Risk and Sustainability: Is Viability that far from Optimality

Keywords: Sustainability, uncertainty, multicriteria, viability

Economic analysis addresses risk and long-term issues with discounted expected utility, focusing on optimality. Viability theory is based on sustainability constraints to be satisfied over time, focusing on feasibility. We make a bridge between these two approaches by showing that viability is equivalent to an array of degenerate intertemporal optimization problems. This makes the approach more interpretable in economic terms, and especially regarding efficiency. First, the deterministic case is examined. A particular emphasis is put on the connections between the viability kernel and the minimal time of crisis function. Then, we present stochastic viability with the notions of viable scenario and maximal viability probability. We show that the maximal viability probability shares dynamic programming properties with optimal discounted expected utility. Thus, both exhibit time-consistency, which may be a basis for an axiomatization of criteria under risk and long run for public decision-making.

Risque et durabilité : la viabilité est-elle si loin de l'optimalité ?

Michel De Lara * , Vincent Martinet † et Luc Doyen ‡

Résumé :

L'analyse économique aborde les questions de risque et de long terme dans le cadre de l'utilité espérée actualisée, dans une perspective d'optimalité. La théorie de la viabilité est basée sur des contraintes de soutenabilité, dans une perspective de faisabilité. Nous proposons ici un pont entre ces deux approches en montrant que la viabilité est équivalente à un ensemble de problèmes d'optimisation intertemporelle dégénérés. Cela rend l'approche plus facilement interprétable en termes économiques, et en particulier du point de vue efficacité. Le cas déterministe est examiné tout d'abord. Nous soulignons les connections entre le noyau de viabilité et la fonction de temps de crise minimal. Nous présentons ensuite la viabilité stochastique, avec les notions de scénario viable et de probabilité maximale de viabilité. Nous montrons que la probabilité maximale de viabilité partage des propriétés de programmation dynamique avec l'optimum de l'utilité espérée actualisée. Les deux approches sont donc cohérentes dynamiquement, ce qui pourrait servir de base pour une axiomatisation de critères pour la prise de décision de long terme en présence de risque.

Mots-clés : Durabilité, incertitude, multicritère, viabilité.

Classifications JEL : Q01, D81, D63

Introduction

Dealing with environmental issues such as climate change, biodiversity preservation, or managing natural resources, requires to account for conflicting objectives, dynamics, uncertainty and long-term.

The issue of decision under risk has been widely addressed in the economic literature, back to the foundation of expected utility theory axiomatized by von [START_REF] Neuman | Theory of games and economic behaviour[END_REF]. However, the expected utility framework is known to exhibit serious limitations in certain situations (e.g., Allais and Ellsberg paradoxes), and these limitations are often highly relevant in environmental and resource economics [START_REF] Shaw | Why environmental and resource economists should care about non-expected utility models[END_REF].

Dynamic and long-term issues also received a particular focus in the economic literature, especially in growth theory [START_REF] Koopmans | On the concept of optimal economic growth[END_REF]. Regarding environmental issues, the sustainability debate has shed a new light on the discounted utility approach. In the literature addressing optimal growth theory with environment, it has been shown that the usual neo-classical criterion may lead to unsustainable economic trajectories [START_REF] Chevé | La croissance optimale d'une économie polluante : durabilité économique versus durabilité écologique[END_REF], mainly because of discounting. [START_REF] Fleurbaey | Quelques réflexions sur la croissance optimale[END_REF] emphasize that, in a finite horizon framework, a criterion without discounting can easily be defended, but that, in infinite horizon, discounted approach is used more for practical than for theoretical reasons. The discounted utility criterion has mainly been criticized because it neglects long-run utility, being qualified as a "dictatorship of the present" by [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF]. The sustainability debate has thus been marked by the introduction of other criteria [START_REF] Rotillon | Économie des ressources naturelles[END_REF].

Given these important contributions on risk on the one hand, and dynamic/longrun issues on the other hand, one could be surprised to note that joint issues of risk and sustainability have received less attention as to the axiomatic foundations. An exception is the book Sustainability: Dynamics and Uncertainty by [START_REF] Chichilnisky | Sustainability: Dynamics and Uncertainty[END_REF] in which risk and dynamics are addressed mainly in the discounted expected utility framework. In his textbook, The Economics of Risk and Time [START_REF] Gollier | The Economics of Risk and Time[END_REF], Gollier provides two arguments to justify the use of discounted expected utility with exponential discounting. First, it is time-consistent, which may be considered as a fine property for public decision making [START_REF] Cohen | How should control theory be used to calculate a time-consistent government policy?[END_REF]. Second, it accounts for pure preference for the present (impatience) via discounting, which may have a sense in individual decision making but is criticized in the sustainability debate on long-run issues. The discounted expected util-ity has not been designed to address sustainability issues but, as far as we know, usual sustainability criteria have not yet been applied in a stochastic framework, providing no alternative. However, as stressed in Stern's Review for climate change, environmental issues are characterized by both risk and dynamic/long-run, and these issues should be addressed in a single framework. [START_REF] Howarth | Sustainability under uncertainty: A deontological approach[END_REF] emphasizes that, under uncertainty, a deontological approach should be used to address the sustainability issue, and sustainability conditions should be imposed as prior constraints on the maximization of a social welfare function.

According to [START_REF] Gerlagh | Efficiency of conservationist measures: an optimistic viewpoint[END_REF], conservationist policies can be Pareto efficient, and strict resource conservation is equivalent to nondictatorship of the present. This echoes the following quotation of Marcel Boiteux:

(. . . ) pour les modèles à long terme, l'approche par les prix n'est pas la meilleure (mieux vaut travailler sur les quantités et trouver les prix par dualité pour orienter ensuite les choix décentralisés des acteurs) [START_REF] Boiteux | À propos de la "critique de la théorie de l'actualisation telle qu'employée en France[END_REF] Using quantities to deal with the sustainability issue, due to its long-term perspective, is thus an alternative approach to expected utility [START_REF] Mäler | Environment, uncertainty, and option values[END_REF].

When sustainability objectives are defined using indicators (quantitative measurement of economic or physical meaning) and thresholds, the problem of the regulator in coping with all the objectives simultaneously is to avoid crisis situations, and sustainability appears closer to a 'satisficing' problem (bounded rationality [START_REF] Simon | A behavioral model of rational choice[END_REF]) than to an optimizing problem [START_REF] Krawczyk | Satisficing solutions to a monetary policy problem: a viability theory approach[END_REF]. Actually such a 'satisficing' approach is rather the norm in practice: natural resource management issues are often addressed using indicators and associated thresholds. For example, the global change issue is addressed with a GHG concentration upper limit, and emission reductions are also defined in quantity terms. The concept of "stewardship" mentioned in Stern's Review also stresses similar issues:

The notion of "stewardship" can be seen as a special form of sustainability. It points to particular aspects of the world, which should themselves be passed on in a state at least as good as that inherited from the previous generation.

Let us also mention the generalized capacity of Solow:

If sustainability means anything more than a vague emotional commitment, it must require that something be conserved for the very long run. It is very important to understand what that thing is: I think it has to be a generalized capacity to produce economic well-being. [START_REF] Solow | An almost practical step toward sustainability[END_REF] If, in an intergenerational equity perspective, the constraints defined by these indicators have to be satisfied throughout time, such sustainability problems can be studied in the viability framework [START_REF] Martinet | Defining sustainability objectives[END_REF]Doyen, 2007, Baumgärtner and[START_REF] Baumgärtner | Ecological-economic viability as a criterion of strong sustainability under uncertainty[END_REF]. A major mathematical instrument of the viability analysis is the so-called viability kernel [START_REF] Aubin | Dynamic Economic Theory ; A Viability Approach[END_REF]. It is composed of all initial states from which economic development paths respecting the constraints can start, under appropriate sequences of decisions. Focusing on such a geometrical tool is not traditional in economic analysis.

