationship to the risk premium and the probability premium. These latter concepts have intuitive economic appeal as risk aversion measures. A similar justification is required in n-dimensions, i.e., the generalized definition of risk aversion should be connected to economically meaningful measures of risk aversion. This is done in Section 2 of the present paper. In that section an n-dimensional analogue of the risk premium is introduced and related to the proposed definition of “more risk averse.”

The major result in the paper is Theorem 2 which can be viewed as a generalization of the Arrow-Pratt portfolio result. This theorem demonstrates that the major determinant of the effect of risk aversion on savings in an uncertain environment is the relationship between savings and the return to savings in a certain environment. This result plays the same role in the n-dimensional risk aversion theory that the Arrow-Pratt portfolio theorem pays in the one-dimensional case. It provides a second justification for the proposed definition of n-dimensional risk aversion, by demonstrating the usefulness of that definition.

A difficulty encountered in generalizing the Arrow-Pratt theory of risk aversion is that n-dimensional von Neumann-Morgenstern utility functions may represent different preference orderings on the set of commodity bundles. As is pointed out in Section 2, attempts to compare the risk averseness of utility functions representing different ordinal preferences are confounded by the differences in these preferences. The approach to n-dimensional risk aversion carried out in this paper circum- vents this difficulty by allowing comparisons of only those von Neumann- Morgenstern utility functions which represent the same ordinal preferences over non-random commodity bundles. The proposed definition of “more risk averse” for n-dimensional utility functions is presented in Section 2. The main result of this section, which has already been referred to, is a proposition which relates the definition to an n-dimensional analogue of the risk premium. Section 2 also introduces a two dimensional analogue of the risk aversion measure r. It is shown that risk aversion comparisons

1 When the utility function has its domain in the positive real line, utility functions are distinguished by subscripts. This makes it possible to denote derivatives with a prime. However, when the utility function has its domain in the positive orthant of Euclidean n-space, it is convenient to distinguish utility functions by superscripts. This allows us to use the subscript notation for denoting partial derivatives.

RISK AVERSION WITH MANY C~~M~~iT~ES 363

made using this measure are identical with those using the proposed definition. As an introduction to the n-dimensional generalizations of Section 2, a restatement and summary of Pratt’s Theorem I is presented in Section 1.

The relationship between the approach introduce approach of Stiglitz [6], who compares the risk averseness of ~-dimeRsio~a1 utility functions by comparing the risk averseness of the corresponding one-dimensional utility of income functions and applying the Arrow- Pratt results, is discussed in Section 3.1. Section 3.2 places 5ur generalization in the framework introduced by Yaari [7].

The importance of the concept of risk aversion introduced by Arrow and Pratt is the opportunity it provides to study the influence of risk on economic behavior. Using the Arrow-Pratt concept it is possible to show that more risk averse people choose less of a risky asset in a portfolio. Similarly the generalization of risk aversion is of interest ~rirna~i~~ because of the possibility it creates for studying the effect of risk aversion on economic behavior when utility depends on more than one variable.

Thus the final sections of the paper apply the generalized definition of “more risk averse” to describe the effect of risk aversion on the co~~s~rn~~ tion-savings choice. The major results are Theorem 2 of Section 4 and its corollary which is contained in Section 5, These results can be sum- marized as follows. First, the effect of risk aversio.n on the consumption savings choice is not unique; there are some situations for which savings rise if the consumer becomes more risk averse, in other cases savings decrease. The factor which determines whether savings rise or fall when the consumer becomes more risk averse is a property of the ordinal preferences represented by the utility functions. The crucial property is the relationship between savings and the return to savings when that return is certain. Specifically, if certain savings increase ( the return to savings increases then uncertain savings decrea when the consumer becomes more risk averse. This property o preferences can also be stated in terms of the elasticity In fact, if the elasticity of substitution is greater (less) tha savings decrease (increase) when the utility function representing ordinal preferences becomes more risk averse.

1. The Arrow-Pratt Results in One Dimension

The risk aversion function r(x) = -(u”(x)/$(x)) plays an important role in the Arrow-Pratt theory. In fact, their definition of risk aversion is framed in terms of the function Y. Specifically, the utility function y


is said to be more risk averse than u2 if rl > r2 2 everywhere. The intuitive justification for this definition is provided by Pratt’s Theorem 1 which relates Y to an economically motivated measure of risk aversion-the risk premium. If 2 is a random variable representing the outcome of a lottery, the risk premium is the non-random value rr which makes the individual indifferent between the certain sum E(2) - rr and the lottery 1. For any wealth x and gamble 2, ~(x, E) is defined formally by the equation,

u(x + E(Z) - 7(X, 2)) = E(u(x + 2)).

Pratt also shows in Theorem 1 of [5] that one utility function, u1 , is more risk averse than another, u2 , if and only if u1 is obtained from u2 by a concave transformation. This result serves as the basis for our generalization of the concept of risk aversion to more than one dimension. Theo- rem 1 summarizes the main results of Pratt.

THEOREM 1 (Pratt). Let r,-(x) and rri(x, 5) be the risk aversion finction and the risk premium corresponding to the twice continuously dzferentiable and monotonically increasing utility function ui(x), i = 1, 2. The following conditions are equivalent,

(4 rdx) a(>> r2(x)

@I T(X, g> a(>> r,(x, 3

(c) There exists an increasing, (strictly) concave, twice continuously difirentiable function k such that ul(x) = k(u2(x)).3

A complete statement and proof of this result appears in Pratt [5], and will not be reproduced here. Rather an alternative proof (due to Vernon Smith) that (a) is equivalent to (c) is presented. The idea of this proof is used below to prove the analogous n-dimensional result.

Proof of (a)+(c). Because of the assumptions made about u1 and ZQ, , there exists a monotonically increasing and continuously differentiable function k = u,(z&r) such that ur = k(u,). Differentiating we get

Ul ' = kb;, and u; = k”(uzy)2 + k’u; (1.1)

Using equations (I. 1) to solve for k”, we find that

u; - k’u; k”=oZ=

1

2 Recall that.r+(x) = -u:(x)/u;(x), i = 1,2; this notation will be used throughout. 3 The function k is (strictly) concave if k(tx + (1 - t)x’)(>) > tk(x) + (1 - t)k(x’)

for all x and x’ and all t E (0, 1).

