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ScienceDirect Journal of Economic Theory 157 (2015) 606–623 www.elsevier.com/locate/jet

Decreasing aversion under ambiguity Frédéric Cherbonnier a,∗ , Christian Gollier b,1 a Toulouse School of Economics (IEP, LERNA and IDEI), France b Toulouse School of Economics (LERNA and IDEI), France

Received 20 May 2014; accepted 3 January 2015 Available online 16 January 2015

Abstract Under which condition does the set of desirable uncertain prospects expand when wealth increases? We show that the decreasing concavity (DC) of the utility function u is necessary and sufficient in the α-maxmin expected utility model. In the smooth ambiguity aversion model with the ambiguity valuation function φ, the DC of u and of φ ◦ u is necessary and sufficient. An alternative classical definition of decreasing aversion is based on the hypothesis that the investment in a risky asset is increasing in wealth. We show that this hypothesis does not hold in general under ambiguity aversion, and that one needs to constrain the structure of ambiguity to obtain unambiguous results. © 2015 Elsevier Inc. All rights reserved. JEL classification: D81 Keywords: Decreasing concavity; Portfolio choice; α-MEU; Smooth ambiguity aversion; Maxmin

1. Introduction One of the most ubiquitious assumption in the economics of risk is that wealthier people are less risk-averse. Various definitions of the concept of decreasing aversion exist in the literature. For example, an agent is said to have decreasing aversion if any risk that is undesirable * Corresponding author.

E-mail addresses: [email protected] (F. Cherbonnier), [email protected] (C. Gollier). 1 This research was supported by the Chair of the Financière de la Cité at TSE, by the Chair SCOR-IDEI, and by the

European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) Grant Agreement no. 230589. http://dx.doi.org/10.1016/j.jet.2015.01.002 0022-0531/© 2015 Elsevier Inc. All rights reserved.

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at some specific wealth level is also undesirable at any smaller wealth level. Another definition of decreasing aversion is that in the one-risk-free-one-risky-asset portfolio choice problem, the demand for the risky asset is an increasing function of initial wealth. In the classical expected utility model, these two definitions of decreasing aversion are equivalent, and the necessary and sufficient condition is expressed by the decreasing nature of the Arrow–Pratt index of absolute risk aversion (DARA). Since Arrow [2] postulated decreasing absolute risk aversion, numerous studies have confirmed this hypothesis using both experimental (Levy [22], Guiso and Paiella [18]) and econometric (Bar-Shira et al. [3]) methods. This universally accepted property of individual risk preferences plays a crucial role in many applications of the expected utility theory, as illustrated in Gollier [14]. Risk aversion is indeed essential to understand individual choices about wealth accumulation, retirement, portfolio accumulation and insurance. Somehow, how risk aversion shapes economic decision is still an open issue. In particular, the observed level and share2 of savings invested in risky assets is not well predicted by existing models. Recent works enrich these models by taking into account labor income (e.g. Viceira [29], Cocco et al. [8]) or housing (e.g. Cocco [7], Yao and Zhang [30]), or by refining preferences with persistent habits (Gomes and Michaelides [16], Brunnermeier and Nagel [5]) and more recently ambiguity aversion (Campanale [6]). In this paper, we explore the concept of decreasing aversion in the context of ambiguity and ambiguity aversion. In most cases, the probability distribution of the risk is not perfectly known, i.e., it is ambiguous. Examining a simple thought experiment, Ellsberg [10] suggested that economic agents do not behave accordingly to the subjective expected utility model. Under ambiguity, contrary to Savage [27] theory of subjective expected utility, they do not use a subjectively chosen probability distribution to compute the expected utility of the set of possible acts to determine their optimal strategy. Many experiments have confirmed Ellsberg’s hypothesis that in the absence of an objective probability distribution, individuals tend to favour a relatively pessimistic plausible distribution to measure their welfare ex ante. Gilboa and Schmeidler [11] were the first to propose a decision criterion (thereafter refered to as maxmin) that is compatible with Ellsberg’s hypothesis, and that generalizes the expected utility model. In short, agents are assumed to have multiple priors whose formation is a characteristic of the preferences of the agent. The agent’s ex ante welfare associated to an act is the smallest expected utility generated by this act over the different possible priors. More recently, two models have been proposed to account for ambiguity attitude. Ghirardato, Maccheroni and Marinacci [12] have proposed the α-maxmin expected utility (or α-MEU) family of preferences in which the agent’s ex ante welfare is measured by an α-weighted average of the smallest and the largest expected utility levels among a convex, compact set of probability distributions. The alternative approach (thereafter refered to as KMM) provided by Klibanoff, Marinacci and Mukerji [21] represents the agent’s welfare under uncertainty by the certainty equivalent of the different prior-dependent expected utility levels. This certainty equivalent is 2 Another property central to this literature is relative risk aversion (RRA), that determines how the share invested in the risky asset evolves with initial wealth in the one-risk-free-one-risky-asset portfolio choice problem. Here, we focus on absolute rather than relative risk aversion, since the DARA property has been validated by most empirical studies whereas empirical evidence regarding RRA are not conclusive: some empirical studies support decreasing RRA (Cohn et al. [9], Morin and Suarez [23], Levy [22], Guiso et al. [17], Ogaki and Zhang [24]), other studies are in favor of increasing RRA (Siegel and Hoban [28], Bar-Shira et al. [3], Barksy et al. [4]). It must be noted that decreasing relative risk aversion is sufficient but not necessary to decreasing absolute risk aversion.