Replaced in our discussion on risk, viability can be interpreted in a stochastic framework as the high probability to be above thresholds, following the lead of De [START_REF] Lara | Sustainable Management of Natural Resources[END_REF]. This is related to the psychological process of aspiration assessment: Lopes claims that "sensible people often base their choices on the probability of coming out ahead" [START_REF] Lopes | When time is of the essence: Averaging, aspiration, and the short run[END_REF]. When stakes are high, as life and death issues illustrated in [START_REF] Dubbins | How to gamble if you must[END_REF], this is a fairly "reasonable" approach.

In this paper, we examine how the stochastic viability approach addresses conflicting objectives, uncertainty, dynamic processes and long-term issues. The contribution of this paper is twofold.

First, from a theoretical point of view, we provide a criterion-like description of the viability approach, which allows us to stress its links with the usual economic approaches. In fact, the equivalence is not with one but with several degenerate optimization problems: we provide a description of time additive and time multiplicative criteria, and interpret each kind of formulation. This allows us to draw parallels between the discounted utility criterion and the viability approach. The viability problem being expressed as a dynamic optimization problem appears closer to usual economic representations.

Second, we argue that stochastic viability is a pertinent complementary approach to deal with dynamic problems under uncertainty, and thus to define sustainable management of natural resources or to cope with long-run environmental issues such as climate change. In particular, we show that if viability in a deterministic framework does not rank all trajectories, in a stochastic framework the probability to achieve the sustainability objectives, represented by viability constraints, is a natural currency to rank decision rules.

The paper is organized as follows. In Sect. 2, we describe both the usual discounted utility and the viability approach. We emphasize their conceptual differences, and the implications of these differences for natural resource management. In Sect. 3, we describe how both approaches read in a stochastic framework. We show that, when time and risk interact, the previously exhibited differences reduce and are sources of complementarity, at least to address natural resources management or environmental issues. We conclude on research perspectives on sustainability criteria mixing risk and time in Sect. 4.

Deterministic intertemporal optimal choice problem

As stated in the introduction, natural resources management and environmental issues are dynamic in nature. To address these issues, we use a dynamic modeling framework from control theory. We shall consider management of dynamic systems in discrete time. We indeed want to avoid technical difficulties related to continuous time, and concentrate on conceptual issues. In all the sequel, we shall deal with stationary problems without explicit dependence upon time, for the sake of simplicity of notations.

Dynamic economic model

The economy is represented by the following discrete-time control dynamical system

x(t + 1) = G x(t), c(t) , t = t 0 , . . . , T -1 , x(t 0 ) = x 0 , (1)
where the time index t is discrete (t ∈ N is an integer), with t 0 the initial time and T the horizon, which may be finite (T < +∞) or infinite (T = +∞).

The state x(t) is a vector belonging to X := R n ; each capital stock may represent man-made reproducible capital, natural resources (renewable or not) or pollution stocks while x 0 ∈ X is the initial state for the initial time t 0 . The control c(t) ∈ C := R p may represent investment, consumption, catches or harvesting effort, or emissions. The mapping G : X × C → X stands for the dynamics representing the evolution of the various stocks through time.1 It may include economic models with capital and labor, population dynamic models of natural resources, or pollution accumulation-absorption models.

Outputs/indicators

Main outputs of this system are given by so-called indicators The parallel can be drawn with Lancaster's consumer theory ( [START_REF] Lancaster | A new approach to consumer theory[END_REF]) as decisions and states are not necessarily a matter of interest in and of themselves, but impact utility by their effects on various characteristics, which can be arguments of a multi-attribute utility function. In our case, one could define a function U (I 1 , ..., I K ) of the indicators.

I k x(t), c(t) , k = 1, . . . , K. An indicator I k : X × C → R

Maximal intertemporal discounted utility

In an intertemporal framework, the purpose of economic analysis is to define optimal intertemporal decisions to be applied from a given initial state.

The usual criterion in economics is the discounted return, which is the discounted sum of present and future benefits or values:

max c(•) T -1 t=t 0 1 (1 + δ) t U I 1 x(t), c(t) , . . . , I K x(t), c(t) .
(

) 2
where c(•) denotes a control path c(•) := c(t 0 ), . . . , c(T -1) . Here, the utility function U depends on the outputs I 1 x(t), c(t) , . . . , I K x(t), c(t)

of the system, hence is implicitly relying on state and control.2 Such a program defines an optimal growth trajectory in the terminology of neo-classical economics [START_REF] Koopmans | On the concept of optimal economic growth[END_REF]. In the long run, it can lead to unsustainable situations, with utility decreasing toward zero, in particular in models with exhaustible natural resources [START_REF] Dasgupta | The optimal depletion of exhaustible resources[END_REF] or pollution [START_REF] Chevé | La croissance optimale d'une économie polluante : durabilité économique versus durabilité écologique[END_REF].

The viability approach

Following stewardship or satisficing concerns related to sustainability, suppose now that the decision maker's goal is not to maximize the discounted utility but to maintain some stocks, some aggregate capital or, more generally, some indicators above viability thresholds:3

I k x(t), c(t) ≥ τ k , ∀k = 1, . . . , K . (3)
Recall that I k : X × C → R is an indicator,4 namely a state and control function having economic or biological meaning; the real τ k is a threshold. Using these indicators and associated thresholds acting as constraints, a sustainable development path is an economic trajectory that meets all the constraints (eq. 3) at all times t. Sustainability being defined this way, viability theory determines the conditions for economic trajectories to be sustainable [Martinet and Doyen, 2007]. In particular, the viability analysis describes the conditions on states (economic endowments) and controls (economic decisions) for the resulting trajectory to be viable, that is, to respect all the constraints at all times, given the dynamics of the system. The main mathematical instrument of the viability analysis is the so-called viability kernel [START_REF] Aubin | Dynamic Economic Theory ; A Viability Approach[END_REF]. It is composed of all initial states from which viable trajectories can start, i.e., all states from which there are intertemporal decisions resulting in trajectories which satisfy the constraints. From the mathematical point of view, the viability kernel at initial time t 0 reads

Viab(t 0 ) =                x 0 ∈ X there exist controls c(t 0 ), . . . , c(T -1) such that ∀k = 1, . . . , K , ∀t = t 0 , . . . , T -1 , I k x(t), c(t) ≥ τ k x(t + 1) = G x(t), c(t) x(t 0 ) = x 0                . (4)
A basic viability problem consists in characterizing this set. Handling such a geometrical tool is not usual in economic analysis, which favors optimization approaches. Nevertheless, we show in next section how a viability problem can expressed as a (degenerate) dynamic optimization problem, closer to usual economic representations and efficiency.

Viability as a degenerate optimization problem

Numerous formulations of viability problems in terms of optimality have been provided. In the continuous case, [START_REF] Aubin | Dynamic Economic Theory ; A Viability Approach[END_REF] especially focuses on exit time functions together with support or indicator functions for Hamiltonian characterizations. In Martinet and Doyen [2007] links with maximin criterion are pointed out. We shall now present alternate equivalent forms of the viability problem (3) and give their interpretation. These forms will be formulated as optimization problems. However, these latter are degenerate optimization problems with no unique solution in general. Indeed, they involve characteristic functions as follows.

Reformulating viability with characteristic functions

The viability kernel (4) is the set of initial states from which start trajectories satisfying conditions (3) at every period. These states are called viable states. They can be characterized by solving optimization problems, using characteristic functions.

