Meaning of more risk averse when preferences are over mean andvariance∗

Timothy Mathews1

1Department of Economics, California State University-Northridge, 18111Nordhoff St., Northridge, CA 91330-8374, USA (e-mail: tmathews@csun.edu)

Forthcoming, The Manchester School

Summary. Building upon the intuition of Ross, a definition of more riskaverse is proposed for situations in which preferences are over mean andvariance. If agents can be compared by this definition, the more risk averseagent will choose a less risky alternative. If this definition cannot be applied,it is not clear which agent will choose a riskier alternative. The definitionapplies whenever agents are ordered according to Ross’ notion of more riskaverse. The definition may or may not be consistent with the Arrow-Prattnotion of more risk averse (and therefore, may apply when Ross’ notion doesnot).

Keywords and Phrases: Choice Under Risk, Risk Aversion, Comparisonof Risk Attitudes Across Agents

JEL Classification Numbers: D8, D81

∗I would like to thank James Dow, Qihong Liu, Soiliou Namoro, Konstantinos Serfes,Andriy Zapechelnyuk, and an anonymous referee for helpful comments.

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1 Introduction

Consider a choice under risk from a fixed set of alternatives. When all agentsagree on the relative riskiness of the alternatives from which they mustchoose, it would be useful to be able to infer when one agent will choosea less risky alternative than another agent. Utilizing the intuition presentedby Ross (1981), a definition of more risk averse is proposed in such environ-ments. The proposed definition incorporates not only the preferences of theagents, but potentially accounts for the class of random variables from whichagents choose. If two agents can be compared by the proposed definition,the agent that is more risk averse will optimally choose an alternative thatboth agents view as less risky. However, if the proposed definition cannotbe applied, then for any agent it is possible to appropriately construct a setof risky outcomes from which the agent will choose an alternative that bothagents view as riskier than that which is chosen by the other agent. Thatis, given a choice from a fixed set of alternatives, whenever the proposeddefinition does not apply it is not clear which agent will choose a riskieralternative.

This analysis is motivated by the fact that when standard notions ofwhat it means for one agent to be more risk averse than another are applied,counterintuitive outcomes can occur. In particular, consider a choice from afixed set of risky alternatives by two agents that can be ordered in regardsto their degree of absolute risk aversion. It is possible for the agent withthe higher degree of absolute risk aversion to optimally choose an alternativewhich both agents view as riskier than the alternative optimally chosen bythe agent with the lower degree of absolute risk aversion. Ross illustrates thatsuch choices can result, by examining a simple portfolio selection problem aswell as a partial insurance problem (in which an agent cannot avoid all risk).

Another economic example, arising in Mathews (2004b), is related tothe random revenue resulting from offering a buyout option in an auction.1

When given a choice over different distributions of revenue which result fromdifferent buyout option prices plus a finite constant, it is possible for anindividual with a higher degree of absolute risk aversion to optimally choosean alternative which all individuals view as riskier than that which is chosenby an agent with a lower degree of absolute risk aversion.

Previous notions of what it means for one agent to be more risk aversethan another are discussed in Section 2. In Section 3 an intuitive definition ofmore risk averse is proposed for situations in which preferences can be statedover level of risk and expected payoff alone. The implications of this definition

1This example is presented in Subsection 5.3.2.

2

on the optimal level of risk are examined in Section 4, where it is shown thatthe proposed definition eliminates the possibility of counterintuitive choices.In Section 5 the proposed definition is related to the notions due to Arrow-Pratt and Ross.

2 The Meaning of More Risk Averse

Consider an expected utility maximizing agent. The level of utility for anagent i from a realized payoff of x is obtained by evaluating the Bernoulliutility function, ui(·), at x. Given a distribution of payoffs F (x), the vonNeumann-Morgenstern expected utility of this agent is

Ui (x) =∫

ui(x)dF (x).

The most well known, and widely used, notion of more risk averse is theArrow-Pratt measure of risk aversion. Simply stated, agent a is more riskaverse than agent b (in the Arrow-Pratt sense), if ra(x) ≥ rb(x) for all possiblex, where ri(x) = −u′′i (x)/u′i(x) is the coefficient of absolute risk aversion.

