O Yousfi
email: oyousfi@u-paris10.fr

Leveraged Buy Out and Tax saving advantage: a double-sided moral hazard model

Keywords: LBO, tax advantage, double moral hazard, debt, capital structure. JEL classi…cation: G23, G24, G32

This paper o¤ers an explanation for the prevalent use of debt in LBO …nance. We consider a double sided moral hazard model with three agents: the entrepreneur, the LBO fund and the bank. The entrepreneur and the LBO fund have to provide e¤orts in order to improve the productivity of their project; e¤orts are not observable.

Under some restrictive conditions, the debt-equity contracts induce the entrepreneur and the LBO fund to invest e¢ ciently. In the sense, without constraining the debt's payments, we show that the e¤orts depend on the project's quality. If the project is not very risky, the entrepreneur and the LBO fund provide the …rst best e¤orts and they share equally the bene…t. If it is highly risky, they provide the second best e¤orts and the bene…t's share given to each agent depends on the impact of his e¤ort on the project's performance. When the bank's payments are constrained to be non-decreasing with the project's payo¤, the agents'e¤orts do not depend on the project's quality. Whether the project is …nanced through a mixture of debt and equity or solely through equity, the entrepreneur and the LBO fund provide the same levels of e¤orts.

We show that the excessive use of debt is explained by the tax saving advantages: the interests of the debt are tax-deductible which creates additional revenues. But these revenues have no impacts on the agents'incentives.

Introduction

Leveraged Buy Out commonly known as LBO1 accounts for a signi…cant part of the corporate …nance and plays a major role in structuring mergers, acquisitions and transmissions.

There are three main facts about LBO …nance:

First, the level of debt is signi…cantly high. Moreover, there are many kinds of debt, such as the mezzanine debt, the subordinate debt and the convertible debt. These debts have di¤erent levels of risk.

Second, the LBO fund (hereafter he) is an active investor and he is well connected with many industries: he is engaged in the day to day operations of the …rm, he helps to recruit key personnel, he negotiates with the suppliers, the bank (hereafter he) and the other …nancial partners and he advises the entrepreneur (hereafter she) on all the strategic decisions.

Finally, the use of convertible securities becomes prevalent in LBO …nance. It is surprising because these securities are very rarely issued in the presence of banks or passive outside equity holders who …nance more established and less risky companies.

This paper provides a theory of LBO …nancing based on a contractual approach. The theoretical model that we present deals with the two …rst facts. The model describes the relationship between the LBO fund and the entrepreneur engaged in the acquisition of a …rm.

The success of these acquisitions is explained not only by the use of debt but it depends also on the market conditions, the performance of the Op Co and the partners'abilities. The entrepreneur is endowed with technical skills and he knows well the acquired company while the LBO fund plays a dual role: he is a …nancier and an adviser at the same time.

The question raised in this paper is the following: why does the entrepreneur prefer asking for …nancing and for advice from the LBO fund and the bank, while the LBO fund alone is able to advise her and to …nance the project?

The LBO fund is usually not wealth-constrained, the private equity funds are well established …nanciers: these funds participate in the formation of large companies and they issue high level of equity. In Venture capital2 , the venture capitalist fund (hereafter VC fund) is usually the only …nancier in the project and he is also providing advice.

To answer the question, we consider three agents: the entrepreneur, the LBO fund 3 and the bank. The entrepreneur is the manager of the Op Co and she wants to acquire it. The entrepreneur asks for advise and for funds …rst from the LBO fund. These two partners sign a …rst contract: the holding contract and they establish the holding company (hereafter the holding). They can also ask for additional funding from the bank and then they sign a second contract: the debt contract. The entrepreneur and the LBO fund have to exert non observable e¤orts which induce a double-sided moral hazard problem.

The optimal …nancial contracts have to meet three objectives: (1) each agent gets at least the cost of his initial investment (2) to determine the payments of each agent when the project succeeds and when it fails and (3) to incite the entrepreneur and the LBO fund to exert e¢ cient e¤orts.

We study the impacts of …nancial capital structure on the e¤orts when there is a double-sided moral hazard problem. To our knowledge, there are no papers exploring this issue in LBO. In the opposite with LBO, a number of papers study this question on the venture capital.

Several papers focus on the relationship between capital structure of the start-ups and the incentives of e¤orts in the presence of a double-sided moral hazard problem. [START_REF] Bergemann | Venture Capital Financing, moral hazard, and learning[END_REF] present a dynamic agency model with learning and moral hazard problems. They show that short-term re…nancing4 is never optimal but long-term contract allowing for intertemporal risksharing such as the stage …nancing5 is optimal. This …nancial regime constrains the entrepreneur to provide optimal e¤ort so that she could obtain funding for his project. If not, the VC fund will not invest additional capital and will abandon the project. [START_REF] Cornelli | Stage Financing and the Role of convertible securities[END_REF] show that the stage …nancing may induce a window dressing problem: the entrepreneur is tempted to announce a good short-term performance. Her aim is to reduce the probability that the project will be abandoned or liquidated. They conclude that the use of the convertible debt solves the "window dressing" problem: when the project looks too pro…table, the VC fund will convert his debt into equity which reduces the entrepreneur's pro…t. [START_REF] Schmidt | Convertible Securities and Venture Capital Finance[END_REF]Schmidt ( , 2003) ) does not study the stage …nancing regime but he focuses on the incentive properties of the convertible securities. He proposes a model where the pro…t that can be generated by the entrepreneur and the VC fund depends on three factors: the quality of the project and/or the abilities of the entrepreneur (the state of the world), their e¤orts, and further …nancial investment of the VC fund. The state of the world has an impact on the results of the project. It is unknown to both parties at date 0 and can be observed only after the initial investment has been sunk. Schmidt shows that there is no debt-equity contract that induces both parties to invest e¢ ciently. He joins Cornelli and Yosha and he deduces that the use of convertible securities induces both agents to exert optimal e¤orts. When he considers a multi-dimensional investment6 , the convertible debt contract still implements the …rst best investments of both parties.

My model is closely related to those of [START_REF] Casamatta | Financing and Advising: Optimal …nancial Contracts with Venture Capitalists[END_REF] and [START_REF] Repullo | Venture Capital Finance: A security design approach[END_REF]. They present a double-sided moral hazard model with a pure …nancier.

Casamatta considers a model with three agents: an entrepreneur endowed with an innovative idea, an adviser and a pure …nancier. When the adviser invests money in the project he is a VC fund, and he is a consultant when he only exerts e¤ort. The entrepreneur and the advisor have to provide substitute e¤orts. When these e¤orts are observable, whether the advisor is a consultant or a VC fund is irrelevant: there are many ways to implement the …rst best. This is no longer true when e¤orts are unobservable. The consultant's e¤ort is less e¢ cient than her proper e¤ort. Thus, she prefers asking for the fund's advice.

In order to solve the double moral hazard problem, she shows that all agents must participate …nancially in the project. When the project is not very risky, the presence of the pure …nancier induces both agents to exert …rst best e¤orts. On the contrary, when it is very risky, they need high powerful incentives to make e¢ cient e¤orts. Casamatta proposes to pledge the revenue to the pure …nancier when the project fails and to let the entrepreneur and the VC fund share the revenue in case of success: it is a "live or die" contract (Iness, 1990). This is still not su¢ cient to induce them to provide the …rst best e¤orts; the moral hazard problem induces them to provide the second best e¤orts.

Repullo and Suarez consider a stage …nancing model. The entrepreneur is wealth-constrained and she asks for advice and for money from two VC funds. One of them does not provide e¤ort so he may be considered as a pure or passive …nancier. The entrepreneur and the other fund have to exert non observable e¤orts. They conclude that the entrepreneur must ask for advice and fund from the partner who provides both money and advice. The project gives no gains when it fails and ' otherwise, where ' is a continuous random variable. ' is non observable when they sign the contract, they will learn more about it at the end of the …rst period. When the project is pro…table, it is continued, otherwise it is abandoned. When it is continued, the VC fund must invest additional capital.

We consider a single-period model and that all the cost of the project is invested when the contracts are signed.

We determine endogenously the …nancial contribution of each agent and their revenues'shares. We focus particularly on the presence of the bank and its impact on the agents'incentives. We …nd that the LBO fund must issue a signi…cant part of equity and that the bank's payments decrease with the project's revenues. We join [START_REF] Casamatta | Financing and Advising: Optimal …nancial Contracts with Venture Capitalists[END_REF], [START_REF] Jensen | Agency costs of free cash- ‡ow, corporate …nance, and takeovers[END_REF][START_REF] Jensen | Eclipse of the Public Corporation[END_REF], [START_REF] Jensen | Theory of the …rm: Management Behavior, Agency Costs and Ownership Structure[END_REF] and show that the debt induces the agents to provide the highest e¤orts. Furthermore, the way the project's outcome is shared between the agents, the bank's payments and the provided e¤orts depend on the project's quality but they do not depend on the …nancial capital structure. Our results show that: When the project is not very risky, we generalize the Casamatta's model and we show that both agents exert the …rst best e¤orts. We conclude that they must get equal shares of the project's bene…t.