Denote by 1 A the characteristic function of the set A, which is equal to one when its argument belongs to A, and to zero otherwise. Using such a tool, the quantity

1 [τ k ,+∞[ I k x(t), c(t)
represents the effectiveness of the economy to satisfy sustainability objective k at time t, i.e., whether the constraint k is satisfied at the given time or not. Using this characteristic function formulation, condition (3) can be described in multiplicative 5 form, as K k=1 1 [τ k ,+∞[ I k x(t), c(t) = 1. If any of the indicators is below the associated threshold, the characteristic function is equal to zero for this indicator, and the product is nil. If all of the constraints are respected, it is equal to one.

There are at least two different ways to describe the viability kernel in optimality terms using the characteristic functions. On the one hand, one can use a time-additive form, the criterion being to minimize the following value 6

T -1

t=t 0 1 - K k=1 1 [τ k ,+∞[ I k x(t), c(t) , (5)
which counts the number of time periods during which (at least) one constraint is not respected along a given trajectory. In our sustainability issue, it can be interpreted as the number of generations that do not achieve the sustainability objectives. It is equal to zero when all the constraints are respected at all times along the trajectory defined by the given controls. This time-additive form has an easy interpretation based on the time of crisis criterion developed by [START_REF] Doyen | Scale of viability and minimum time of crisis[END_REF], [START_REF] Béné | A viability analysis for a bio-economic model[END_REF], Martinet et al. [2007]. The optimal control problem associated with this criterion termed minimal time of crisis corresponds to 7

C(t 0 , x 0 ) = min c(•) T -1 t=t 0 1 - K k=1 1 [τ k ,+∞[ I k x(t), c(t) .
It turns out that the viability kernel of the problem is composed of all initial states where the minimal time of crisis is nil:

Proposition 1 x 0 ∈ Viab(t 0 ) ⇐⇒ C(t 0 , x 0 ) = 0.
On the other hand, one can use a time-multiplicative form, the criterion being to maximize the product upon time of the product of characteristic 5 An equivalent additive form is

K k=1 1 -1 [τ k ,+∞[ I k x(t), c(t)
= 0, and the min- 6 As we are summing nonnegative numbers, the given sum is mathematically welldefined either the time horizon is finite, i.e., T < +∞, or infinite, i.e., T = +∞.

imum form is min k=1,...,K 1 [τ k ,+∞[ I k x(t), c(t) = 1.
7 See footnote 6.

functions8

V (t 0 , x 0 ) = max c(•) T -1 t=t 0 K k=1 1 [τ k ,+∞[ I k x(t), c(t) . (6)
As soon as one of the constraints is not respected at some time period, the criterion is equal to zero. It is equal to one when all the constraints are respected at all times, characterizing a viable state:

Proposition 2 x 0 ∈ Viab(t 0 ) ⇐⇒ V (t 0 , x 0 ) = 1 .
Even if it has little economic meaning in the deterministic case, this form will appear useful in the stochastic case that we shall address in the next section.

Both the criteria ( 5) and ( 6) will provide the same information by characterizing the viability kernel of the problem. Nevertheless, the multiplicative form only gives a boolean information, whereas the minimum time of crisis indicates, for states characterized by a strictly positive value function, the minimal number of crisis period the economy is going to face. It thus provides a meaningful information on what happens outside the viability kernel, and how to reach it [Martinet et al., 2007].

Altogether, the viability approach focuses on feasibility, and not on optimality, defining efficient trajectories with respect to the given objectives. Hence, an economic objection to this approach is that it does not rank all the various trajectories, or, more specifically, that all viable trajectories have the same value. Note however that one can apply another (economic) criterion to select an optimal trajectory among those which are efficient in meeting the viability constraints. In the next section, we show that this critic vanishes in a stochastic framework.

Stochastic intertemporal optimal choice problem

In this section, we proceed to the same description of both discounted utility and viability in a stochastic framework. We show how a dynamic programming structure occurs in both approaches.

Dynamic economic model

Consider the following discrete-time control dynamical system

x(t + 1) = G x(t), c(t), w(t) , t = t 0 , . . . , T -1 , x(t 0 ) = x 0 , (7)
where w(t) ∈ W := R q denotes an uncertainty or disturbance which affects the dynamics at time t. The initial state x 0 is supposed to be deterministic and known; however, if needed, x 0 could be added to the uncertainties. We define

Ω := W T -t 0 (8)
as the set of scenarios, the notation for a scenario being w(•) := w(t 0 ), . . . , w(T -1) . ¿From now on, we shall assume that the set Ω is equipped with a probability9 P which measures the likelihood of subsets of scenarii. The notation w(•) = w(t 0 ), . . . , w(T -1) still denotes a generic point in Ω; however, it may also be interpreted as a sequence of random variables when w(•) is identified with the identity mapping from Ω to Ω. The mathematical expectation with respect to P is denoted by E.

Decision rules

Let us define a decision rule as a mapping c : N × X → C. A decision rule is a (state) feedback which assigns a control c = c(t, x) ∈ C to any state x for any time t. With such a definition, we implicitly assume that the state is (at least partially) measured.10 Given a decision rule c, a scenario w(•) ∈ Ω, an initial state x 0 ∈ X and an initial time t 0 , the solution state x(•) = (x(t 0 ), . . . , x(T )) is well defined as the solution of the difference equation , c(T -1) is the associated decision path where c(t) = c t, x(t) .

x(t + 1) = G x(t), c t, x(t) , w(t) with x(t 0 ) = x 0 The solution control c(•) = c(t 0 ), c(t 0 + 1), . . .

Discounted expected utility

There are plenty of criteria to make choices under risk (see for instance [START_REF] Savage | The Foundations of Statistics[END_REF]), but less attention has been paid to criteria for dynamical problems mixing time and risk. The usual criterion in economics is the discounted expected return, which is the expected discounted sum of present and future benefits or values:

max c E T -1 t=t 0 1 (1 + δ) t U I 1 x(t), c(t) , . . . , I K x(t), c(t) . (9)
Criterion ( 9) is built upon two well axiomatized theories, the discounted intertemporal utility ( [START_REF] Koopmans | On the concept of optimal economic growth[END_REF]) and the expected utility [START_REF] Neuman | Theory of games and economic behaviour[END_REF]). This approach is widely used and offers interesting properties as time consistency and dynamic programming. However, discounting makes this criterion less suitable for long-run issues such as resources management.

Viable scenarios and viability probability

We now describe how the viability approach described in the previous Sect. 2 can be extended to the stochastic framework of this section. 11We impose viability constraints in this uncertain case as follows

I k x(t), c(t), w(t) ≥ τ k , k = 1, . . . , K.
For these constraints to make sense in the stochastic case, we need to introduce the notion of a viable scenario. A scenario w(•) is said to be viable under a given decision rule c if the state and control trajectory driven by the decision rule c, introduced in §3.2, satisfies the constraints. It means that, if the given scenario occurs, the economic trajectory defined by this decision rule is viable.

For any decision rule c, initial state x 0 , and initial time t 0 , let us define the set of viable scenarios by:

Ω c,t 0 ,x 0 :=                w(•) ∈ Ω x(t 0 ) = x 0 x(t + 1) = G t, x(t), c(t), w(t) c(t) = c t, x(t) I k x(t), c(t), w(t) ≥ τ k k = 1, . . . , K t = t 0 , . . . , T -1                . ( 10
)
This is the set of scenarios for which the given decision rule would result in viable economic trajectories. The larger the set, the larger the number of scenarios in which the given decision rule succeeds in meeting the viability constraints. Once a probability is defined on the set of scenarios, the viability problem becomes one of probability maximization, as follows.

We say that P (Ω c,t 0 ,x 0 ) is the viability probability associated to the initial time t 0 , the initial state x 0 and the decision rule c. Given initial time t 0 and initial state x 0 , the maximal viability probability is defined by

PV(t 0 , x 0 ) = sup c P (Ω c,t 0 ,x 0 ) .
From that point of view, the stochastic viability approach aims at characterizing the decision rules which maximize the probability to satisfy the constraints over time.