RISK AVERSION WITH MANY COMMODITIES 365

k” <(<) 0 if and only if Y]. a-(>) Y, .

2. Extensions of the Arrow-Pratt Results to ~-~~~e~~io~s

2 .: The purpose of this section is to generalize the Arrow-Pratt concept of risk aversion by providing a formal meaning for the expression “z.2 is more risk averse than u2” when u1 and u2 are functions of IZ real variables. One of the desiderata for such a generahzation is that it be consistent with the important properties of the Arrow sional) risk aversion measure.

Moving from one to n dimensions, however, introduces d~~c~lt~ which make it unreasonable to expect that all properties can be generalize The problem is that, in zz-dimensions, two different von Weumann- Morgenstern utility functions will, in general, represent two different preference orderings; i.e., the utility functions will not lead to the same indifference curves over the set of commodity bundles. The nature of the

difficulty can easily be illustrated. Let ~2 and U* be two distinct utihty functions representing two different preference orderings on the set -Q2 4

4 In this paper R, will represent Euclidean n space and .C?% will represent the non- negative orthant of R, , i.e., x E 0, , then x > 0. Here x > 0 means x, > 0 for all i, x > 0 means xi > 0 for all i, and xi > 0 for some i.


of commodity bundles. Let G? = (a, , a,) and X = (X, , ZZ) be two distinct points in Q2, with the property that u’(X) > u’(g) and u”(i) > u2(Z), as is illustrated in Fig. 1. Now suppose that consumer 1, having utility function ~8, and consumer 2, having utility function u2, are both faced with the choice of receiving E with certainty or a gamble between .Y and 2. Clearly consumer 1 prefers X with certainty to any gamble between $ and E. However, consumer 2 prefers any gamble between f and X to E with certainty. In such a situation it is impossible to compare the behavior of these two individuals toward risk. Consumer 1 acts more risk aversely than consumer 2. However, the explanation is not that consumer 1 is more risk averse than consumer 2. This behavior occurs because of the differences in the ordinal preferences represented by u1 and zP. This difficulty never arises in one dimension since in that special case all monotonically increasing utility functions represent the natural ordering5 on the real line; i.e., all utility functions represent the same ordinal preferences.

In the face of this difficulty the most natural approach, and the one taken here, is to limit comparisons of risk averseness to utility functions which represent the same ordinal preferences. Thus in the definition of ‘W is more risk averse than u2,” u1 and u2 are both assumed to represent the same ordinal preferences, >, over the set of commodity bundles.

Motivated by condition (c) of Theorem 1, the following definition generalizes the concept of risk aversion to utility functions of more than one variable. In this definition, and throughout the paper, it is assumed that z& is strictly concave and has continuous second derivatives. Thus the definition and results will apply only to risk averse utility functions. Also, we assume that uji E &&/I%~ > 0, i = I, 2, j = 1, 2 ,..., n. Finally, since u1 and u2 are assumed to represent the same preferences, there exists a function k such that u1 = k(u2), and k’ > 0. Note that the differentiability assumptions made about u1 and u2 imply that k” exists and is continuous.

DEFINITION. u1 is at least as risk averse (a representation of >) as u2[u1Ru2] if u1 = k(u2) where k’ > 0 and k is concave. u1 is more risk averse (as a representation of >) than u2[u1Pu2] if k is strictly concave.

Notice that the use of this definition makes it impossible to compare utility functions which represent different ordinal preferences. For if u1 is more risk averse than u2 in the sense of the definition then u1 and u2 must represent the same ordinal preferences. Also note that for y1 = 1, Theorem 1 establishes that this definition is equivalent to the Arrow-

s The natural ordering is a preferred to b if a > b.

RISK AVERSION WITH MANY COMMQDIT6ES 367

Pratt definition. Thus it makes sense to use the notation ar,Ru, when U; has its domain in Q, .

The propositions of Sections 2.2 and 2.3 are n-dimensional analogues of Theorem 1. These propositions require the introduction of n-dimensional analogues of the risk premium and the risk aversion function. The “directional risk premium,” which is defined in Section 2.2 generalizes the one-dimensional risk premium. Propositions 1 and 2 of Section 2.2 provide the analogue of the result that (b) is equivalent to (c) in Theorem I. Proposition 1 shows that the size of the directional risk premium is no larger for u2 than for u1 if uIRu*. The two-dimensional risk aversion function is introduced in Section 2.3. Proposition 3 in Section 2.3 is the analogue of the result that (a) is equivalent to (c) in Theorem I.

2.2 Let x, y E Gn,, (possibly x = (O,..., 0)) and let I be a random variable which takes values in [0, co). Consider gambles in 52, of the form x + 5y. These gambles lie on the line, shown in Fig. 2, originating at x, through the point x + y. The choice of y is essentially a choice of direction. Only gambles x + Zy, f > 0, with outcomes greater than x bttt constrained to be in the direction of x + y are considered. For the utility function z& we will study the risk premium +(x, y, .%) associated with the random variable ,5?, the point x and the direction y.

Analogous to the Arrow-Pratt risk premium, ri(x, y, 5) is defined formally by the equation


The following corollary of Theorem 1 demonstrates the strength of our definition of more risk averse.

PROPOSITION 1. If u1 is at least as risk averse as (more risk averse than) u2 then, for every x, y E Q, and every gamble I > 0, the risk premium &(x, y, 2) for u1 is at least as large as (larger than) the corresponding risk premium 9(x, y, 5) yor u2. Formally, u1Ru2(u1Pu2) implies that, for all x, y and f, T?(x, y, 5) a(>) T~(x, y, 5).

Proof. For z E [0, co), let v~,~(z) be defined by v&(z) = z&(x + zy). When x and y are fixed vi,,(z) is a function of the one dimensional variable z, and +(x, y, 5) is the analogous one dimensional Arrow-Pratt risk premium. The assumption that u1Ru2(u1Pu2) implies that

vi&> = k(d,,tzN

where k is (strictly) concave. By Theorem 1 the (strict) concavity of k implies that zJ(x, y, 5) >(>) rr2(x, y, .Z). 1

The next Proposition is the converse of Proposition 1 and is again a corollary of Theorem 1.

PROPOSITION 2. Suppose that u1 and u2 both represent the preference ordering >. If there exists a y such that, for all 2, ~~(0, y, 2) >( >) ~~(0, y, .%), then u1Ru2(u1Pu2).