When the project is very risky, the entrepreneur and the LBO fund provide the second best e¤orts. Double moral hazard problem induces them to make less e¢ cient investment decisions. They share the revenue in the good state of nature and pledge the entire revenue to the bank in the bad state of the nature. The sharing rule of the bene…t depends on the impact of each e¤ort on the project's performance.

If the debt's payments are non-decreasing with the results'project, the presence of the bank has no impacts on the agents' incentives: whether the project is …nanced through the LBO fund and the bank or only through the fund, the agents provide equal levels of e¤orts: these e¤orts do not depend on the …nancial capital structure.

The taxation creates additional revenues which explain why the entrepreneur asks for advice and for money from the bank and the LBO fund while the latter is not wealth constrained. But, we show that the tax saving advantage does not in ‡uence the e¤orts provided by the entrepreneur.

The model and the assumptions are presented in the section 1. Section 2 solves the model and derives the properties of the optimal …nancial contracts when all agents can invest …nancially in the entrepreneur's project. In section 3, we suppose that the debt's payments are non-decreasing with the revenues of the project and study the agents' incentives to e¤orts. Section 4 compares these contracts with those provided when the LBO fund does not contribute …nancially to the acquisition. In Section 5, we introduce the tax advantage in the model and deduce the optimal …nancial capital. Concluding remarks are in Section 6. All the proofs are presented in the Appendix.

The model

We consider an entrepreneur E who wants to acquire a company where she is the manager. The cost of the project is equal to K. She asks …rst for advice and for funds from the LBO fund A. If the latter participates in the project, he issues the amount of equity i in exchange of share's payo¤: 1 (0 1). The two partners may ask for further funds from the bank. Let I denote the amount of the debt. If the amount of equity and debt is not enough to cover the acquisition's price, the entrepreneur invests all the residual capital W = K (i + I).

The project is risky and generates an observable random revenue . It depends on the quality of the project, the entrepreneur skills, the market conditions... It can take two values: u when the project succeeds and d when it fails where u > d . When the project fails, d is equal to its liquidation value. The probability of success is denoted p(e; a), where e and a 2 R + are the e¤orts provided respectively by the entrepreneur and the LBO funds. The entrepreneur's e¤ort is related to technical skills. The LBO fund may exert a technical e¤ort or a managerial e¤ort such as the monitoring. We assume that the probability function p(e; a) is increasing and concave. We add the condition @ 2 p(e;a) @e@a > 0 so that we ensure that e¤orts are complementary. Furthermore, p(e; 0) = p(0; a) = 0: it means that both agents must provide strictly positive e¤orts so that the probability of success would be strictly positive. Making e¤orts (e; a) = (0; 0) is a Nash equilibrium but it means that the project fails with probability one.

There is a continuum of LBO fund and banks. All agents are risk neutral and the risk-free rate is normalized to 0. When contracts are signed, agents cannot abandon the project before the date of exit.

The sequence of events in the model

The sequence of events is summarized in the following …gure: At the date 0, E and A negotiate and sign a …rst contract: the holding contract. If they need further …nancial investment, they can ask for required money from the bank. Then, the holding and the bank B sign a second contract: the debt contract.

At the date 1, E and A have to exert respectively the non contractible e¤orts e and a. Let u(e) and v(a) denote their respective cost functions. These functions are increasing, convex and satisfy u(0) = v(0) = u 0 (0) = v 0 (0) = 0.

At the date 2, the project is completed. If it succeeds, the bank gets the payment D and the two agents share the residual amount; they get respectively ( u D) and ( 1)( u D). Otherwise, the bank perceives the collateral H (H d ). If H = d , the entrepreneur and the LBO fund get zero payo¤s. If H < d , they obtain respectively ( d H) and ( 1)( d H).

Financial contracts

Two …nancial contracts must be speci…ed in this model:

1. The holding contract: it determines the amount of equity that must be issued by the LBO fund and his share of bene…ts. He participates in the project if his revenue is positive. This constraint is written:

EU A = (1 )[p(e; a)( u D) + (1 p(e; a))( d H)] v(a) i 0 (P C A )
where (P C A ) is the participation constraint of the LBO funds. The agent E asks for money and for advice from many LBO funds. He deals with the one who makes the best o¤er (i; ). This o¤er gives him a zero payo¤; (P C A ) is binding which implies that EU A = 0. where (P C B ) is the participation constraint of the bank. Because of the competition between the banks, we deduce that EU B = 0.

The …rst best e¤orts

The social value of the project V (e; a) is given by:

V (e; a) = p(e; a)

+ d u(e) v(a) K where = u d .
The …rst best e¤orts e F B and a F B are deduced from the …rst order conditions of V (e; a). They are given respectively by: @V (e, a)

@e e F B , a

F B = p e (e F B ; a F B ) u 0 (e F B ) = 0 (1) @V (e, a) @a e F B , a F B = p a (e F B ; a F B ) v 0 (a F B ) = 0 (2)
The equations (1) and (2) can be written:

= u 0 (e F B ) p e (e F B ; a F B ) = v 0 (a F B ) p a (e F B ; a F B ) (3)
The ratios of the marginal cost to the marginal probability of e¤orts are equal to the di¤erence between the revenues of the bad and the good states of nature. The probability of success when the agents provide …rst best e¤ort is denoted p F B = p(e F B , a F B ). Then, the optimal social value of the project is given by:

V F B = p F B + d u(e F B ) v(a F B ) K
V F B is assumed to be strictly positive.

The …rst best solution can be implemented in a number of ways: the entrepreneur and the LBO fund may ask money from the bank or the LBO fund may invest i = K W . Without transaction costs, there are many ways to implement the …rst best solution: the entrepreneur can ask for advice and for money from the LBO fund (they sign one contract: the holding contract) or she can rely on the LBO fund and the bank (they sign two contracts: the holding and the debt contracts). She may also ask for advice from the consultant and for money from the bank. In this case, the entrepreneur is indi¤erent between a consultant and an LBO fund.

2. The optimal …nancial contracts with a double sided moral hazard problem

The reaction functions

We suppose now that e¤orts are unobservable. The entrepreneur (respectively the LBO fund) exerts the level of e¤ort e (respectively a) that maximizes her (respectively his) expected pro…t given the e¤orts'costs and the established contracts: hence, each e¤ort is an increasing function of the other one.

The reaction functions of E and A are given by their incentive constraints. They are respectively written:

e(a) 2 arg max e2R + [p(e; a)( u D) + (1 p(e; a))( d H)] u(e) W (IC E ) a(e) 2 arg max a2R + (1 )[p(e; a)( u D) + (1 p(e; a))( d H)] v(a) i (IC A )
These functions e(a) and a(e) are respectively the solutions of the following equations:

( D + H) = u 0 (e) p e (e; a) (4)
(1

)( D + H) = v 0 (a) p a (e; a) (5)
By di¤erentiating the equations (4) and (5), we obtain: The e¤ort of the entrepreneur (respectively the LBO fund) is an increasing function of the e¤ort of the LBO fund (respectively of the entrepreneur): the e¤orts are strategic complements.

The optimal …nancial contracts when the e¤orts are unobservable

Competition among the LBO funds and the banks induces A and B to propose respectively holding and debt contracts that maximize the expected pro…t of the entrepreneur. The latter must maximize her expected gain under their participation constraints and the incentive constraints. The program to be solved is therefore given by:

( ; i ; I ; D ; H ) 2 arg max , i, I, D, H EU E = [p(e; a)( D + H) + d H] u(e) K + i + I s.t (P C A ) , (P C B ) , (IC A ) and (IC E )
with the additional conditions:

0 D u , 0 H d and 0 1 (6) -9 -
The participation constraints (P C A ) and (P C B ) are binding, so the amounts of equity and debt issued respectively by the LBO fund and the bank are given by:

i = (1 )[p(e; a) ( D + H) + d H] v(a) (7)
I = p(e; a) (D H) + H (8)
We replace (7) and (8) in the objective function. The optimal …nancial contracts induce the entrepreneur to maximize the social value of the project under the incentive constraints (IC E ) and (IC A ). Hence, the optimal …nancial contracts induce the entrepreneur to maximize the social value of the project under the incentive constraints: with the conditions (6).

The project is not very risky

The social value of the project is optimal if the agents exert …rst best e¤orts, in other words, the e¤orts e F B and a F B must satisfy the incentive constraints (4) and (5).