The concept of viability kernel can then be expanded through the maximal viability probability. At a given confidence level β ∈ [0, 1], it reads as follows:

Proposition 3 x 0 ∈ Viab β (t 0 ) ⇐⇒ PV(t 0 , x 0 ) ≥ β .

Dynamic programming and time-consistency

As the viability probability can be written in the following expected intertemporal form,

P [Ω c,t 0 ,x 0 ] = E T -1 t=t 0 K k=1 1 [τ k ,+∞[ I k x(t), c(t), w(t) (11)
a stochastic dynamic programming structure can be derived [START_REF] Lara | Sustainable Management of Natural Resources[END_REF]. Whenever the probability P is a product of its marginals (independence), the maximal viability probability satisfies the following backward induction

PV t, x) = max c E w K k=1 1 [τ k ,+∞[ I k x, c, w PV t + 1, G(x, c, w . (12)
Such dynamic programming is connected to time consistency. Define, loosely, time consistency as the following property. Today, at time t 0 , I formulate an optimization problem with criterion π t 0 , x(•), c(•), w(•) ; this yields a sequence of optimal decision rules c t 0 (t 0 , x), c t 0 (t 0 + 1, x), . . . Tomorrow, at time t 0 + 1, I will formulate an optimization problem with criterion π t 0 + 1, x(•), c(•), w(•) ; this will yield a sequence of optimal decision rules c t 0 +1 (t 0 + 1, x), c t 0 +1 (t 0 + 2, x), . . . Time consistency holds true whenever my today rule for tomorrow coincides with my tomorrow rule for tomorrow : c t 0 +1 t 0 + 1, x = c t 0 (t 0 + 1, x). This happens to be the case when optimal decision rules are given by a dynamic programming equation. In the economic literature, time consistency is rather defined with reference to temporal lotteries as in [START_REF] Hammond | Consistent plans, consequentialism, and expected utility[END_REF], and consistency may also be related to stationarity of policy design [START_REF] Cohen | How should control theory be used to calculate a time-consistent government policy?[END_REF]. This is not our point of view here, which may explain discrepancies.

Conclusion

Managing natural resources or dealing with environmental issues imply to consider both risk and dynamics/long-term. The main economic approach combining risk and time is the discounted expected utility, which is criticized in the sustainability debate as long-run issues are neglected due to exponential discounting.

The present paper points out how viability, and especially stochastic viability approach, can contribute to this debate. Viability aims at defining the conditions for sustainability constraints to be satisfied over time. Using the stochastic viability approach to address sustainability issue is interpreted as maximizing the probability to be above thresholds. In "life and death" or irreversibility situations, this can be a reasonable view as in the famous casino case of Dubbins and Savage. 12 Imagine yourself at a casino with $1,000. For some reason, you desperately need $10,000 by morning; anything less is worth nothing for your purpose. What ought you do? The only thing possible is to gamble away your last cent, if need be, in an attempt to reach the target sum of $10,000. [START_REF] Dubbins | How to gamble if you must[END_REF] Stochastic viability defines the decision rules that minimize the risk to violate the constraints, and thus aims at avoiding related catastrophic outcomes. For example, in the climate change issue, it would consist in favoring decision rules which result in thin tail temperature distributions, thus reducing the probability of climate disasters. It echoes the conclusion by Weitzman [2007].

We have shown that viability can be formulated as a degenerate optimization problem, either in a time-additive form (which defines the minimum time of crisis of an economic development path) or in a time-multiplicative form (well-adapted to probabilities). These two formulations share common interesting properties with the discounted expected utility (time-consistency, dynamic programming).

In discounted expected utility, the utility function embodies the relative importance of objectives I 1 , . . . , I k . In the viability framework, the relative importance of objectives is embodied in the definition of the thresholds τ 1 , . . . , τ k . The definition of such sustainability objectives is behind the scope of the present paper. However, we conclude on different ways to deal with this selection issue.

One can define a preference relationship on the thresholds on the one hand, and describe the necessary trade-offs between thresholds on the other hand. Choosing among them requires is a deontological approach to sustainability [START_REF] Martinet | Defining sustainability objectives[END_REF].

Another way, which strengthens the links between viability and optimality emphasized in this paper, consists in replacing the indicator functions by more regular functions by "smoothing" thresholds. This can be done by making τ 1 , . . . , τ k random [START_REF] Gollier | The Economics of Risk and Time[END_REF] or by concavification. 13 In a sense, this amounts to going from quantiles toward more regular risk measures with better properties, such as the coherent risk measures [Artzner et al., 1999, Föllmer and[START_REF] Föllmer | Stochastic Finance. An Introduction in Discrete Time[END_REF].

The dynamics G might explicitly depend upon time t. However, we shall treat only the stationary case for the sake of notational simplicity.

The utility function U might explicitly depend upon time t. However, we shall treat only the stationary case for the sake of notational simplicity.

Without loss of generality: a "bad" indicator, such as pollution, can be represented by its negative value.

The indicator I k might explicitly depend upon time t. This can include absence of constraints (take I k having constant value greater than τ k , or final target constraint (take I k (t, x, c) ≥ τ k for all t = t 0 , . . . , T -1 but not for I k (T, x, c)).

As we are multiplying numbers within [0, 1], the given product is mathematically well-defined either the time horizon is finite, i.e., T < +∞, or infinite, i.e., T = +∞.

The probability P is defined on the Borel product σ-field of W T -t0 . The mappings G, I 1 , . . . , I k , and all decision rules c (see below) are supposed to be measurable.

As a consequence, we shall not consider the case where only a corrupted observation of the state is available to the decision-maker (as it is nevertheless the case in practical situations).

Mathematical materials for stochastic viability can be found in[START_REF] Aubin | The viability theorem for stochastic differential inclusions[END_REF],[START_REF] Buckdahn | Stochastic control with exit time and contraints, application to small time attainability of sets[END_REF] but they focus on the continuous time case. Contributions for discrete time systems are[START_REF] Doyen | Sustainability of exploited marine ecosystems through protected areas: a viability model and a coral reef case study[END_REF],[START_REF] Béné | Contribution values of biodiversity to ecosystem performances: A viability perspective[END_REF], De[START_REF] Lara | Sustainable Management of Natural Resources[END_REF],[START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF].

Imagine, in the following tale, that a contract has been put on your head and that you desperately need money to pay the killer before!

A characteristic function 1 [τ k ,+∞[ (I k ) being both non concave and discontinuous, one might think of concavify and smoothing it by replacing 1 [τ k ,+∞[ (I k ) by min (τ k , I k ).


O Yousfi
email: oyous…@u-paris10.fr

Exit routes in LBO projects

Keywords: LBO, moral hazard, excessive taking risk, …nancial structure, Exits. JEL Classi…cations: G15, G23, G32

The current paper studies the …nancial structure in buyout …rms under moral hazard due to unobservable e¤orts and an excessive risk-taking.

The choice of the exit route may lead to agency con ‡icts between the entrepreneur and the LBO …rm: the former may take very risky decisions to increase the probability of IPO exit. If the target is taking public, he gets a non transferable and private bene…t. The opportunistic behavior of the entrepreneur decreases the probability of sale exit; the preferred exit route of the LBO …rm.

Without moral hazard, there are many ways to …nance the project and the two agents exert strictly positive e¤orts. With moral hazard, the entrepreneur, the LBO …rm and the bank must …nance jointly the buyout. Financing the project through standard debt-equity contracts does not implement the …rst-best solution.

Only a set of projects can be …nanced through both the LBO fund and the bank at the macroeconomic level. If the entrepreneur is not wealthy enough, her project is not undertaken.