As illustrated by Ross, the Arrow-Pratt measure of risk aversion may betoo weak when attempting to make inferences regarding how agents that aremore risk averse will behave. In particular, Ross examines a situation inwhich agents can only partially insure against risk. Comparing the choice ofwhether or not to purchase insurance by two individuals whose risk attitudescan be compared in the Arrow-Pratt sense, Ross obtains the counterintuitiveresult that the more risk averse agent may be willing to pay less for insurance.Ross proposes a new definition of more risk averse, as well as an alternativedefinition of an asset being riskier, and shows that this counterintuitive out-come cannot occur if his definitions are applied.2 Ross also analyzes a simpleportfolio selection problem and illustrates that, when comparing the optimalportion of wealth to invest in two assets that can be compared in terms ofriskiness (using Ross’ definition of one asset being riskier than another), it ispossible for an individual that is more risk averse in the Arrow-Pratt senseto optimally invest more in the riskier asset. This counterintuitive outcomecannot occur if Ross’ definition of more risk averse is applied. Hadar andSeo (1990) examine a similar portfolio selection problem and show that sucha counterintuitive result can occur even when Ross’ new definition of morerisk averse is applied, for assets that can be compared in terms of riskiness

2Ross states that an asset Y that offers a higher return than an asset X is a riskierasset if E (Y −X|x) ≥ 0 for all x ∈ X.

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based upon the traditional definition of Rothschild and Stiglitz (1970).3

Like the Arrow-Pratt concept, Ross’ notion of more risk averse is statedas a condition directly on the Bernoulli utility functions of agents. UnderRoss’ definition agent a is more risk averse than agent b if there exists λ > 0such that for all x and y: u′′a(x)/u′′b (x) ≥ λ ≥ u′a(y)/u′b(y). This definition isstrictly stronger than the Arrow-Pratt concept.

3 Definition of More Risk Averse

Suppose an agent’s preferences over a class of random variables X can bestated as preferences over expected payoff and level of risk alone.4 When thiscan be done, all agents agree on the relative riskiness of the random variablesin X: the level of risk of a random variable is completely indexed by thevariance of the random variable. That is, if asked to rank the alternativesin terms of riskiness, all agents, regardless of their type or degree of riskaversion, would rank them in the same order. If agents agree on the riskinessof the alternatives, those with lower variance are viewed as less risky.

If the preferences of agents cannot be stated over mean and variancealone, then all risk averse agents do not agree on the relative riskiness ofthe random variables in the class X. In such instances there is no clearranking of the available alternatives in terms of riskiness. As a result, itwould be meaningless to attempt to compare the level of riskiness of theoptimal choices of different agents.

Throughout this discussion, the variance of a random variable will bedenoted by ρ in order to reinforce the fact that variance is a complete indexof riskiness. Consider x ∈ X. The variance and expected value of x are givenby the pair (ρx, µx).

Let �i denote the preferences of agent i. Assume that the preferences ofthis expected utility maximizing agent can be represented by a continuousfunction ui(x), such that u′i(x) > 0 and u′′i (x) < 0. As a result, the vonNeumann-Morgenstern expected utility of this individual, that is Ui (x), isstrictly increasing in µx and strictly decreasing in ρx.

3Mathews (2004a) reexamines the simple portfolio selection problem considered byHadar and Seo and argues that the outcome is not so counterintuitive, since the result issimply that an agent that is more risk averse will choose a less risky portfolio.

4This is always the case for agents with quadratic Bernoulli utility functions. For riskaverse expected utility maximizing agents: Bigelow (1993) shows that this is the case ifand only if the class of random variables from which agents must choose is normalized riskcomparable; Meyer (1987) shows that this is the case if the class of random variables fromwhich agents must choose satisfies a location and scale parameter condition.

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When agents agree upon the relative riskiness of the random variablesfrom which they are choosing, not only is there a clear notion of whichoutcomes are riskier, but also the notion of what it means for one agent to bemore risk averse than another is intuitively clear. Consider a second randomvariable in this class, y ∈ X, with ρy < ρx and µy < µx (y is characterizedby “less risk” and “less return” than x). Suppose an agent b prefers y to x.Drawing upon the intuition presented by Ross in his Application 1, it canbe recognized that when comparing the random alternatives x and y, agentb does not find the additional expected payoff associated with x (that is,µx− µy) to be large enough to compensate for the additional risk associatedwith x (that is, ρx − ρy). Ross properly notes that if a second agent a isto be considered more risk averse than b, then a should also find that theadditional expected payoff associated with x is not large enough to offset theadditional risk associated with x.

Now consider a third random variable from this class, z ∈ X, with ρz > ρx

and µz > µx (z is characterized by “more risk” and “more return” than x).Suppose agent a prefers z to x. For individual a, the reduction in expectedpayoff associated with x in comparison to z (that is, µz − µx) is too large tocompensate for the reduction in risk (that is, ρz−ρx). By the same intuitionas above, if b is to be considered less risk averse than a, then b should alsofind that the reduction in expected payoff associated with x is too large tocompensate for the reduction in risk.

These intuitive notions are easily stated after generalizing the idea ofacceptance sets to alternatives characterized by positive levels of risk. Theconcept of an acceptance set as defined by Yaari (1969) refers to the set ofrisky alternatives which an agent is willing to accept over receiving a certainpayment of zero. The definition of an acceptance set below extends this con-cept for comparisons to random payoffs with non-zero expectations.