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computed by using a function φ that is increasing and concave, and whose degree of concavity is an index of ambiguity aversion. For these two decision criteria under ambiguity, we determine the conditions under which wealthier people are less averse to uncertainty, under the two standard definitions for this concept. Consider first the definition of decreasing aversion based on the shrinkage of the set of desirable lotteries when wealth is reduced. In the α-MEU family of preferences, this property is obtained under the necessary and sufficient condition that the utility function exhibits DARA. In the case of smooth ambiguity aversion, the shrinkage of the set of desirable lotteries when wealth decreases prevails if and only if both u and φ ◦ u exhibit decreasing concavity à la Arrow–Pratt. This condition is weaker than the sufficient condition that u and φ are decreasingly concave. The definition of decreasing aversion based on the increasing demand for the risky asset when wealth increases is more complex to characterize. In the maxmin model, we show that the decreasing concavity of the utility function is not enough to guarantee the desired comparative statics property, except in the small. Different sufficient conditions are derived. For example, a sufficient condition is that the utility function belongs to the HARA class with decreasing aversion. Another sufficient condition is that all priors can be ranked according to the Jewitt’s order, and relative prudence is smaller than relative risk aversion plus one. A similar condition is obtained in the KMM’s smooth ambiguity aversion model, under the addictional condition that φ is decreasingly concave. The condition relating relative prudence and relative risk aversion may be removed at the cost of replacing the Jewitt’s order by the monotone likelihood ratio order, which is stronger. 2. Acceptance of risk In this section, we characterize the conditions under which wealthier people have a larger set of desirable loteries. Definition 1. We say that an agent is Decreasingly Averse if a reduction in wealth can never makes an undesirable uncertain investment desirable. In other words, by contraposition, the set of undesirable lotteries shrinks when wealth increases. In that case, we say that the agent exhibits Decreasing Aversion (DA). The benchmark model is expected utility. Under this model, decreasing aversion is translated into the following condition. For any initial wealth z, and for any random variable x so that the support of z + x is in the domain of the utility function u, we have that Eu(z + x ) ≤ u(z)

⇒

Eu z + x ≤ u z

∀z ≤ z.

(1)

This means that if an agent with utility u dislikes lottery x at wealth level z, she must also dislike it at all wealth levels smaller than z. We know since Pratt [25] that this is true if and only if the utility function u exhibits decreasing absolute risk aversion (DARA). We formalize this by using the following definition. Definition 2. We say that a function f : R → R satisfies Decreasing Concavity (DC) if −f /f is non-increasing.

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It is easy to check that f DC means that there exists a concave function g such that −f = g ◦ f . In the expected utility model, decreasing aversion holds if and only if u exhibits DC. Notice that condition (1) is equivalent to the single-crossing from below of function v defined as v(z) = Eu(z + x ) with respect to function u. Thus, condition (1) is equivalent to Eu(z + x ) = u(z)

⇒

Eu (z + x ) ≥ u (z).

(2)

Suppose now that there is some ambiguity about the true distribution of the payoff of the lottery. For the sake of simplicity, suppose that the payoff of the lottery has n possible distributions, corresponding to random variables x1 , . . . , xn . Departing from the EU model, we hereafter examine two decision models under uncertainty: α-MEU and smooth ambiguity aversion. Under the α-MEU model with α ∈ [0, 1], the agent measures his/her welfare under uncertainty by an α-weighted average of the smallest and the largest expected utility levels. In this context, dexn ) so that the creasing aversion requires for any z, and for any set of random variables ( x1 , . . . , support of z + xθ is in the domain of the utility function u for all θ = 1, . . . , n, α min Eu(z + xθ ) + (1 − α) max Eu(z + xθ ) ≤ u(z) θ

(3)

θ

implies that ∀z ≤ z α min Eu z + xθ + (1 − α) max Eu z + xθ ≤ u z . θ

(4)

θ

Proposition 1. The α-MEU criterion implies decreasing aversion if and only if u exhibits decreasing concavity. Proof. Suppose by contradiction that there exists z and z ≤ z such that α min Eu(z + xθ ) + (1 − α) max Eu(z + xθ ) ≤ u(z)

(5)

α min Eu z + xθ + (1 − α) max Eu z + xθ > u z .

(6)

θ

θ

and θ

θ

Let us define a = arg min Eu(z + xθ ) θ

and b = arg max Eu z + xθ . θ

Because α ∈ [0, 1], these definitions imply that αEu(z + xa ) + (1 − α)Eu(z + xb ) ≤ α min Eu(z + xθ ) + (1 − α) max Eu(z + xθ ) θ

θ

(7)

and αEu z + xa + (1 − α)Eu z + xb ≥ α min Eu z + xθ + (1 − α) max Eu z + xθ (8) θ

θ

Let us define the random variable X by its distribution (α, xa ; 1 − α, xb ). Combining inequalities (5) and (7) implies that Eu(z + X) ≤ u(z). Similarly, combining inequalities (6) and (8)

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implies that Eu(z + X) > u(z ). In short, a reduction in wealth from z to z makes the initially undesirable lottery X desirable. This is a contradiction. 2 We can conclude from this result that the multi-prior α-MEU model has the property of decreasing aversion if and only if the corresponding expected utility model has it also, which is the case if −u /u is non-increasing. This result applies in particular in the special cases of the maxmin and the maxmax models. We now examine the model of smooth ambiguity aversion, as introduced by Klibanoff, Marinacci and Mukerji (KMM) [21]. A new ingredient is the vector (q1 , . . . , qn ) of non-negative scalars that sum up to unity. Parameter qθ can be interpreted as the (subjective) probability that xθ describes the true distribution of the lottery. Another ingredient is a new real-valued function φ that is increasing and concave, so that the welfare if the lottery is accepted is measured by the certainty equivalent of the conditional expected utility Eu(z + xθ ): n W (z) = φ −1 qθ φ Eu(z + xθ ) θ=1

Because φ is assumed to be concave, the existence of ambiguity reduces welfare since, by Jensen’s inequality, W (z) ≤ Eu(z + x ), where x is distributed as (q1 , x 1 ; . . . ; qn , xn ). If the lottery is rejected, the agent enjoys welfare φ −1 (φ(u(z))) = u(z). Decreasing aversion as defined in this section requires in this case that W single-crosses u from below. This requires that the following property be satisfied: n −1 φ qθ φ Eu(z + xθ ) = u(z) ⇒ θ=1 n

xθ ) Eu (z + qθ φ Eu(z + xθ ) ≥ u (z)φ u(z) .