Given the equations (3), we replace in (4) and (5). We obtain:

= ( D + H) = (1 )( D + H) (9)
It means that the bene…t's share of the entrepreneur is equal to = 1 2 . The revenues of the project are …xed, if the optimal contracts give powerful incentives to one agent, it will reduce the incentives of the other agent. The optimal …nancial contracts must boost the incentives of both agents such that they provide the …rst best e¤orts; so they must get equal pro…t shares. We notice that this sharing rule depends neither on the capital structure, nor on the e¤orts'e¢ ciency.

When we substitute = 1 2 in (9), we get:

H = + D This means that + D < d () D < 2 d u
, but D 0, in other words, the project must be not very risky in the sense u 2 d . We conclude that the bank payments are decreasing with the outcome of the project: the collateral is larger than his payment when the project succeeds.

The following proposition presents the properties of the optimal …nancial contracts:

Proposition 1 If the project is not very risky u 2 d , under the conditions d 2 [l+ maxfv(a F B ) p F B ; 0g, 2l p F B ] and u 2 [ l, 2l ], the …nancial contracts that implement the …rst best e¤ orts are given by:

= 1 2 i = V F B +W + u(e F B ) v(a F B ) W 2 [0 , K[ and D = 2[K W + v(a F B )] u H = 2[K W + v(a F B )] d I = 2[K W + v(a F B )] p F B d where l = K W + v(a F B ).
The proof of this proposition is presented in the appendix A.

This proposition states that there is an in…nity number of …nancial contracts that induce the entrepreneur and the LBO fund to provide the …rst best e¤orts. However, the LBO fund and the bank should propose contracts which maximize the expected gain of the entrepreneur. Accordingly, EU E is optimal if the entrepreneur does not issue equity; W = 0 which implies that i + I = K. We conclude therefore that under the conditions d 2 [l 0 + maxfv(a F B ) p F B ; 0g, 2l 0 p F B ] and u 2 [ l 0 , 2l 0 ], the optimal …nancial contracts are given by:

= 1 2 i = V F B +u(e F B ) v(a F B )
and

D = 2[K + v(a F B )] u H = 2[K + v(a F B )] d I = 2[K + v(a F B )] p F B d where l 0 = K + v(a F B ).
When the project is not very risky, the LBO fund issues a high amount of equity. Moreover the LBO fund and the entrepreneur provide the …rst best e¤orts and they get equal shares of the bene…t. Although that the entrepreneur does not contribute …nancially into the acquisition, their payments are equal: they get u K v(a F B ), if the project succeeds and d K v(a F B ) when it fails. The di¤erence between these payments is equal to the di¤erence between the revenues of the project < d .

The intuition of the proposition 1 is the following: in order to induce the agents to provide optimal e¤orts, the bank payments must be decreasing with the outcome of the project. Indeed, despite the fact that is not very large, they are constrained to provide …rst best e¤orts in order to perceive the highest payments; the success payments.

We conclude that when the project is not very risky in the sense de…ned in proposition 1, the presence of the bank constrains the agents to exert optimal e¤orts. In addition, the entrepreneur captures the whole social value of the project while the LBO fund and the bank recoup the costs of their investments.

The project is very risky

When the project is very risky ( u > 2 d ), the general model does not enable us to deduce the properties of the optimal …nancial contracts. This is why, we rely in the following speci…cation:

p(e; a) = e 1 a where 2 ]0; 1[ measures the impact of the e¤ort of the LBO fund on the success'probability. In other words, it is the LBO's e¤ort elasticity of the success probability. The functions of cost are given by:

u(e) = e 2 2 and v(a) = a 2 2
where e and a take values on [0 , 1] and > > 0: the LBO's e¤ort is more costly than the entrepreneur's e¤ort. We substitute e 1 a , e 2 2 and a 2 2 in the expected gain of the entrepreneur and the LBO fund.

The optimal solution

The social value of the project is given by:

V (e , a) = e 1 a e 2 2 a 2 2 + d K
The …rst best e¤orts are given by the …rst order conditions of V (e , a):

e F B = ( ) (10)
a F B = (1 ) 1 2 ( ) (11)
where

( ) = [ (1 )] 1 =2 [ ] =2
. These e¤orts are increasing with the di¤erence between the revenues of the project. When the project is very risky, the levels of the …rst best e¤orts become very large. Furthermore, when the impact of the e¤ort of the LBO fund on the success probability is high such that + , his …rst best e¤ort is larger than the e¤ort provided by the entrepreneur. Otherwise, the entrepreneur provides the highest e¤ort.

We denote by p

F B = 2 ( ) (1 )
the probability of success of the project when agents provide the …rst best e¤orts. To ensure that this probability is inferior to 1, we assume that:

(1 ) 2 ( ) -12 -
The optimal social value of the project is given by:

V F B = [ ( ) ] 2 2 (1 ) + d K
When is not high enough such that 2]0; + [, this value decreases with and it increases when becomes high 2[ + ; 1[ .

The solution when the agents'e¤orts are unobservable

When there is an information asymmetry, the reaction functions (4) are (5) give the following e¤orts in equilibrium:

e = [ (1 ) ( )] 1=2 ( D + H) (12) a = [ (1 ) ( )] 1=2 ( D + H) (13)
where

( ) = [ (1 )] 1 [ (1 )] .
It is easy to check that the di¤erence between the bank's payments in cases of success and failure must be inferior to the di¤erence between the project's revenues d and u , i.e.

D H (14)

Otherwise, the e¤orts (12) and (13) are negative. Besides, the agents'e¤orts are increasing with the di¤erence between the project's revenues. The more high the bank's payment in case of success, the less the entrepreneur and the LBO fund will be induced to provide e¤orts. However, the increase of H increases the e¤orts e and a.

Then, the project succeeds with the following probability:

p = ( )( D + H).
We add the following assumption:

( D + H) 1 ( )
to ensure that this probability is inferior to 1.

If the …nancial contracts attribute all the gain to the entrepreneur or to the LBO fund, the project will fail with probability equal to 1 ( (0) = (1) = 0). This enables us to conclude that: 0 < < 1.

The expected gain of the entrepreneur is given by:

EU E = 1 2 (1 + ) ( ) ( D + H) 2 + d H W (15)
Because of the competition among the LBO funds and the bank, the participation constraints (CP A ) and (CP B ) are binding. We deduce that:

i = (1 1 2 )(1 ) ( )( D + H) 2 + (1 ) d H (16) I = ( )( D + H) (D H) + H (17)
We substitute that (12), ( 13), ( 16) and ( 17) in (15). Consequently, the entrepreneur is induced to maximize the social value of the project:

max , D, H V (:) = ( ) ( D + H) + d K 1 2 ( + 2 ) ( )( D + H) 2
with the conditions:

0 < < 1, 0 D u and 0 H d ( 18
)
The following proposition summarizes the properties of the optimal …nancial contracts when the project is very risky:

Proposition 2 If the project is very risky u > 2 d , the optimal …nancial contracts are given by:

^ ( ; k) = ( (1 )(2 ) k p 2(1 2 ) if 2 ]0 , 1[ =f 1 2 g 1 2 if = 1 2 { = (1 1 2 )(1 ^ ) ( ^ )( u ) 2 and D = 0 , Ĥ = d and Î = d (1 u ( ^ )) Ŵ = K { Î
where k = d u and = (1

) u (1 + )(2 ) u 6 d + ( d ) 2 :
The proof of this proposition is presented in the appendix B.

In contrast to the proposition 1, the proposition 2 shows that the optimal …nancial contracts are unique in the speci…ed model. The debt contract is a live or die contract: the bank is paid only in case of failure. in this case, he gets all the failure's revenue. The success revenue of the project is shared between the entrepreneur and the LBO fund.

Moreover, the optimal sharing rule ^ is a function of . When > 1 2 , the fund's share of bene…t is larger than the entrepreneur's share. Despite the fact that the e¤ort of LBO fund is more costly than the entrepreneur's e¤ort, it has the most important impact on the probability of success. The optimal …nancial contracts should provide powerful incentives to induce him to spend high e¤ort; for example giving him the highest share of the project's outcome. When the impact of the fund's e¤ort on the success' probability is low, it is e¢ cient to give powerful incentives to the agent who provides the less costly e¤ort.

When the project is highly risky, the agents make the e¤orts:

ê = [ ^ (1 ) ( ^ )] 1 2 u and â = [ (1 ^ ) ( ^ )] 1 2 u (19)
-14 -If we substitute ( ^ ) for its expression in (19), we can write:

ê = ^ 1 2 (1 ^ ) 2 ( ) u and â = ^ 1 2 (1 ^ ) 1+ 2 h (1 ) i 1 2 ( ) u (20)
These e¤orts depend on .