Proof. Since u1 and u2 represent the same preferences there exists a function k such that u1 = k(u2). For this same k, v:,,(z) = k(vi,,(z)). (Recall that vi,,(z) = ui(x + zy)). Applying Theorem 1, ~~(0, y, 5) >(>) +‘(O, y, 5). for all i, implies that k is (strictly) concave. 1

Proposition 2 states that if there exists one direction in which u1 always has a larger risk premium than u2, then u1 is more risk averse than u2. A corollary of these two results is the following: u1 always has a larger risk premium than u2 in one direction if and only if it always has a larger risk premium in every direction.6

Thus in spite of the fact that the value of the directional risk premium varies with the direction, risk aversion comparisons of utility functions, obtained by comparing directional risk premium are independent of direction. This means that the directional risk premium is, in a very important sense, a directionless measure of risk aversion. These strong

6 Note that if ~‘(0, y, 5) > ~“(0, y, 5:) for some y and all I, then Proposition 2 implies u1Ru2. By Proposition 1, uZRul implies ~‘(0, y, 5) > ~~(0, y, .Z?) for all y > 0, and all I.

results are obtainable only because comparisons are restricted to utility functions representing the same ordinal preferences.

2.3 One of the more important features of the Arrow-Wyatt risk aversion theory is the usefulness of the risk aversion function r(x). In view of the importance of this measure it seems natural to try to develop a generalization of the function T(X) for higher dimensional utility functions.

In order to introduce the generalization it is necessary to reduce the set of ordinal preferences allowed. Let

To this point it has been assumed that all utility functions are strictly concave. This assumption implies that

(-1)” 4,b 3 0.’ (2.3. rg

In this section it is necessary to make the stronger assumption that a strict inequality holds in (2.3.1). If the strict inequality is satisfied at 61 ,‘--, xn) then

p(x1 ,.‘.> xn) = (-1)” A,

((-1)” dnb)nln+l (2.3.2)

is well defined. Proposition 3 demonstrates that when IZ = 2, p is an appropriate measure of risk aversion.

PRQPOSTION 3.s Suppose 12 and z.8 represent the preferences a. Then u1Ru2(u~Pu2) if and only if pl(xl , x2) >( >) p2(xi , x2) fir all x1 , x2 .

Proof. Since u1 and u2 represent the same preferences, u1 = k(u’) where k’ > 0. Thus

and (2.3.3a)

7 The following example, due to Katzner, [4], shows that u may be strictly concave with zero determinant. Let u = (x1x,” + x13x2) r/* and compute the determinant on the line xX = x2 _

8 In this paper only the two-dimensional case is considered. Preliminary investigations suggest that an n-dimensional risk aversion function can be defined and an n-dimensionai generaiization of Proposition 3 obtained. Specifically, we conjecture that in I? dimensions p is the appropriate risk aversion measure. Notice that when n = 1, p = Y.


Now let

and

Ci = (d)” 42 ,

/ I

Di =

i i u2 4

i u22

Straightforward computations verify that

and

Di = 0, (2.3.5)

pi = (- [Bi 2 Ci]}2’3 ’ (2.3.6)

Combining equations (2.3.3) and (2.3.4) yields

LB2 + C21 = [$I” P, + Gl - [$I [~2~, -u121

x [up]2 u12u22

[

2422 u22u12 [u2”]” I[ 1 -u12 *

Another simple calculation shows that the quadratic form in the second term is zero; so that

f-P2 + C21)2’3 = WI2 I-& + Cd2’3 (2.3.7)

Using (2.3.3b), (2.3.5) and the fact that the determinant is a linear function of each column, we get

Al = [k’j2 A, + k’k”[B2 + C,].

Solving for k” yields

k” = AI - WI2 A2

W& + C21

( - [B2-:‘C2]j1i3 [ [k’12 ( - [ii; + C2]}2/3 - {-[B, “+z C2]}2’3 ’ 1


Substituting (2.3.7) and then (2.3.6) in this equation we get

k” = -k’

1-B + CP3 i (-LB, 41 Cr])2’3 -

= (-- [B2-$,1)‘/” bl - Pzl-

Since k’l(- [B, + C2]}1/3 > 0, k” <( <) 0 if and only if p2 - p1 <CC> 0.

Proposition 3 demonstrates that risk aversion comparisons of utility functions, obtained using p, agree with comparisons obtained using the ordering R. This provides the primary justification for calling p a measure of risk aversion. It should also be noted that p possesses another requisite property of a useful risk aversion measme, viz., p is invariant under linear transformations. To show this let w = ~2 + bu. Then

It is interesting that p does not have a unique claim to being the two- dimensional analogue of r. In fact it is possible to introduce other two- dimensional risk aversion measures, and to provide the same j~sti~~atio~ for these measures as has been given for p, Specifically, for any z F G, and y E L& , let the directional risk aversion measure rO,,,(z) be defined by

4 Yc4 ro,y(z) = - L =

CZ=l Cf=l %CzY> YiYi 9

vb,,(z> CT=1 ‘it’Y> Yi ’

where v,,,(z) = u(x + zy). The measure yO,* is simply the Arrow-Pratt risk aversion measure of

the one-dimensional utility function v~,~(z) = Z&J). Hence r,,, is invariant under linear transformations.

A strong justification for calling Y~,~ a two-dimensional risk aversion measure is provided by the following corollary to Propositions 1 and 2.

COROLLARY 1. sUppOSe U1 and u2 both represent the preference ordering >. Let y be any vector in ,iz, . Then u~~~2~u1P~2~ if and only $

&AZ) >(>I ro2,&),for all z.

9 rjote that rO,y is defined only if Xi=, u,yi # 0. This sum is non-zero if the marginal utility of all commodities is positive and if y > 0.


Proof. Theorem 1 implies that r;Jz) a(>) r&(z), for all z, if and only if 7+&Z) a(>) $JZ), f or all 5. Thus Propositions 1 and 2 imply the desired result. 1

Corollary 1 shows that the risk aversion comparisons of utility functions, obtained from r,,y agree with the comparisons obtained from the ordering R. Recall that Proposition 3 plays a similar role for the measure p. Thus, for any y > 0, r,,V is, like p, an appropriate two-dimensional risk aversion measure.

