Elyès Jouini

Clotilde Napp

Diego Nocetti

Jouini Elyès

Clotilde Ceremade

Napp

Economic Consequences of Nth-Degree Risk Increases and Nth-Degree Risk Attitudes

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

risk? and 2) What is the interpretation of these conditions, which frequently involve establishing the sign of two or more high-order partial derivatives of the payo¤ function? The focus of the early literature analyzing these questions was on establishing the impact of a fairly narrow class of risk changes -mostly, increases in risk as de…ned by [START_REF] Rothschild | Increasing risk I: A de…nition[END_REF]-and on interpreting the resulting conditions on the payo¤ function within the speci…c context of the problem being analyzed. More recently, the work of Louis Eeckhoudt has advanced our knowledge about these important questions in at least two fundamental ways. First, Eeckhoudt and colleagues have evaluated the impact of a larger class of risk increases -namely, Nth-degree stochastically dominated shifts [START_REF] Jean | The geometric mean and stochastic dominance[END_REF] and Nth-degree risk increases as de…ned by [START_REF] Ekern | Increasing Nth-degree risk[END_REF]-in di¤erent frameworks, including the classical problems of precautionary saving [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF], precautionary labor supply [START_REF] Chiu | The e¤ects of stochastic wages and non-labor income on labor supply: update and extensions[END_REF], portfolio choice [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF], and production under output-price uncertainty [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF]). 1 Second, Eeckhoudt and colleagues have introduced a new approach to characterize risk attitudes via preferences over pairs of simple lotteries [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF][START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF][START_REF] Eeckhoudt | Apportioning of risks via stochastic dominance[END_REF], Denuit et al., 2010a[START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF]. In particular, we now know that the signs of the derivatives of the utility function are closely tied to a preference to disaggregate harms across lottery outcomes.

In this way, this new approach has established a link between the decision maker's attitudes towards risk (i.e. preferences over simple lotteries) and his or her optimal response to changes in risk.

The focus of the recent work and much of the earlier literature has been on determining unambiguous comparative statics results for a given set of individuals. In this type of exercise, it is established that all expected utility maximizers within a given set (i.e. a set of utility functions) will select the optimal response to an increase in risk (or will rank lotteries) in a particular unambiguous manner. An alternative line of inquiry, taken on this paper, is to determine the maximal set of expected utility maximizers for which an increase in risk induces a particular behavior. This alternative type of exercise not only establishes unambiguous comparative statics results for a given set of individuals, but it also reveals how a particular behavioral response to a risk increase is related to the preferences of the decision maker. It is only by performing this type of analysis that behavior in di¤erent settings can be pinned down to a maximal set of individuals and, in particular, that the optimal response to changes in risk in a speci…c problem can be linked directly to the decision maker's choice between lottery pairs. In this paper, we use this latter approach to study a large class of problems that involve 1) a bivariate utility function, 2) a linear constraint that links the two attributes that enter the decision maker's payo¤ function, and 3) Nth-degree stochastically dominated shifts, with particular emphasis on increases in Nth-degree risk à la [START_REF] Ekern | Increasing Nth-degree risk[END_REF].

We begin the analysis in the next section by revisiting [START_REF] Ekern | Increasing Nth-degree risk[END_REF] results. Following the methods of [START_REF] Rothschild | Increasing risk I: A de…nition[END_REF], [START_REF] Ekern | Increasing Nth-degree risk[END_REF] characterized Nth-degree risk increases.

We prove a dual result along the lines of [START_REF] Denuit | Extremal generators and extremal distributions for the continuous s-convex stochastic orderings[END_REF]. This result is the main ingredient for characterizing necessary and su¢ cient conditions for unambiguous comparative statics of risk within our framework and also for the interpretation of our results.

In Section 3 we perform the comparative statics analysis within the mentioned framework and for two di¤erent scenarios, one in which the risk is additive and another one in which the risk is multiplicative in the decision variable. In each case, we provide necessary and su¢ cient conditions for an unambiguous impact of Nth-degree risk increases and we compare our conditions with existing conditions in the literature. As an illustration, we show that no decision maker that views the attributes as goods (i.e. with a positive marginal utility) will always increase the level of the decision variable when faced with a …rst-degree multiplicative-risk increase. Similarly, no decision maker with diminishing marginal utility (i.e. risk averse) will always increase the level of the decision variable when faced with a Rothschild-Stiglitz multiplicative-risk increase. While these results are intuitive in the classical portfolio choice problem, we show that they hold much more generally within our framework and that they generalize to increases in Nth-degree risk.

Section 4 develops a simple and intuitive interpretation of the previously obtained conditions.

We propose concepts of additive and multiplicative directional Nth-degree risk aversion; these are characterized via preferences for harms disaggregation across outcomes of 50-50 bivariate lotteries.

The harms we consider involve unidimensional increases in Nth-degree risk together with bidimensional non-stochastic shifts in the attributes. These concepts of directional Nth-degree risk aversion include as special cases the concept of prudence analyzed by [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] and the concept of cross-prudence analyzed by [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF]. For a …xed level of one of the attributes, the lotteries we study are isomorphic to a set of lotteries studied by [START_REF] Eeckhoudt | Apportioning of risks via stochastic dominance[END_REF] in the case of additive risks (generalized by Denuit et al. 2010a to the multivariate case) and to the lotteries studied by [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF] in the case of multiplicative risks. Our main contribution is to characterize the preference for harms disaggregation (i.e. to establish necessary and su¢ cient conditions). Speci…cally, we characterize the situations where Nth-degree changes in risk have an unambiguous impact on the optimal decision in terms of Nth-degree preference for harms disaggregation.

We provide some concluding remarks in Section 5. The proofs for all the results in the paper in terms of Nth-degree risk increases are collected in Appendix A. In Appendix B, we establish and prove analogous results for more general Nth-degree stochastically dominated shifts.

A Preliminary Result

Let us start by revisiting the notion of increases in risk proposed by [START_REF] Ekern | Increasing Nth-degree risk[END_REF]. e i the distribution functions and, for k = 1; 2; ::: we de…ne the functions

F [k+1] e i on R + by F [k+1] e i (x) = Z x 0 F [k] e i (t)dt for x 2 R + :
We say that e 2 is an increase in Nth-degree risk over e 1 ; and we denote it by e 2 < N e 1 ; if

F [N ] e 2 (x) F [N ]
e 1 (x) for all x 2 [0; B] where the inequality is strict for some x and

F [k] e 2 (B) = F [k]
e 1 (B) for k = 1; :::; N: For example, an increase in 2nd-degree risk coincides with [START_REF] Rothschild | Increasing risk I: A de…nition[END_REF] mean preserving increase in risk, while an increase in 3rd-degree risk coincides with a mean and variance preserving increase in risk that [START_REF] Menezes | Increasing Downside Risk[END_REF] labeled 'increase in downside risk'.

An increase in 4th-degree risk is equivalent to what [START_REF] Menezes | Increasing outer risk[END_REF] call an 'increase in outer risk'. [START_REF] Ekern | Increasing Nth-degree risk[END_REF] characterizes increases in Nth-degree risk: he establishes that e 2 is an increase in Nth-degree risk over e 1 (as de…ned above) if and only if2 E [q(e 2 )] > E [q(e 1 )] for all N times continuously di¤erentiable real valued function q such that ( 1) N q (N ) > 0 where q (N ) = d N q d N : In particular, this means that [START_REF] Ekern | Increasing Nth-degree risk[END_REF] shows that if a function q is such that ( 1) N q (N ) > 0 then E [q(e 2 )] > E [q(e 1 )] for all pair (e 1 ; e 2 ) such that e 2 < N e 1 : The following Lemma, established by [START_REF] Denuit | Extremal generators and extremal distributions for the continuous s-convex stochastic orderings[END_REF] in a somewhat di¤erent form, extends this last result by also showing the reverse implication. 3 It characterizes the set of N times continuously di¤erentiable functions for which E [q(e 2 )] E [q(e 1 )] for all pair (e 1 ; e 2 ) where e 2 is an increase in Nth-degree risk over e 1 and shows that these functions are exactly those such that ( 1) N q (N ) 0: 4 Lemma 1 (Denuit, De Vylder, and Lefevre) Let q be a given real valued function that is N times continuously di¤ erentiable on R + . The following are equivalent.

1. For all pair (e 1 ; e 2 ) such that e 2 < N e 1 ; we have E [q(e 2 )] E [q(e 1 )] :

2. For all x 0; we have ( 1) N q (N ) (x) 0:

This result pins down how expected utility maximizers in a given set rank increases in risk and it also reveals how a particular behavioral response to a risk increase is related to the preferences of the decision maker. As an illustration, for N = 2; we retrieve the classical result that a risk averse agent (i.e. an agent who dislikes mean preserving spreads) is an agent whose utility function is concave. Lemma 1 will prove to be essential in Sections 3 and 4.

Remark. Lemma 1 can be generalized to Nth-degree stochastically dominated shifts. Such shifts do not restrict the …rst N 1 moments of the distributions to be equal. Following [START_REF] Jean | The geometric mean and stochastic dominance[END_REF], and using the notation in De…nition 1, e 2 is dominated by e 1 in the sense of Nth-degree stochastic dominance; and we denote it by e 2 < N SD e 1 ; if

F [N ] e 2 (x) F [N ]
e 1 (x) for all x 2 [0; B], where the inequality is strict for some x; and

F [k] e 2 (B) F [k]
e 1 (B) for k = 1; :::; N 1: Lemma 1 then applies to Nth-degree stochastic dominance shifts when 1) e 2 < N e 1 is replaced by e 2 < N SD e 1 and 2) the condition ( 1) N q (N ) (x) 0 is replaced by the condition ( 1) k q (k) (x) 0 for k = 1; :::; N . As an illustration, for N = 1; 2, we retrieve the classical result that 1st-degree stochastic dominance is characterized by non-decreasing utility functions and 2nd-degree stochastic dominance is characterized by non-decreasing and concave utility functions. Appendix B provides a precise statement and a proof of these results. 3 Ekern's (1980) result, but not Lemma 1, also implies that e 2 <N e 1 is necessary to have unanimity of ranking within the class of utility functions q such that ( 1) N q (N ) > 0, so the information provided by the two results is somewhat di¤erent. We thank an anonymous referee for pointing this out. 4 Denuit et al.'s (1999) result is expressed in terms of maximal generators for s-convex orders and established for non-necessarily regular functions by the introduction of the concept of Nth degree divided di¤erence. The authors remark that for regular functions the conditions on Nth-degree divided di¤erence are equivalent to conditions on the Nth derivative. In Appendix A we propose a direct and simpler proof of the result when the functions are assumed to be regular.