We focus on the particular case where = 1 2 . Hence ê and â are written:

ê = 1 4 3 4 1 4 u = 1 2 u e F B and â = 1 4 1 4 3 4 u = 1 2 u a F B (21)
The double moral hazard induces the agents to make less e¢ cient e¤orts: the levels of …rst best e¤orts are high, we need powerful incentive mechanisms to constrain the entrepreneur and the LBO to exert these e¤orts. Note that the revenue should be shared between them in the case of success but the revenue should be paid to the bank in case of failure. However, this is still not enough to induce them to spend the …rst best e¤orts.

Notice that the LBO fund issues a signi…cantly high amount of equity i = 3 32 ( )

1 2 ( u ) 2 . Under the condition u < 4( ) 1 2 , the amount of debt is given by Î = d h 1 1 4 ( ) 1 2 u i
. The entrepreneur cannot get the whole optimal social value, her expected gain is given by:

EU E = 2 u d 2 ( ) 2 (1 ) u + d K < V F B
To get the highest revenues, the entrepreneur and the LBO fund may be tempted to announce a success whether the project succeeds or fails. If the project's result is a failure, they sell the assets and the equipment, pay D to the bank and share the remaining amount of money.

To avoid such behavior, hereafter we consider that the debt's payments are non-decreasing with the project's revenues.

3. The …nancial structure when the debt' s payments are non-decreasing with the project' s revenues

Hereafter, we assume that the bank's payment is the highest in case of success. In other words, we add the following condition:

D H: (22)
to the program described in the previous section.

The objective of the entrepreneur is to maximize the social value of the project:

max , D, H V (:) = ( ) ( D + H) + d K 1 2 ( + 2 ) ( )( D + H) 2
with the following conditions:

(18) and ( 22) :

-15 -

The optimal …nancial contracts are characterized in the following proposition:

Proposition 3 When the debt's payments are non-decreasing with the project's revenues, the following …nancial contracts induce the entrepreneur and the LBO fund to provide the same levels of e¤ orts:

( ) = ( (2 )(1 ) p (1 2 )(2 ) 2(1 2 ) if 2 ]0, 1[ = 1 2 1 2 if = 1 2 (23) { = (1 1 2 )(1 ) ( )( ) 2 W 2 [0 , K[ and D = H = I = K W (1 1 2 )(1 ) ( )( ) 2
The proof of this proposition is presented in the appendix C.

As explained in proposition 1, there are many …nancial contracts that induce the agents to exert the following e¤orts:

e = [ ( ) (1 ) ( ( ))] 1 2 and a = [ (1 ( )) ( ( ))] 1 2 (24)
These e¤orts are also written:

e = h ( ) i 1 2 [(1 ( )] 2 e F B and a = h ( ) i 1 2 [(1 ( )] 1+ 2 a F B (25)
They induce the success of the entrepreneur's project with the following probability:

p = ( ( )) (26)
The proposition 3 states that the acquisition can be …nanced through a mixture of debt and equity or solely equity. In the latter case, the entrepreneur should issue the amount of equity W = K {. However, in order to maximize the expected gain of the entrepreneur, the LBO fund and the bank o¤er the best contracts such that W = 0, the acquisition is therefore …nanced through the equity issued by the LBO fund and the debt. It is straightforward to see that the amount of equity issued by the LBO fund does not depend on the …nancial contributions of the other agents which means that the bank invests higher amount of debt, such that:

D = H = I = K {
Whether the project is very risky or not, the non-decreasing revenue constraint imposes to give to the bank at least as much in the good state as in the bad state. He perceives a …xed payment in cases of success and failure. This payment is equal to the amount of the issued debt which implies that the interest rate of the bank is null. Consequently, the amount of issued debt is equal to his payment. Lending money to the entrepreneur and the LBO fund is not a risky investment for the bank. He gets his money back whatever the project's result.

The optimal sharing rule ( ) depends neither on the …nancial capital structure nor on the project's revenues. But, it depends on the impact of each e¤ort on the project's performance:

If the LBO's e¤ort has a low impact on the success probability, in the sense 2]0; 1 2 [, the LBO fund perceives the lowest bene…t's share. At the opposite, the e¤ort a is more e¢ cient than e, this is why the optimal …nancial contracts must induce him to provide the highest e¤ort. So, he gets the highest bene…t's share.

If

= 1 2 , both e¤orts have equal impacts on the success probability: given that e and a are complementary e¤orts and the revenues are …xed, the optimal …nancial contracts must boost simultaneously the agents'incentives.

Let us now compare the sharing rules ( ) and ^ ( ; k) given by the proposition 2: when k converges to 0, whatever the value of , we …nd ^ converges to . However, when k and converge respectively to 1 2 and 1, we get ( ) < ^ ( ; k). On the contrary, if k and converge respectively to 1 2 and 0, ( ) > ^ ( ; k).

Financing the acquisition with a mixture of debt and equity or just with equity does not in ‡uence the agents'incentives. The e¤orts do not depend on the …nancial capital structure: the entrepreneur and the LBO fund provide the same levels of e¤orts.

According to these results, under the condition of a non-decreasing debt's payments, the debt has no impact on the agents'incentives: the levels of e¤orts (25) are lower than the …rst best levels and even than those provided by the entrepreneur and the LBO when the project is very risky and the debt's payments are decreasing with the project's payo¤ (see proposition 2). When = 1 2 , the entrepreneur and the LBO fund provide the following e¤orts:

e c = 1 4 3 4 1 4 < e F B and a c = 1 4 1 4 3 4 < a F B (27)
In this case, the expected gain of the entrepreneur is written:

EU E = [ ( ) ] 2 2 (1 ) + d K < V F B (28)
4. What is achieved with a consultant?

In this section, we consider that the entrepreneur asks for advice from a consultant and for further funds fom the bank. Assume that the LBO fund does not contribute …nancially to the project but he still spend the e¤ort a such that the LBO fund becomes a consultant. The entrepreneur' …nancial investment is given by W = K I.

-17 -The participation constraints of the consultant and the bank (CP A ) and (CP B ) are written:

EU A = (1 )[p(e; a) ( D + H) + d H] v(a) 0 (29) EU B = p(e; a) (D H) + H (K W ) 0 (30)
The e¤orts that maximize the expected gains of the entrepreneur and the LBO fund are given by (4) and (5).

The optimal …nancial contracts maximize the expected gain of the entrepreneur under the participation constraints of the bank and the consultant and the incentive constraints. Consequently, the program to be solved is given by: max , e, a, D, H

EU E = [p(e; a) ( D + H) + d H] u(e) W s.t (
4) , (5) , ( 29) and (30)

with the conditions (6).

The participation constraints of the LBO fund and the bank give

(1 ) [p(e; a) ( D + H) + d H] v(a) 0 (31)
W = K H p(e; a) (D H) (32)
In contrast with the participation constraint of the bank, the participation constraint of the consultant may not be binding.

When we substitute (32) into the objective function of the entrepreneur, we get the following program: max with the conditions (6).

It is easy to check that the social value of the project is optimal if the agents provide the …rst best e¤orts, in other words, when = 1 2 and the bank's payments are decreasing with the revenues' project, i.e. H D = .

The following proposition summarizes the properties of the optimal …nancial contracts.

Proposition 4 When the project is not very risky ( u 2 d ) and i = 0, the following …nancial contracts induce the entrepreneur and the LBO fund to provide the …rst best e¤ orts:

~ = 1 2 K + p F B d W K 1 p F B -18 - and D = K W 1 p F B H = K W + p F B Ĩ = K W
The proof of this proposition is presented in the appendix D.

These contracts exhibit the properties of the optimal …nancial discussed in the proposition 1: when the project is not very risky, the entrepreneur and the consultant provide the …rst best e¤orts and share equally the revenues of the project. These results show that the identity of the agent providing advice is irrelevant for the entrepreneur. In other words, it is the debt which constrains both agents to provide optimal e¤orts. Note that in these contracts, the entrepreneur issues the smallest amount of equity she can invest:

W = K + p F B d
Consequently, the bank should o¤er the following debt contract:

D = 2 d u , H = d and Ĩ = d p F B
In contrast with a live or die contract whereby the bank gets all the failure's revenue, he gets a strictly positive payment is case of success. Whatever the impact of each e¤ort on the success probability and its cost, the entrepreneur and the LBO fund get equal payments ( ).

When the project is very risky, as shown previously the failure revenue of the bank is given by H = d . The following proposition presents the properties of the optimal …nancial contracts in the speci…ed model.

Proposition 5 If u > 2 d and i = 0, under the condition u < +2 d , the optimal …nancial contracts are given by:

~ = 2 2 + W = K d + 1 2 2 ( ) ( ) 2 and D = 1 2 + 1 d 1 2 u and H = d I = d 1 2 2 ( ) ( ) 2
The proof is presented in the appendix E.