The measure p has one interesting and potentially important property that is not shared with r,,u or with rO,r . The value of p is independent of direction. Thus p is direction invariant in a rather strong sense. Neither r,,, nor rrO,y possess this property since the values of these measures vary with y. However, recall from the discussion in Section 2.2, that the risk aversion comparisons based on z-~,~ are independent of direction; i.e., they are independent of y. Similarly Corollary 1 and Proposition 3 can be interpreted as showing that the comparisons based on rg,V and p respec- tively are also direction invariant. Thus T,,~ and r,,y are direction invariant in a weaker, but nevertheless, important, sense than p. The measure p is direction invariant in both the weak and the strong sense. Since p is strongly directionally invariant, it is potentially useful for extending the concepts and results of Arrow and Pratt to two dimensions. One concept in particular which would be interesting to extend is the notion of increasing (and decreasing) risk aversion. In one dimension, r plays the central role in defining this concept. The measure p is an obvious candidate for a similar role in two dimensions.

3. Relationship to Other Approaches

3.1 It has been pointed out above that differences in ordinal preferences confound attempts to compare the risk averseness of two utility functions. The approach taken to avoid this problem was to compare the risk averseness of two utility functions only if they represent the same ordinal preferences.

It may, however;be useful to compare the risk averseness of two utility functions which represent different ordinal preferences. To do so the discussion can be restricted to a class of gambles, call it r, that excludes those troublesome gambles for which the ordinal preferences disagree about the relative ranking of the prizes. It may then be possible to give a formal meaning to the expression ‘W is more risk averse than z? relative to the class of gambles r.”

As an example, consider a competitive consumer, and let r be the class of gambles for which income is random. Assume that prices are

RISK AVERSION WITH MANY COMM~ODITIES 373

fixed and that for each possible income IeveS the consumer chooses consumption to maximize utility subject to the resulting budget constraint, For such gambles the risk averseness of all utility functions can be compared (at least locallylo) regardless of the preferences they represent. The reason is that the consumers’ actions will be determined by their utility of income function (the indirect utility function with prices fixed) which is a function of one variable. In this case the Arrow results apply. This approach, which is essentially th in [6], allows a comparison between utility functions which do not represent the same ordinal preferences.

DEFINITION. If u(x) is a utility function on Q, the corresponding indirect utility function, U(p, Z), (where y E .Q, and Z E X2,) is defined by the equation

where x(p, Z) maximizes U(X) subject to p . x < Z.

Note that for fixed p, U(p, Z) is a function of income, Z. Let p and p’ be two price vectors (p may equal p’). The Arrow-Pratt results can be used to compare the risk averseness of Ul(p, Z) and U’(g’, Z). Note that as income varies randomly, consumer 1 chooses a gamble in which the

FIGURE 3

I0 The risk averseness of two utility functions can be compared locally if there is a neighborhood in which the risk averseness of the utility functions can be compared. This terminology is also used in Section 3.2.


prizes are consumption bundles on his Engel curve xl(p, 1) (see Fig. 3), but the prizes in the gamble which consumer 2 chooses are bundles on the Engel curve x2($, I) (again see Fig. 3). In other words, the gamble chosen by consumer 1 is not the same as the gamble chosen by consumer 2. Hence risk aversion comparisons between Ul(p, I) and U2(p’, 1) are in fact comparisons of the risk averseness of u1 as consumption varies randomly along xl(p, I) to the risk averseness of u2 as consumption varies randomly along x2($, 1).

The troublesome gambles discussed in Section 2.1 are ruled out by this approach. To see why, refer to Fig. 4 and suppose that both consumers are forced to consume xl(p, 1) when I = f and x2@, 1) when I = f.ll The consumers now disagree about the relative rankings of the possible outcomes of f and 1. Consumer 1 will prefer the lower income 1, while consumer 2 prefers the higher income 1. Then consumer 1 prefers 1 with certainty to any gamble between 1 and r^. Consumer 2 prefers the gamble to f with certainty. This is essentially the same situation that arose in the example of Section 2.1. Risk aversion comparisons are confounded by the fact that the consumers’ relative rankings of the outcomes 1 and f differ.

FIGURE 4

I1 In Fig. 4, p’ is assumed to equa1 p.

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Suppose now that each individual is able to choose any consumption bundle on his budget constraint. Consumer i will choose x”(p, 1) when B= 1 and xi(p, 1) when I = 1. Both consumers will prefer the higher income f to the lower outcome f. Risk aversion comparisons are now possible because the consumers no longer disagree about 1 and a.

The next proposition shows that when the utility functions being compared represent the same ordinal preferences, risk aversion cQrn~~~~so~s obtained from the approach just discussed agree with

PROPQSITION 4. z.2 is at least as risk averse as (more risk averse than) u2 if and only if for every p E 52, , Ul(p, I) is at least as risk auerse as (~Q~@ risk averse than) U2(p, I).

Proof. The assumption that u1 and u2 represents the same preferences implies that x(p, 1) is the same for u1 and ~3. The fact that ~1~ = k(9), with k’ > 0, implies

WP, 1) = ~WP, 0) = k(u2(x(p, I))) = k(U2(p, 1)).

If ulJW then k is concave and U1 is more risk averse than U2 for every p by Theorem 1. On the other hand, if U1 is more risk averse than U*@ some p then k is concave, again because of Theorem 1, and GRu2.

Note that this is again a very strong result. If U1 is more risk averse than U2 for a single p, then u1 is more risk averse than u2 and s/l is more risk averse than U2 for all p.

3.2 An approach to risk aversion which has the Arrow-Pratt theory as a special case was introduced by Yaari, 171, who uses the concept of an acceptance set as the basis of his definition

Consider two mutually exclusive events, E and +E, such that the probability of E is 4 E (0, 1). A gamble is then a pair (zl , z2) E G?, x L?, such that the player is awarded z1 if E occurs and z2 if NE occurs.

DEFINITION. For any gamble (zl, z2), the acceptance set A(z, j z.J, associated with the utility function U, is the set of gambles which yield

at least as high an expected value of u as (zl ) z2). Formally,

Yaari uses the acceptance sets Ai(z, z) corresponding to ui to show the equivalence of his and the Arrow-Pratt definition of risk aversion in the one-dimensional utility case. It is this definition of risk aversion which we shall attribute to Yaari.