Definition 1. Ai (x) = {y|(ρy, µy) �i (ρx, µx)} is individual i’s accep-tance set to x.

The set Ai (x) is the set of risky outcomes that agent i strictly prefers tox. Decompose Ai (x) into two sets: the increased risk acceptance set and thedecreased risk acceptance set as follows.

Definition 2. AHi (x) = {y|(ρy, µy) �i (ρx, µx); ρy > ρx} is individual i’s

increased risk acceptance set to x.

Definition 3. ALi (x) = {y|(ρy, µy) �i (ρx, µx); ρy < ρx} is individual i’s

decreased risk acceptance set to x.

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AHi (x) is the set of risky outcomes with higher levels of risk which i

prefers to x; ALi (x) is the set of risky outcomes with lower levels of risk

which i prefers to x. The proposed definition of more risk averse is stated interms of the relationship of the acceptance sets for two different agents.

Definition of More Risk Averse. Agent a is more risk averse thanagent b over the class of random variables X, denoted aRb, if for all x ∈ X:AL

b (x) ⊆ ALa (x) and AH

b (x) ⊇ AHa (x).

This definition simply states that if preferences can be stated over ex-pected payoff and level of risk alone, a is more risk averse than b if thefollowing two conditions are met. First, for any fixed reduction in risk, ais willing to accept a larger decrease in expected payoff than b will accept.Second, to be compensated for any fixed increase in risk, a requires a greaterincrease in expected payoff than b requires. This relationship is illustratedin Figure 1, by sketching the indifference curves through an arbitrary pointx for agents a and b such that aRb.5 The acceptance set to the point x foreach individual is the area above the indifference curve through x.

Note that the first part of this definition was stated by Ross when compar-ing the risk premium for two individuals in environments where only partialinsurance is available. Ross noted that if a is to be considered more riskaverse than b, then for any reduction in risk a should be willing to incur agreater reduction in expected payoff (that is, should be willing to pay a largerpremium) than b. This implies AL

b (x) ⊆ ALa (x).

Further, the second part of the definition above is equivalent to the firstpart. In order to see this, suppose that AL

b (x) ⊆ ALa (x) for all x ∈ X.

Consider an arbitrary random variable x ∈ X, characterized by µ and ρ.Now consider a higher level of risk ρ > ρ. Let x (with variance ρ andexpected value µ > µ) denote the random variable in the class X for whichUb(x) = Ub(x). Since AL

b (x) ⊆ ALa (x) for all x ∈ X, it follows that AL

b (x) ⊆AL

a (x). This implies Ua(x) ≥ Ua(x). Since this is true for any arbitrary x, itfollows that AH

b (x) ⊇ AHa (x) for any such x. That is, if AL

b (x) ⊆ ALa (x) for

all x ∈ X, then AHb (x) ⊇ AH

a (x) for all x ∈ X.Meyer (1987) notes that when considering a choice over a class of random

variables satisfying the location and scale parameter condition, if a is morerisk averse than b in the Arrow-Pratt sense, then at any particular pointthe indifference curve of a must be steeper than the indifference curve of b.

5For this illustration of preferences to be valid, it must be that each (ρ, µ) throughwhich an indifference curve is drawn could result from some x ∈ X.

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An implication of this observation is that the definition proposed here canbe applied when analyzing a choice over such a class of random variables.However, the location and scale parameter condition is a sufficient, but notnecessary, condition for being able to consistently state preferences over ex-pected payoff and level of risk. Thus, the definition developed here need notalways apply whenever preferences can be stated over mean and variance.

4 Implication on the Optimal Level of Risk

The definition of more risk averse proposed here is only of use if it allows usto draw inferences about the behavior of agents making decisions under risk.

The optimization problem of such an agent can be stated as

maxy

U(y)

subject toy ∈ Θ

where Θ represents the set of feasible alternatives. x∗i ∈ Θ is an optimalchoice if and only if Θ ∩ Ai (x

∗i ) = ∅. That is, a feasible alternative x∗i is

optimal for i if there is no other feasible alternative which i strictly prefers.Let ρx∗i

denote the level of risk associated with x∗i .Note that the solution to this optimization problem for agent i need not

be unique. Of all the x∗i which solve this problem, let x∗i denote the optimalchoice with the highest level of risk and let x∗i denote the optimal choicewith the lowest level of risk. In general, these choices lead to ρx∗i

and ρx∗irespectively, with ρx∗i

≤ ρx∗i. If there is a unique solution to this problem for

i, then x∗i = x∗i = x∗i and ρx∗i= ρx∗i

= ρx∗i.

Theorem 1 characterizes the relationship between the proposed definitionof more risk averse and the solution to the problem stated above.

Theorem 1. Given a pair of agents a and b and a class of randomvariables X: aRb if and only if for any set of feasible alternatives Θ ⊆ X,ρx∗a ≤ ρx∗

b.