Introduction

Despite the fact that buyout investments represent a considerable part of private equity investments, few papers has been done related to the …nancial structure and exit routes in LBO projects. The main topic often mentioned in the literature is the impact of LBO …rm (hereafter he) on the performance of the acquired …rm (called also Op Co or target). In the late 1980's, many papers focused on the investigation of other determinants of the target's performance (after the exit of the LBO …rm) 1 .

An important element of the buyout investment is the contractual agreement to end up the project and to repay the parties within a spec-i…ed period of time. The exit is the most important and last way the LBO …rm can realize a high positive return on the investment: he invests in buyouts with the aim of exiting after 3-5 years. He wants to get his money back quickly in order to invest it in a new deal. Consequently, there must be a clear route for him to exit the buyout.

In this paper, we shed light on the topic of exit choices in buyout investments and aim at investigating the following question: does the …nancial structure in LBO solve the agency con ‡icts between the entrepreneur (hereafter she) and the LBO …rm?

We focus on the agency con ‡icts related to the choice of the exit route under moral hazard. The exit route and timing are crucial for …nancing. The entrepreneur must know that the LBO …rm will eventually want to exit the buyout, and that very often this means that the project will be sold to another company (trade sale) or to another LBO …rm (secondary LBO). If the target turns out to be non-performing, it is abandoned showing the ability of the LBO …rm to …lter out good from bad investments.

The choice of the exit date and route may lead to agency con ‡icts between the entrepreneur and the LBO …rm. The three main possible exit channels are initial public o¤erings IPO, sales 2 and write-o¤s (typically called quick- ‡ips).

An IPO results in the highest valuation of the target and is very often the preferred exit vehicle. The entrepreneur favors an IPO because it preserves the target's independence and ensures the liquidity of its securities. If the company is taking public, the entrepreneur keeps the control and gets private bene…ts 3 because she shares the control with a large number of investors who usually face information asymmetry. In order to get an IPO exit, the entrepreneur may take very risky decisions which can, for example be asking for additional funds from the bank (hereafter he) and other …nanciers, investing in new mergers and acquisitions which are not valuable for the buyout, hiding important information to the LBO …rm, recruiting new sta¤ to control the majority of voting rights and to get more private advantages. However, in contrast with a sale, such exit does not end the LBO …rm's involvement with the target. He may be restricted for many reasons, from selling any or a portion of his shares in the o¤ering. 4 Sale has very di¤erent consequences; it is more attractive for the LBO …rm since it o¤ers many advantages: fast and full exit opportunities, less restrictions than in IPO, quick payment in cash or marketable securities and ends the partnership's involvement with the buyout. It is unwelcome for the entrepreneur because the target may be merged with or acquired by another company and cannot remain independent.

Less pro…table or underperforming projects are abandoned: the LBO …rm writes the buyout o¤ if it cannot cover its investment costs. Keeping such projects in the fund's portfolio would signal his inability to di¤erentiate between good and bad projects. This exit channel is considered as a constraint rather than an option.

The present model is related to the …nancial literature on exit in private-equity investments. To our knowledge, most of the papers study the exit in the venture capital but few papers, mostly empirical studies, focus on the topic of buyouts.

These papers argue that the exit route is a function of the …rm's performance. Indeed, IPO is the exit route of highly pro…table buyouts: these …rms are called high- ‡yers. But only the …rms which are in a mature development stage (high growth prospects and strong future cash- ‡ows) are taken public which suggests that the probability of an IPO exit in LBO is smaller compared to the other private equity investments: the target has very often low growth prospects and generate important cash- ‡ows.

The most common exit route in LBO is sales which are subdivided into trade sales, secondary sales and buybacks. Less pro…table buyouts are sold to a strategic investor (trade sale) or a private equity fund (secondary sale) or the entrepreneur (buyback). The probability of exit through buyback is the highest in France since LBO projects are very often LMBO projects which are contracted to save a family business or call the project a Leveraged Management Buy Out LMBO)... 4 to sell the buyout to the heirs of the founding family. Then, the buyout organizes the …nancing of an ownership change (Fenn et al., 1995 and[START_REF] Chérif | Aspects …nanciers[END_REF].

There are many papers dealing with the choice of the exit route in private equity investments; mostly in venture capital. However, few papers mostly empirical, explain the choice of the exit vehicle in buyout investments. They argue that the exit decision and route depend on the target's performance.

For instance, [START_REF] Giot | IPOs, trade sales and liquidations: modelling venture capital exits using survival analysis[END_REF] argue that the exit decision depends on the type of exit strategy and on the timing. For instance, biotechnology and Internet projects are the fastest in exiting through IPO. Unpro…table Internet …rms are written o¤ quickly. The level of innovation in the venture may depend on the choice the exit route. In fact, going public is more pro…table than trade sale when the project is very innovative. The IPO exit enables the entrepreneur to remain in the …rm, keep its control and get private bene…t. Consequently, she is tempted to distort the innovation strategy so that the IPO looks the preferred exit route [START_REF] Schwienbacher | Innovation and Venture capital Exits[END_REF]. Empirical analyses show that IPOs are the most pro…table, followed by secondary sales, buybacks and write-o¤s for less pro…table projects [START_REF] Cumming | Venture Capital Exits in Canada and the United States[END_REF]. However, sales'probability is the highest in the buyout stage [START_REF] Das | The pri-vate equity discount: An empirical Examination of the Exit of Venture Backed Companies[END_REF]. Groh and Gottschalg (2008) point out that the US buyout investments clearly outperform the market benchmark.

More recently, Cao and Lerner (2009) conduct a study about the performance of 496 reverse LBO (RLBO) in the USA. Their results indicate that the IPOs that had been bought by LBO investors outperform other IPOs and the stock market as a whole, and that quick ‡ips5 perform much worse than the …rms kept longer than one year by the LBO …rm. [START_REF] Schmidt | Exit strategies of Buyout Investment-An empirical Analysis[END_REF] analyze the determinants in ‡uencing the choice of the exit option. They consider a sample of 666 buyouts in Europe and in the United States between 1990 and 2005. Their results show strong support for signaling e¤ect. If the return is very poor, the LBO …rm writes the project o¤ early instead of holding it in his portfolio as living-dead buyout: he is able to di¤erentiate between good and bad investments quickly. Only most pro…table projects are taking public. Nikoskelian and Wright (2007) consider a sample of 321 UK buyouts, exited between 1995 and 2004. They …nd a positive relationship between the value increase and the management ownership. [START_REF] Ick | Private equity Returns: Is there really a Bene…t of low Comovement with Public Equity Markets?[END_REF] investigates the risk and return relationship of private equity relative to public market equity and …nds that the private equity returns depend on the stage of the investment. Later stage investments achieve higher risk adjusted returns.

These papers do not consider agency con ‡icts between the entrepreneur and the LBO fund when they raise the question of exit. Both of them are supposed to have the same criterions to choose the exit route; the preferred exit route must be valuable for the target whatever the revenue-sharing rule and the gains of each party.

My model is also related to the …nancial literature on the …nancial structure when there is asymmetric information due to unobservable efforts; these e¤orts are supposed to increase the project's performance. For instance, Bergemann and Hege (1998), [START_REF] Cornelli | Stage Financing and the Role of Convertible Securities[END_REF], [START_REF] Repullo | Venture Capital Finance: A Security Design Approach[END_REF], [START_REF] Schmidt | Convertible Securities and Venture Capital Fi-nance[END_REF]...These papers highlighted the importance of adequate incentive-rewarding schemes, the role of the stage …nancing and convertible securities to mitigate the moral hazard problem. Most of these papers consider a double sided moral hazard model with two agents: the entrepreneur and the VC fund. Only [START_REF] Casamatta | Financing and Advising: Optimal Financial Contracts with Venture Capitalists[END_REF] adds a third partner (the pure …nancier) and shows that under speci…c conditions, the entrepreneur and the VC fund exert the …rst-best levels of e¤orts.