DEFINITION. u1 is [Yaari] at least as risk averse as u2[u1Yu2] if for all z E Q, , A2(z, z) I &(z, z). ~2 is [Yaari] more risk averse than u2[z$Vu2] if for all 2 E Q2, , A2(z, 2) 3 Al(z, z).

Yaari’s definition of risk aversion generalizes the Arrow-Pratt definition of risk aversion for one dimensional utility functions since risk aversion can be defined for all preference orderings over gambles in the Yaari definition, whereas only preference orderings derived from expected utility functions are comparable under the Arrow-Pratt definition.12

From a geometrical point of view, if ~2 is more risk averse than u2 in Yaari’s sense, the hyperplane supporting the set A2(z, z) at (z;z) also supports the set Al(z, z) at (z, z). Thus if the risk averseness of ~2 and of u2 are to be compared at (z, z) using Yaari’s approach, the supporting hyperplanes to the sets Al(z, z) and A2(z, z) must be the same.

As might be expected, in one dimension all utility functions are locally comparable using the Yaari definition. The situation when u is one- dimensional is shown in Fig. 5. The cross hatched region indicates the acceptance set Al(z, z) associated with z.2 while the shaded region indicates

FIGURE 5

I2 Notice that the statement of Yaari’s definition presented above implicitly assumes a von Neumann-Morgenstern utility function by using expected utility. It can, however, be stated more generally.

RISK AVERSION WITH MANY C~~~DIT~E~ 377

the acceptance set A2(z, z) associated with u2. These regions are convex because of the assumption that ui is concave. The slope of the line T which supports both A+, z) and A2(.z, z) at (z, Z)I is --q/(1 - CJ), where 4 is the probability of the event E. Indeed for all one-dimensional utility functions the slope, at (z, z), of the frontier of the acceptance set A(?, z) equals -q/(1 - 4). Thus u1 and 9, and, in fact, all o~e-d~rn~~si~~a~ utility functions, are locally comparable using Yaari’s definition. In the particular case of Fig. 4, there is a neighborhood in which A1(zz, z) is contained in A2(z, z) so that u1 is locally [Yaari] more risk averse than u2.

On the other hand it is not the case that all utility functions in higher dimensions are locally comparable in the Yaari sense. unexpected since, as has been discussed above, not a are comparable in terms of risk aversion in higher precisely, in the one dimensional case every gamble i if E occurs and z2 if -J-E occurs. However, in the n-d payoff is the vector (z,l, z21,..., z,l) in case E occurs and (zr*, zzl,.~., zn2) if WE occurs. Silence the Yaari acceptance set is a 2n-dimensional subset of 2n-dimensional space. The marginal rates of substitution between the n commodities determine the hyperplane which supports the acceptance set. These marginal rates of substitution are different unless t preferences represented by the utility functions are the same. utility functions are not even locally comparable in the Yaari. sense unless the ordinal preferences are the same.

Proposition 5 provides a formal justification for our assumption that u1 and u2 must represent the same preferences to be comparable in terms of their risk averseness. Proposition 6 connects u1 Yu2 to u1 them to be equivalent.

PROPOSITION 5. If u1Yu2 then u1 and u2 must represeat the same ~~d~~a~ preferences.

P$VJO$ Suppose u1 and u2 represent different preference orderings. Then there exist x and y such that U’(X) > u”(v) but u”(x) < u”(y). Then G-> Y> E AYY, Y> but k Y) $ A2(y, Y>.

The following proposition shows the equivalence of the Uaari definition and the definition introduced in this paper. Before stating this result recall that the certainty equivalent of a gamble (zX , zz) is the certain payo z which yields the same expected utility as the gamble. Note that in the Yaari framework the certainty equivalent for a particular gamble (zr ) z2) is the point (z, z) on the frontier of the acceptance set A(z, J z.J.

PROPOSITION 6. u1Yu2 if and only if u1


Proof. First suppose dRu2, and suppose (y, , u2) E Al(z, z). Then

mu2(Yl) + (1 - 4) u”cY2>) 3 qk(u2b5N + (1 - 4) WU2(Y2)) 3 W2(4)-

Since k is monotonically increasing this inequality implies ( y1 , yz) EA~(z, z). Now suppose u1Yu2. Then, by Proposition 5, z2 and u2 represent the

same preferences, hence z2 = k(u2), k’ > 0. Suppose k is not concave over the range of values taken by u2. Then there exists y1 and y2 such that

k(qu2(yd + (1 - d 54”(u2>> -=c qk(u2(ylN + (1 - d k(U2(yJ)-

Let z be the certainty equivalent of ( y1 , y2) for ~2. Then ( y1 , J$) E Al(z, z). However,

Nu2W = &u2hN + (1 - 4) k(u2h)) > k(qu204 + (1 - 4) u”b2)>,

which implies that (ul, v2) $ A2(z, z), a contradiction. I

4. Risk-Aversion and the Consumption-Savings Choice

Arrow [I] used the concept of risk aversion to derive results comparing the portfolio choices of an individual at different levels of wealth. He showed that an individual whose utility of wealth function is increasingly (decreasingly) risk averse, in the sense that r(x) is an increasing (decreasing) function of x, invests less (more) in the risky asset as his wealth increases. A similar result showing that a more risk averse individual always invests less in the risky asset and more in the safe asset, was proved, as Theorem 7, by Pratt in [S].

The existence of an n-dimensional generalization of the concept of risk aversion makes it possible to ask similar questions in a more general context. In particular, the effect of risk aversion on consumption-savings choices when the return to saving is uncertain can be studied. In this context savings play a role like that of the risky asset in the portfolio model. Similarly, engaging in present consumption is analogous to investing in the safe asset. The Arrow-Pratt results suggest that an increase in risk aversion will lead to less savings. There is, however, an important complication in the consumption-savings model which fails to arise in the portfolio choice problem. In the portfolio model the income earned from the risky asset is a perfect substitute for the income obtained by investing in the safe asset. In the consumption-savings model present consumption is not, in general, a perfect substitute for future consumption. Indeed, the nature of the ordinal preferences for present and future consumption plays a crucial role in determining how risk aversion affects saving when the interest rate is uncertain. Using the term “uncertain

RISK AVERSION WITH MANY ~O~~~~~T~~S 379

savings” (“certain savings”) to refer to the arnount saved when the return to savings is uncertain (certain), the major result in this section is that the effect of risk aversion on uncertain savings is determined by the nature of the relationship between the interest rate and certain savings. Section 5 demonstrates that the relationship between the interest rate and certain savings is in turn determined by the elasticity of substitution of the ordinal preferences for present and future consumption. Thus the effect of risk aversion on uncertain savings ultimately depends on the elasticity of substitution.