Optimal Decision and Increasing Risk

The problem that we analyze follows closely the setup in [START_REF] Dardanoni | Optimal choices under uncertainty: the case of two-argument utility functions[END_REF]. The decision maker has an increasing, strictly concave and in…nitely di¤erentiable two-dimensional utility function U (y; z) de…ned for y and z positive. As usual, we denote by U (i;j) (y; z) the (i; j) th cross partial derivative of the utility function. Uncertainty is described by a probability space ( ; F; P ) where describes the set of possible states of the world, F is the set of measurable events and P is a probability measure.

The decision maker has the possibility to buy a quantity x of an asset that has a random payo¤ ~ 0 in terms of the second attribute at a (deterministic) cost p > 0 in terms of the …rst attribute.

The initial endowment of the decision maker in terms of the …rst attribute is deterministic and denoted by K: The initial endowment in terms of the second attribute is random and denoted by ~ 0: Both random variables ~ and ~ are assumed to be bounded above. The decision maker's problem P U;K;p (~ ; ~ ) is then the following

M ax x E [U (K xp; x~ + ~ )] : (1)
The …rst-order necessary condition and the second-order su¢ cient condition for optimality, assuming an interior solution, are given by

E [g (x; ~ ; ~ )] = 0 (2) E [g x (x; ~ ; ~ )] < 0 (3)
with g (x; ; ) = pU (1;0) (K xp; x + ) + U (0;1) (K xp; x + ) and where g x (x; ; ) = p 2 U (2;0) (K xp; x + ) 2p U (1;1) (K xp; x + ) + 2 U (0;2) (K xp; x + ) : By strict concavity of U; this last quantity is always negative. The …rst order condition is then necessary and su¢ cient.

In the analysis that follows we will alternatively need the following Inada-type conditions:

Assumption A1 The utility function U is such that lim z!0 U (0;1) (y;z)

U (1;0) (y;z) = 1 and lim z!1 U (0;1) (y;z) U (1;0) (y;z) = 0.
Assumption A2 The utility function U is such that z U (0;1) (y;z) U (1;0) (y;z) is unbounded. It is classical to assume that lim z!0 U (0;1) (y; z) = 1 and lim z!1 U (0;1) (y; z) = 0 which means that the second attribute is necessary for the agent's survival and that the agents approach satiation for large quantities of that attribute. Assumption A1 is a little bit stronger and limits the possibilities for substitutability between the two attributes. A low level of the second attribute increases the marginal utility for that attribute faster than for the other one and satiation with respect to the second attribute does not mean satiation with respect to the …rst one. For example, Assumption A1 is satis…ed by homothetic utility functions satisfying the classical Inada conditions in the direction of both attributes, i.e. lim z!0 U (0;1) (y; z) = 1, lim z!1 U (0;1) (y; z) = 0, lim y!0 U (1;0) (y; z) = 1 and lim y!1 U (1;0) (y; z) = 0. With a separable utility function U (y; z) = u(y) + v(z), Assumption A1 is equivalent to the classical Inada conditions on v.

Assumption A2 means that U must satisfy either lim z!0 z U (0;1) (y;z) U (1;0) (y;z) = 1 (which is a stronger condition than the …rst condition in Assumption A1) or lim z!1 z U (0;1) (y;z) U (1;0) (y;z) = 1. For example, with a separable utility function, Assumption A2 can be rewritten as zv 0 (z) is unbounded. This is the case for all CRRA functions except log.

The problem P U;K;p has found many important applications, including The 2-date optimal saving model with either time-non-separable utility (e.g. [START_REF] Leland | Saving and uncertainty: The precautionary demand for saving[END_REF][START_REF] Sandmo | The e¤ect of uncertainty on saving decisions[END_REF] or time-separable utility (e.g. [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF][START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF][START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF] and uncertainty surrounding the rate-of-return on saving or the future income endowment.

The canonical portfolio problem with one risky asset and one risk-free asset (e.g. Rothschild andStiglitz, 1971, Chiu et al., 2012). 5The optimal allocation of income to medical expenditures and consumption of non-medical goods (e.g. [START_REF] Dardanoni | Uncertainty and the demand for medical care[END_REF], and the optimal allocation of a national health budget [START_REF] Courbage | Priority Setting in Health Care and Higher Order Degree Change in Risk[END_REF], when either the return on medical expenditures or the consumer's health status is uncertain.

The trade-o¤ between leisure and consumption, with wage income or non-wage income risks (e.g. [START_REF] Block | The allocation of e¤ort under uncertainty: the case of risk-averse behavior[END_REF], Tressler and Menezes, 1980[START_REF] Chiu | The e¤ects of stochastic wages and non-labor income on labor supply: update and extensions[END_REF].

The trade-o¤ between (dirty) production and environmental-quality, with uncertainty surrounding the damages that the productive activity generates or the level of environmental quality itself (e.g. [START_REF] Baiardi | Pigouvian Tax, Abatement Policies and Uncertainty on the Environment[END_REF].

The private provision of public goods under uncertainty surrounding the contributions of others [START_REF] Sandler | Free riding and uncertainty[END_REF][START_REF] Keenan | The Private Provision of Public Goods under Uncertainty: A Symmetric-Equilibrium Approach[END_REF].

The leisure/production trade-o¤ of an entrepreneur facing price uncertainty in a competitive environment (Rothschild andStiglitz, 1971, Chiu et al., 2012).

In some of the above cases, like in the 2-date optimal saving model or the canonical portfolio problem, the asset is a classical …nancial asset. In other cases, the asset corresponds to the mechanism that transforms money into health (health care) or leisure into money (labor) or production into environmental quality, etc.

The objective in these articles has been the evaluation of how increases in risk a¤ect the optimal value of the choice variable. While the earlier literature focused almost exclusively on 2nd-degree stochastic dominance shifts and, in particular, on mean preserving increases in risk as de…ned by [START_REF] Rothschild | Increasing risk I: A de…nition[END_REF], a number of recent papers have evaluated the e¤ect of Nth-degree stochastically dominated shifts and, in particular, of increases in Nth-degree risk as de…ned by Ekern ). In the present paper, we will also focus on the general Nth-degree risk framework, considering separately the case where the endowment in terms of the second attribute ~ is random and the case where the asset's payo¤ ~ is random. Analogous results for Nth-degree stochastic dominance shifts can be found in Appendix B.

Uncertainty Over the Endowment

Suppose that e = is deterministic and remains unchanged while considering a change in the second attribute initial endowment from e 1 to e 2 , where e 2 is an increase in Nth-degree risk over e 1 : Furthermore, let x 1 and x 2 respectively denote the solutions of P U;K;p (e 1 ; ) and P U;K;p (e 2 ; ):

In order to provide some intuition for the results that follow, let us consider the classical 2-date precautionary saving problem with a separable utility function U of the form U (y; z) = u(y) + v(z):

In such a setting, it is well known that an agent raises his optimal saving when adding a zero-mean risk (to a deterministic second period initial endowment) if and only if his marginal utility of future consumption is convex, v (3) 0 (see e.g. [START_REF] Gollier | The Economics of Risk and Time[END_REF]. In our more general setting where U is not separable and where the initial endowment e 1 is not necessarily deterministic, the analysis should lead us to introduce conditions that involve cross derivatives as well as the cost p of the second attribute in terms of the …rst attribute in order to re ‡ect the trade-o¤ between these two attributes. The following proposition provides a characterization of the utility functions for which any increase in Nth-degree risk increases the optimal level of the choice variable.

Proposition 1 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A1. Let us consider p and as given. The following properties are equivalent:

1. For all initial endowment (K; e 1 ); any increase in Nth-degree risk over the second attribute initial endowment from e 1 to e 2 increases the optimal level of the choice variable, i.e. x 2 x 1 .

2. For all (y; z) ; we have ( 1) N pU (1;N ) (y; z) + U (0;N +1) (y; z) 0.

Therefore, for example, for e 1 = k 1 2 R + , if e 2 is an increase in …rst-degree risk over e 1 of the form e 2 = k 2 , for a positive constant k 2 < k 1 , and if U (1;1) 0 then x 2 x 1 (remark that, by concavity of U; U (0;2) is nonpositive). As a second example, for e 1 = k 1 2 R + , if e 2 is an increase in second-degree risk over e 1 of the form e 2 = k 1 + e ", where e " is a mean zero random variable, and if U (1;2) 0 and U (0;3) 0 then x 2 x 1 .

If the problem under consideration corresponds to the classical 2-date precautionary saving problem with endowment risk and with a separable utility function U of the form U (y; z) = u(y) + v(z), we obtain that x 1 x 2 for all pair (e 1 ; e 2 ) such that e 1 4 N e 2 if and only if ( 1) N v (N +1) (z) 0: The "if" part of this result has been established by [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF]. We emphasize that our condition ( 1) N v (N +1) (z) 0 is necessary and su¢ cient. 6

Uncertainty Over the Asset' s Payo¤

Suppose now that e = is deterministic and remains unchanged while considering a change from e 1 to e 2 , where e 2 is an increase in Nth-degree risk over e 1 : Let x 1 and x 2 respectively denote the solutions of P U;K;p ( ; e 1 ) and P U;K;p ( ; e 2 ): Furthermore, it will be useful to de…ne a measure of Nth-degree relative risk aversion (in attribute z) as follows (see e.g. [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF])

R N (y; z) z U (0;N +1) (y;z) U (0;N ) (y;z) :
When the utility function U (y; z) is separable we will write this function as R N (z) ; which for N = 1 corresponds to the standard measure of relative risk aversion and for N = 2 corresponds to the measure of relative prudence [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF].