(9)

θ=1

One can easily extract two necessary conditions for decreasing aversion in this framework. The first is that u must be DC, since one possibility is that all xθ be identically distributed. Indeed, in that case, condition (9) is equivalent to (2). The second condition can be derived in the xθ takes value yθ almost surely, for special case in which all xθ are degenerated. Suppose that all θ = 1, . . . , n. Let random variable y be distributed as (y1 , q1 ; . . . ; yn , qn ). Then, condition (9) can be rewritten as Eφ ◦ u(z + y ) = φ ◦ u(z)

⇒

E(φ ◦ u) (z + y ) ≥ (φ ◦ u) (z).

As recalled above, this condition holds for all z and all random variable y if and only if φ ◦ u exhibits DC. This observation is just a restatement of the fact that, when the multiple priors are all degenerated, the KMM model simplifies to the EU model with utility function φ ◦ u. This means that both conditions u DC and φ ◦ u DC are necessary for decreasing aversion in the KMM model. We hereafter show that this joint condition is also sufficient.3 3 In the small, this proposition is a consequence of Maccheroni, Marinacci and Ruffino [26], who show that, in the KMM model, the risk premium is approximately equal to the sum of the Arrow–Pratt indexes of both φ and φ ◦ u respectively multiplied by two different variance parameters.

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Proposition 2. Consider the KMM model characterized by functions (u, φ). It exhibits decreasing aversion if and only if u and φ ◦ u both exhibit decreasing concavity. Proof of sufficiency. Let yθ be the certainty equivalent of xθ under function u, i.e., u(z + yθ ) = Eu(z + xθ ). Let random variable y be distributed as (y1 , q1 ; . . . ; yn , qn ). Suppose that n −1 W (z) = φ qθ φ Eu(z + xθ ) = u(z), θ=1

or equivalently, that Eφ ◦ u(z + y ) = φ ◦ u(z). Because u is DC, Eu(z + xθ ) = u(z + yθ ) implies xθ ) ≥ u (z + yθ ). It implies in turn that that Eu (z + n

n xθ ) Eu (z + qθ φ Eu(z + xθ ) ≥ qθ φ u(z + yθ ) u (z + yθ ) = E(φ ◦ u) (z + y ).

θ=1

θ=1

Because φ ◦ u is DC and Eφ ◦ u(z + y ) = φ ◦ u(z), we know that E(φ ◦ u) (z + y ) ≥ (φ ◦ u) (z). We conclude from the previous equation that n

xθ ) Eu (z + qθ φ Eu(z + xθ ) ≥ φ u(z) u (z) = φ W (z) u (z).

θ=1

It implies that the right condition in (9) is satisfied. This concludes the proof of the sufficiency of u and φ ◦ u being DC. 2 An interesting case arises when u exhibits constant absolute risk aversion and φ is a power function: u(c) = −A−1 exp(−Ac) and φ(u) = −(−u)1+γ /(1 + γ ), with γ ≥ 0. Function φ is increasing and concave in the relevant domain of u. Observe that −φ (u)/φ (u) = −γ /u, which is increasing in u. Thus, we have that u and φ exhibit respectively constant concavity and increasing concavity. However, these preferences satisfy the necessary and sufficient condition of Proposition 2, since (φ ◦ u)(c) = −

e−A(1+γ )c , (1 + γ )A1+γ

is weakly DC. This shows that this necessary and sufficient condition does not require that φ exhibits DC. Only the DC of u and φ ◦ u is required. A sufficient condition is that u and φ be DC. Indeed, we have that −

(φ ◦ u) (z) u (z) φ (u(z)) (z) − = − u . (φ ◦ u) (z) φ (u(z)) u (z)

(10)

Because u is increasing and u is decreasing, the right-hand side of this equality is decreasing in z if both −φ /φ and −u /u are decreasing. This yields the following corollary. Corollary 1. Consider the KMM model characterized by functions (u, φ). It exhibits decreasing aversion if both u and φ exhibit decreasing concavity. When the subjective distribution is uniform, i.e. qθ = 1/n, then the certainty equivalent W (z) of KMM is nothing but the generalized φ-mean of the values Eu(z + xθ ). We can then apply the previous corollary when φ(z) = ln(z), φ(z) = −1/z or φ(z) = zn with n ≤ 1 since those

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functions are DC. Hence, the geometric mean, the harmonic mean and the power mean with exponent lower than one exhibit decreasing aversion when u exhibits decreasing concavity (and takes positive values). 3. Portfolio allocation In this section, we examine an alternative decision model. The decision maker with initial wealth z can invest in two assets. There is a risk free asset whose return is normalized to zero, and a risky asset whose return is expressed by random variable x . The agent must choose the size α of her investment in the risky asset. In the expected utility model, this decision problem can be written as max Eu(z + α x ). α

(11)