Suppose that consumer i has a twice continuously differentiable utihty function uz’(cl , c2) where ci is consumption in period j. Let W denote the consumer’s wealth and let 3i: represent the random rate of return on savings, S. Assume that there is no safe asset. Let p represent the probability distribution of f. We assume that ,u(? >, O> = 1, and denote this assumption by writing p 2 0. Consumer i then chooses P(p) to maximize

quy w - s, SC}, subject to the constraint 3 s 3 0.13

Note that in this model uncertainty enters only in the second period through the returns on savings. However, the effect of uncertainty is felt on the consumption decision in the first period. Also note that saving corresponds to consuming the “risky commodity,” S. Thus, the Arrow- Pratt result might lead one to expect SI(yl) < P(p) if ~2 is more risk averse than ~2. As pointed out above, however, when there are two commodities the situation is not so simple. To see why, s first that there is no uncertainty; i.e., assume x is known. Since both consumers have the same ordinal preferences and thus will behave in the same way.

Formally, ui chooses F(x) to maximize hi(S, x) = u”( W - S, Sx) an F(x) = P(x). So we can let S(x) = ,9(x). Equivalently in the certainty case z& will choose (c,(x), cZ(x)) to maximize ui(cl , c2) subject to plcl + p2ce = W where p1 = 1 and pz = l/x.14 Here S(x) = W - cl(x).‘” An increase in x is effectively a decrease in the price of cZ . So, depending on the ordinal preferences, c1 may rise or fall when x rises. Equiva~e~t~y, since S(x) = W - c,(x), S may rise or fall as x rises. In fact, as shown in Section 5, if u is homothetic, then &,/ax = -(l/x2)(2c,/apZ) >(<) 0 and i;S/ax <(>) 0 if the elasticity of the substitution, ci, is always less

l3 Note that SQL) actually depends on Was we11 as p. For convenience, our notation suppresses this dependence.

I4 Note that the c1 and c2 do not need superscripts since the ordinal preferences, and, therefore, the choices made under certainty, are the same for both utility functions.

l5 Again the reader should be aware that our notation suppresses the dependence of S(x) and q(x) on W.


(greater) than one. (Note that under our assumptions about U, S(x) and cl(x) are differentiable functions of both Wand x.)

Turning to the uncertain situation, consider the effect of an increase in risk aversion on savings. A consumer who becomes more risk averse can be thought of as viewing 2 as though it were, in some sense, more risky (Diamond and Stiglitz [2] show that this can, in fact, be formalized.). For risk averters, an increase in the riskiness of 4, in the case of uncertainty, is like a decrease in x, in the case of certainty. This intuitive reasoning leads one to expect that an increase in risk aversion will have the same effect in an uncertain situation as decreasing x has in the certain situation. Thus we are led to the following conclusion which will be proved as Theorem 2 below.

Suppose that the preferences, >, are such that S(x) is always an .

increasing (decreasing) function of x. In this situation, u1 more risk averse than u2 should imply S+) <(>) S2(p), for all Wand TV. Theorem 2, below, verifies this statement and proves that the converse is also true.

Before stating and proving Theorem 2, we prove the result for three examples. Example 1 is the Cobb-Douglas case where S(x) is independent of x. In Example 2, c1 and c2 are perfect substitutes; in Example 3, c1 and c2 are perfect complements. These examples warrant special treatment for several reasons. First, by providing concrete illustrations they help shed some light on the reasons underlying the main result. Second, they are extreme cases. In Example 2, the elasticity of substitution is infinite, and (certain) savings are increasing in x. Example 3 is the opposite extreme where the elasticity of substitution is zero, and (certain) savings are decreasing in x. Example 1 is the boundary case where the elasticity of substitution is one and the (certain) saving decision is independent of x. Third, the examples are of independent interest. Example 1 is a case where uncertainty about x and how the consumer views it (his risk averseness) has no effect on his (uncertain) saving decision. This is exactly what one would expect when (certain) savings are independent of x. Example 2, is the case studied in portfolio theory. From the present perspective the Arrow-Pratt theorem about the effect of risk aversion on risky investment is a special case in which the conclusion of Theorem 2 holds. The final reason for studying Examples 2 and 3 separately is the fact that the general proof provided for Theorem 2 requires differentiability assumptions not satisfied in these extreme cases.

EXAMPLE 1. Suppose u represents Cobb-Douglas preferences, i.e., u(cl , c2) = $(c~%~~), where $’ > 0. Then

RISK AVERSION WITH MANY C5~~5~IT~ES 381

i.e., S(X) is independent of X. When x is uncertain, the first order condition for maximizing I?(+([ W - Sl* Saxs)> is

[P/S - (a/[ w - S])][ w - s]o! SaE{$‘([ w - sl” 9x”) xq = 0.

Since [ W - S]” SBE{ (b’( [ W - Sp SBxo) x”> > 0 when S > 0 and [ W - S] > 0, SW = wq(~ + p>. I n other words, with @ebb--Douglas preferences, the introduction of uncertainty has no effect on the consumption-savings decision What is more important in the present context is that savings is given by S(p) = Wp/( a + p) independent of the utility fu~ctiorz u whiclz represents these preferences.

EXAMPLE 2. Let u(cI , cJ = $(cl + c.J, where # > 0 and 6” < 0. Then

iw c1(-4 = (0 if x<l,

and S(X) = (Ow if x&l

if x>l; if x29.

This is an extreme case of S increasing with X. This example is equivalent to the portfoho problem considered by

Arrow and Pratt, where c1 represents investment in the non-risky asset and S represents investment in the risky asset. Thus Pr applies to give S(p) < Pfj~) for all p 3 0 if and only if

EXAMPLE 3. Finally, suppose that

u2(c1 , c2) = sb(minb , c,>),

where 4’ > 0 and 4” < 0. In this case

44 = W/Cl + wx)), and S(x) = W/(x + I)

i.e., savings decrease as x increases. Let gi(S) = 2%. We assume for simplicity that p is absolutely continuous, then

and S2(p) is the solution of the equation

g2’(S) = -+( w - S) )x{x: w - s < Sx)

+ L: w-S>.sd x$bySx) p(dx) = 0.