Proof of Theorem 1. Consider two agents a and b such that aRb. Forany fixed set Θ, let x∗b denote the optimal choice of b with the lowest level ofrisk and let x∗a denote the optimal choice of a with the highest level of risk.x∗b is such that Θ ∩ Ab (x∗b) = ∅, implying Θ ∩ AH

b (x∗b) = ∅.Since aRb, AH

b (x∗b) ⊇ AHa (x∗b). As a result, Θ ∩ AH

a (x∗b) = ∅, implyingthat there is no feasible point with a higher level of risk that a prefers to x∗b .Thus, ρx∗a ≤ ρx∗

b.

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Now consider two agents such that a is not more risk averse than b overX. As a result, there exists an x ∈ X such that either AL

b (x) 6⊆ ALa (x)

or AHb (x) 6⊇ AH

a (x). If ALb (x) 6⊆ AL

a (x), then there exists wl ∈ X withρwl

< ρx such that wl �b x while x �a wl. Given a choice over Θ = {wl, x},x∗b = x∗b = wl and x∗a = x∗a = x. As a result, ρx∗a > ρx∗

b. If AH

b (x) 6⊇ AHa (x),

then there exists wh ∈ X with ρx < ρwhfor which x �b wh and wh �a x. If

Θ = {x, wh}, then x∗b = x∗b = x and x∗a = x∗a = wh. Thus, ρx∗a > ρx∗b. Q.E.D.

By Theorem 1, if the proposed definition of more risk averse applies,the more risk averse agent will never choose an alternative characterizedby a higher level of risk. As a result, the proposed definition eliminatesthe possibility of counterintuitive outcomes like those identified by Ross (inwhich an agent that is more risk averse chooses an alternative which is riskierthan that which is chosen by the less risk averse agent).

A further implication of Theorem 1 is that in order to be able to unequiv-ocally state that one agent will optimally choose a lower level of risk thananother agent (when choosing from any set Θ ⊆ X) the proposed definitionof more risk averse must be applicable. To see this, consider two agents suchthat (over the class of random variables X) a is not more risk averse thanb and b is not more risk averse than a. For these two individuals, in (ρ, µ)space there must exist at least one indifference curve for a that intersects asingle indifference curve for b more than once.6 Such a pair of indifferencecurves is depicted in Figure 2.7 As illustrated, it is clear that: if Θ = {y, x},then x∗a = y and x∗b = x, while if Θ = {x, z}, then x∗a = z and x∗b = x. Thus,when given a choice between y or x agent b chooses a higher level of risk, andwhen given a choice between x or z agent a chooses a higher level of risk.8

As a result, whenever the definition stated in Section 3 cannot be applied, itis not clear which agent will choose a riskier alternative.

Further, when the proposed definition is not applicable, if any competingnotion of more risk averse is used counterintuitive choices can arise. Thatis, suppose that according to a competing notion of more risk averse it isclaimed that a is more risk averse than b. For Θ = {x, z} we arrive at thecounterintuitive conclusion that a riskier alternative is chosen by the agentthat is supposedly more risk averse. Similarly, suppose that according to a

6This is clear once it is noted that for two such agents we cannot have AHa (x) ⊂ AH

b (x)and AL

a (x) ⊂ ALb (x) for all x ∈ X, since this would lead to a crossing of indifference curves

for each individual agent.7Again, for this illustration of preferences to be valid, it must be that each (ρ, µ)

through which an indifference curve is drawn could result from some x ∈ X.8It should be stressed that both agents agree that y is the least risky and z is the most

risky of these alternatives, since variance is a complete index of riskiness and ρy < ρx < ρz.

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competing notion of more risk averse it is claimed that b is more risk aversethan a. For Θ = {y, x} we arrive at the counterintuitive conclusion that ariskier alternative is chosen by the agent that is supposedly more risk averse.

5 Applying the Definition

The proposed definition of more risk averse may or may not coincide withRoss’ definition or the standard Arrow-Pratt notion of more risk averse. Foragents with quadratic Bernoulli utility functions, the three definitions areequivalent. However, since the concept defined here is not a global statementon the Bernoulli utility functions of agents, but rather may depend upon theclass of random variables X, for agents with non-quadratic Bernoulli utilityfunctions it need not exactly coincide with either of the competing definitions.

When considering agents with non-quadratic Bernoulli utility, attentionmust be restricted to classes of random variables for which preferences can bestated over mean and variance alone. As previously stated, Meyer identifieda sufficient condition for consistency between expected utility and mean-variance analysis. From the results of Meyer it follows that when consideringa class of random variables which differ only by location and scale, aRb solong as a is more risk averse than b in the Arrow-Pratt sense.