The following Proposition establishes the analog of Proposition 1 for an increase in multiplicative risk.

Proposition 2 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A2. The following properties are equivalent:

1. For all initial endowment (K; ) and all asset cost and payo¤ (p; e 1 ) such that x 1 0, any increase in Nth-degree risk over the asset's payo¤ from e 1 to e 2 increases the optimal level of the choice variable, i.e. x 2 x 1 .

2. For all (y; z) ; we have ( 1) N U (1;N ) (y; z) 0, R N (y; z) N and ( 1) N U (0;N ) (y; z) 0.

The necessary and su¢ cient conditions in Proposition 2 appear to be similar to a number of results established previously in the context of di¤erent applications of our model. 7 There is, however, a crucial di¤erence. Consider again the 2-date saving problem with separable utility, but now with rate-of-return risk. This problem was analyzed by [START_REF] Rothschild | Increasing risk II: Its Economic Consequences[END_REF] in the context of increases in 2nd-degree risk and more recently by [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF] for more general increases in risk. In both papers, the authors show that when future labor income is zero (in our setting = 0, so z = x ) and the consumer is risk averse (v (2) (z) < 0), he will save more in response to an increase in 2nd-degree risk if R 2 (z) 2: As our proof makes it clear, this condition is indeed necessary and su¢ cient for an increase in savings when = 0. But Proposition 2 also shows that the assumption of zero labor income does not come without loss of generality.

Once we consider the more general case, the necessary and su¢ cient conditions for an increase in savings become R 2 (z) 2 and v (2) (z) 0. In other words, any risk-averse consumer will decrease savings in response to higher risk for some initial endowment levels. 8 In essence, this result corresponds to Rothschild and Stiglitz's (1971, p. 72) conclusion that "no risk averse investor will always increase his holdings of risky assets when their variability increases." Proposition 2 implies that this important result holds much more generally in the 8 Note that the condition R N (y; z) N is coupled with ( 1) N U (0;N ) (y; z) 0 which, for N = 2; excludes risk averse consumers. In the special case that = 0 this additional condition is no longer necessary. Therefore, assuming risk aversion, ( 1) 2 U (0;2) (y; z) < 0, leads to a change in the direction of the inequality of the …rst condition, which becomes R 2 (y; z) 2, as established by [START_REF] Rothschild | Increasing risk II: Its Economic Consequences[END_REF] and [START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF] in the context of the 2-date saving problem with rate-of-return risk.

context of problem P U;K;p ( ; ~ ) and generalizes the conclusion by stating that no agent for which

( 1) N U (0;N ) (y; z) < 0 will always increase the demand for the asset in problem P U;K;p ( ; ~ ) when facing an increase in Nth-degree risk.

As another illustration, consider the static labor supply problem analyzed by Chiu and Eeckhoudt (2010). In this context y represents leisure, the choice variable x is labor supply, represents wage income, and represents non-wage income. [START_REF] Chiu | The e¤ects of stochastic wages and non-labor income on labor supply: update and extensions[END_REF] show that the conditions ( 1) N U (1;N ) (y; x + ) 0 and ( 1) N x U (0;N +1) (y; x + ) + N U (0;N ) (y; x + ) 0 are su¢ cient for an increase in Nth-degree risk in wage income to increase the supply of labor. Note that the second condition does not correspond to our second condition in Proposition 2. Indeed, the derivatives are not taken at the same point and the two conditions coincide only for = 0:

In fact, if we divide by U (0;N ) (y; x + ) the condition introduced by Chiu and Eeckhoudt (2010

)
we obtain a condition on the concept of proportional N-th degree relative risk aversion, which is de…ned as z U (0;N +1) (y;z+ ) U (0;N ) (y;z+ ) . Instead, our condition relies on the concept of N-th degree relative risk aversion, which is more natural and easier to interpret. Our Proposition implies, in particular, that imposing the Chiu and Eeckhoudt (2010) condition on the proportional N-th degree relative risk aversion for all is equivalent to imposing a condition on the usual concept of N-th degree relative risk aversion as well as imposing ( 1) N U (0;N ) (y; z) 0: This means that Proposition 2 of Chiu and Eeckhoudt (2010) cannot be applied in order to characterize the situations where the supply of labor is increased in response to a risk increase at all initial endowment levels . Indeed, the authors assume that ( 1) N U (0;N ) (y; z) 0, which is not consistent with the necessary condition

( 1) N U (0;N ) (y; z) 0. This implies, for instance, that a mean preserving spread in wage income cannot always increase the supply of labor for a consumer with diminishing marginal utility of consumption.

These results then suggest that, when considering a risk increase on the asset's payo¤, it may be more natural to analyze the conditions under which the agent decreases his level of exposure to the asset's risk. While Proposition 2 characterized the situations where any increase in risk on the asset's payo¤ implies an increase in the choice variable, the following result characterizes the situations where any increase in risk on the asset's payo¤ implies a decrease in the choice variable.

Corollary 1 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A2. The following properties are equivalent:

1. For all initial endowment (K; ) and all asset cost and payo¤ (p; e 1 ) such that x 1 0, any increase in Nth-degree risk over the asset's payo¤ from e 1 to e 2 decreases the optimal level of the choice variable, i.e. x 1 x 2 .

2. For all (y; z) ; we have ( 1) N U (1;N ) (y; z) 0, R N (y; z) N and ( 1) N U (0;N ) (y; z) 0.

For example, if we consider the 2-date optimal saving problem with rate-of-return risk and a separable utility function, we obtain that any increase in N-th degree risk leads to a decrease of the optimal saving (i.e. A risk averse agent decreases his optimal demand for the risky asset at all wealth levels and when facing an increase in 1st-degree risk in the asset's payo¤ if the degree of relative risk aversion is no greater than one, R 1 (z) 1 (or equivalently, 0 R 1 (z) 1 since the utility function is assumed to be nondecreasing and concave).

A risk averse agent decreases his optimal demand for the risky asset at all wealth levels and when facing an increase in 2nd-degree risk in the asset's payo¤ if the degree of relative prudence is no greater than two, R 2 (z) 2 (or equivalently, 0 R 2 (z) 2 when the utility function is assumed to be prudent).

In summary, this section establishes precisely the necessary and su¢ cient conditions for unambiguous comparative statics of changes in risk in a large class of problems. Our next objective is to develop a simple and intuitive approach to interpret these conditions.

Lottery Choices and Optimal Exposure to Risk

In this section we show that the optimal response to changes in risk can be characterized via preferences over particular classes of lottery pairs. For any two bidimensional lotteries A and B, we use the notation A B to denote the individual's preference relation "lottery A is preferred to lottery B:" We …rst consider the case of additive risks and then evaluate the case of multiplicative risks.

Additive Risks

Consider two attributes with nonnegative initial quantities y and z, and imagine a lottery in which with a 50 percent chance risk e 1 is added to z and with a 50 percent chance risk e 2 is added to z, where e 2 is an increase in Nth-degree risk over e 1 : Denote this lottery by [(y; z + e 1 ) ; (y; z + e 2 )].

Now consider the following location experiment: the consumer is told that she must accept the bundle x 1 y ; x 1 z in tandem with one of the lottery outcomes of her choice and the bundle x 2 y ; x 2 z in tandem with the other lottery outcome, where x 1 ; x 2 ; y ; z are constants and

x 2 > x 1 ; to which outcome will she a¤ect each bundle?

The preceding question consists in evaluating the following pair of lotteries,

A a = y + x 2 y ; z + x 2 z + e 2 ; y + x 1 y ; z + x 1 z + e 1 (4)
B a = y + x 1 y ; z + x 1 z + e 2 ; y + x 2 y ; z + x 2 z + e 1 :

Clearly, the answer depends on the magnitude and the direction of y ; z . We therefore propose the following de…nition, De…nition 2 We say that preferences display Nth-degree risk aversion in the direction of y ; z if, for all (y; z; x 1 ; x 2 ) 2 R 4 + such that x 2 > x 1 and for all pair of random variables (e 1 ; e 2 ) such that e 2 is an increase in Nth-degree risk over e 1 ; we have A a B a .

To understand these concepts it might be useful to consider a few special cases that were analyzed by [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] and [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF]. 10 Suppose …rst that y = 0, z > 0 and N = 1 or 2, in which case, we retrieve the univariate de…nition of risk aversion and of prudence in attribute z as in [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF]. Still with N = 1 or 2 and taking y > 0 and z = 0, we recover Eeckhoudt et al.'s (2007) de…nitions of correlation aversion and of cross-prudence. As explained by [START_REF] Eeckhoudt | A good sign for multivariate risk taking[END_REF], in all of these cases the consumer views the risks as 'mutually aggravating,'so she prefers to disaggregate the harms across outcomes of the lotteries. Similarly, for general N and general directions, our lottery ordering generalizes these concepts by capturing a preference to disaggregate harms across lottery outcomes, where the harms are represented by an Nth-degree risk increase (e 2 < N e 1 ) and by a shift from x 2 to x 1 < x 2 in the direction of y ; z (notice that we refer to the bundle x 1 y ; x 1 z as a harm, in the direction of y ; z ; relative to the bundle x 2 y ; x 2 z even though the latter bundle would be perceived as more favorable in the usual sense when y and z are negative).

The following proposition establishes precisely what this lottery preference means in an expected utility framework.

Proposition 3 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function on R2 + . The preferences represented by U display Nth-degree risk aversion in the direction of y ; z if and only if ( 1) N f (0;N ) (y; z) 0 where f is de…ned on R 2 + by f (y; z) = y U (1;0) (y; z)+ z U (0;1) (y; z):

There are, in fact, two e¤ects that determine the consumer's bivariate preference for harm disaggregation. The condition for risk aversion in the direction of y ; z can be decomposed into 1) a 'Nth-degree risk aversion in z e¤ect', captured by the term z U (0;N +1) , and 2) a 'Nth-degree correlation aversion'e¤ect, captured by the term y U (1;N ) . y > 0 and z > 0: 11 The contribution of our Proposition is twofold. First, and most obvious, our notion of directional Nth-degree risk aversion is more encompassing. More importantly, by making use of Lemma 1, the proposition characterizes the set of expected utility maximizers that display Nth-degree risk aversion in the direction of y ; z . This characterization enables us to establish a direct link between lottery choices and optimal exposure to risk, as we do in the following corollary.