/


If ~2 = k(u2) where k’ > 0 and k” < 0; i.e., if u1Ru2; then

g”(S) = --k’($b(W - S)) I$‘( w - s) p{x: w - s < Sx)

Since k” < 0, k’(+( W - S)) < k’(+(Sx)) for all x such that W - S > Sx. Hence

g1’(W4 3 k’(qW - S20>> s”‘Wpl) = 0.

Since g1 is a concave function of S, gl’ is a decreasing function of S, and hence S1&) > S2(p).

THEOREM 2.16 Assume that ui is strictly concave, t& exists and is continuous, and that ut > 0 for i, j, k = 1,2. Suppose that, for all W, S(x) is a strictly increasing (decreasing) function of x.l’ Then 9(p) <(>) S2(p) for all E.L 2 0 and all W > 0 if and only if u1Pu2.1s Also S(p) <( 2) Sz(,u) for all p > 0 and all W > 0, if and only ifu1Ru2.

Note. The hypothesis that S(x) is strictly increasing (or decreasing) is a restriction on the ordinal preferences, >, which u1 and u2 represent.

Outline of the Proof. The continuity of S(x) which follows from the assumptions about ui assures that there exists an X such that S(Z) = S2&). The assumption that S(x) is strictly increasing in x implies that, at S = S2&) = S(X), h12(S, x) < 0 if x < E and h12(S, x) > 0 if x > Z. (Recall that hi@, x) = ui( W - S, Sx) and hli(S(x), x) = 0 for i = 1,2.) In proving these inequalities, the strict concavity of u2 and the fact that h12(S(x), x) = 0 is used. In the proof that u1Ru2 implies S1&) < P(p), the essential idea is that taking a concave transformation of u2 is equivalent to multiplying the measure p, of x, by (k’/J k’p(dx)), hence creating a new measure which shall be called $. By the concavity of k this measure shifts probability from x’s above X to x’s below X. Thus, when evaluated at S = S2(p) = S(X), the expectation s h12@(dx) will be smaller than the expectation s h12,u(dx) which equals zero. By the definition of $, J hI1~(dx) = J” k’p(dx) J- h12j’I(dx). S ince J k’&dx) is positive, J h,$(dx)

I6 Diamond and Stiglitz [2] have independently discovered the sufficiency part of this result. In [2] it is assumed that risk aversion varies smoothly with a continuous param- eter p, an assumption which is not made in this paper. This assumption makes it possible for Diamond and Stiglitz to prove the result by differentiating with respect to p.

I7 To be perfectly correct S(x) is assumed to be strictly increasing (decreasing) when S(x) is greater than 0 but less than IV, allowing S(x) to be non-decreasing when corner solutions arise.

I8 We also assume implicitly that p is such that W > P(p) > 0. This assumption insures that the set of x’s where corner solutions arise has probability zero.


must be negative. The strict concavity of hlp(dx) then insures that Syp) < S2@).

The converse is proved by contradiction. It is assumed that S(~.L) < Sz&) for all p and Wand that k is not strictly concave. Since k has continuous second derivatives, if k is not strictly concave then it must be convex on some interval. A set X is then constructed with the property that, for x and x’ in X, u2( W - S(x), ~‘S(X)) lies in the interval where k is convex. Only those gambles 3i: for which ,u(X) = 1 are considered. For these gambles it is possible to adapt the proof that u1Ru2 implies S-I&) < S2&), by reversing the inequalities, because k is convex rather than concave, to prove that S(U) > S2&). This contradicts the assertion that S+) < S2(,u) for all p and W, so k must be strictly concave.

Proof. The proof is for the case u1Pu2 and S(x) strictly increasing in x. A similar proof works when u1Ru2 and/or S(x) is strictly decreasing in 3t.

First note that under our assumptions about u2, ,!P(LL) is the unique solution to the equation Eh12(S, p) = 0, where Ehi(S, ,u) = St&( W- S, Sx) ,LL(~x). Moreover since S(x) is a continuous function of x, for each W, there exists an 2 such that S(Z) = S2(p). Also since S(x) is a strictly increasing function of x,

S(x) < S(Z) if x < Z, (4l.a)

FIGURE 6

384

and

KIHLSTROM AND MIRMAN

S(x) > S(X) if x > E. (4.1.b)

Let x --c X; we wish to show that h12(S2~), x) < 0. Suppose this is not the case; i.e., suppose h12(S2&), x) > 0. Since h12(S(x), x) = 0 and S2&) = S(X), 0 < h12(S2(p), x) implies hr2(S(x), x) G h,2(S(Z), x). The concavity of h2 then implies S(x) >, S(Z) which contradicts (4.1.a). Thus (4.1 .a) implies

0 > h2(S2cpl, 4 if x < X. (4.2.a)

Similarly, (4.1 .b) implies

0 < h2(S20, x> if x > 2. (4.2.b)

Figure 6 illustrates this point. In Fig. 6, h12(S, x) is downward sloping because h2(S, x) is concave. Note that in Fig. 6 we let 2 -=c E < 2.

Now suppose uxPu2. Then U* = k(u2), with k strictly concave. The assumption that u22 is positive and the strict concavity of k imply that for all 5,

k’(h2(S, x)) > k’(h2(S, Z)), if x < X, (4.3.a)

FIGURE 7


and

k’(KyS, x)) < k’(l@(S, Z)), if x>K

Combining (4.2.a) with (4.3.a) and (4.2.b) with (4*3-b) yields

(4.3.b)

= k’(P(S, 2)) E/y(S, p) = 0, when S = Pi (4.49

Since E/z1 is concave, this implies 9(p) < Sz&), as shown in Fig. 7. Now suppose S+) < P(p) for all p > 0 and all W > 0, but that k is

not strictly concave. Since k is twice continuously differentiable there exists an interval (a, b) on which k’ is non-decreasing. By the continuity of S(x) there exists a wealth level, W, and a non-empty open subinterval X = (x’, 2) of [O, co) such that, for all x, x’ E 1, uz( W - S(x’), xS(x’)) E (a, 5).