In this model, it is easy to check that an increase in the initial wealth raises the dollar investment in the risky asset if and only if u is decreasingly concave, as first shown by Arrow [1]. Indeed, because the objective function is concave in α, this requires to show that the cross derivative of the objective function in (11) with respect to α and z evaluated at the optimal α is positive. We normalize the unit of the risky asset in such a way that it is optimal to invest exactly one monetary unit in the risky asset, so that the first order condition of the above program yields E x u (z + x ) = 0. Because u DC means that −u is a concave function g of u, it implies that x ) u (z + x u (z + E x u (z + x ) = −E x g u(z + x ) ≥ −g u(z) E x ) = 0. The inequality above comes from the observation that for all x, −xg (u(z + x)) ≥ −xg (u(z)). This demonstrates that the DC property in the expected utility model is necessary and sufficient for two testable results. Namely, the DC of u means that when wealth increases, the set of acceptable lotteries inflates, and the demand for the risky asset increases. In the following, we examine whether this property is still satisfied under ambiguity aversion. As in the previous section, we examine two models of ambiguity aversion: the α-MEU model and the KMM model. For expository reasons, we limit the analysis of the α-MEU model to its special case of the maxmin criterion. The polar case of the maxmax criterion is discussed in a technical appendix available online, whose results are briefly summarized at the end of the next section. 3.1. The maxmin model Suppose that the distribution of the return of the risky asset is ambiguous. This ambiguity xn ). Under the maxmin criterion, the is characterized by n possible random variables ( x1 , . . . , decision program can be rewritten as max min Eu(z + α xθ ). α

θ

(12)

It would be natural to expect that the DC of u would be enough to induce the wealthier maxmin investor to invest more in the risky asset. This is not true in general, as shown in the following example.

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Fig. 1. Maxmin welfare as a function of the investment in the risky asset α for two different wealth levels z, in the case described in Example 1. The two curves have been translated vertically for the sake of comparison.

Example 1. Consider function u(c) = c − 1/c in R+ . This function is increasing, concave and x2 ∼ (−9, 0.10719; 2, 0.86155; 3, 0.03126). DC. Consider x1 ∼ (−8, 0.47437; 9, 0.52563) and In Fig. 1, we show that an increase in wealth around z = 10 reduces the demand for the risky asset. This counterexample tells us that some additional restrictions are necessary to obtain the intuitive comparative static result of DA in the maxmin porfolio problem. The next lemma provides a general condition under which an increase of wealth raises the demand for the risky asset in the maxmin model. This condition states that whenever the agent with utility u is indifferent between xj , but would prefer to invest more than one unity in xi and less than one unity in xj , then xi and xj . the expected marginal utility under xi is smaller than under Lemma 1. Suppose that the utility function u is DC. In the maxmin model, an increase in wealth increases the demand for the risky asset if, for all (i, j ) ∈ {1, . . . , n}2 , ⎫ ⎪ Eu(z + x i ) = Eu(z + xj ) ⎬

E x i u (z + x i ) ≥ 0 ≥ E x j u (z + xj ) x i ) ≤ E u (z + xj ) . (13) ⇒ E u (z + ⎪ ⎭ E x i u (z + x i )E x j u (z + x j ) = 0 Proof. Let’s consider the optimal investment α ∗ that solves the maximization problem (12). xθ ) is reached at only one prior θ = i ∈ {1, . . . , n}, then locally If the minimum minθ Eu(z + α ∗ xi ) and the decreasing aversion around z, the decision program can be rewritten as maxα Eu(z +α property is a consequence of u DC. Hence, in order to check the decreasing aversion property, we only need to consider cases where there exist (i, j ) ∈ {1, . . . , n}2 such that the optimum of the decision problem is such that (14) Eu z + α ∗ xi − Eu z + α ∗ xj = 0, where α ∗ is normalized to unity. An optimality condition for α ∗ = 1 at this kink of the objective xi ) and E xj u (z + xj ) cannot be both either strictly positive or strictly function is that E xi u (z + negative. Without loss of generality suppose that E x i u (z + x i ) ≥ 0 ≥ E x j u (z + x j ).

(15)

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Fig. 2. Effect of an increase in wealth from z to z > z on the optimal portfolio allocation in the maxmin model.

If those two terms are zero, the solution α ∗ of the maxmin program coincides with the solutions x j . When z increases slightly, the solution of the maxmin of the EU decision program for x i and program still belongs to the interval delimited by such solutions. Since u is DC, the solutions of the two EU programs increase with z, so is it for α ∗ . Now, suppose that at least one of those two terms is non-zero, and observe that condition (14) characterizes α ∗ locally around z. Fully differentiating this equality with respect to z yields Eu (z + xi ) − Eu (z + xj ) dα ∗ =− . dz E xi u (z + α ∗ xi ) − E xj u (z + α ∗ xj ) This is positive if E[u (z + x i )] ≤ E[u (z + x j )].

2

In fact, sufficient condition (13) is also necessary if n = 2, or if the set of priors can be modified so that only priors i and j drive the solution, i.e., if the minimal expected utility is a corner solution with Eu(z + α ∗ xi ) = Eu(z + α ∗ xj ). In that case, as shown in the proof, a reversal in the right inequality in (13) would imply that α ∗ would be locally decreasing in wealth. In Fig. 2, we have illustrated condition (13). At wealth level z, the objective function h(α) = min(Eu(z + α xi ), Eu(z + α xj )) has a maximum at the kink that is characterized by the left conditions in (13). We also see on the picture that the right condition in (13) is satisfied. Indeed, the increase in wealth has a larger effect on Eu(z + α ∗ xj ) than on Eu(z + α ∗ xi ). It implies that α ∗ (z ) > α ∗ (z). Condition (13) is linked to the DC property of u. Indeed, suppose that one of the two priors is degenerated. Suppose in that case, and without loss of generality, that xi = 0 almost surely. In that special case, condition (13) can be rewritten as follows:

Eu(z + x j ) = u(z) x j ) ≥ u (z). ⇒ E u (z + E x j u (z + xj ) < 0 Observe first that Eu(z + x j ) = u(z) implies that E x j u (z + x j ) ≤ 0. Indeed, function H (α) = Eu(z + α xj ) is concave in α. Thus, if H (1) = H (0), it must imply that H (1) ≤ 0.