The contribution of the proposed definition is most clearly recognizedin the case in which agents have non-quadratic Bernoulli utility functions,but nonetheless the random alternatives under consideration are such thatpreferences can be consistently stated over mean and variance alone (whichcan be done when the alternatives can be ordered in terms of riskiness in thesense of Rothschild and Stiglitz after being normalized to have zero mean, asis the case in Ross’ Application 1). For such classes of random variables whichare normalized risk comparable (but do not necessarily satisfy the locationand scale condition of Meyer), it is shown that the current notion of more riskaverse is applicable whenever agents are ordered by Ross’ definition of morerisk averse. However, the proposed definition may apply even when Ross’definition does not. Specifically, considering agents with constant degreesof absolute risk aversion (for which the Arrow-Pratt notion of more riskaverse does apply, but Ross’ notion does not): a class of random variables isidentified for which the proposed definition does apply, and a class of randomvariables is identified for which the proposed definition does not apply. Thus,Ross’ definition of more risk averse is stronger than the definition proposedhere in that (whenever preferences can be stated over mean and variancealone): the proposed definition of more risk averse applies whenever agentsare ordered according to Ross’ notion of more risk averse, however classes of

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random variables can be identified for which the proposed definition applieseven when agents are not ordered by Ross’ notion of more risk averse. Whilethe proposed definition is weaker than Ross’ definition, as shown in Section4 the proposed definition still eliminates the possibility of counterintuitivechoices. This suggests that when preferences can be stated over mean andvariance alone, Ross’ solution to the partial insurance problem is too strong.

5.1 Quadratic Bernoulli Utility

Consider the quadratic Bernoulli utility function

ui(x) = K0i + K1ix−K2ix2/2

with K1i > 0 and K2i > 0.9 Without any restrictions on the distributionof x, this Bernoulli utility function leads to the von Neumann-Morgensternutility function

Ui(x) = K0i + K1iµx −K2iµ2x/2−K2iρx/2

where µx is the expected value and ρx is the variance of x.The degree of absolute risk aversion for agent i is ri(x) = −u′′i (x)/u′i(x) =

K2i/(K1i −K2ix). a is more risk averse than b in the Arrow-Pratt sense solong as ra(x) ≥ rb(x) for all possible x. This condition is satisfied if and onlyif

K1b/K2b ≥ K1a/K2a.

a is more risk averse than b as defined by Ross, if and only if K2a/K2b ≥(K1a −K2ay)/(K1b −K2by). This condition can be expressed as

K1b/K2b ≥ K1a/K2a.

For the proposed definition, aRb if and only if for every point in (ρ, µ)space the indifference curve of a is at least as steep as the indifference curveof b. The slope of an indifference curve in (ρ, µ) space for such an agent i is:

MRSρx,µx = −∂Ui

∂ρx

∂Ui

∂µx

=12K2i

K1i −K2iµx

.

Thus, aRb if and only if12K2a

K1a−K2aµx≥

12K2b

K1b−K2bµx, or equivalently

K1b/K2b ≥ K1a/K2a.

As a result, for agents with quadratic Bernoulli utility, the three notionsof more risk averse are equivalent.

9In order for the marginal utility of income to always be positive, it should be assumedthat K1i/K2i is greater than the largest possible realization of revenue.

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5.2 Relation to Ross’ Definition

For agents with non-quadratic Bernoulli utility, the proposed definition ofmore risk averse may only apply if the class of random variables under con-sideration is normalized risk comparable. Theorem 2 states that over anyclass of random variables which is normalized risk comparable, the proposeddefinition of more risk averse applies whenever Ross’ definition is applicable.

Theorem 2. Consider the preferences of two agents a and b over anyclass of random variables which is normalized risk comparable. If there ex-ists λ > 0 such that for all x and y u′′a(x)/u′′b (x) ≥ λ ≥ u′a(y)/u′b(y), then aRb.

Proof of Theorem 2. This result can be obtained by relying upon thearguments presented by Ross in the discussion of his Application 1.

Consider a class of random variables X which is normalized risk compa-rable. Choose a random variable x ∈ X, with variance ρ > 0 and expectedvalue µ. Let x denote an arbitrary random variable from X with varianceρ < ρ and expected value µ < µ for which Ub(x) > Ub(x).

Since X is normalized risk comparable, the random variables within thisclass can be ordered in terms of riskiness in the sense of Rothschild andStiglitz, after being normalized to have zero mean. From here it follows thatx can be represented as x = x + v + ε, where v ≥ 0 represents an additionalreturn and ε represents an additional risk such that E (ε|x + v) = 0.

Ross showed that if a is more risk averse than b according to his definition(in which case there exists λ > 0 such that for all x and y u′′a(x)/u′′b (x) ≥ λ ≥u′a(y)/u′b(y)), it must be that there exists a function G(·) (such that G′(·) ≤ 0and G′′(·) ≤ 0) and λ > 0 such that ua(x) = λub(x) + G(x). Therefore, ifRoss’ definition holds, Ua (x) = λUb (x) + E (G (x)) . From here,

λUb (x) + E (G (x)) > λUb (x) + E (G (x))

≥ λUb (x) + E (G (x))

= Ua (x) .