Generalizing the work of

Corollary 2 Let U be a given increasing, strictly concave, in…nitely di¤ erentiable utility function on R 2 + , satisfying Assumption A1. Let us consider (p; ) 2 R 2 + as given. The following properties are equivalent:

1. For all initial endowment (K; e 1 ); any increase in Nth-degree risk over the second attribute initial endowment from e 1 to e 2 increases the optimal level of the choice variable, i.e. x 2 x 1 .

The preferences represented by U display Nth-degree risk aversion in the direction of ( p; ).

This result is quite intuitive. It establishes that, in response to an increase in risk, the consumer's optimal level of the choice variable (e.g. the level of savings, of labor supply, of medical care) will re ‡ect her preferences towards harm disaggregation as de…ned in this paper. Consider, for instance, the case of a mean preserving spread and, to be concrete, the problem of precautionary saving with time-non-separable utility. As explained above, two e¤ects operate. First, a consumer that is prudent (in second period consumption) would like to allocate a higher level of savings to second period consumption to mitigate the increase in risk. Second, a higher level of savings implies a lower level of …rst period consumption, which a cross-prudent consumer dislikes to match with the higher risk. As a result, the higher level of risk implies a higher level of savings if the prudence e¤ect is stronger than the cross-prudence e¤ect, i.e. pU (1;2) (y; z) + U (0;3) (y; z) > 0, where in this context p = 1 and is the non-stochastic rate of return. Equivalently, savings increase in response to the higher risk if the consumer displays 2nd-degree risk aversion in the direction of ( p; ), so he or she always prefers the lottery [(y px 2 ; z + x 2 + e 2 ) ; (y px 1 ; z + x 1 + e 1 )] over the lottery [(y px 1 ; z + x 1 + e 2 ) ; (y px 2 ; z + x 2 + e 1 )] ; where x 2 > x 1 , e 2 < 2 e 1 , and the attributes of the lotteries represent consumption at two di¤erent dates.

Similarly, consider a decrease in second period income with certainty. A consumer that is risk averse (in second period consumption) would like to mitigate this harm by increasing savings. The harm will then be present when …rst period consumption is lower, which a correlation averse individual dislikes. Savings will increase when the risk aversion e¤ect is stronger than the correlation aversion e¤ect, i.e.

pU (1;1) (y; z) + U (0;2) (y; z) < 0. Equivalently, savings will increase if the consumer displays 1st-degree risk aversion in the direction of ( p; ), so he or she always prefers the lottery [(y px 2 ; z + x 2 + 2 ) ; (y px 1 ; z + x 1 + 1 )] over the lottery [(y px 1 ; z + x 1 + 2 ) ; (y px 2 ; z + x 2 + 1 )] ; where x 2 > x 1 and 2 < 1 .

Clearly, a similar intuition holds for all of the above-mentioned applications and for increases in risk of any degree. A decision maker that views risks as mutually aggravating would like to compensate the higher risk in attribute z with a higher level of the choice variable. Alternatively, he could compensate the higher risk by reducing the level of the choice variable and, as a result, by increasing the level of the other attribute. Whether the level of the choice variable increases or decreases in response to an increase in risk then depends on the relative strengths of the two opposite forces. Corollary 2 then shows that evaluating the strength of the two opposite forces is equivalent to establishing the sign of the function f (0;N ) (y; z), or equivalently, characterizing the decision maker's preference over the lotteries A a and B a (with y = p and z = ).

Multiplicative Risks

As before, consider the following experiment: the consumer needs to locate x 1 y ; x 1 z and x 2 y ; x 2 z to each of the two (random) outcomes of the lottery (y; z + e 1 ) and (y; z + e 2 ) where

x 2 > x 1 0 and where e 2 0 is an Nth-degree risk increase over e 1 0. Now, however, the e¤ect of x 1 z and x 2 z is to scale the risks e 1 and e 2 : In particular, the consumer evaluates the following lotteries:

A m = y + x 2 y ; z + x 2 z e 2 ; y + x 1 y ; z + x 1 z e 1 (5)
B m = y + x 1 y ; z + x 1 z e 2 ; y + x 2 y ; z + x 2 z e 1 :
We propose the following de…nitions.

De…nition 3

We say that preferences display Nth-degree multiplicative-risk attraction (resp. aversion) in the direction of y ; z if, for all (y; z; x 1 ; x 2 ) such that x 2 > x 1 0 and for all pairs of random variables (e 1 ; e 2 ) such that e 2 is an increase in Nth-degree risk over e 1 ; we have

A m B m (resp. B m A m ).
To understand the di¤erent forces at play in such preference ordering, consider …rst the case with y = 0 and a …xed value of y analyzed by [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF]. On the one hand, since x 2 > x 1 , the higher risk in A m is scaled up, which hurts a consumer that dislikes higher risks. On the other hand, x 2 > x 1 also implies that the distribution of z + x 2 z e 2 is shifted upwards relative to z + x 1 z e 1 : As in the previous section, this implies that an individual that prefers to disaggregate harms would like to match this higher level of z with the higher risk, making A m relatively more attractive than B m . Therefore, as stated by [START_REF] Chiu | On relative and partial risk attitudes: Theory and Implications[END_REF], the choice of A m over B m will depend on the relative strengths of these two opposite e¤ects. If we now allow y to di¤er from zero we have, as in the previous section, another e¤ect that arises from the consumer's preference to match the higher risk with a higher level of the other attribute (y). If y > 0 this additional e¤ect will make A m more desirable than B m for an individual that prefers to disaggregate harms, while the opposite will be true if y < 0. In the next, we will focus on the case y 0 and this tends to make B m more desirable than A m for an individual that prefers to disaggregate harms. 12 It is in reference to this situation that we choose to call Nth-degree risk aversion the fact that B m is preferred to A m and to call Nth-degree risk-attraction the opposite behavior.

The following Proposition establishes precisely the di¤erent forces at play in an expected utility model and for general increases in Nth-degree risk.

Proposition 4 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function on R 2 + and let y 0 and z 0 be given. The preferences represented by U display Nth-degree multiplicative-risk attraction (resp. aversion) in the direction of y ; z if and only if

( 1) N U (1;N ) (y; z) 0 (resp. 0); ( 1) N U (0;N ) (y; z) 0 (resp. 0) and R N (y; z) N:

Note that the characterizations depend on y ; z only through their signs: Therefore, we will say that the preferences represented by U display Nth-degree multiplicative-risk aversion (attraction) in the direction of R R + in order to say that they display Nth-degree multiplicative-risk aversion (attraction) in the direction of some y ; z 2 R R + or equivalently in the direction of all y ; z 2 R R + : We have then the following immediate corollary.

Corollary 3 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A2. The following properties are equivalent:

1. For all initial endowment (K; ) and all asset cost and payo¤ (p; e 1 ) such that x 1 0, any increase in Nth-degree risk over the asset's payo¤ from e 1 to e 2 increases (resp. decreases) the optimal level of the choice variable, i.e. x 2 x 1 (resp. x 1 x 2 ).

2. The preferences represented by U display Nth-degree multiplicative-risk attraction (resp. aversion) in the direction of R R + :

In other words, an individual will always decrease the demand for the asset in problem P U;K;p ( ; e ) when the asset's payo¤ is subject to an increase in Nth-degree risk if and only if he or she always selects lottery B m over lottery A m : Similarly, an individual will always increase the demand for the asset in problem P U;K;p ( ; e ) when the asset's payo¤ is subject to an increase in Nth-degree risk if and only if he or she always selects lottery A m over lottery B m . However, as it is clear from Proposition 4, no individual for which ( 1) N U (0;N ) (y; z) 0 (e.g. for N = 2; risk averse individuals) can 1 2 The case with y 0 and z 0 is the relevant scenario to interpret the conditions found in Section 3. also display Nth-degree multiplicative-risk attraction in the direction of R R + . Therefore, no such individual will always prefer lottery A m over lottery B m , or equivalently, no such individual will always increase the demand for the asset in problem P U;K;p ( ; e ). We emphasize again that the "only if" part of Proposition 4 is the critical ingredient to provide a direct link between lottery choices and the optimal exposure to risk.

As an example, suppose that P U;K;p ( ; e ) represents the classic labor supply problem with labor income risk, and consider a 2nd-degree risk increase. Given the higher risk, prudent individuals would like to mitigate the harm by increasing the supply of labor. Doing so, however, scales up the risk, which all risk-averse consumers dislike. In addition, a higher supply of labor implies that the higher risk is coupled with a lower level of leisure, and this is disliked by all cross prudent individuals.

Whether the supply of labor increases or decreases then depends on the relative strength of these di¤erent forces. For consumers that are both risk averse and cross prudent, the supply of labor will always decrease if the coe¢ cient of 2nd-degree relative risk aversion (i.e. relative prudence) is smaller than two. Equivalently, these consumers will always prefer lottery B m over lottery A m ;

where the attributes of the lotteries represent leisure and consumption, y ; z = ( 1; 1), and e 2 < 2 e 1 . For all other risk averse individuals, the higher risk will have an ambiguous e¤ect on the supply of labor, and their preference over lotteries B m and A m will depend on the initial endowments of leisure and consumption.

Remark. As mentioned above, Chiu et al. (2012) analyzed the univariate case with y = 0 and y …xed. They conclude that (Theorem 2): (using our notation) given ( 1) n U (0;n) (y; z) 0 for n = N; N + 1, then A m ( ) B m if and only if ( 1) N xU (0;N +1) (y; x + z) + N U (0;N ) (y; x + z) ( ) 0 for all x 0. As it is clear from the proof of Proposition 4, the second part of this statement is equivalent to our results when y = 0. We remark that the "only if" part follows from Lemma 1.

Furthermore, our proof also clari…es that the assumption ( 1) N U (0;N ) (y; z) 0 is not consistent with the condition ( 1) N xU (0;N +1) (y; x + z) + N U (0;N ) (y; x + z) 0. In fact, as stated in Proposition 4, the condition ( 1) N U (0;N ) (y; z) 0 is necessary for an individual to always prefer A m over B m in an expected utility framework.