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This inequality is strict unless u coincides with a linear function on the domain spanned by z + x j . This means that the above condition can be rewritten as

Eu(z + x j ) = u(z) ⇒ E u (z + x j ) ≥ u (z), which is the standard DC condition. We can conclude from this observation that if one of the two priors is degenerated, the decreasing concavity of the utility function is sufficient to guarantee that, in the maxmin model, wealthier people invest more in the risky asset. Another way to see the link between DC and condition (13) is to examine the case in which xj are small. In the small, the indifference between xi and xj is equivalent to condiboth xi and tion m1i − 0.5Am2i = m1j − 0.5Am2j , mki = E xik

(16) A = −u (z)/u (z)

where is the k-th moment of random variable xi , and is absolute xj are respectively larger and smaller risk aversion. The condition that the demands for xi and for than unity is equivalent to the following condition: m1i − Am2i ≥ 0 ≥ m1j − Am2j .

(17)

Replacing m1i by its expression derived from (16) in the first inequality allows us to rewrite these two inequalities as follows: A

m2i + m2j ≤ m1j ≤ Am2j . 2

(18)

These two inequalities imply in particular that m2i be smaller than m2j .4 Intuitively, if two small risks i and j yield the same expected utility, but the demand for i is larger than the demand for j , it must be that the riskiness of i is smaller than the riskiness of j (and the expected payoff of i is smaller than the expected payoff of j ). x i )] ≤ E[u (z + x j )] is equivalent to Now, observe that in the small, condition E[u (z + m1i − 0.5P m2i ≥ m1j − 0.5P m2j ,

(19)

where P = −u (z)/u (z) is absolute prudence. Using condition (16), this inequality holds if and only if (m2i − m2j )(A − P ) ≥ 0. Because m2i ≤ m2j from (18), this condition holds if and only if P ≥ A, i.e., if and only if u is DC. To sum up what we have at this stage in the maxmin model, the decreasing concavity of the utility function is sufficient for the demand for the risky asset to raise with wealth when one of the two priors is degenerated, or when the different possible priors entail small risk. However, Example 1 tells us that DC is generally not sufficient. The Jewitt order is useful for this purpose. Definition 3. We say that xi dominates xj in the sense of Jewitt if the following condition holds: xj , then all agents more riskfor all increasing and concave u, if agent u weakly prefers xi to xj . averse than u also weakly prefer xi to 4 Notice that the same result holds when condition E x i u (z + x i ) ≥ 0 ≥ E x j u (z + x j ) is relaxed to E x i u (z + xi ) ≥ x j ). E x j u (z +

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This is denoted xi J xj . Jewitt [19] fully characterizes this stochastic order. A sufficient condition is that the cumulative distribution function of xi crosses the cumulative distribution function of xj only once, from below. Now suppose that we can rank the set of priors { x1 , . . . , xn } according to Jewitt’s order. Without loss of generality, suppose that x1 J x2 J ... J xn . Define function ψ such that ψ(x) = xu (z + x) for all x. Suppose that both −u and ψ are more concave than u in the sense of Arrow–Pratt. Consequently, for any (i, j ) ∈ {1, . . . , n}2 with i > j , if Eu(z + xi ) = Eu(z + xj ), then Eu (z + xi ) ≤ Eu (z + xj ) and E xi u (z + xi ) ≥ E xj u (z + xj ). This is stronger than requested by condition (13). The conditions in the following proposition guarantee that −u and ψ are more concave than u. Proposition 3. Normalize the demand for the risky asset at wealth level z to unity. Suppose that the set of priors { x1 , . . . , xn } can be ranked according to Jewitt’s order. In the maxmin model, an increase in wealth around z raises the demand for the risky asset if the utility function u is DC and for all x in the joint support of { x1 , . . . , xn },

x P (z + x) − A(z + x) ≤ 1, (20) where A and P are the indices of absolute risk aversion and of absolute prudence of u, respectively. Proof. It remains to prove that −u and ψ are two concave transformations of u. We already know that u DC just means that −u is more concave than u. Concerning ψ, let us define function k such that ψ(x) = k(u(x)) in the joint support of { x1 , . . . , xn }. By definition of ψ, fully differentiating this equality twice yields u (z + x) + xu (z + x) = k u(z + x) u (z + x), and 2 2u (z + x) + xu (z + x) = k u(z + x) u (z + x) + k u(z + x) u (z + x). Eliminating k from these two equations allows us to write 2 (u (z + x))2 k u(z + x) u (z + x) = u (z + x) + x u (z + x) − . u (z + x) This implies that k is concave in the relevant domain of x if condition (20) is satisfied.

2

Observe that this proposition implies that u DC is sufficient if the possible priors are small risk, that is, if the joint support of priors is in a small neighborhood of 0. Observe also that condition (20) is quite general, as we show now. Define relative risk aversion and relative prudence respectively as Ar (c) = −cu (c)/u (c) and P r (c) = −cu (c)/u (c). Condition (20) can then be rewritten as

P r (z + x) − Ar (z + x) − z P (z + x) − A(z + x) ≤ 1. Assuming that the domain of possible wealth levels is in R+ , DC implies that the last term in the left-hand side of the above inequality is always positive. This implies that a sufficient condition for this inequality is that P r (z + x) is weakly lower than Ar (z + x) + 1 for all z + x. This is the case for example when u has constant relative risk aversion.