In another brand of the literature, it is showed that the managers' shareholding reduces the agency costs and increases therefore the buyout's performance [START_REF] Bruton | Corporate restructuring and performance: An agency perspective on the complete buyout cycle[END_REF]. Going private through a buyout acquisition is a mean to consolidate the …rm and to solve agency con- ‡icts before going public [START_REF] Cornett | Going Private Buyouts and Determinants of Shareholders'Returns[END_REF]. Only the …rms in a mature development stage can be …nanced through both the LBO …rm and the bank. [START_REF] Andres | Going private via LBO -Shareholders gains in the European markets[END_REF] conducted a study in the European market. They claimed that the debt solves the agency con ‡icts in LBO projects, particularly when the managers are not su¢ ciently monitored and the buyout is very pro…table. However, these papers do not consider the …nancial structure of the target under asymmetric information.

Yous… (2009) considers a model with three agents: the entrepreneur, the LBO …rm and the bank, and argues that the write-o¤ threat leads the two …rst agents to exert high levels of e¤orts but does not implement the …rst best solution with standard debt-equity contracts.

In the current model, we consider that there are two sources of moral hazard. First, the entrepreneur and the LBO …rm exert costly and unobservable e¤orts to increase the probabilities of exit through IPO and sale. Second, to take to target public, the entrepreneur may take very risky decisions in order to get a non transferable and private bene…t. These decisions are costly and decrease the probability of the sale exit. The LBO …rm prefers a sale exit rather than an IPO. For the stake of simplicity, we assume that this channel does not induce further costs (in contrast with an IPO). This may be explained by the fact that the LBO …rm is well connected with many industries and was deeply involved in many past investments. Consequently, he has the power through his extensive contacts to bring in other industrialists, …nanciers and other private equity …rms. The entrepreneur has very often no past experience in LBO acquisitions. Consequently, the LBO …rm is usually more engaged in the sale exit than the entrepreneur. The opportunistic behavior of the latter induces therefore agency con ‡icts.

Without moral hazard, the project can be implemented in a number of ways. The entrepreneur and the LBO …rm must exert strictly positive levels of e¤orts. Consequently, the entrepreneur is indi¤erent between hiring an LBO …rm who contributes …nancially and technically to the project and asking for advice from a consultant who does not put money into the target. If the entrepreneur is wealth constrained, the target is …nanced. This is no longer true under moral hazard: all agents must …nance jointly the target. We show that standard debt-equity contracts do not implement the …rst best solution: the agents exert non optimal levels of e¤orts.

Moreover, relying on a consultant is too costly for the entrepreneur: when the LBO …rm contributes …nancially, he reduces the cost of his advice. The main insight is that the outside equity raised by the privateequity investor compensates the entrepreneur for the rent that she would leave to induce him to exert high level of e¤ort.

When the …nancial contracts are chosen to maximize the expected gain of the entrepreneur, we show that risk-taking does not worsens the moral hazard problem in the sense it increases the agents'incentives and induce them to exert high levels of e¤orts. This result is such a break with pervious literature.

The model and assumptions are presented in the next section. Section (3) derives the optimal …nancial contracts. Here I analyze the optimal provision of e¤ort and level of outside …nancing when the entrepreneur may make risky decisions to take the target public. Section (4) discusses the implementation of these contracts with convertible securities. Concluding remarks are in the last section and proofs in the Appendix.

The model

Consider a market with a continuum of risk-neutral entrepreneurs. Each entrepreneur is endowed with the opportunity to acquire a …rm which costs K, and an initial wealth w K, which can, for example, be the heritage or the fortune generated by her past investments. If she needs outside …nancing, she asks …rst for …nancing from the LBO …rm. If the latter accepts to undertake the investment with the entrepreneur, he issues the amount of equity i in exchange of an outcome's share 1 where 0 1. If they still need further funds, they can turn to the bank to raise the residual capital I = K (w + i). The revenuesharing rule and the outside …nancing i and I are endogenous to be characterized in the …nancial contracts. The riskless interest rate r is normalized to zero.

The target's revenues are given by X = We assume that the entrepreneur's e¤ort is less costly than the LBO's e¤ort : < and is assumed to be signi…cantly high. This captures the idea that the entrepreneur's e¤ort is more e¢ cient than the LBO's e¤ort: the project relies more on the entrepreneur's expertise.

Timing

The time line of the game is presented in the following …gure:

Fig 1-
The sequence of events in the model.

At date 0, the entrepreneur, the LBO …rm and the bank sign the holding and debt contracts.

At date 1, the project is completed. In case of failure, the bank gets the whole liquidation value C. Otherwise, the bank is paid (1 + r B ) I where r B is the interest rate of the bank (0 r B 1).

-If the entrepreneur takes very risky decisions, the probability of IPO exit increases by " > 0 which …rst is supposed to be exogenous. Risk-taking costs b > 0 to the entrepreneur and the LBO …rm. The entrepreneur gets a non transferable private bene…t B > 0 in addition to her outcome's share

[X (1 + r B ) I b]. The residual amount (1 ) [X (1 + r B ) I b
] is paid to the LBO …rm. However, the probability of sale decreases by ". The project is sold 6 with probability " to …nanciers 7 and/or industrialists. The entrepreneur will retain [X (1 + r B ) I] for herself and the LBO …rm gets the remainder (1

) [X (1 + r B ) I].
-If there is no risk-taking, the project is sold or taking public with the same probability (" = 0). If the exit route is an IPO, exit cost is null (b = 0) and the entrepreneur cannot get private bene…t (B = 0).

Contracting with the LBO and the bank

1. The holding contract ( , i)

The LBO …rm issues the equity i only if:

A = (1 ) f2 [X (1 + r B ) I] ( + ") bg v (a) i 0 (P C A )
There is a continuum of LBO …rms, the competition induces the LBO …rm to make the best o¤er possible to the entrepreneur. Consequently, his expected gain is null.

The debt contract (r B , I)

The participation constraint of the bank is written:

B = [2 (1 + r B ) 1] I + (1 2 ) C 0 (P C B )
Because of the competition among the banks, the constraint (P C B ) is binding.

The expected gain of the entrepreneur is written:

E = 2 [X (1 + r B ) I] + ( + ") [B b] u (e) w (1)
Given the fact that participation constraints of the LBO …rm and the bank are binding, we replace w = K i I in (1). The expected revenue of the entrepreneur is therefore written:

E = (A + B) + " (B b) + C u (e) v (a) K (2)
where A = [2 (X C) b].

The …rst best

As a reference point, let us de…ne the …rst best e¢ cient investments when there is no moral hazard and when the entrepreneur abstains from taking excessive risks. Consequently, the project will be …nanced. This corresponds to the …rst best solution. Let V (e, a) = 2 (X C) + C u (e) v (a) K be the social value of the project. The …rst best e¤orts are given by the …rst order conditions of V (e, a). They are written:

e F B = (X C) and a F B = (X C) (3)
It is straightforward to see that the entrepreneur provides the highest e¤ort because it is less expensive than the LBO's one. But, as expected, the increase of the parameters and leads to a decrease of the e¤orts' costs which increases e F B and a F B . Moreover, they are increasing with the di¤erence between the revenue X obtained if the target is taking public (IPO) or private (sale), and the failure revenue C.

The optimal social value of the project is therefore given by

V F B = ( + ) (X C) 2 + C K
There are many ways to implement the …rst best: the identity of the agent providing funds and e¤orts is irrelevant. The entrepreneur and the LBO …rm may rely only on debt …nancing to fund the acquisition (K = I). They can also …nance the project through a standard equity contract: one of them or both can provide …nancing, in the sense K = i + w. Accordingly, the entrepreneur between asking for …nance and advice from the LBO …rm and hiring a consultant who provides only advice. This is no longer true under moral hazard.