The proposition 5 states that the bank's payments are decreasing with the revenues of the project. In contrast with the proposition 2, the debt contract is not a "live or die" contract. But, 4.1. What happens if the bank' s payments are constrained to be non-decreasing with the project' s outcome?

The participation constraint of the consultant is binding (see appendix D and E). Then, we can write n e 1 a (

D + H) + d H o = n e 1 a ( D + H) + d H o a 2 2
Substituting the …rst term of this equation and the e¤orts (12) and (13) in the expected gain of the entrepreneur gives:

max , D, H EU E = ( ) ( D + H) 1 2 [ + (1 2 ) ] ( ) ( D + H) 2 + d K s.t D H and (6)
We deduce the following lemma:

Lemma 1 When the bank's payments are non-decreasing with the project's revenues and the LBO fund does not contribute …nancially to the project, the following contracts induce the entrepreneur and the LBO fund to provide the optimal e¤ orts:

0 = ( (2 )(1 ) p (1 2 )(2 ) 2(1 2 ) if 2 ]0, 1[ = 1 2 1 2 if = 1 2 D 0 = H 0 = I = K W W 2 [0 , K]
The proof is presented in the appendix F .

The …nancial contracts that induce the entrepreneur and the LBO fund to provide the e¤orts (24) are not unique. The debt's payments do not depend on the project's revenues: whether the project succeeds or fails, the bank gets the same amount of money. This implies that the bank interest is null.

If the entrepreneur is not wealth-constrained, she can implement the project without asking for debt from the bank: W = K. However, the optimal …nancial structure is …nanced only through debt:

W 0 = 0 and I 0 = K When the debt's payments are non-decreasing with the revenues, the identity of the agents providing the e¤ort a is irrelevant to the entrepreneur: the consultant and the LBO fund provide the same levels of e¤orts whatever the amount of the debt. This result enables to generalize the result of the proposition 3. It was established that …nancing the acquisition with a mixture of debt and equity or only with equity does not in ‡uence the agents' incentives. The lemma 1 states that the use of debt induces the entrepreneur and the LBO fund to provide also the e¤orts (24). Accordingly, we conclude that when the bank's payments are nondecreasing, the incentives of the entrepreneur and the LBO fund does not depend on the …nancial structure of the acquisition.

The …nancial structure in LBO and Tax saving advantage

In this section, we consider that the debt's interests are tax-deductible and study the impact of the tax saving advantage on the agents'incentives and on the …nancial capital structure.

Let denotes the corporate income tax, 0 1.

Hereafter the success payment of the bank D is equal to (1 + r) I where r is the interest rate of the bank.

The success revenue after taxation is given by:

(1 ) [ u (1 + r) I] + (1 + r) I = (1 ) u + (1 + r) I
where (1 ) [ u (1 + r) I] is the revenue shared between the entrepreneur and the LBO fund.

But in case of failure, the revenue after taxation is given by:

(1

) d H + H = (1 ) d + H
where (1 ) d H is the failure's revenue shared between the entrepreneur and the LBO fund.

If the project is liquidated H = d , the entrepreneur and the LBO fund have no payments.

5.1. The optimal …nancial contracts when the e¤orts are observable

Hereafter, we study the impact of taxation on the agents'incentives …rst when the e¤orts are observable. The expected gain of the entrepreneur after taxation is written:

EU E = (
-22 -If we substitute the second term of the equation (37) in ( 35) and (36), the expected gain of the entrepreneur and the amount of the equity issued by the LBO can be written respectively:

EU E = (1 ) n p(e; a) u + [1 p(e; a)] d I o u(e) W (38)
i = (1 ) (1 ) n p(e; a) u + [1 p(e; a)] d I o v(a) (39)
Given the …nancial constraint K = W + i + I, if we substitute (39) in (38), the expected gain of the entrepreneur becomes:

EU E = (1 ) h p(e; a) + d i u(e) v(a) K + I ( 40
)
EU E is strictly increasing with the level of debt. This means that it is optimal to …nance the project only through the debt. We conclude that:

I P I = K i P I = W P I = 0
As a result, the entrepreneur is indi¤erent toward the identity of the agent providing the e¤ort a. Whether her partner is consultant or LBO fund, they do not issue equity.

When the e¤orts are observable, the levels of e¤orts e P I and a P I which maximize the expected gain of the entrepreneur are given by:

(1 ) = u 0 (e) p e (e; a)

= v 0 (a) p a (e; a) (41)
These conditions (41) imply that the agents cannot provide the …rst best e¤orts.

If we consider the speci…ed model, the e¤orts e P I and a P I are written:

e P I = ( ) (1 ) (42)
a P I = (1 ) 1 2 ( ) (1 ) (43)
These e¤orts are increasing with the corporate income tax : if converges to 0, ( 42) and (43) converge to the …rst best e¤orts. On the contrary, when converges to 1, the revenues after taxation are very low, consequently, the agents are not induced to provide e¤orts.

The e¤orts (42) and (43) induce the success with the following probability:

p P I = 2 ( ) (1 ) (1 ) (44)
The bank's payment of success is equal to

D P I = (1 + r) K.
The bank's payment of failure is given by the participation constraint (37), it is written:

H P I = 1 (1 + r) p P I 1 p P I K (45)
H P I exists under the following condition:

< (1 ) (1 + r) (1 ) 2 ( )
This condition ensures also that the success probability p P I is inferior to 1.

It is easy to check that the debt's payments are non-decreasing with the project's revenues

D P I H P I = rK 1 p P I > 0
We substitute (42), ( 43), ( 44) and i = 0 in the participation constraint of the LBO fund. The optimal sharing rule is therefore given by:

P I = 1 v a P I (1 ) p P I + d K
When the information is perfect, the expected gain of the entrepreneur is given by: EU E = (1 ) p P I + d u e P I v a P I (1 ) K 5.2. The optimal …nancial contracts when the e¤orts are unobservable

The pure equity contract

The expected gain of the entrepreneur and the LBO fund are written:

EU E = (1 ) h p(e; a) + d i u(e) W 0 EU A = (1 ) (1 )
h p(e; a)

+ d i v(a) i = 0
The e¤orts in equilibrium are given by the …rst order conditions of EU E and EU A . They are written:

e c T = [ (1 ) ( )] 1 2 (1 ) (46)
a c T = [ (1 ) ( )] 1 2 (1 ) (47)
Given 0 < < 1, these e¤orts are strictly lower than the …rst best e¤orts. The success probability is therefore given by p c T = (1 ) ( ) .

The optimal …nancial contracts induce the entrepreneur to maximize her expected gain:

max EU E = 1 2 [2 (1 2 ) ] ( ) (1 ) 2 2 + (1 ) d K s.t 0 < < 1
The …rst order condition of EU E gives the results of the following lemma:

Lemma 2 When the debt's interests are tax deductible and the project is …nanced only through equity, the optimal holding contract is given by:

T = ( (2 )(1 ) p (1 2 )(2 ) 2(1 2 ) if 2 ]0, 1[ = 1 2 1 2 if = 1 2 i T = 1 1 2 (1 T ) ( T ) (1 ) 2 ( ) 2 + (1 ) (1 T ) d W T = K i T
The holding contract is unique. The …nancial contribution of the LBO fund is decreasing with the corporate income tax . Consequently, if increases, the amount of equity issued by the entrepreneur should increase.

If the project is …nanced only through the equity issued by the LBO fund, the agents provide the following e¤orts:

e c T = [ T (1 ) ( T )] 1 2 (1 ) (48)
a c T = [ (1 T ) ( T )] 1 2 (1 ) (49)
It is straightforward to see that the e¤orts (48) and (49) are inferior to (24). There are no tax advantages when the entrepreneur and the LBO fund issue only equity to fund the acquisition. Accordingly, their net revenues are reduced due to taxation. This is why the agents'incentives are decreasing with the corporate income tax. Given 0 < T < 1, it is easy to check that the entrepreneur and the LBO fund provide lower e¤orts than those provided when there is no information asymmetry.