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Corollary 2. Normalize the demand for the risky asset at wealth level z to unity. Suppose that the xn } can be ranked according to Jewitt’s order. In the maxmin model with a set of priors { x1 , . . . , utility function DC, a marginal increase in wealth from z raises the demand for the risky asset if relative prudence is uniformly smaller than relative risk aversion plus one. To complete this section, let us consider the special case of HARA utility functions, i.e., functions with linear absolute risk tolerance: c 1−γ u(c) = ξ η + , (21) γ for some scalars ξ , η, and γ . The consumption domain of this utility function is such that η + c/γ is positive. We assume that ξ(1 − γ )/γ is positive to insure that u is increasing and concave in its domain. DC holds if γ is positive. The HARA set includes power, log, exponential and quadratic functions. It is easy to check that xi xi −γ Eu(z + xi ) = ξ E ηˆ + ηˆ + γ γ −γ xi ξ xi −γ = ξ ηE ˆ ηˆ + + E xi ηˆ + γ γ γ γ 1 = ηˆ xi ) + xi ) Eu (z + E xi u (z + 1−γ 1−γ where ηˆ = η + z/γ > 0. This implies that Eu (z + xi ) =

1−γ 1 xi ). Eu(z + xi ) − E xi u (z + ηγ ˆ ηγ ˆ

A symmetric condition holds for xj . This implies that Eu (z + xi ) − Eu (z + xj ) =

1−γ xj ) Eu(z + xi ) − Eu(z + ηγ ˆ

1 xj ) − E xi u (z + xi ) . E xj u (z + + ηγ ˆ

Suppose that the left conditions in (13) hold. This implies that the first term in the RHS of the above equality vanishes, and that the second term is negative if γ is positive, i.e., if u is DC. This yields the following proposition. Proposition 4. Suppose that the utility function u is HARA and DC, i.e., that condition (21) holds with γ > 0. In the maxmin model, an increase in wealth always increases the demand for the risky asset. It is noteworthy that in the HARA subset with increasing concavity (condition (21) with γ < 0) as the quadratic utility function, the demand for the risky asset is always decreasing in wealth.5 5 In the case of HARA with γ = −1, i.e. quadratic utility functions, the Arrow–Pratt approximation is exact so that the optimal α is given by the mean-variance criterion α ∗ (z) = E( x )/A(z)σ 2 ( x ) where A(z) is the Arrow Pratt index, and the increasing aversion is a direct consequence of the increasing concavity of u.

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Fig. 3. Optimal investment α ∗ (z) as a function of wealth in Example 2.

Regarding the polar case of the maxmax model, we show in the technical appendix available online that the previous proposition still holds: when the utility function is HARA and DC, an increase in wealth increases the demand for the risky asset for a maxmax expected utility maximizer. Somehow, the maxmax case is more complex: in particular, a utility function DC is not enough to ensure decreasing aversion in the small. 3.2. The KMM model Let’s now consider the portfolio problem in the KMM smooth ambiguity framework. The decision problem becomes α ∗ (z) = argmaxα Eφ Eu(z + α xθ ) . (22) Let us first explore the special case of small uncertainty. Maccheroni, Marinacci and Ruffino [26] show in this case that the optimal size of investment in the risky asset can be approximated by a formula similar to the classical mean-variance Markowitz solution, with an additional term proportional to the Arrow–Pratt index for φ ◦ u, that is α ∗ (z)

E x Au (z)Var( x ) + Aφ◦u (z)Var(Ex θ)

where Af (z) denotes the Arrow–Pratt index Af = −f /f evaluated at z, and Var(Ex θ ) is x 1 ; . . . ; qn , E x n ). This implies that, in the small, a the variance of the random variable (q1 , E sufficient condition is that both φ and φ ◦ u are decreasingly concave. This result is not robust in the large, as shown in the following example. Example 2. Consider function u(c) = 1 − e−c + 0.00001c in R+ . This function is increasing, concave and strictly DC. Consider also function φ(z) = 1 − e−z and the couple of priors x2 ∼ (−0.0198151, 0.5; 0.0208059, 0.5). In Fig. 3, x1 ∼ (−0.0837022, 0.5; 0.100443, 0.5) and we show that an increase in wealth around z = −5 reduces the demand for the risky asset. This counterexample shows that some additional restriction is required either on φ, or on the structure of ambiguity. The following result tells us that the strategy to restrict φ is a dead end.

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Proposition 5. Suppose that u is CARA and φ is DC. Then, there exists an ambiguous asset whose demand is locally decreasing in wealth. Proof. See Appendix A.

2

It is thus necessary to restrict the structure of ambiguity to derive an unambiguous comparative static property of an increase in wealth in the KMM portfolio problem. More specifically, we will characterize in this section some structures of ambiguity for which the DA property holds when both φ and u are decreasingly concave. We normalize again the unit of the risky asset in such a way that α ∗ (z) = 1. The first order condition is therefore

x E φ Eu(z + x x (23) θ ) E θ u (z + θ ) = 0, We examine the condition under which the demand for the risky asset is increasing in wealth. Because the objective function in (22) is concave in α, this decreasing aversion property is satisfied if and only if

x x E φ Eu(z + x x x x x θ ) Eu (z + θ )E θ u (z + θ ) + E φ Eu(z + θ ) E θ u (z + θ) ≥ 0 (24) Given a utility function u, we obtain a necessary and sufficient condition for a set of priors to satisfy decreasing aversion for any ambiguity aversion φ that is DC. The next lemma is similar in spirit to Lemma 1 of the previous section. Lemma 2. Consider the KMM portfolio problem and suppose that u is DC. Given a set of priors P = ( x 1 , . . . , x n ), the demand for the risky asset described by {P} increases with wealth for any DC ambiguity aversion function φ if and only if for any i = j , xj ) Eu(z + x i ) ≥ Eu(z + E x i u (z + x i ) > 0 > E x j u (z + x j ) ⇒ (25) Eu (z + x i ) ≤ Eu (z + xj ) Proof. See Appendix A.