The first inequality above follows from the fact that b prefers x to x; thesecond inequality above follows from the fact that G(·) is such that G′(·) ≤ 0and G′′(·) ≤ 0. As a result, Ua (x) > Ua (x), implying AL

b (x) ⊆ ALa (x) for

any such arbitrary x.In Section 3 it was argued that if AL

b (x) ⊆ ALa (x) for all x ∈ X,

then AHb (x) ⊇ AH

a (x) for all x ∈ X. Thus, it immediately follows thatAH

b (x) ⊇ AHa (x), implying aRb. Q.E.D.

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From Theorem 2 (along with the discussion in Subsection 5.1) we havethat whenever preferences can be stated over mean and variance alone, aRbif agent a is more risk averse than agent b according to Ross.

5.3 Applicability when Ross’ Definition is Violated

Consider two agents a and b with preferences that can be represented by theBernoulli utility functions ua(x) = 1 − e−ax and ub(x) = 1 − e−bx.10 Agenta has a constant degree of absolute risk aversion equal to a; agent b has aconstant degree of absolute risk aversion equal to b. a is more risk averse thanb in the Arrow-Pratt sense if a ≥ b. Ross has shown that his definition ofmore risk averse is not satisfied for such agents. By way of example it is shownthat, depending upon the class of random variables under consideration, itmay or may not be that aRb. The first of these examples illustrates thatagents may be ordered by the proposed definition of more risk averse even ifthey are not ordered by Ross’ definition.

5.3.1 Distribution for which a is more risk averse than b

Consider the random variable xd ∼ U [0, d], with d ∈ [0,∞), and the ran-dom variable yd,c = xd + c, where c is an arbitrary finite constant. It isstraightforward to show that the class of random variables yd,c is normalizedrisk comparable, so that preferences over the random variables within thisclass can be stated over mean and variance alone. For this class of randomvariables, those characterized by lower values of d are less risky. Further, itcan be shown that for this class of random variables, aRb for agents a and bwith constant absolute risk aversion.

In order to see this, consider an arbitrary point in (ρ, µ) space resultingfrom a particular value of d along with a particular value of c. As a functionof d and c, the von Neumann-Morgenstern utility of agent a is

Ua(d, c) = 1−(e−ac/ad

) (1− ead

).

If d is changed to d, an additional payment of Ka is required to maintainthis initial level of von Neumann-Morgenstern utility for agent a. That is,Ka is such that Ua(d, c) = Ua(d, c + Ka). Similarly, let Kb be such thatUb(d, c) = Ub(d, c + Kb). Comparing agents a and b, aRb if and only ifKa ≤ Kb for d < d and Ka ≥ Kb for d > d.

10Recall that for risk averse expected utility maximizing agents, preferences can bestated over mean and variance alone if and only if the class of random variables fromwhich agents must choose is normalized risk comparable as defined by Bigelow (1993).

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Solving for Ka, we have

Ka = −1

aln

{d

d

(1− e−ad

1− e−ad

)},

which can be written as Ka = −η(a)a

with η(a) = ln{

dd

(1−e−ad

1−e−ad

)}.

To see how Ka changes as a changes, examine ∂Ka

∂a= η(a)−aη′(a)

a2 . The sign

of ∂Ka

∂ais the same as the sign of τ(a) = η(a)−aη′(a). It must be shown that

τ(a) < 0 for d < d and τ(a) > 0 for d > d.Since

η′(a) =d

ead − 1− d

ead − 1

we have

τ(a) = ln

{d

d

(1− e−ad

1− e−ad

)}− ad

ead − 1+

ad

ead − 1.

As a result, τ(a) → 0 as a → 0. It remains to show that τ(a) is decreasingin a for d < d and increasing in a for d > d.

Differentiating τ(a), we have

τ ′(a) = −aη′′(a) =d(ad)ead

(ead − 1)2− d(ad)ead

(ead − 1)2.

It must now be argued that τ ′(a) < 0 for d < d and τ ′(a) > 0 for d > d.Since τ ′(a) = 0 if d = d, it is sufficient to show that τ ′(a) is increasing in d.Treating this as a function of d, we can write

Λ(d) =d(ad)ead

(ead − 1)2− d(ad)ead

(ead − 1)2.

Defining λ(x) = 2ex − 2− xex − x we have

Λ′(d) = −

adead

(ead − 1)3

λ(ad).