The e¤orts in equilibrium

When the e¤orts are unobservable, each agent chooses the level of e¤ort which maximizes his expected gain but he/she must take into account the level of e¤ort exerted by the other agent. These e¤orts are solutions of the following programs:

ê 2 arg max E e and â 2 arg max A a
The …rst order conditions of E and A give the results summarized in the following lemma: Lemma 1 The equilibrium levels of e¤orts ê and â are given respectively by:

ê = 2 f2 [X (1 + r B ) I] bg + 2 B ( 4
) â = 2 (1 ) f2 (1 + r B ) I] bg (5)
We assume [ + ( 1)] f2 [X (1 + r B ) I] bg + B < 1 (A:1) to ensure that the probability is inferior to 1. If there is no risk taking, is the probability that the LBO …rm will exit through IPO or sale. Notice that B and b a¤ect the agents'incentives, in contrast with ". The assumption that risk-taking increases the probability of IPO exit and decreases the probability of sale exit with the same probability " is an important one. The two e¤ects cancel each other from an incentive point of view.

The entrepreneur's e¤ort increases (respectively decreases) with the amount of the private bene…t B (respectively the cost exit b) and with her share of revenue . In contrast, the LBO's e¤ort is a decreasing (respectively increasing) function of b (respectively his share of revenue ( 1)). At a …rst sight, the e¤ort â does not depend on the private bene…t B since it is not transferable but the optimal sharing rule, as showed below, is a function of B.

The optimal …nancial contracts should induce simultaneously both parties to exert high levels of e¤orts: if we give high outcome's share to one of them, this will increase his or her incentives to exert high level of e¤ort but it will decrease the incentives of the other one. Notice that these e¤orts are independent: the entrepreneur and the LBO …rm have dominant strategies.

If there is no excessive risk taking problem, we replace B = b = 0 in (4) and (5) to get the agents' e¤orts: ê = [X (1 + r B ) I] and

â = (1 ) [X (1 + r B ) I].
To induce the entrepreneur and the LBO …rm to exert the …rst-best levels of e¤orts, the debt's payments should be decreasing with the project's outcome and C < X 2C (in other words, the buyout should be not very risky) 8 . But this is not a feature of a standard debt contract since the bank is expecting the highest payment in the good state of the nature.

Adding excessive taking risks is supposed to worsen the moral hazard problem: in the sense, the entrepreneur will exert high level of e¤ort in contrast with the LBO …rm. Now, the …rst best e¤orts are implemented only if the e¤orts ê and â satisfy respectively (3) which gives:

^ = 2 (X C) B 4 (X C) B Î = ( + ) (X C) [B b 2 (X C)] + C rB = 1 2C X + 1 2 (B b) Î { = ( + 2 ) (X C) 2 2 (X C) 4 (X C) B "b ŵ = K { Î
Notice that these contracts exist under the condition C + 1 2 (B b) < X 2C + 1 2 (B b) which means that the project is not very risky and the bank's payments should be decreasing with the project's outcome:

(1 + r B ) I < C. Consequently, we conclude that there are no debt-equity contracts that induce the entrepreneur and the LBO …rm to provide optimal e¤orts.

Optimal …nancial contracts

The optimal …nancial contracts maximize the expected gain of the entrepreneur (1) subject to the participation constraints of the LBO …rm and the bank (P C A ) and (P C B ), and the incentive constraints (4) and (5). Furthermore, we add the feasible condition K = w + i + I and constrain the bank's payments to be nondecreasing with the project's outcome:

(1 + r B ) I C (6)
We replace i and I by their expressions given by (P C A ) and (P C B ) in E . The entrepreneur's program is to maximize the net present value of the project:

arg max , I E = (A + B) + " (B b) + C u (e) v (a) K s.t 0 1, C (1 + r B ) I X, (4) and (5)
In the …rst best, we showed that the entrepreneur may not invest money. This is no longer true when the levels of e¤orts are not observable.

Proposition 1 Under moral hazard, the entrepreneur maximizes her expected gain by hiring both the LBO …rm and the bank.

The proof is presented in the appendix A. Under moral hazard, all agents must contribute …nancially to the target. In contrast with the previous section, the entrepreneur now prefers hiring an LBO …rm rather than a consultant; the latter is too costly for her. In fact, the rent that she would leave to the consultant is too high compared with the LBO …rm. The amount of equity { issued by the LBO …rm comes as a compensation for the "agency" rent left to the LBO …rm for incentive motive. In other words, when he contributes …nancially to the buyout, he reduces the cost of the e¤ort a.

The optimal debt contract does not depend neither on the e¤orts' problem nor on risk taking. The bank is paid the liquidation value C not only if the …rm fails but even if it is taken public or sold. Lending money to the entrepreneur and the LBO …rm is not a risky task since the bank will get his money back. The interest rate r B must be null. In contrast, given high and the assumption (A:1), the amount of outside equity is larger when the entrepreneur chooses an opportunistic behavior. Also, according to the condition K = w + i + I, the entrepreneur issues now smaller amount of equity.

If the entrepreneur is wealth constrained, in the sense w < ŵ (where ŵ is de…ned in the appendix A), only the projects where the entrepreneur is wealthy enough (has an initial wealth superior to the minimal level ŵ) are undertaken and …nanced through both the LBO …rm and the bank: the other projects have to look for other ways of …nancing.

Let us see under what conditions the entrepreneur would take very risky decisions:

2 ( + 2 ) + 2 ( + 2 ) 4 ( + ) 2 (A + B) 2 + " (B b) + C K > 2 ( + 2 ) + 2 ( + 2 ) ( + ) 2 (X C) 2 + C K
where the left-hand side is the expected gain of the entrepreneur if there is risk-taking, and the right-hand side is her gain otherwise. This condition is satis…ed only if the amount of the private bene…t is larger than the risk-taking cost B > b (A:2). This is a surprising result since she would pay only b if the project is taking public and the residual cost (1 ) b is paid by her partner. But, for incentive motive, she must surrender to the LBO …rm an additional outcome's share 9 + B A which is increasing with both the private bene…t and the risk-taking cost. In the remainder of the analysis, (A:2) will be assumed to hold.

If there is risk-taking, the optimal …nancial contracts show that the amount of equity issued by the entrepreneur decreases with B and increases with b and ". The explanation is that the …nancial contribution of the LBO …rm increases with B and decreases with b and ", in contrast with the amount of raised debt which depends only on the liquidation value and the interest rate of the bank.

The increase of the private bene…t B and the cost of risk-taking b lead to a decrease (respectively an increase) in the outcome's share given to the entrepreneur (respectively the LBO …rm). This result is very intuitive since an increase of the risk-taking cost b a¤ects in a negative way the LBO's incentives. Consequently, in the optimal contracts, the entrepreneur must surrender to him a higher share's outcome. In contrast with the LBO …rm, these contracts must assign to the entrepreneur a lower outcome share to make risk-taking not too attractive.