Concluding Remarks

Given the ubiquitous presence of uncertainty in most economic decisions, it is not surprising that a large amount of research has been devoted towards understanding the economic consequences of changes in risk. The work of Professor Eeckhoudt has been of fundamental importance towards that goal, raising and answering new questions and also looking for new answers to classical questions.

Professor Eeckhoudt's work has also stimulated a large amount of research in this area and this paper has been a result of such inspiration.

This paper complements Professor Eeckhoudt's work in two important ways. First, we establish the minimum set of necessary and su¢ cient conditions for unambiguous comparative statics of changes in risk in the large class of problems involving bidimensional consequences. Second, we link these conditions with more primitive attitudes towards risk in the form of preferences over simple lottery pairs. In particular, we show that making unambiguous statements about the ordering of a particular class of lottery pairs is equivalent to making unambiguous statements regarding the optimal response to changes in risk in the problems under consideration.

In this paper, we considered separately the case in which the risk is additive and the case in which the risk is multiplicative. It would be interesting to analyze the case in which both risks are present and to evaluate the impact on optimal choice of bivariate stochastic dominance shifts. In Appendix A. Increases in Nth-degree Risk

Proof of Lemma 1

The fact that 2. implies 1. results directly from [START_REF] Ekern | Increasing Nth-degree risk[END_REF]. For the sake of completeness we rederive it. Let (e 1 ; e 2 ) be such that e 2 < N e 1 : We have

E [q(e 2 )] E [q(e 1 )] = Z B 0 q(x)dF e 2 (x) Z B 0 q(x)dF e 1 (x) = N X k=1 ( 1) k 1 q (k 1) (B) h F [k] e 2 (B) F [k] e 1 (B) i + Z B 0 ( 1) N q (N ) (x) h F [N ] e 2 (x) F [N ] e 1 (x) i dx:
Since q and its N …rst derivatives are continuous on R + , they are bounded on all compact subsets and all our integrals are well de…ned.

By de…nition, all the terms in the sum are equal to 0 and

F [N ] e 2 (x) F [N ]
e 1 (x) 0 on [0; B] : By 2., the integral is then nonnegative and E [q(e 2 )] E [q(e 1 )].

Let us now prove that 1. implies 2. Let q be an N times continuously di¤erentiable function on R + such that E [q(e 2 )] E [q(e 1 )] for all pair (e 1 ; e 2 ) such that e 2 < N e 1 : Let, if it exists, be a nonnegative real number such that q (N ) ( ) 6 = 0 and let " denote a positive real number such that q (N ) (t) 6 = 0 for t 2 [ ; + "] : The sign of q (N ) remains then constant on [ ; + "] : Let e 1 and e 2 be two nonnegative bounded above random variables such that e 2 < N e 1 : Let B be a common upper bound for e 1 and e 2 and let e 1 = " B e 1 + and e 2 = " B e 2 + : The random variables e 1 and e 2 take their values in [ ; + "] and it is easy to check that F

[k] e i (t) = " B k 1 F [k] e i t "
B for k = 1; 2::: and i = 1; 2. Therefore, e 2 < N e 1 which implies, by 1., that E [q(e 2 )] E [q(e 1 )] : We have E [q(e 2 )] E [q(e 1 )] = Z +" q(x)dF e 2 (x)

Z +" q(x)dF e 1 (x) = N X k=1 ( 1) k 1 q (k 1) ( + ") h F [k] e 2 ( + ") F [k] e 1 ( + ") i + Z +" ( 1) N q (N ) (t) h F [N ] e 2 (t) F [N ] e 1 (t) i dt:
By de…nition, we have F e 2 (t) F

[N ] e 1 (t) 0 and the inequality is strict for some t in [ ; + "] ; and even on a neighborhood of t by continuity of F

[N ] e 1 and F

[N ] e 2 : Since the sign of q (N ) remains constant on [ ; + "] ; this gives that ( 1) N q (N ) (t) > 0 on [ ; + "] and ( 1) N q (N ) (x) > 0 for all x such that q (N ) (x) 6 = 0 which completes the proof.

Proof of Proposition 1

Let us prove that 2. implies 1. Let (e 1 ; e 2 ) be such that e 2 < N e 1 and let q( ) = g(x 1 ; ; ): We have @ N g @ N (x 1 ; ; ) = pU (1;N ) (K x 1 p; x 1 + ) + U (0;N +1) (K x 1 p; x 1 + ) : By 2., we have ( 1) N @ N g @ N (x 1 ; ; ) 0 and ( 1) N q (N ) ( ) 0: Since e i is bounded above (i = 1; 2), x 1 + e i is bounded and bounded away from zero and U and its N + 1 …rst derivatives are bounded on the convex hull of the set of values taken by (K x 1 p; x 1 + e i ). The same applies to q on the convex hull of the set of values taken by e i ; i = 1; 2: By Lemma 1, this leads to E [q(e 1 )] E [q(e 2 )] : By de…nition, we have E [q(e 1 )] = 0; which gives E [q(e 2 )] 0 or E [g(x 1 ; e 2 ; )] 0: By concavity of U; it is easy to check that E [g(x; e 2 ; )] is a decreasing function of x: Since x 2 is characterized by E [g(x 2 ; e 2 ; )] = 0; we obtain that x 2 x 1 :

Let us prove 1. implies 2. As in the proof of Lemma 1, we consider e 1 and e 2 two nonnegative random variables with a common upper bound B such that e 2 < N e 1 : As above, we introduce the random variables e 1;" = " B e 1 + and e 2;" = " B e 2 + for some > 0 and some " > 0: The random variables e 1;" and e 2;" take their values in [ ; + "] and e 2;" < N e 1;" : Let us consider a given real number and let us de…ne the random variables e i;" ; i = 1; 2; as lotteries giving e i;" with probability " and with probability 1 ": The constant provides an additional degree of freedom that will prove useful in order to control the pair (K x 1 p; x 1 + ) : We have e 2;" < N e 1;" as an immediate consequence of the stability of this order relation under probability mixtures. Let

x 1;" and x 2;" be respectively the solutions of P U;K;p (e 1;" ; ) and P U;K;p (e 2;" ; ): By 1., we have x 2;" x 1;" : By de…nition, we have E g(x 1;" ; e 1;" ; ) = 0 and E g(x 2;" ; e 2;" ; ) = 0: By concavity of U; g is a decreasing function of x and we have E g(x 1;" ; e 2;" ; ) 0: Let q be de…ned by q( ) = g(x 1;" ; ; ): We have E q(e 2;" ) E q(e 1;" ) = (E [q(e 2;" )] E [q(e 1;" )]) "

= N X k=1 ( 1) k 1 q (k 1) ( + ") h F [k]
e 2 ;P ( + ") F

[k] e 1 ;P ( + ")

i "

+"

Z +" ( 1) N q (N ) (t)
h F

[N ] e 2 ;P (t) F

[N ] e 1 ;P (t) i dt:

By construction, the left side of the equality is nonnegative, all the terms in the sum are equal to zero and F

[N ] e 2 ;P (t) F

[N ] e 1 ;P (t) is nonnegative and nonzero. Note that e i;" is bounded and bounded away from zero for i = 1; 2. Therefore q and its …rst N derivatives are bounded on the convex hull of the set of values taken by e i;" ; i = 1; 2: Therefore, ( 1) N q (N ) (t) is nonnegative at least on a given subinterval of [ ; + "] : Let " be in [ ; + "] such that ( 1) N q (N ) ( " ) 0: We have then ( 1) N +1 pU (1;N ) K x 1;" p; x 1;" + " + ( 1) N U (0;N +1) K x 1;" p; x 1;" + " 0

where x 1;" satis…es E h pU (1;0) K x 1;" p; x 1;" + e 1;" + U (0;1) K x 1;" p; x 1;" + e 1;" i = 0

or "E h g(x 1;" ; " B e 1 + ; ) i + (1 ")g(x 1;" ; ; ) = 0:

Remark that, until now, ; K and have been arbitrarily chosen. Let us now choose them carefully in order to derive our result. Let (Y; Z) be arbitrary in R + 2 . By our Inada condition,

lim z!0 U (0;1) (Y;z) U (1;0) (Y;z) = 1 and lim z!1 U (0;1) (Y;z)
U (1;0) (Y;z) = 0 which gives that there exists some z > 0 such that U (0;1) (Y;z ) U (1;0) (Y;z ) = p or pU (1;0) (Y; z )+ U (0;1) (Y; z ) = 0: We choose x > 0 such that x < inf(Z; z ) and and K are taken such that = Z x > 0 and K = Y + px : Let be given by = z x > 0. We have g(x ; e ; ) = pU (1;0) (K x p; x + ) + U (0;1) (K x p; x + ) = pU (1;0) (Y; z ) + U (0;1) (Y; z ) = 0:

The solution of Equation [START_REF] Dardanoni | Uncertainty and the demand for medical care[END_REF] for " = 0 is then given by x : In a well chosen neighborhood of

x , x 7 ! K xp and x 7 ! x + " B e 1 + are bounded and bounded away from 0. The functions U (1;0) and U (0;1) being continuously di¤erentiable, the function

(x; ") 7 ! "E h g(x; " B e 1 + ; ) i + ( 1
")g(x; e ; ) is then di¤erentiable with respect to x at (x ; 0): Furthermore, the derivative of this last function with respect to x at (x ; 0) is nonzero (concavity of U ). The solution x 1;" of Equation ( 7) is then continuous with respect to " in a neighborhood of 0 which gives lim "!0 x 1;" = x :

Furthermore, we clearly have lim "!0 " = : Taking the limit in Equation ( 6) when " tends to 0;

we obtain

( 1) N +1 pU (1;N ) (K x p; x + ) + ( 1) N U (0;N +1) (K x p; x + ) 0 or, by construction

( 1) N +1 pU (1;N ) (Y; Z) + ( 1) N U (0;N +1) (Y; Z) 0

Proof of Proposition 2

Let us prove that 2. implies 1. Let (e 1 ; e 2 ) be such that e 2 < N e 1 and let q( ) = g(x 1 ; ; ):