2

In other words, the DA property is satisfied if, whenever an agent with utility u prefers to xj , then this agent, as well as the agent invest more than one unity in xi and less than one unity in with utility −u , does prefer xi to xj . We can use this lemma to obtain two propositions. Let us first show that combining condition φ DC to the condition on priors presented in Proposition 3 is sufficient. Proposition 6. Consider the KMM portfolio problem. Suppose that u and φ are DC, and normalize the demand for the risky asset at wealth level z to unity. If Eu(z + x1 ) ≤ . . . ≤ Eu(z + xn ) x2 J . . . J xn , then an increase in wealth around z raises the demand for the risky and x1 J asset provided condition (20) holds for all x in the joint support of { x1 , . . . , xn }. xθ+1 Proof. Because u is DC, we have that −u is more concave than u. Because u prefers to xθ , xθ+1 J xθ implies that the agent with utility function −u also prefers xθ+1 to xθ , i.e., xθ+1 ) ≥ E − u (z + xθ ). Moreover, condition (20) implies that ψ(x) = xu (z + x) E − u (z + is more concave than u, so that Eψ( xθ ) = E xθ u (z + xθ ) is increasing in θ as well. Applying Lemma 2 yields the result. 2

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In the next proposition, we consider an alternative restriction on the set of priors based on another stochastic dominance order defined by Gollier [13]. Let us introduce the locationweighted-probability function Tθ as follows: x Tθ (x) =

tdFθ (t),

(26)

where Fθ is the cumulative distribution function of xθ . Following Gollier [13], we say that xθ+1 dominates xθ in the sense of Central Dominance (CD) if there exists a non-negative scalar m such that Tθ (x) ≤ mTθ+1 (x) for all x in the joint support of xθ and xθ+1 . SSD-dominance is neither necessary nor sufficient for CD-dominance. In the next proposition, we assume that the set of priors can be ranked by the two orders symmetrically. Proposition 7. Suppose that u and φ are DC. In the KMM model, an increase in wealth raises the demand for the risky asset if x1 SSD . . . SSD xn and x1 CD . . . CD xn . A sufficient condition is that the priors can be ordered according to the Monotone Likelihood Order. Proof. Because u and −u are increasing and concave, we know from the SSD ordering that the right properties in (25) are satisfied for any pair (i, j ) such that i > j . We also know from Gollier [13] that x1 CD . . . CD xn implies that E xθ u (z + xθ ) has the single-crossing property property in θ . In other words, the left condition in (25) implies that i > j , which implies in turn that the right conditions in (25) are satified. The sufficiency of the MLR order comes from the fact that it is a special case of the SSD and CD orders. 2 This proposition is similar to the main result in Gollier [15], who shows that an increase in ambiguity aversion reduces the demand for the risky asset in the KMM model if the priors can be ranked according to SSD and CD. A special case is when the set of priors can be ranked according to the MLR order. In that case, the DC of both u and φ is sufficient to guarantee that wealthier investors will take more ambiguous risk. 4. Conclusion Two basic hypotheses prevail in decision theory with a large consensus in the profession. The first one is that decision makers are averse to uncertainty. The second one is that they are decreasingly averse to uncertainty. In the classical expected utility framework, these properties of human behavior prevail respectively if the utility function u is concave, and if it is decreasingly concave in the sense that −u /u is decreasing. In this paper, we have focused our attention to this concept of decreasing aversion, by examining two different decision problems when the decision maker is not ambiguity-neutral. We first define decreasing aversion by the property that the set of desirable uncertain prospects expands when wealth increases. In the smooth ambiguity aversion model, we have shown that the classical conditions of decreasing risk aversion and of decreasing ambiguity aversion imply this property, and that an intuitive weaker condition is necessary and sufficient. In the α-MEU model, the standard DARA condition is necessary and sufficient. Another definition of decreasing aversion is that the demand for a risky asset is increasing with wealth. The introduction of the ingredient of ambiguity-sensitive preferences into the picture implies much more complexity than in the above discrete choice problem. Even in the simpler

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maxmin criterion, the decreasing concavity of u is not sufficient to get this result, except in the case of small risks. As in the smooth ambiguity aversion model with a decreasingly concave ambiguity-related function φ, the unambiguous comparative static result requires some assumptions on the structure of ambiguity. As in Jewitt and Mukerji [20], this paper illustrates once again the fact that even the most intuitive departures from the classical subjective expected utility model introduce much richness to our decision models. This is at the cost of a non-marginal increment in the complexity of the analysis. Appendix A Proof of Proposition 5. The first part of the proof works for any utility function u. We start with a binary random variable x1 taking both a positive and a negative value. The portfolio problem max Eu(z + α x1 ) has a solution, and we can normalize it so that E x1 u (z + x1 ) = 0. It is then easy to see that if we slightly increase the negative value of x1 , we obtain a new binary random variable x with a higher expected utility and such that E x u (z + x ) > 0: the ∗ portfolio problem max Eu(z + α x ) has a solution with α > 1. If we set x2 = α ∗ x , we then have Eu(z + x1 ) < Eu(z + x2 ), and E xi u (z + xi ) = 0 for i = 1, 2. It means that an investor would invest 1 share of xi for i = 1, 2 and that the investor strictly prefers x2 to x1 . We can then replace xi by λi xi with λ1 < 1 and λ2 > 1 close enough to one so that we still have Eu(z + λ1 x1 ) < Eu(z + λ2 x2 ). We then have E xi u (z + xi ) positive when i = 1 and negative when i = 2, so that we can find a probability p such that the investor would invest optimally 1 in the ambiguous distribution {(p, x1 ), (1 − p, x2 )}, i.e. the first order condition (23) hold: pφ Eu(z + x1 ) E x2 ) E x1 u (z + x1 ) + (1 − p)φ Eu(z + x2 u (z + x2 ) = 0 It remains to show that when wealth increases, the investor will be willing to invest less than one, i.e. that the relation (24) is no satisfied. We do that when u is CARA, so that the second part of the term in inequality (24) is null since it is equal to the first order condition up to a multiplicative factor. The other term can be written as −E[ξi φ (Eu(z + xi ))E xi u (z + xi )] where ξi = Aφ (Eu(z + xi ))Eu (z + xi ) with Aφ (u) = −φ (u)/φ (u). This is very similar to the first order condition, and we just need to show that the weight ξi are higher on the negative term, i.e. when i = 1. When u is CARA, the inequality Eu(z + λ1 x1 ) < Eu(z + λ2 x2 ) implies Eu (z + λ1 x1 ) > Eu (z + λ2 x2 ), and when φ is DC, the function Aφ is decreasing so that ξ1 > ξ2 as required. 2 Proof of Lemma 2. We first prove sufficiency. The second term in the left-hand side of inequality (24) is positive if u is DC. Indeed, u DC means that A(c) = −u (c)/u (c) is decreasing, which implies in turn that