The desired result will follow if λ(x) < 0 for all x > 0.λ(0) = 0. λ′(x) = ex − xex − 1, implying λ′(0) = 0. λ′′(x) = −xex,

which is negative for all x > 0. Thus, λ(x) < 0 for all x > 0. As aresult, aRb for agents with constant absolute risk aversion choosing from theclass of all uniformly distributed random variables.11 Note that Theorem 2

11As Meyer points out, the class of all uniformly distributed random variables satisfiesthe location and scale parameter condition. The fact that the proposed definition of morerisk averse is applicable also follows from this observation.

13

and this example together imply that the proposed definition is weaker thanRoss’ definition, in that the proposed definition not only applies wheneveragents are ordered by Ross’ definition, but may apply even when agentsare not ordered by Ross’ definition. Further (since the proposed definitionstill eliminates the possibility of counterintuitive choices like those identifiedby Ross), it follows that when agents have non-quadratic Bernoulli utilityfunctions but preferences can consistently be stated over mean and variance,the intuition suggested by Ross in his Application 1 may still apply, evenwhen his proposed definition of more risk averse does not apply.

5.3.2 Distribution for which a is not more risk averse than b

Let xv denote the random variable defined by the distribution function

Fv(x) =

{x(1 + v)− x2, 0 ≤ x < v − v2

2

1− v + x(1 + v)− x2, v − v2

2≤ x ≤ v

with v ∈ [0, 1].12 This function is illustrated in Figure 3 for v = .5. Consideryv,c = xv + c, where c is an arbitrary finite constant. It is shown in Mathews(2003) that the class of random variables yv,c is normalized risk comparable,with those members characterized by lower v being less risky. As a result,preferences over the random variables within this class can be stated overmean and variance alone. Further, for agents with preferences that can berepresented by Bernoulli utility functions of the form ua(x) = 1 − e−ax, thenotion of more risk averse developed here is not applicable.

To ease the calculations and to highlight the fact that this example is notunique, the result is shown for a similar class of random variables. Let xp

denote the random variable defined by the distribution function

Hp(x) =

{ √px, 0 ≤ x < 2

√p/3

1− p +√

px, 2√

p/3 ≤ x ≤ √p

with p ∈ [0, 1]. This random variable can be thought of arising from asituation in which: 2

√p/3 is realized with probability 1 − p, while a ran-

dom variable uniformly distributed between zero and√

p is realized withprobability p. This function is illustrated in Figure 4 for p = .5. Consideryp,c = xp + c, where c is an arbitrary finite constant. It is straightforward toshow that the class of random variables yp,c is normalized risk comparable,with those members characterized by lower p being less risky.

12This is the random revenue resulting in Mathews (2004b) when a seller facing two riskneutral, non-discounting bidders in an auction with a buyout option sets a buyout priceB = v − v2

2 .

14

For an agent with ua(x) = 1−e−ax, the von Neumann-Morgenstern utilityfor an alternative yp,c can be expressed as a function of p and c as

Ua(p, c) = 1− e−ace−a2√

p/3 + pe−ace−a2√

p/3 −√

pe−ac

a

(1− e−a

√p).

Evaluating this function at p = 1 and c = 0 we have

Ua(1, 0) = 1−(1− e−a

)/a.

Since the certainty equivalent of a risky alternative decreases as the degreeof absolute risk aversion increases, in order to illustrate that for this class ofrandom variables a is not more risk averse than b, it is sufficient to identifya pair (p, c) such that for some a > b we have Ua(1, 0) > Ua(p, c) whileUb(1, 0) < Ub(p, c).

Consider a = 25, b = 12, and the pair (p, c) = (.5,−.02). Evaluating thefunctions above at these values:

.960000 ≈ U25(1, 0) > U25(.5,−.02) ≈ .953361

and.916667 ≈ U12(1, 0) < U12(.5,−.02) ≈ .922886.

Thus, in comparison to the random outcome resulting from (p, c) = (1, 0),there exists a point in (ρ, µ) space with less risk that is preferred by b but notpreferred by a. That is, there exists a less risky prospect which is preferredby the individual that is less risk averse in the Arrow-Pratt sense, but notpreferred by the individual that is more risk averse in the Arrow-Pratt sense.This counterintuitive result is similar in nature to those identified by Ross.

For this example, the certainty equivalent to the risky outcome charac-terized by (p, c) = (1, 0) is C12 ≈ .207076 for agent b and C25 ≈ .128755 foragent a. Choose an arbitrary certain outcome of C such that C25 < C < C12.This point would result from p = 0 and c = C. It is clear that