Fora =a,b, I~={ic,,c*):u*1c,,c2)=“}.

FIGURE 8

6421813-9


The general proof of the existence of X is omitted, however, the reasons for the existence of X are suggested by Fig. 8. In Fig. 8, X = (Z,2) = (g, 9). Note that in Fig. 8, u2( W - S(Y), x”S(x”)) = 01 for 01 = a, b. Also note that S(X) is strictly increasing in Fig. 8.

Let f be a random variable such that ~{a E (k, i)} = 1. Since S(Z) 3 S2(p) 3 s(G), and S(X) is increasing and continuous in X, there exists an X E X such that S2(p) = S(Z). Thus h2(S2(,u), X) E (a, b) for all x E X. Hence by the fact that k’ is non-decreasing,

k’(h2(S2W, 4) < k’(h2(S2(p), 3) if x < X, XE X, (4.5.a) and

k’(h2(S2b4L), 4) 2 k’(h2(~2(pl.), 3) if x > X, x E X. (4.5.b)

Combining (4.2.a) with (4.5.a) and (4.2.b) with (4.5.b) yields

Then the concavity of Ehl implies Sl(p) 3 S2&), a contradiction. 1

Comments on the Proof of Theorem 2. In obtaining the inequalities in (4.3) the fact that k’(h2(S, x)) is a decreasing function of x when S is fixed was used. At first, it appears that the assumption that S(X) is an increasing function of x implies that h12(S, x) is an increasing function of x. If this were true we could then prove that 9(p) < S2(p) by applying the following useful

Inequality. If f (x) and g(x) are both monotonically increasing or both monotonically decreasing functions of x then

Ef(x)g(x) > EfW -Q(x); i.e., cov(.L g> 3 0.

If one of the functions is increasing and the other decreasing then

Ef (x) g(x) < Ef (4 &b); i.e., cov(f, g) < 0.

This result appears as Theorem 236 in Hardy, Littlewood and Polya [7]. They, in turn, attribute it to Tchebychef.

This approach would have been more straightforward than the one actually given. It fails because the hypothesis that S(X) is an increasing function of x does not imply that h12(S, x) is everywhere an increasing function of x. It only implies that in a neighborhood of S(x), h12(S, x) is an increasing function of x. This means that h12(S, X) and h12(S, x) may cross as is shown in Fig. 6. This problem arises for the example u2(c1, c2) = (c;l + CL’)-‘. For this example S(x) decreases in x, but h12(S, X) is not monotonically decreasing in x.


5. Risk Aversion and the Elasticity of Substitution

The preceding section investigated the effect of risk aversion on uucer- tain savings. Theorem 2 showed that the direction of this effect is determined by a property of the consumer’s ordinal preferences. Specifically, if the ordinal preferences are such that certain savings increase (decrease) with the certain interest rate, then a more risk averse representation of the ordinal preferences leads to less (more) uncertain savings. The purpose of this section is to demonstrate that the behavior of certain savings is determined by the elasticity of substitution of the ordinal preferences. In other words, the elasticity of substitution is the crucial factor in determining the effect of risk aversion on savings under uncertainty.

The basic result, which has already been referred to, is that certain savings are a decreasing (increasing) function of the interest rate if and only if the elasticity of substitution is less (greater) than one. This result, which is stated as Proposition 7, can then be combined with Theore to prove that if the ordinal preferences have the property that the elasticity of substitution is always less (greater) than one then an increase in risk aversion leads to an increase (decrease) in uncertain savings. This is stated formally as a corollary to Theorem 2 below.

Only homothetic preference orderings which can be represented by a strictly concave utility function U(C 1 , cZ) with continuous second derivatives are considered in this analysis. The demand function (c&~ ) pZ ) P)? q&1 ) pZ , I)) maximizes u(cl , cJ subject to plc, + pze, = I. The certain savings function is S(x) = W - c,(p, ,pz , I> where ppl = 1, pZ = l/x3 and I = W.

PROPOSITION 7. The elasticity of substitution, uY is less (greater) than *ne at (4~~ , pa ,I), c,(P, , p2 , I)> if and only ilf

Proof. Let q be the marginal rate of substitution, (tc&J, and let r = (cZ/cl). When u is homothetic, q and r satisfy the equation

which is independent of cL . Then

and


Since (cl(pl , pz , I), C&J, , pz , I)) satisfies the first order conditions, ui--Ap,=Q,i=1,2,andI-pp,cl-pp,c,=O,

where

J = --ullP22 -I- 2%,P,P, - u,,p,2 > 0, and h > 0.

Equations (5.1) and (5.2) imply &,/a~, <( >) 0 if and only if u <( >) 1. 1

COROLLARY TO THEOREM 2. Suppose the preferences are homothetic and the elasticity of substitution, o, is uniformly greater (less) than one. Then u1Pu2 if and only if S(p) -K(>) S2&) for all ,u > 0 and all W > 0.

Proof. Proposition 7 implies 0 >(<) 1, if and only if (LLS/&) = (l/x2)(~c,/&J >(-c) 0. When (%?/a~) >(<) 0, Theorem 2 implies that u1Pu2 if and only if ,9(p) -c(>) S2(p), for all p 2 0 and all W > 0. 1

REFERENCES

1. K. J. ARROW, “Essays in the Theory of Risk Bearing,” Markham, Chicago, 1971. 2. P. A. DIAMOND AND J. E. STIGLITZ, “Increasing Risk and Risk Aversion,“unpublished

paper presented to the NSF-NBER Conference on Decision Rules and Uncertainty, University of Iowa, (May, 1972).

3. G. H. HARDY, J. E. LITTLEW~~D AND G. POLYA, “Inequalities,” Second Edition, Cambridge University Press, London, 1964.

4. D. W. KATZNER, “Static Demand Theory,” MacMillan, New York, 1970. 5. J. W. PRATT, Risk aversion in the small and in the large, Econometrica 32 (1964),

122-136. 6. J. E. STIGLITZ, Behavior towards risk with many commodities, Econometrica

37 (1969), 660-667. 7. M. E. YAARI, Some remarks on measures of risk aversion and on their uses, 1. Econ.

Theory 1 (1969), 315-329.

sournal qf economic theory 8, 361-388...

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