We have q (N ) ( ) = p (x 1 ) N U (1;N ) (K x 1 p; x 1 + ) + (x 1 ) N U (0;N +1) (K x 1 p; x 1 + ) + N (x 1 ) N 1 U (0;N ) (K x 1 p; x 1 + ).
Since we assumed that x 1 0, by 2, we have

( 1) N p (x 1 ) N U (1;N ) (K x 1 p; x 1 + ) 0 and (
1) N x 1 U (0;N +1) (K x 1 p; x 1 + ) + N U (0;N ) (K x 1 p; x 1 + ) = ( 1) N x 1 x 1 + (x 1 + ) U (0;N +1) (K x 1 p; x 1 + ) + N U (0;N ) (K x 1 p; x 1 + ) + ( 1) N N x 1 + U (0;N ) (K x 1 p; x 1 + ) 0
which gives ( 1) N q (N ) ( ) 0: Since e i is bounded above (i = 1; 2), x 1 e i + is bounded and bounded away from zero and U and its N + 1 …rst derivatives are bounded on the convex hull of the set of values taken by (K x 1 p; x 1 e i + ). The same applies to q on the convex hull of the set of values taken by e i ; i = 1; 2: By Lemma 1, this leads to E [q(e 1 )] E [q(e 2 )] : By de…nition,

we have E [q(e 1 )] = 0 which gives E [q(e 2 )] = E [g(x 1 ; ; e 2 )] 0: By concavity of U; it is easy to check that g(x; ; ) is a decreasing function of x: From there we derive that x 2 x 1 :

Let us prove that 1. implies 2. As in the proof of Lemma 1 and Proposition 1, we consider e 1 and e 2 two nonnegative random variables with a common upper bound B such that e 2 < N e 1 :

As above, we introduce the random variables e 1;" = " B e 1 + and e 2;" = " B e 2 + for some > 0 and some " > 0: The random variables e 1;" and e 2;" take their values in [ ; + "] and e 2;" < N e 1;" : Let us consider a given and let us de…ne the random variables e i;" ; i = 1; 2; as a lottery that takes the value e i;" (!) with probability " and the value with probability 1 ": As previously, we have e 2;" < N e 1;" : Let x 1;" and x 2;" be respectively the solutions of P U;K;p ( ; e 1;" ) and P U;K;p ( ; e 2;" ): If x 1;" is nonnegative then, by 1., we have x 2;" x 1;" : By de…nition, we have E g(x 1;" ; ; e 1;" ) = 0 and E g(x 2;" ; ; e 2;" ) = 0: By concavity of U; g is a decreasing function of

x and we have E g(x 1;" ; ; e 2;" ) 0: Let q be de…ned by q( ) = g(x 1;" ; ; ): We have E q(e 2;" ) E q(e 1;" ) = E q(e 2;" ) E q(e 1;" ) "

= N X k=1 ( 1) k 1 q (k 1) ( + ") h F [k]
e 2 ;P ( + ") F

[k] e 1 ;P ( + ")

i "

+"

Z +" ( 1) N q (N ) (t) h F [N ]
e 2 ;P (t) F

[N ] e 1 ;P (t) i dt:

By construction, the left side of the equality is nonnegative, all the terms in the sum are equal to zero and F

[N ] e 2 ;P (t) F

[N ] e 1 ;P (t) is nonnegative and nonzero. Note that e i;" is bounded and bounded away from zero for i = 1; 2. Therefore q and its …rst N derivatives are bounded on the convex hull of the set of values taken by e i;" ; i = 1; 2: Therefore, ( 1) N q (N ) is nonnegative at least on a given subinterval of [ ; + "] : Let " be in [ ; + "] such that ( 1) N q (N ) ( " ) 0: We have then ( 1) N +1 p x 1;" N U (1;N ) K x 1;" p; x 1;" " +

+( 1) N " x 1;" N U (0;N +1) K x 1;" p; x 1;" " + +( 1) N N x 1;" N 1 U (0;N ) K x 1;" p; x 1;" " + 0 where x 1;" satis…es E h pU (1;0) K x 1;" p; x 1;" e 1;" + + e 1;" U (0;1) K x 1;" p; x 1;" e 1;" + i = 0

or "E h g(x 1;" ; ; " B e 1 + ) i + (1 ")g(x 1;" ; ; ) = 0:

Remark that, until now p; ; K; and have been arbitrarily chosen. Let us now choose them carefully in order to derive our result. We assume …rst that is equal to 1. Let x > 0 be given and

let (M; Y; Z) in (R + ) 3 such that M < Z: We take p = U (0;1) (Y;x +Z M ) U (1;0) (Y;x +Z M ) , K = Y + px ; = Z M and = M x :
By construction, we have > 0 and we have

g(x ; ; ) = pU (1;0) (K x p; x + ) + e U (0;1) (K x p; x + ) = pU (1;0) (K x p; x + ) + U (0;1) (K x p; x + ) = pU (1;0) (Y; x + Z M ) + U (0;1) (Y; x + Z M ) = 0:
The solution of Equation ( 9) for " = 0 is then given by x : In a well chosen neighborhood of x ,

x 7 ! K xp and x 7 ! x + " B e 1 + are bounded and bounded away from 0. The functions U (1;0) and U (0;1) being continuously di¤erentiable, the function (x; ") 7 ! "E h g(x; ; " B e 1 + ) i +

(1 ")g(x; ; ) is then di¤erentiable with respect to x at (x ; 0): Furthermore, the derivative with respect to x at (x ; 0) is nonzero. The solution x 1;" of Equation ( 9) is then continuous with respect to " in a neighborhood of 0 which gives lim "!0 x 1;" = x and guarantees that x 1;" > 0 for " small enough: Since we clearly have lim "!0 " = ; taking the limit in Equation [START_REF] Denuit | Extremal generators and extremal distributions for the continuous s-convex stochastic orderings[END_REF] when " tends to 0;

we obtain We assumed that z U (0;1) (Y;z) U (1;0) (Y;z) is unbounded. We then have either lim z!0 z U (0;1) (Y;z) U (1;0) (Y;z) = 1 or lim z!1 z U (0;1) (Y;z) U (1;0) (Y;z) = 1. If lim z!0 z U (0;1) (Y;z) U (1;0) (Y;z) = 1, it su¢ ces to take x = 2 and M = Z for arbitrarily small to make the quantity px = U (0;1) (Y;x +Z M ) U (1;0) (Y;x +Z M ) x = 1 2 U (0;1) (Y; ) U (1;0) (Y; ) arbitrarily large. Since M; Y and Z are bounded, Equation [START_REF] Denuit | Some consequences of correlation aversion in decision science[END_REF] gives then ( 1) N U (1;N ) (Y; Z) 0. If lim z!1 z U (0;1) (Y;z) U (1;0) (Y;z) = 1; it su¢ ces to take x su¢ ciently large to make the quantity x +Z M suf-…ciently large and U (0;1) (Y;x +Z M ) U (1;0) (Y;x +Z M ) (x + Z M ) arbitrarily large. Since Z is kept …xed,

x x +Z M is arbitrarily close to 1 and px arbitrarily large. Since M; Y and Z are bounded, Equation [START_REF] Denuit | Some consequences of correlation aversion in decision science[END_REF] gives then ( 1) N U (1;N ) (Y; Z) 0.

The Proof of Corollary 1 follows directly by reversing the inequalities above.

Proof of Proposition 3

In an expected utility framework, Nth-degree risk aversion in the direction of y ; z is equivalent to + : Since we want this inequality to be true for all (e 1 ; e 2 ) such that e 2 is an increase in Nth-degree risk over e 1 ; this inequality is equivalent, by Lemma 1, to ( 1) N f (0;N ) 0:

Proof of Proposition 4

In an expected utility model, Nth-degree multiplicative-risk attraction in the direction of y ; z is equivalent to e 2 (x) F

1 2 E U
[N ] e 1 (x) for all x 2 [0; B] where the inequality is strict for some x and F Jean (1980) characterizes Nth-degree stochastic dominance: He establishes that e 2 is dominated in the sense of Nth-degree stochastic dominance by e 1 if and only if E [q(e 2 )] > E [q(e 1 )] for all N times continuously di¤erentiable real valued function q such that ( 1) k q (k) > 0 for k = 1; :::; N where q (k) = d k q d k : The following Lemma characterizes the set of N times continuously di¤erentiable functions for which E [q(e 2 )] E [q(e 1 )] for all pair (e 1 ; e 2 ) where e 2 < N SD e 1 :

Lemma 2 Let q be a given real valued function that is N times continuously di¤ erentiable on R + .

The following are equivalent.

1. For all pair (e 1 ; e 2 ) such that e 2 < N SD e 1 ; we have E [q(e 2 )] E [q(e 1 )] :

2. For all x 0; we have ( 1) k q (k) 0 for k = 1; :::; N:

Proof. The fact that 2. implies 1. results directly from [START_REF] Jean | The geometric mean and stochastic dominance[END_REF]. Let us now prove that 1. implies 2. Let q be an N times continuously di¤erentiable function on R + such that E [q(e 2 )] E [q(e 1 )] for all pair (e 1 ; e 2 ) such that e 2 < N SD e 1 : Let k be in f1; :::; N g and let (e 1 ; e 2 ) such that e 2 < k e 1 : By de…nition, we have F

[k] e 2 (x) F

[k] e 1 (x) for all x 2 [0; B] where the inequality is strict for some x and F e 1 (B) for i = 1; :::; k 1: By integration, we obtain F

[j] e 2 (x) F

[j] e 1 (x) for all j k and all x in [0; B] and, in particular, e 2 < N SD e 1 hence E [q(e 2 )] E [q(e 1 )] : By Lemma 2 we have ( 1) k q (k) 0 on [0; B] for k = 1; :::; N:

The following Proposition establishes the e¤ect on optimal choice of an Nth-degree stochastically dominated shift on the initial endowment .

Proposition 5 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A1. Let us consider p and as given. The following properties are equivalent:

1. For all initial endowment (K; e 1 ), a switch from e 1 to e 2 such that e 2 < N SD e 1 increases the optimal level of the choice variable, i.e. x 2 x 1 .