The debt-equity contracts

The e¤orts of equilibrium e T and a T are given by the …rst order conditions of EU E and EU A given by (38) and (39):

(1 ) = u 0 (e) p e (e , a)

(1

) (1 ) = v 0 (a) p a (e , a) (51)
-25 -When the e¤orts are unobservable, the entrepreneur and the LBO fund do not provide the …rst best e¤orts. In the speci…ed model, the conditions (50) and (51) enable us to deduce the following e¤orts:

e T = [ (1 ) ( )] 1=2 (1 ) (52)
a T = [ (1 ) ( )] 1=2 (1 ) (53)
Then, the project succeeds with the probability:

p T = ( ) (1 ) (54)
To ensure that p T 1, we assume:

1 ( ) (1 ) (55)
The optimal …nancial contracts induce the entrepreneur to maximize her expected gain given by: arg max

, I EU E = 1 2 [2 (1 2 ) ] ( ) [(1 ) ] 2 + (1 ) d K + I s.t 0 < < 1 and 0 I K W
The …rst order conditions of EU E give the results presented in the lemma 3:

Lemma 3 In the presence of taxation, the optimal …nancial contracts are given by:

T = ( (2 )(1 ) p (1 2 )(2 ) 2(1 2 ) if 2 ]0, 1[ = 1 2 1 2 if = 1 2 i T = W T = 0 and I T = K H T = 1 ( T ) (1 ) (1 + r) 1 ( T ) (1 ) K
This lemma states that under the condition (55), without constraining the debt's payments, these contracts state that these payments are non-decreasing with the project's payo¤ which is consistent with the features of the LBO …nance.

However, the identity of the agent providing advice is irrelevant: whether he is a consultant or an LBO fund, the two partners do not contribute …nancially to the acquisition. Moreover, because of the tax saving advantage of the debt, the acquisition is …nanced only through the debt.

According to the lemmas 1 and 3, we conclude that the …nancial structure of the project does not depend on the nature of the e¤orts (observable or not). In other words, the double sided moral hazard problem does not in ‡uence the …nancial structure of the acquisition.

In this case, the entrepreneur and the LBO fund provide the following e¤orts:

e T = [ T (1 ) ( T )] 1=2 (1 ) (56)
a T = [ (1 T ) ( T )] 1=2 (1 ) (57)
Given 0 < < 1, the e¤orts (56) and (57) are lower than e P I and a P I but equal to the e¤orts provided when the project is …nanced through equity. This result shows that the tax advantage of the debt does not in ‡uence the agents'incentives but it induces them to get high level of debt.

These results are in line with those of Modigliani and Miller (1963) who correct that the tax advantage of debt …nancing is greater than what they originally suggested in their …rst paper in 1958. They show that under some restrictive conditions, the value of the levered company is equal to the amount of the tax-deductible interests and the value of the unlevered company. The relationship between the …nancial capital structure and the investment decision is therefore established when there is tax.

In contrast with Modigliani andMiller (1963), Miller (1977) distinguished between the corporate and the personal income taxes and …nd that for individual, the e¤ects of the two types of taxes end up cancelling each other. He suggests that there is a tax saving advantage only in macroeconomic level. In the current model, when we add the personal income tax, it is easy to check that the Miller's result does not hold and that the tax saving advantages are still persistent.

In LBO, leveraged also potentially increases …rm value through the tax-deductibility of interests. [START_REF] Lowenstein | Management Buyouts[END_REF] mentions incentive e¤ects of debt, but argues tax e¤ects play a major role in explaining the value increase of the …rm's value. Kaplan (1989a) consider a sample of 76 LMBO7 projects exited between 1980 and 1986, their results show that there is a tax saving advantage in LBO acquisitions: the median value of tax bene…ts estimated at the time of going private via LBO has a lower bound of 21% and an upper bound of 143% of the premium paid to the pre-buyout shareholders. Notice that their estimated value depend on the rate of the debt of the acquired company paid and on the tax rate applied to the interest deductions. In a second paper (Kaplan, 1989b), he estimates that interest tax shield can explain from 4% to 40% of the …rm's value.

However, [START_REF] Kaplan | Leveraged Buyouts and Private Equity[END_REF] explain that the value of this tax shield is notoriously di¢ cult to calculate because it requires restrictive assumptions of the tax advantage of debt (net of personal taxes), the expected permanence of the debt, and the riskiness of the tax shield.

"It is safe to say, therefore, that taxes create some value, but di¢ cult to say exactly how much". [START_REF] Jensen | Eclipse of the Public Corporation[END_REF] notices that usually the other tax revenues are ignored in LBO acquisitions. He counts …ve sources of additional tax revenues generated in these projects:

1. Capital gains taxes paid by pre-buyout shareholders; 2. Capital gains taxes paid on post-buyout asset sales; 3. Tax payments on the large increases in operating earnings generated by e¢ ciency gains; 4. Tax payments by creditors who receive interest payments on the LBO debt; 5. Taxes generated by more e¢ cient use of the company's total capital. He explains that these taxes compensate the tax saving advantage of the LBO debt (as an example, see the RJR Nabisco, in the same paper).

Conclusion

This paper has studied the properties of the optimal …nancial contracts in LBO. We have focused on the incentive e¤ects of debt in the presence of a double-sided moral hazard problem. The model explains why the entrepreneur prefers asking for money and for advice from the bank and the LBO fund, despite that the latter is not wealth-constrained.

We show that all agents have to participate …nancially in the project and that the payments of the bank are decreasing with the outcome of the project. The agents' e¤orts depend on the quality's project. If the project is not very risky, the entrepreneur and the LBO fund provide the …rst best e¤orts and they get equal shares of the outcome of the project. This is no longer true when the acquisition is very risky; the moral hazard problem induces them to provide the second best e¤orts. They must share the revenue in case of success and pledge the entire revenue to the bank when the project fails.

However, constraining the debt's payments to be non-decreasing with the project's outcome has no impacts on the agents'incentives. Whether the project is …nanced through the bank and the LBO fund or solely through the LBO fund, the e¤orts do not depend on the …nancial capital structure.

To explain the excessive use of debt in LBO acquisitions, we show that the tax saving advantage creates extra revenues which explains the high level of debt in LBO acquisitions. However, we show that tax advantage does not in ‡uence the agents'incentives.

Exploring in full details how these optimal contracts are implemented in LBO is an important topic for future research. The analysis of the model yields the following empirical predictions: First, the model predicts that the entrepreneur's …nancial participation is lower in more-pro…table/not very risky buyouts than in less-pro…table/very risky buyouts. On the contrary, the LBO fund issues more equity if the project is not very risky.

Second, consultant services should not very frequent in LBO acquisitions. The entrepreneur relies on the advising and …nancing of the LBO fund. 1. 0 < D < u , 0 < H < d and 0 < < 1 According to the conditions (B10), (B11), (B12), (B13), (B14) and (B15), all the multipliers are null: j = 0, j = 1::6. We substitute them in the equations (B1), (B4) and (B7). We get the following system:

2 1 (1 ) ( ) (1 2 ) ( )( D + H) ( + 2 ) 1 (1 ) ( )( D + H) = 0 (B16) ( ) + ( + 2 ) ( )( D + H) = 0 (B17) ( ) ( + 2 ) ( )( D + H) = 0 (B18)
Notice that the equations (B17) and (B18) can be written

( ) = ( + 2 ) ( )( D + H) (B19)
We substitute ( ) for its expression in (B16), we get = 1 2 . Substituting = 1 2 in (B19), given our starting hypothesis, we deduce that the bank's payments are decreasing with the project's revenues: D H = < 0. This is true only when the project is not very risky.

2. 0 < D < u , H = d and 0 < < 1

According to (B10), (B11), (B12), ( B14) and (B15), 1 = 2 = 3 = 5 = 6 = 0. Substituting them in the …rst order conditions (B1), ( B4) and (B7) gives:

2 0 ( ) (1 2 ) ( )( u D) ( + 2 ) 0 ( )( u D) = 0 (B20) ( ) = ( + 2 ) ( )( u D) (B21) 4 ( ) ( + 2 ) ( )( u D) (B22)
If we substitute (B21) in (B20) and (B22), we deduce that = 1 2 and the Kuhn-Tucker multiplier 4 0. To deduce the bank's payments, we substitute = 1 2 in (B21) which leads to a contradiction D = u 2 < 0.

3. 0 < D < u , H = 0 and 0 < < 1

According to (B10), (B12), (B13), (B14) and (B15), the multipliers @ 1 = 0 @L @ 2 = D 0 , 2 0 , 2 D = 0 (D13) @L @ 3 = H 0 , 3 0 , 3 H = 0 (D14) @L @ 4 = u D 0 , 4 0 , 4 ( u D) = 0 (D15) @L @ 5 = d H 0 , 5 0 , 5 d H = 0 (D16) @L @ 6 = 0 , 6 0 , 6 = 0 (D17) @L @ 7 = 1 0 , 7 0 , 7 (1 ) = 0 (D18) -37 -Consider that 0 < D < u , 0 < H < d and 0 < < 1. Given the complementary slackness conditions, we deduce that j = 0, j = 2::7. The …rst order conditions (D1), (D4) and (D7) give We conclude therefore that = 1 2 and the bank's payments are decreasing with the project's revenues such that H D = .