x x E φ Eu(z + x x x x x θ ) E θ u (z + θ ) = −E φ Eu(z + θ ) E A(z + θ ) θ u (z + θ)

x x x ≥ −A(z)E φ Eu(z + θ ) E θ u (z + θ ) = 0. Using the same method, we can rewrite the first term in inequality (24) as follows:

x E φ Eu(z + x x x θ ) Eu (z + θ )E θ u (z + θ)

= −E ξ( θ )φ Eu(z + x x x θ ) E θ u (z + θ) with ξ(θ) = Aφ (Eu(z + xθ ))Eu (z + xθ ) where Aφ (u) = −φ (u)/φ (u) is the absolute measure of ambiguity aversion. Let us rank the θ s such that E xθ u (z + xθ ) is positive if and only if θ ≥ θ

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for some θ . Condition (25) implies that ξ(θ) ≤ ξ( θ ) for all θ ≥ θ , and ξ(θ) ≥ ξ( θ ) for all θ ≤ θ. From the above equality, this implies that

x E φ Eu(z + x x x θ ) Eu (z + θ )E θ u (z + θ)

≥ −ξ( θ )E φ Eu(z + x x x θ ) E θ u (z + θ ) = 0. Thus, the two terms in (24) are positive, thereby proving sufficiency. Reversely, if property (25) is not satisfied, i.e. if one of the two inequalities on the right is not true for a pair (i, j ) such that E x i u (z + x i ) > 0 > E x j u (z + x j ), there exist a function φ and a distribution {pi , pj } of those priors such that there is increasing aversion. Indeed, it is straightforward for a given function φ to find {pi , pj } such that the first-order condition (23) is satisfied. We thus just need to pick φ such that the relation (24) cannot be true. The relation (24) is equivalent to E[B( θ )φ (Eu(z + x x x θ ))E θ u (z + θ )] ≤ 0 where B( θ) = −

φ (Eu(z + E x x x θ )) θ u (z + θ) (z + x ) − Eu . θ φ (Eu(z + x E x x θ )) θ u (z + θ)

When the second condition is not satisfied, i.e. E[−u (z+ x i )] < E[−u (z+ x j )], then we choose −λz φ(z) = −e with λ big enough so that Bi > Bj > 0. When the first condition is not satisfied, i.e. Eu(z + x i ) < Eu(z + x j ), then we pick φ enough decreasingly concave, for instance φ(z) = λ/zn e with λ and n large enough, so that we still have Bi > Bj . In both cases, this condition combined with the first order condition (23) implies that the relation (24) is not satisfied. 2 Appendix B. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/ j.jet.2015.01.002. References [1] K.J. Arrow, Liquidity preference, in: The Economics of Uncertainty, in: Lecture Notes for Economics, vol. 285, 1963, pp. 33–53, Lecture VI, undated, Stanford University. [2] K.J. Arrow, Aspects of the theory of risk-bearing, Yrjö Jahnssonin Säätiö, 1965. [3] Z. Bar-Shira, R.E. Just, D. Zilberman, Estimation of farmers’ risk attitude: an econometric approach, Agr. Econ. 17 (2) (1997) 211–222. [4] R.B. Barsky, F.T. Juster, M.S. Kimball, M.D. Shapiro, Preference parameters and behavioral heterogeneity: an experimental approach in the health and retirement study, Quart. J. Econ. 112 (2) (1997) 537–579. [5] M.K. Brunnermeier, S. Nagel, Do wealth fluctuations generate time-varying risk aversion? Micro-evidence on individuals, Amer. Econ. Rev. 98 (3) (2008) 713–736. [6] C. Campanale, Learning, ambiguity and life-cycle portfolio allocation, Rev. Econ. Dynam. 14 (2) (2011) 339–367. [7] J.F. Cocco, Portfolio choice in the presence of housing, Rev. Finan. Stud. 18 (2) (2005) 535–567. [8] J.F. Cocco, F.J. Gomes, P.J. Maenhout, Consumption and portfolio choice over the life cycle, Rev. Finan. Stud. 18 (2) (2005) 491–533. [9] R.A. Cohn, W.G. Lewellen, R.C. Lease, G.G. Schlarbaum, Individual investor risk aversion and investment portfolio composition, J. Finance 30 (2) (1975) 605–620. [10] D. Ellsberg, Risk, ambiguity, and the Savage axioms, Quart. J. Econ. 75 (1961) 643–669. [11] I. Gilboa, D. Schmeidler, Maximin expected utility with non-unique prior, J. Math. Econ. 18 (1989) 141–153. [12] P. Ghirardato, F. Maccheroni, M. Marinacci, Differentiating ambiguity and ambiguity attitude, J. Econ. Theory 118 (2004) 133–173. [13] C. Gollier, The comparative statics of changes in risk revisited, J. Econ. Theory 66 (1995) 522–535. [14] C. Gollier, The Economics of Risk and Time, MIT Press, Cambridge, 2001.

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