If the entrepreneur abstains from risk-taking, the optimal revenuesharing rule assigns to her the highest outcome's share = + . The e¤ort e is less costly than a, so it is e¢ cient to give her high incentives. This is no longer true when she takes very risky decisions : if the amount of the private bene…t is too large in the sense B > 2 A, the optimal contracts give the highest share to the LBO …rm to let him exert high level of e¤ort. Despite the fact that the two e¤orts are perfect substitutes, whether there is excessive risk-taking or not, the entrepreneur exerts higher level of e¤ort than the LBO …rm. 9 Without risk-taking, the LBO'share is + otherwise he gets + + + Consequently, the optimal …nancial contracts are supposed to maximize E under the participation constraint of the LBO …rm, the incentive constraints [START_REF] Casamatta | Financing and Advising: Optimal Financial Contracts with Venture Capitalists[END_REF] and the feasible condition K = w + i. The entrepreneur's program becomes:

arg max E = (A + B) + " (B b) + C u (e) v (a) K s.t 0 1 and (7)
We replace the e¤orts in the objective function. Given the fact that w = K i, the expected revenue of the entrepreneur is written

arg max E = 1 2 [ + (1 )] A + 2 B (A + B) + " (B b) 4 ( A + B) 2 4 (1 ) 2 A 2 + C K
The …rst order condition of E enables us to deduce that ^ = A B ( + )A . Then, it is easy to check that the entrepreneur and the LBO …rm exert the levels of e¤orts ( (BB5) and ((BB6). Whether there is risk-taking or not, equity contract does not implement the …rst best but leads them to exert the same levels of e¤orts as with debt-equity contracts. Financing the project through the LBO …rm and the bank or solely through the LBO …rm lead the two agents to exert the same levels of e¤orts. Debt …nancing is not used to lead the entrepreneur and the LBO …rm to exert the …rst-best levels of e¤orts. This result does not join the …nancial literature which highlights the disciplinary value of debt (see among others [START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure[END_REF][START_REF] Jensen | Agency costs of free cash ‡ow, corporate …nance, and the market of takeovers[END_REF], 1989, Kaplan and Strömberg, 2008...). One explanation is that debt and equity are substitutes, in the sense we ignore some features of the debt …nancing, like for example, the deductibility of the debt's interests (see appendix B). Another explanation is that the bank is paid the whole liquidation value in the bad state of nature, and simultaneously her payments are supposed to be nondecreasing with the project's outcome. The bank is therefore a passive …nancier. The presence of such …nancier along with the LBO …rm and the entrepreneur is irrelevant from an incentive point of view since it does not increase the levels of e¤orts. 11

Conclusion

This paper analyzed the …nancial structure when there is an excessiverisk taking. The entrepreneur and the LBO …rm provide simultaneously non observable e¤orts which improve the project's performance. They face a double moral hazard problem. We show that there are no debtequity contracts that solve the moral hazard.

Adding the excessive taking risk worsens the moral hazard, in the sense, it leads the entrepreneur to provide higher level of e¤ort, in contrast with the LBO …rm. The opportunistic behavior of the entrepreneur decreases his incentives to exert e¤ort and leads him to raise a small capital.

In the current paper, we studied the features of debt-equity contracts. However, the use of convertible securities becomes prevalent in buyout investments. These securities are rarely issued in the presence of passive …nanciers such as banks and other outside equity holders. In practice, the LBO …rm can convert his debt/stocks into equity in order to get the majority of the voting rights. It would be interesting to study how the use of these securities in ‡uences the agents'incentives as well as the …nancial structure, namely the debt to equity ratio.

Despite the fact that the strip …nancing and the debt syndication are commonly used to mitigate agency con ‡icts between the entrepreneur and the LBO …rm, there are no academic papers analyzing how they solve such problems under asymmetric information.

where [START_REF] Andres | Going private via LBO -Shareholders gains in the European markets[END_REF] [X (1 + r) I b] is shared between the entrepreneur and the LBO …rm. If the exit route is a sale, the revenue after taxation is where (1 ) [X (1 + r) I] is shared between them. Otherwise, the project fails and they have no payments.

After some manipulations, the straight application of the Kuhn-Tucker theorem shows that debt …nancing is the only source of funds (the entrepreneur and the LBO …rm does not contribute …nancially to the project). The use of debt does not improve the agents' incentives but it creates an incentive to be "heavily" in debt.

.

X

  + B b with the probability + " X with the probability " C with the probability (1 2 ) where X C, C is the liquidation value of the project. 0 " 1, B 0 and 0 b X. = min fe + a; 1g and e and a 2 [0 , 1] are the e¤orts provided simultaneously and respectively by the entrepreneur and the LBO …rm. These e¤orts are unobservable and costly. Let u (:) denote the entrepreneur's disutility of e¤ort and v (:) the LBO's disutility of e¤ort such that: u (e) = e 2 and v (a) = a 2

  r) I] + (1 + r) I = (1 ) X + (1 + r) I

See among others,[START_REF] Jensen | Stockholder, Manager, and Creditor Interests: Applications of Agency Theory[END_REF],[START_REF] Jensen | Agency costs of free cash ‡ow, corporate …nance, and the market of takeovers[END_REF][START_REF] Jensen | Active Investors, LBO's, and the privatization of bankruptcy, …nancial management[END_REF],[START_REF] Kaplan | The E¤ects of Management Buyouts on Operating Performance and Value[END_REF],[START_REF] Lehn | Free cash ‡ow and stockholder gains in going private transactions[END_REF],[START_REF] Kaplan | The Evolution of buyout Pricing and Financial Structure (Or, What Went Wrong) In the 1980's[END_REF]...

Sales encompass trade sales, secondary sales and buybacks.

Examples of the private bene…ts might be the entrepreneur's consumption perks, her desire to be a business owner or to keep a family business going (In this case, we

Private equity …rms sell o¤ their investments within a year after acquisition.

The entrepreneur may acquire the shares of the LBO fund. Here, the exit vehicle is buyback. For simpli…city, we assume that sales include trade sales, secondary sales and buybacks.

They may be private equity funds such as LBO funds: the project is then called secondary LBO.

If the bank was a pure …nancier, in the spirit of[START_REF] Casamatta | Financing and Advising: Optimal Financial Contracts with Venture Capitalists[END_REF], it would be possible to reach the …rst-best levels of e¤orts (only if the project is not very risky). The pure …nancier is regarded as a budget breaker. This is not enough when the project is very risky (X > 2C) because the levels of e F B and a F B become very high and we need high powered incentive scheme.

Whether the collateral C is shared between the entrepreneur and the LBO …rm or assigned to one of them, our results do not vary.

See Appendix B to explain why buyouts are levered.

With standard debt-equity contracts, the two agents provide non optimal e¤orts:

It is easy to check that they increase with the entrepreneur private ben-e…t B and decrease with the risk-taking cost b. Given the condition (A:2), it is straightforward to see that these e¤orts are higher than those provided when there is no risk-taking (B = b = 0). The e¤orts' problem worsens moral hazard in contrast with risktaking. This result is such a break with previous literature that argue that risk-taking worsens moral hazard. Our result is explained by the fact that we consider …nancial contracts that maximize the expected gain of the entrepreneur and a competitive LBO market. Accordingly, LBO expected gain is null.

Does leverage help to reduce risk-taking?

In order to answer the question, we consider hereafter that the target is …nanced only through equity. The LBO …rm is the only outside …nancier. He issues the residual capital i = K w.

In contrast with previous sections, the collateral C is shared between the entrepreneur and the LBO …rm. 10 Consequently, the participation constraint (P C A ) is written:

The expected revenue of the entrepreneur is given by:

The incentive constraints of the LBO …rm and the entrepreneur are given by the derivative of A and E with respect to a and e. Their e¤orts in equilibrium are therefore given by:

A Proof of proposition 1

The straight application of the Kuhn-Tucker theorem enables to deduce the following …nancial contracts:

Assume A B to ensure that the optimal revenue-sharing rule ^ > 0. The interest rate of the bank is therefore null r B = 0. The buyout is not risky for the bank since he will get his money back.

It is easy to check that the entrepreneur and the LBO …rm will not provide the …rst-best levels of e¤orts. Their e¤orts are given by

Then, the expected gain of the entrepreneur is:

More formal proof is available upon request.

B Why buyouts are heavily in debt?

One among the explanation for the involvement of bank in LBO acquisitions is a mean to save taxes. The tax deductibility of the debt's interests seems to be a convincing theoretical rationale for the involvement of banks in buyout acquisitions. Let denotes the corporate income tax, 0 1. If the exit route is an IPO, the revenue after taxation is