1 = 3 = 4 = 5 = 0 (D7) H 0 (D8) H @L @H = 0 (
E. Proof of the proposition 5

We substitute p(e; a) and v(a) for e 1 a and a 2 2 in the participation constraints of the consultant and the bank (29) and (30) which gives: 1) The expected gain of the entrepreneur is therefore written:

EU A = (
EU E = n e 1 a ( D + H) + d H o e 2 2 W (E3)
The e¤orts of the entrepreneur and the LBO fund in the equilibrium are given respectively by (12) and (13).

Because of the competition among the banks, the participation constraint of the bank (CP B ) is binding which implies that:

W = K H e 1 a (D H) (E4)
At the opposite, the participation constraint of the consultant (E1) may not be binding.

We substitute (12), ( 13) and (E4) in the objective function (E3), the …nancial contracts induce the entrepreneur to solve the following program: The Lagrangian is therefore given by: L : = L( , D, H, j , j = 1::6) = where j , j = 1::7 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions are written:

@L @ = 1 2 (1 + ) 2(1 )
(1 ) and the complementary slackness conditions are written:

( ) ( D + H) 2 + d H + 1 (1 ) ( ) ( D + H) (D H) + 5 6 + 1 n 1 1 2 1 2 ( ) ( D + H) 2 d + H o 0 (E5) 0 (E6) @L @ = 0 (E7) @L @D = ( ) [ 2 (D H)] (1 + ) ( ) ( D + H)
@L @ 1 = (1 ) h 1 1 2 ( ) ( D + H) 2 + d H i 0 1 0, 1 @L @ 1 = 0 (E14)
@L @ 2 = D 0 , 2 0 , 2 D = 0 (E15) @L @ 3 = H 0 , 3 0 , 3 H = 0 (E16) @L @ 4 = u D 0 , 4 0 , 4 ( u D) = 0 (E17) -39 -@L @ 5 = d H 0 , 5 0 , 5 d H = 0 (E18) @L @ 6 = 0 , 6 0 , 6 = 0 (E19) @L @ 7 = 1 0 , 7 0 , 7 (1 ) = 0 (E20)

Given the fact that the e¤orts e and a are complementaries, both agents have to provide strictly positive e¤orts otherwise the projects fails with probability equal to 1. This is satis…ed only if 0 < < 1 which implies that 6 = 7 = 0. Furthermore, the following cases are not considered:

-D = u and H = d : in this case, the agents' e¤orts are null and the project fails with probability equal to 1.

-D = u and 0 H < d because the e¤orts are strictly negative.

1. Consider the case 0 < D < u and 0 < H < d . According to the conditions (E15), (E16), (E17) and (E18), the Kuhn-Tucker multipliers 2 , 3 , 4 and 5 are null.

Consider …rst that 1 > 0, the condition (E14) is satis…ed @L @ 1 = 0 which implies that:

1 2 (2 ) ( ) ( D + H) 2 + d H = 0 (E21)
Notice that if 0 < < 1, ( ) > 0. The condition (E21) is satis…ed only if D+H = 0 and H = d which gives D = u and H = d and leads to a contradiction.

Consider now that 1 = 0, the conditions (E5), (E8) and (E11) give the following system: If 1 > 0, the condition (E14) is satis…ed: @L @ 1 = 0 which gives:

(1 ) 1 1 2 ( ) ( u ) 2 = 0 (E25) But given 0 < < 1, the latter condition is never satis…ed.

FIG. 1 -

 1 FIG. 1 -The sequence of events in the model.

e 0

 0 (a) = u 0 (e)p ea (e; a) u 00 (e)p e (e; a) u 0 (e)p ee (e; a) 0 a 0 (e) = v 0 (a)p ea (e; a) v 00 (a)p a (e; a) v 0 (a)p aa (e; a) 0

V

  (e; a) = p(e; a) + d u(e) v(a) K s.t (4) and (5)

,

  e, a, D, H EU E = [p(e; a) ( D + H) + d H] u(e) K +H + p(e; a) (D H) s.t (4) , (5) and (31)

  1 ) p(e; a) u + [1 p(e; a)] d p(e; a) (1 + r) I [1 p(e; a)] Hg u(e) W (35) Because of the competition among the LBO funds and the banks, after taxation the participation constraints of the agents A and B enable us to write: e; a) u + [1 p(e; a)] d p(e; a) (1 + r) I [1 p(e; a)] Hg v(a) (36) I = p(e; a) [(1 + r) I H] + H

  p e (e; a) ( D + H) u 0 (e) + p e (e; a) (D H) + 1 (1 ) p e (e; a) ( D + H) p a (e; a) ( D + H) + p a (e; a) (D H) + 1 f(1 ) p a (e; a) ( D + H) v 0 (a)g = 0 (D11) and the complementary slackness conditions are written:

  This system is satis…ed if and only if 1 = 1 > 0. The condition (D12) implies that the participation constraint of the consultant is binding. Besides, the conditions (D10) and (D11) become p e (e; a) u 0 (e) = 0 p a (e; a) v 0 (a) = 0

  EU B = e 1 a (D H) + H (K W ) 0 (E2)

  ) 2 + d H +H K + ( )( D + H) (D H)

  (E23) in (E24), we obtain therefore the following contradiction: now the case D = 0 and H = d , the multipliers 3 = 4 = 0.

  2. The debt contract: the bank lends I at the date 0. At the date 3, he is paid D when the project's result is success, otherwise he gets H. The bank is willing to lend money only if he will recoup what he would get if he makes a risk-free investment. It means that:

	EU B = p(e; a)D + (1 p(e; a))H I 0	(P C B )

LBO is the acquisition of a company, called the Op Co, using mostly debt and a small amount of equity. The debt is secured by the Op Co assets. The acquiring company, called the holding company or the New Co, uses these assets as collateral for the debt in hopes that the future cash- ‡ows will cover the debt's payments.

The venture capital is a type of the private equity capital provided by professional, outside investors to new high-potential-growth companies. The objective is to take the company to an IPO (Initial Public O¤ ering ).

In the last section, we discuss the case where the LBO fund does not contribute …nancially into the project: he will be considered as a consultant.

The entrepreneur sign a new contract with a new VC fund, but only for one period. The next period, she is looking for another one; the VC market is supposed to be competitive.

While the commitment is to fund the entire amount, in venture capital projects, the funding is contingent on the company attaining his short-term objectives. At each stage, the entrepreneur asks for additional capital from the VC fund.

First, he assumed that entrepreneur's investment is one-dimensional. Then, he supposed that the entrepreneur has to choose a multi-dimensional investment vector such as to invest in R&D, to spend e¤ort in order to organize the …rm, to hire the key sta¤ and to invest in marketing, in supplier and customer relations.

LMBO (Leveraged Management Buy Out) is a LBO project where the entrepreneur(s) is/are the manager(s) and/or the employees of the acquired company.

= 0: We substitute H = 0 and 1 = 3 = 4 = 5 = 6 = 0 in the conditions (B1), (B4) and (B7). We get a contradiction:

the bank gets the entire revenue in case of failure. Notice that the bank's payment in case of success depends on the impact of the LBO's e¤ort on the success probability .

Note that the optimal sharing bene…t is decreasing with . In contrast with the previous propositions, if = 1 2 , the entrepreneur has the highest share of bene…t. Furthermore, if converges to 1, despite the fact that the consultant's e¤ort is the most e¢ cient, the entrepreneur has the highest bene…t's share: converges to 2 3 where 2 2 3 , 1 . The entrepreneur and the consultant provide the following e¤orts:

(1 )

It is straightforward to see that these e¤orts are lower than those provided in the presence of the LBO fund. If he does not contribute …nancially to the acquisition, the entrepreneur must give him a share of his revenue to induce him to provide e¤ort. This a¤ects the entrepreneur's incentives in three ways:

The …rst way is a direct and has negative e¤ect: the entrepreneur's revenue decreases when she pays the consultant, which decreases her e¤ort's incentives.

The second way is not direct but has a negative e¤ect: given that the e¤orts e and a are complements, the increase of one e¤ort leads to an increase in the other e¤ort. Recall that a is more costly than e ( < ). If is large, to boost the LBO's incentives, the entrepreneur has to pay more money. The optimal sharing rule depends on , this means that the entrepreneur must increase her e¤ort ; a costly e¤ort. Otherwise, inducing the LBO fund alone to provide high e¤ort has no impacts on the project's performance.

The third way is positive e¤ect: increasing the levels of e¤orts e and a increases the success probability.

The positive e¤ect is not high enough to compensate the other negative e¤ects. Taking the value = 1 2 boosts both agents'incentives. In the previous sections, when the LBO fund contributes …nancially to the acquisition, he funds him self a part of the cost of his e¤ort and his incentive. Accordingly, we conclude that the entrepreneur prefers asking for advice from the LBO fund rather than the consultant.