2. For all (y; z) ; we have ( 1) k pU (1;k) (y; z) + U (0;k+1) (y; z) 0 for k = 1; :::; N .

De…nition 1 (

 1 Ekern) Let e 1 and e 2 denote two random variables with values in [0; B] : For i = 1; 2, we denote by F[START_REF] Baiardi | Pigouvian Tax, Abatement Policies and Uncertainty on the Environment[END_REF]

(

  1980) (e.g. Eeckhoudt and Schlesinger, 2008, Chiu and Eeckhoudt, 2010, Baiardi and Menegatti, 2011, Chiu et al., 2012, Courbage and Rey, 2012

x 2 x 1 ( 1 )

 11 for all e 1 4 N e 2 ) if and only if, for all z; R N (z) N with N v (N ) (z) 0, which corresponds to the conditions established by[START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF] in a model without labor income ( = 0).9 Again, we remark that our conditions are necessary and jointly su¢ cient and that they hold for all (y; z). Still in a one dimensional setting, we retrieve the following well-known results in the context of the classical portfolio choice problem (see e.g.[START_REF] Rothschild | Increasing risk II: Its Economic Consequences[END_REF][START_REF] Hadar | The e¤ects of shifts in a return distribution on optimal portfolios[END_REF], Gollier, 2001, p. 61):

  Eeckhoudt et al. (2009) (i.e. the case with y = 0 and z > 0 and for a …xed value of y), Denuit et al. (2010a) established the "if" part of this Proposition for the case

  this case, again, the recent work of Eeckhoudt and colleagues on a characterization of multivariate stochastic dominance that is connected with a basic preference to disaggregate harms (e.g. Denuit et al. 2010a) and on the consequences of shifts in the dependence structure of two random variables (Denuit et al. 2010b, Denuit et al. 2011) may provide some hints on how to generalize our results of Sections 3 and 4.

e 1 (

 1 + ") for k = 1; :::; N and F [N ]

( 1 ) 10 )

 110 N (x ) N 1 px U (1;N ) (Y; Z) + x U (0;N +1) (Y; Z) + N U (0;N ) (Y; Z) 0 or ( 1) N px U (1;N ) (Y; Z) + M U (0;N +1) (Y; Z) + N U (0;N ) (Y; Z) 0 (This result being true for all (Y; Z) in R + 2 ; all M 2 (0; Z) all x > 0 and for p = for all (Y; Z) in R + 2 and all M 2 (0; Z) or equivalently( 1) N ZU (0;N +1) (Y; Z) + N U (0;N ) (Y; Z) 0 and ( 1) N U (0;N ) (Y; Z) 0 for all (Y; Z) in R + 2 :

2

 2

e 1

 1 (B) for k = 1; :::; N 1:

  + x 2 y ; z + x 2 z + e 2 1 2 E U y + x 2 y ; z + x 2 z + e 1 R 4 + with x 2 > x 1 . The previous inequation is satis…ed for all x 2 > x 1 if and only if E U y + t y ; z + t z + e 2 E U y + t y ; z + t z + e 1 is increasing in t or, equivalently, if and only if E f y + t y ; z + t z + e 2 is larger than E f y + t y ; z + t z + e 1 for all (y; z; t) 2 R 3 + . This is satis…ed if and only if E [f (y; z + e 2 )] is larger than E [f (y; z + e 1 )] for all (y; z) 2 R 2

	1 2	E U y

> 1 2 E U y + x 1 y ; z + x 1 z + e 2 1 2 E U y + x 1 y ; z + x 1 z + e 1 :

for all (y; z; x 1 ; x 2 ) 2

  y + x 2 y ; z + x 2 z e De…nition 4 (Jean) Let e 1 and e 2 denote two random variables with values in [0; B] : We say that e 2 is dominated in the sense of Nth-degree stochastic dominance by e 1 ; and we denote it by e 2 < N SD e 1 ; if F

	[N ]

2 1 2 E U y + x 2 y ; z + x 2 z e 1 > 1 2 E U y + x 1 y ; z + x 1 z e 2 1 2 E U y + x 1 y ; z + x 1 z e 1

(

[START_REF] Denuit | Correlated risks, bivariate utility and optimal choices[END_REF]

Other work evaluating economic consequences of Nth-degree risk increases includes Baiardi and Menegatti's (2011) analysis of the trade-o¤ between (dirty) production and environmental-quality and Courbage and Rey's (2012) analysis of the optimal allocation of a national health budget.

It is more common to de…ne Nth-degree risk increases in terms of a utility function u ( ) as follows: e 2 is an increase in Nth-degree risk over e 1 if and only if E [u(e 2)] < E [u(e 1)] for all N times continuously di¤erentiable real valued utility function u such that ( 1) (N +1) u (N ) > 0: Our equivalent characterization is somewhat more natural when dealing with the e¤ect of risk increases on optimally chosen variables (i.e. on marginal utility).

The canonical portfolio problem arises as a special case of this framework in which the attributes are perfect substitutes (setting U (y; z) = v(y + z) and rede…ning the variables), making it, in essence, a univariate problem. Assumption A2 is always satis…ed in this setting.

Similarly, in the context of problems with non-separable utility and an additive risk,[START_REF] Chiu | The e¤ects of stochastic wages and non-labor income on labor supply: update and extensions[END_REF] and[START_REF] Baiardi | Pigouvian Tax, Abatement Policies and Uncertainty on the Environment[END_REF] established the su¢ ciency of the condition in Proposition 1.

Suppose, for example, that relative risk aversion is constant: v(z) = (1 ) 1 z 1 : Then, if < 1 an Nth-degree risk increase in the rate-of-return will decrease savings. For > 1; an Nth-degree risk increase in the rate-of-return will increase savings for some wealth levels and it will decrease savings for other wealth levels.

0 Eeckhoudt et al. (2007) restrict themselves to the case x1 = 0. All our results could be generalized further, along the lines of Eeckhoudt et al.'s (2009) results in a univariate framework, by considering the case in which x1 is a …rst order stochastically dominated shift over x2.

[START_REF] Baiardi | Pigouvian Tax, Abatement Policies and Uncertainty on the Environment[END_REF] [START_REF] Eeckhoudt | Apportioning of risks via stochastic dominance[END_REF] showed that if e

<N e 1 and x1 <M x2, then the 50-50 lottery [e 2+ x1, e 1+ x2] is an (N+M)th-degree risk increase over [e 2+ x2, e 1+ x1] in the sense of Ekern (1980), i.e. the second lottery is preferred by all decision makers with ( 1) N +M U (0;N +M ) (y; z) 0. When y = 0 and z > 0, and for a …xed value of y, our Proposition deals with the special case in which M = 1. Denuit et al (2010a) proved a multivariate version of Eeckhoudt et al. (2009)'s results.

gratefully acknowledge the …nancial support of the ANR (RISK project) as well as of the Risk Foundation (Groupama Chair). U (0;1) (Y;x +Z M

for all (e 1 ; e 2 ) such that e 1 4 N e 2 and all (x 1 ; x 2 ) such that 0 < x 1 < x 2 and y + x i y 0; i = 1; 2: This inequality is satis…ed for all x 2 > x 1 if and only if E U y + t y ; z + t z e 2 E U y + t y ; z + t z e 1 is increasing on T (y) = t 2 R + : y + t y 0 or, equivalently, if and only if E [h (e 2 ; y; z; t)] is larger than E [h (~ 1 ; y; z; t)] for all (y; z) 2 R 2 + and for t 2 T (y) where h ( ; y; z; t) = y U (1;0) y + t y ; z + t z + z U (0;1) y + t y ; z + t z . By Lemma 1, this last inequality is satis…ed for every pair (e 1 ; e 2 ) such that e 2 is an increase in Nth-degree risk over e 1 if and only if ( 1) N @ N h @ N 0:

where derivatives of U are taken at y + t y ; z + t z . Since t 0 and z 0, our necessary and su¢cient condition can then be rewritten as ( 1) N t y U (1;N ) + t z U (0;N +1) + N U (0;N ) 0 for all ( ; y; z) 2 R 3 + and for t 2 T (y): Denoting y + t y by Y and z + t z by Z; our necessary and su¢ cient condition is equivalent to

+ and all (y; z) such that y Y and 0 z Z and where the derivatives are taken at (Y; Z). Taking successively y = Y and z = Z; y = Y and z = 0 we obtain that ( 1) N U (0;N ) 0 and ( 1) N ZU (0;N +1) + N U (0;N ) 0: Letting y go to 1, we obtain ( 1) N U (1;N ) 0: Conversely, if these 3 conditions are satis…ed and for a given y Y; we have ( 1) N (Y y)U (1;N ) 0 and

( 1)

0 which gives that our necessary and su¢ cient condition is satis…ed.

A far as Nth-degree multiplicative-risk aversion is concerned, it is characterized by the fact that

or, equivalently, by the fact that E [ h (e 2 ; y; z; t)] is larger than E [ h (~ 1 ; y; z; t)] for all (y; z) 2 R 2 + and for t 2 T (y) or, …nally, by ( 1) N +1 @ N h @ N 0: The rest of the proof is then directly adapted from the multiplicative-risk attraction setting.

Appendix B. Nth-Degree Stochastic Dominance

In this appendix we generalize Lemma 1, Propositions 1 and 2, and Corollary 1. We begin by formalizing the concept of Nth-degree stochastic dominance.

Proof. Let us prove that 2. implies 1. Let (e 1 ; e 2 ) be such that e 2 < N SD e 1 and let q( ) = g(x 1 ; ; ): By 2., we have ( 1) k q (k) ( ) 0 for k = 1; :::; N: By Lemma 2, this leads to E [q(e 1 )] E [q(e 2 )] : By de…nition, we have E [q(e 1 )] = 0; which gives E [q(e 2 )] 0 or E [g(x 1 ; e 2 ; )] 0: By concavity of U; it is easy to check that E [g(x; e 2 ; )] is a decreasing function of x: Since x 2 is characterized by E [g(x 2 ; e 2 ; )] = 0; we obtain that x 2 x 1 :

Let us prove 1. implies 2. As in the proof of the Lemma 2, it su¢ ces to remark that, for k = 1; :::; N; e 2 < k e 1 implies e 2 < N SD e 1 :

Finally, the following Proposition establishes the e¤ect on optimal choice of an Nth-degree stochastically dominated shift on the multiplicative variable .