U25(0, C) > U25(1, 0)

andU12(0, C) < U12(1, 0)

for any such C.Thus, y0,C ∈ AL

a (y1,0) but y.5,−.02 /∈ ALa (y1,0). Similarly, y.5,−.02 ∈ AL

b (y1,0)

but y0,C /∈ ALb (y1,0). As a result, AL

a (y1,0) 6⊆ ALb (y1,0) and AL

b (y1,0) 6⊆

15

ALa (y1,0), implying that a is not more risk averse than b and b is not more

risk averse than a in the sense defined here.13

The preceding examples show that agents with constant degrees of abso-lute risk aversion may be or may not be ordered by the proposed definition.This reinforces the fact that the applicability of the proposed definition de-pends upon the class of random variables under consideration. This is instark contrast to the definitions of Arrow-Pratt and Ross, both of which re-late to the set of all possible lotteries. As a result, for any two agents a and bwith Bernoulli utility functions ua(x) and ub(x), the notions of Arrow-Prattand Ross either apply or do not apply. Since the proposed definition relatesseparately to each class of random variables (that is, relates separately to yd,c

and yp,c), for agents a and b with fixed preferences, the proposed definitionmay apply for one class of random variables and not for another. Further,because of this dependence on the class of random variables, it is possible forthe proposed definition to apply even when the Arrow-Pratt notion of morerisk averse does not.14

6 Conclusion

Building upon the intuition of Ross, a definition of more risk averse has beenproposed for situations in which all agents agree on the relative riskiness ofthe alternatives from which they must choose. When this definition can beapplied it is possible to infer how the optimal level of risk will differ for twoagents. If the definition cannot be applied, it is not clear which agent willchoose a less risky alternative.

For agents with quadratic Bernoulli utility functions, this definition coin-cides with Ross’ definition and the standard Arrow-Pratt notion of more riskaverse. Further, whenever preferences can be stated over mean and variancealone, the proposed definition applies for agents that are ordered by Ross’definition.

Further, the proposed definition may apply even when Ross’ definitiondoes not. For agents with a constant degree of absolute risk aversion (inwhich case Ross’ notion of more risk averse does not apply), the proposeddefinition can be applied when the choice is over uniformly distributed ran-dom variables. Thus, when agents have non-quadratic Bernoulli utility func-

13An identical conclusion can be reached for the class of random variables yv,c, byexamining the preferences of agents with a = 25 and b = 12 for the point in (ρ, µ) spaceinduced by v = .5 and c = −.02.

14This can be illustrated by considering an appropriately restricted class of randomvariables X.

16

tions but preferences can consistently be stated over mean and variance, theintuition suggested by Ross may still apply, even when his proposed defini-tion of more risk averse does not apply. In such instances, Ross’ solution tothe partial insurance problem appears too strong.

Finally, for agents with a constant degree of absolute risk aversion the pro-posed definition does not always apply whenever preferences can be statedover mean and variance. This is shown to be the case for a class of randomvariables which is similar to that which arises in Mathews (2004b).

ReferencesBigelow, J.P. (1993). ‘Consistency of Mean-Variance Analysis and Ex-

pected Utility Analysis. A Complete Characterization’, Economics Letters,Vol. 43, No. 2, pp. 187-192.

Hadar, J. and T.K. Seo (1990). ‘Ross’ Measure of Risk Aversion andPortfolio Selection’, Journal of Risk and Uncertainty, Vol. 3, No. 1, pp.93-99.

Mathews, T. (2003). ‘A Risk Averse Seller in a Continuous Time Auctionwith a Buyout Option’, Brazilian Electronic Journal of Economics, Vol. 5,No. 2.

Mathews, T. (2004a). ‘Portfolio Selection with Quadratic Utility Revis-ited’, Forthcoming, GENEVA Papers on Risk and Insurance Theory.

Mathews, T. (2004b). ‘The Impact of Discounting on an Auction witha Buyout Option: a Theoretical Analysis Motivated by eBay’s Buy-It-NowFeature’, Journal of Economics (Zeitschrift fur Nationalokonomie), Vol. 81,No. 1, pp. 25-52.

Meyer, J. (1987). ‘Two-Moment Decision Models and Expected UtilityMaximization’, The American Economic Review, Vol. 77, No. 3, pp. 421-430.

Ross, S.A. (1981). ‘Some Stronger Measures of Risk Aversion in the Smalland the Large with Applications’, Econometrica, Vol. 49, No. 3, pp. 621-638.

Rothschild, M. and J.E. Stiglitz (1970). ‘Increasing Risk: I. A Definition’,Journal of Economic Theory, Vol. 2, No. 3, pp. 225-243.

Yaari, M.E. (1969). ‘Some Remarks on Measures of Risk Aversion andTheir Uses’, Journal of Economic Theory, Vol. 1, No. 3, pp. 315-329.

17

x

aIC

bIC

FIGURE 1

Illustration of the Relation More Risk Averse

ρ

µ

xρ

xµ

bIC

aIC

aIC

bIC

FIGURE 2

Situation in which the proposed definition is not applicable

ρ

µ

x

z

y

( )µρ ˆ,ˆ

( )µρ ,

FIGURE 3 CDF of vx with 5.=v

FIGURE 4 CDF of px with 5.=p