These results are in line with those of [START_REF] Casamatta | Financing and Advising: Optimal …nancial Contracts with Venture Capitalists[END_REF] who shows that when the e¤orts are perfect substitutes, the entrepreneur does not hire the consultant if the latter does not contribute …nancially into the acquisition.

Third, there should be a positive (respectively negative) correlation between the corporate income tax and the level of debt (respectively the amounts of equity issued by the entrepreneur and the LBO fund).

Fourth, in order to maximize the tax saving advantages of the debt, there should be a positive correlation between the level of debt and the level of risk. The debt may induce an excessive risktaking problem. To test this hypothesis, one could estimate the percentage of very risky buyouts among all buyouts.

Moreover, it would probably be reasonable to study the use of convertible securities in the presence of a passive …nancier as the bank. Another topic for further research is how the cash- ‡ow and control rights are allocated when the pro…tability of the project varies. To our knowledge, research in this direction is still pending.

Note that in a dynamic model, when the LBO fund has convertible securities, the share of bene…t given to the entrepreneur fund should be considered as a quality's signal of the project if there is asymmetric information on the market. When this share is signi…cantly high, it means that the project is pro…table: the LBO fund cannot exert his conversion option. He will be a minority shareholder and he will look for a quick exit of the project (Signal theory, Leland andPyle, 1977 andRoss, 1977).

Appendix

A. Proof of proposition 1

Given the participation constraints of the LBO fund and the bank (P C A ) and (P C B ), the amounts of i and I are written respectively:

Moreover, given K W = i + I, we conclude that:

We replace H and D in the participation constraints (P C A ) and (P C B ), we get the optimal amounts of equity and debt which must be issued respectively by the LBO fund and the bank:

These optimal solutions exist under the following conditions:

The equation

On one hand, the conditions (A4) and (A7) imply that:

where

On the other hand, the conditions (A5) and (A6) give: A9) and (A10), we deduce that:

which completes the proof of the proposition 1.

B. Proof of proposition 2

The Lagrangian is given by:

where j , j = 1::6 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions are given by:

The complementary slackness conditions are written:

This case does not satisfy the condition (14), otherwise the entrepreneur and the LBO fund provide negative e¤orts.

5. D = 0, H = d and 0 < < 1

Given the equations (B11), (B12), (B14) and (B15), the multipliers 2 = 3 = 5 = 6 = 0.

We substitute D = 0, H = d and 2 = 3 = 5 = 6 = 0 in the …rst order conditions (B1), (B4) and (B7). We get the following system:

where

2 , it has two solutions:

We conclude that the optimal sharing rule is given by:

Notice that in this case the multipliers 1 and 4 are positive only if ^ ( , k) 1 k

(1 2 ) which is always satis…ed.

We substitute D = 0, H = d and ^ ( , k) in the expressions of e¤orts in equilibrium. Accordingly and given the participation constraint (CP B ), we deduce the amount of the issued debt I which is substituted in the …nancial constraint i = K W I to deduce the amount of the equity issued by the LBO fund.

C. Proof of the proposition 3

The Lagrangian is given by:

where j , j = 1::7 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions are given by:

The complementary slackness conditions are written:

According to the conditions (C11), (C12), (C13), (C14), (C15) and (C16), the Kuhn-Tucker multipliers j = 0, j = 2; ::7. Substituting j = 0, j = 2; ::7 in the equations (C1), (C4) and (C7) gives the following system:

These equations are also written:

Substituting ( ) for its expression (C20) in (C21) gives

If 1 = 0, we deduce that = 1 2 and/or D H = . Substituting = 1 2 in (C20) does not satisfy (C10) because @L @ 1 = D H = < 0. Let us suppose D H = , the equation (C20) leads to a contradiction because ( ) = 0 only if 2 f0 , 1g.

If 1 < 0, the condition (C10) is not satis…ed.

) We conclude that our program has no solutions when 1 0.

If 1 > 0, the condition (C10) enables us to deduce that D = H. Substituting the latter equation in (C20) gives

This equation has two solutions but one of them is a feasible solution and varies between 0 and 1. It is given by:

The equations (C10), (C11), (C14), (C15) and (C16) enable us to deduce 1 = 2 = 3 = 5 = 6 = 7 = 0. We substitute H = 0 and 1 = 2 = 3 = 5 = 6 = 7 = 0 in the equations (C1), (C4) and (C7) such that we get the following system:

The equations (C25) and (C26) give = 1 2 which is substituted in (C26). As a result, we get the following contradiction D = < 0:

3. D = u , H = 0 and 0 < < 1.

According to the equations (C10), (C11), (C14), (C15) and (C16), the Kuhn-Tucker multipliers j = 0, j = 1, 2, 5, 6, 7. Substituting D = u , H = 0 and j = 0, j = 1, 2, 5, 6, 7 in the equation (C4) gives:

But 3 must satisfy the condition (C12) :

4. 0 < D < u , H = d and 0 < < 1.

We have 2 = 3 = 4 = 6 = 7 = 0. Substituting H = d and 2 = 3 = 4 = 6 = 7 = 0 in the Kuhn-Tucker conditions gives the following system:

The equations (C29) and (C30) give 5 = 0. The equation (C28) is therefore simpli…ed such that

According to (C29), the multiplier 1 satisfy

2 and/or D = u (in the latter case, we have a contradiction). Substituting = 1 2 in (C31), we get:

but this solution exists only if the project is not very risky.

If 1 < 0, we have no solutions.

If 1 > 0, according to (C10), D = H = d which is substituted in (C31). We obtain the equation (C23) which is solved if (C24).

5. D = H = 0 and 0 < < 1

In this case, the bank does not contribute …nancially to the acquisition. The Kuhn-Tucker multipliers 3 , 5 , 6 and 7 are null and the conditions (C1), (C4) and (C7) become

Notice that the equation (C34) is satis…ed if = ( ) given by (C24). Moreover, the conditions (C35) and (C36) imply

Given the fact that the Kuhn-Tucker multipliers are positive, we conclude that (C37) is satis…ed if 2 = 4 = 0 and 1 = [1 (1 2 ) ] ( ) > 0.

D. Proof of the proposition 4

The Lagrangian is written:

where j , j = 1::7 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions are given by:

If 1 = 0, the equations (E5), (E8) and (E11) are written:

This system gives

where k = d u < 1 2 . If we substitute (E29) in (E27), we have 2 0.

3. Consider the latter case 0 < D < u and H = d . Given the conditions (E15), ( E16) and (E17), we obtain 2 = 3 = 4 = 0.

If 1 = 0, the …rst order conditions (E5), (E8) and (E11) are written:

The equations (E30) and (E31) enable us to deduce:

If we substitute and D in (E32), it is easy to check that the multiplier 5 = + 2 is strictly positive.

-41 -4. Consider now the following case D = H = 0, the project is therefore implemented without debt. Consequently, the entrepreneur should issue W = K. The Kuhn-Tucker equations give the following system:

(1 )

If 1 = 0, the equations (E34) and (E35) become:

If 2 is a Kuhn-Tucker multiplier, we should get [1 (1 + )] 0, but this implies that 3 < 0.

If 1 > 0, the participation constraint of the consultant is binding which implies that:

(1 )

However, the latter condition is satis…ed only if = 1.

F. Proof of the lemma 1

We add the following condition to the program to be solved:

D H

The Lagrangian is therefore written:

j , j = 1::8 are the Kuhn-Tucker multipliers.

-42 -

The Kuhn-Tucker conditions are given by:

and the complementary slackness conditions are written:

@L @ 3 = H 0 , 3 0 , 3 H = 0 (F12) @L @ 4 = u D 0 , 4 0 , 4 ( u D) = 0 (F13) @L @ 5 = d H 0 , 5 0 , 5 d H = 0 (F14) @L @ 6 = 0 , 6 0 , 6 = 0 (F15) @L @ 7 = 1 0 , 7 0 , 7 (1 ) = 0 (F16) @L @ 8 = D H 0 , 8 0 , 8 (D H) = 0 (F17) Given 0 < < 1, the conditions (F 15) and (F 16) give 6 = 7 = 0. We will not analyze the following cases: D = u and H = d , D = u and 0 H < d and D = H = 0 because of the reasons explained in the appendix E.

-43 -1. If we consider 0 < D < u , 0 < H < d and D > H, whether 1 = 0 or 1 > 0, we get contradictions (see the appendix E).

2. If we consider 0 < D < u , 0 < H < d and D = H. The Kuhn-Tucker conditions (F 4) and (F 7) give

The latter conditions are satis…ed only if:

According to (F 1), the optimal sharing rule is the solution of the following equation:

It is given by:

p

Substituting D = H in the participation constraint of the bank enables us to deduce:

If we analyze the other cases of the appendix E, we get contradictions. Moreover, the hessian is negative semide…nitive when D = H = K W and = ( ) which means that this solution is a local maxima.