Proposition 6 Let U be a given increasing, strictly concave and in…nitely di¤ erentiable utility function satisfying Assumption A2. The following properties are equivalent:

1. For all initial endowment (K; ) and all asset cost and payo¤ (p; e 1 ) such that x 1 0, and any switch from e 1 to e 2 with e 2 N SD e 1 increases (decreases) the optimal level of the choice variable, i.e. x 2 x 1 .

2. For all (y; z) ; we have ( 1) k U (1;k) (y; z) 0 (resp: 0) R k (y; z) N and ( 1) k U (0;k) (y; z) 0 (resp: 0) for k = 1; :::; N .

Proof. Follows the proof of Proposition 5.


Raphaële Bertho

Aurélie Carimentrand

The small producer: from political commitment to a marketing figure. Analysis of use of the portrait for social communication on products

Raphaële Bertho, Aurélie Carimentrand

Uses of the portrait in the media: the illusion of its value as testimony

Using a portrait for communication purposes is a practice that is as old as advertising itself. Highlighting a figure or face personalises the message, creates the illusion of special dialogue, a close relationship, and therefore captures the receiver's attention. With the development of techniques for copying and diffusing images, the number of portraits has greatly increased, and so have the uses of portraits in the media. Portraits are used as much to sell1 as to provide a recommendation or raise awareness in a reader, who is gradually turned into a spectator, as the portrait has also been used for political mobilisation since the late 19 th century. Among other sources, this social documentary tradition began with the work of the American Lewis Hine , particularly his pictures for the National Child Labor Committee in the 1910s. The captions to the pictures state the name, age and history of each photographed individual. The photographs used in the 1930s by the US government to show the ravages of the economic crisis under the Farm Security Administration (1935)(1936)(1937)(1938)(1939)(1940)(1941)(1942) had a similar motivation, but adopted different strategies. Dorothea Lange, for example, gives more generic captions to her pictures, and as a result the figure becomes a symbol rather than a first-hand account of a specific situation. Fair trade's visual strategy is inherited from this iconography.

Use of the portrait in the media generally contributes to the subject being eclipsed by the genre or category, whatever the medium or channel of diffusion. Yet this observation goes against the ideology underlying the development of social and press photography: in both cases, the aim is to build on the testimony value assigned to photographic records. A portrait should not simply "humanise" the story, but act as proof of its truth. It is important to note that this belief in a form of truth value of images is in fact a cultural construct. Engravings were credited with the same quality before it was attributed to photographic images [START_REF] Gervais | L'illustration photographique, Naissance du spectacle de l'information, 1843-1914[END_REF]. But both types of picture, engravings and photographs, are primarily the work of their creators, who make compromises with reality. A photographic shot is a fragment of space and time, whose conditions of existence depend on a number of material, economic, ethical and political contingencies, and can therefore only be a singular reflection of a reality that is always multiple [START_REF] Michaud | Critique de la crédulité[END_REF]. The image, received as a proof, is first and foremost a reflection of the discourse conveyed by the authors of communication media.

What about the images, and more specifically portraits, used in marketing strategies referring to the figure of the "small producer", namely in North-South fair trade, North-North fair trade and the new short food supply chains. For North-South fair trade, the use of inserts containing a brief description of the producers and their families is indeed a conscious strategy to "market" products [START_REF] Wright | Consuming lives, consuming landscapes: interpreting advertisements for Cafédirect coffees[END_REF]). These strategies are largely based on narratives about the producers [START_REF] Goodman | Reading fair trade: political ecological imaginary and the moral economy of fair trade foods[END_REF]) and contextualisation of the products: they include the name of a cooperative, a place and/or a picture of the producers. The communication conveyed by this marketing is supposed to fill in for the interpersonal relations of traditional networks, by reducing the distance between the producers and consumers. Several studies have shown that in fair trade, representations of producers are built on a shared repertory of symbols [START_REF] Wright | Consuming lives, consuming landscapes: interpreting advertisements for Cafédirect coffees[END_REF][START_REF] Adams | Encountering the Fairtrade Farmer: Solidarity, Stereotypes and the Self-Control Ethos[END_REF]. These visuals construct a generic figure, that of the idealised small farmer or producer as the upholder of traditional values, and this figure masks the singularity of situations. The populations presented on the packaging relate to the "myth of the small producer". Far from being documentary in nature, this figure of the "small producer" is entirely constructed, for the purpose of conveying a specific message. This construction certainly helps to humanise the representation of the production channel, but paradoxically it works against the desire for transparency, instead promoting a myth.

The portrait of the small producer in North/South fair trade

The "myth of the small producer" carried by North/South fair trade has its foundations in a postcolonial heritage that contributes both to naturalisation of their inferiority and emphasis of their authenticity [START_REF] Blanchet | Quand l'Autre est le même… mais jamais tout à fait » : une lecture postcoloniale du commerce équitable[END_REF]. This dual trend associates positive values (authenticity) with negative values (weakness and being "left out" of society). The weakness and sidelined situation of the "small producer" is underlined in the discourse, eliciting compassion and appealing to the ethical values of mutual support and solidarity. The consumer comes to associate his purchase with a political act and the idea of going against the established order. The non-verbal communication, on the other hand, is much less conservative. To enhance the authenticity aspect, the figure of the small producer reuses stereotypes based on the portraits of "natives" distributed since the 19 th century (Maresca 2011). Very early on, even in the 1840s2 , photographic mimesis was used in anthropological expeditions with the aim of building up scientific records of ethnic traits. But with no standardisation tool, these portraits remained portraits of specific individuals, and could not be used to establish any real typology. Ethnographic studies thus developed classification methods that ultimately came to resemble the judiciary method, particularly visible in the frontal/profile photographic system [START_REF] Phéline | L'image accusatrice[END_REF]. The main explanation for this complicity between the anthropometric portrait and anthropologists' racial albums, according to Sylvain Maresca (2011), is their common denial of the portrait as a singular representation, whether in form or title. Individuals are considered merely as specimens, physiological ideal-types of the group to which they belong. The staging of the picture helps to solve the contradiction inherent to using the portrait of a particular flesh-and-blood individual as the representative of a general, hypothetical and abstract type. The individual becomes the representative of the group, and vanishes behind the type. The photographs used by most fair trade brands establish a fantasised vision of the Other, whose singularity is lost in his otherness. An exotic figure mainly recognisable by the adoption of traditional attributes (clothing, canvas bag, animal). This reprise of colonial stereotypes is not specific to fair trade. The figure of the small producer is found in many marketing strategies promoting products from the South. The coffee brand Juan Valdez, for example, constructs its image based on the figure of a producer wearing a wide-brimmed hat and a traditional scarf, who is always accompanied by his mule. This iconography is ultimately similar to the stylised figure used by Max Havelaar until it recently standardised and stylised its logo. Marketing strategies in North/South fair trade, when they spotlight the figure of the "small producer", thus tend to reproduce the visual strategies of conventional trade in a representation that relates to the hierarchies inherited from colonial times. The image contrasts with the text, in an obvious contradiction of the discourse.

The dual nature of the small local producer figure

When fair trade is seeking greater attention for individuals, this relates visually to the type or specimen, or even a "concept-portrait" or "specimen-portrait". With the recent development of North-North fair trade and the marketing of authenticity, the figure of the small producer is being repositioned. As a result, the abstraction is based on different presumptions in the North and South. While the visuals selected for commercial channels in the South tend to naturalise difference and weakness, appealing to a sense of compassion, the visuals used in the North relate to the myth of the self-employed producer working alone, a kind of artist of the modern day (this is the case of the Alter Eco "North-North" fair trade range).

The specificity of fair trade, which wants to raise the profile of production channels, is now encountering the ideology of transparency which "means that everything can be shown and made public, to be submitted to examination by others, and also be the object of monitoring and control procedures" (Zarka, 2006, authors' own translation). This false revealing, this apparent discovery of the processes at work and the actors working in the shadows is found in all sectors of communication (industrial, political, etc). It is materialised in packaging by the photograph exposure, on the products, of actors in the production or distribution chain. Their portraits appear on the packaging itself, in the stores and in the advertising. For promotion of local products that claim implicitly or explicitly to be fair trade products, the producer is given a key place in this emphasis on authenticity. When shown and identified on the packaging, he is a guarantee of the product's origins. He provides situational authentification of the product (Merle and Piotrowski, 2012) and its traceability, either real or symbolic. Development of these marketing strategies thus contributes to a shift, or a split into a dual nature, for the "small producer" figure. This figure is emerging in new brands such as "Le Petit Producteur" or short supply chains such as supermarket chain E. Leclerc's "Local alliances". In both cases, the producer has not only a face, but also a full name, a specific history, sometimes even a body and a voice. The situation is ultimately paradoxical: while fair trade channels are depersonalising producers in favour of stereotypical profiles, the actors of conventional trade are succeeding in giving the small producer a more realistic presence. These strategies are a way of adding relational value to products sold in supermarkets. They create confusion with fair trade products by adopting a similar marketing strategy.

In the end, portrait use occupies an ambiguous position, since the aim is to respond to a desire for documented traceability, a social and political concern to make the invisible people in a commercial channel visible, and a marketing strategy to promote the relational by humanising the connection with the product. It appears that so far, the marketing dimension based on promotion of stereotypes has been dominant in fair trade channels, while paradoxically, channels that are further removed from the ethical principles of fair trade have succeeded in emphasising the documentary value of this visual choice.

As shown by the Félix Potin album designed to hold a collection of 510 photographs of personalities of the time, collected one by one when items were purchased in a Félix Potin store (chain of small grocery stores).

The spread of the photographic process from 1839 sparked a major upheaval in the portrait's reign in the West. It had previously been the preserve of a political, economic and intellectual elite, but then spread to new social classes and new uses. This period forged the social uses and a shared visual culture of the photographic portrait. Photography was used for social purposes just as much as scientific and police purposes.