Yousfi Ouidad

Economix

Leveraged Buy Out: Dynamic agency model with write-o¤ option

Keywords: Leveraged Buy Out, incentives, exit, write-o¤ option, double moral hazard. JEL classi…cation: D82, D92, G32, G33

We present a dynamic agency model in which the LBO fund may write the entrepreneur's project o¤ at the end of the starting stage to invest in a competitive project. The two partners provide unobservable e¤orts in both stages to enhance the productivity of the acquired company.

We show that under restrictive conditions, the debt-equity contracts induce the entrepreneur and the LBO fund to provide the …rst best e¤orts under restrictive conditions in the two stages.

Moreover, the write-o¤ threat boosts the incentives of the entrepreneur and the LBO fund such that they provide high e¤orts.

If the compensation cost is exogenous, the sharing rule of this cost depends on the quality of the competitive project. The entrepreneur and the bank share the amount of compensation if it is not very pro…table. Otherwise, the whole amount of compensation is pledged to the entrepreneur.

If the compensation's amount is endogenous, in order to induce the entrepreneur to provide high e¤ort, the optimal …nancial contracts must give her the entire compensation's revenue.

Introduction 1.Motivation

Despite the fact that the Leveraged Buy Out (LBO) acquisitions accounts for a signi…cant part of the private equity investment, many topics in this area are not explored. In many countries most notably in the United States, these projects are …nanced mostly with debt and a small amount of equity, hence the term leveraged : these projects are typically …nanced with anywhere from 60% to 90% debt [START_REF] Jensen | Agency costs of free cash ‡ow, corporate …nance, and the market of takeovers[END_REF], 1989[START_REF] Kaplan | Leveraged Buyouts and Private Equity[END_REF].

The present paper deals with two facts in LBO …nance:

First, the …nancial capital structure, particularly the excessive use of debt in these acquisitions: this paper shows that the optimal …nancial contracts are consistent with this feature in a dynamic framework with double sided moral hazard. We characterize the optimal …nancial capital structure that results from the solution to an optimal contract problem.

We work out the optimal solution to the …nancing problem of an entrepreneur (hereafter she) who would like to acquire a company: she is a manager but she has no business experience to manage the acquired company alone. She asks …rst for money and for advice from the LBO fund (hereafter he). The latter contributes technically and …nancially into the project and in exchange he gets a share of the project's outcome. The partners sign the holding contract and can ask for additional funds from the bank (hereafter he). Therefore, they sign another contract: the debt contract. The optimal sharing rule of the bene…t may change at the end of the …rst period. The optimal solution and the optimal sharing rule are constrained by the characteristics of the environment such as the information available to both parties. The entrepreneur and the fund have to provide costly e¤orts which improve the results of the project. This information is unobservable which creates a moral hazard problem.

Second, we focus on the decision to exit, particularly on the write-o¤ route. The LBO fund is expecting for a high return, his aim is to get back his money and to exit as soon as possible in order to invest this money in a new project. The exit of the LBO fund is quite likely if there are good projects in the LBO market looking for …nance and for advice. The exit date/route may induce agency con ‡icts between the entrepreneur and the LBO fund. If the company is going public the entrepreneur keeps the control and may get private bene…ts. This is no longer true when the buyout is going private such in the trade sale: if she stays in the company, she must share the control with a new partner. In the current paper, we do not consider the issue of the optimal timing/route of exit but we focus on the impact of the write-o¤ threat on the e¤orts made by the entrepreneur and the LBO fund. To our knowledge, the current paper is the …rst theoretical one that addresses this issue in buyout acquisitions.

Related literature

The present model is related to two lines of literature.

First, various papers consider a dynamic agency model with information asymmetry. They analyze the interaction between the …nancial capital structure and incentives when there is an information asymmetry, due to unobservable e¤orts. For instance, Berge-mann 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 Finance[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.

Bergemann and Hege (1998) consider a dynamic agency model in the presence of learning and moral hazard problems. They show that short-term re…nancing 1 is never optimal but long-term contract allowing for intertemporal risk-sharing such as the stage …nancing is optimal: it induces the entrepreneur to provide optimal e¤ort so that the private equity fund will invest further funds in the following stage. If not, the private equity fund writes the project o¤. However, the stage …nancing may create a "window dressing" problem in order to induce the private equity fund to …nance the second stage of the project. They conclude that with a convertible debt contract, this entrepreneur's behavior becomes non pro…table; the private equity fund will convert his debt into equity if the project looks too pro…table. Consequently, he will reduce the entrepreneur's pro…t [START_REF] Cornelli | Stage Financing and the Role of Convertible Securities[END_REF]. Schmidt (1999[START_REF] Schmidt | Convertible Securities and Venture Capital Finance[END_REF] shows that there is no debt-equity contract that induces both parties to invest e¢ ciently and argues that the use of convertible securities mitigates the double sided moral hazard problem. Moreover, these securities outperform any mixture of debt and equity and they induce both parties to provide optimal e¤orts. His result is robust to renegotiation and to changes in the timing of investments and information ‡ows. [START_REF] Repullo | Venture Capital Finance: A Security Design Approach[END_REF] consider a wealth-constrained entrepreneur asking for advice and for money from two private equity 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 return of the project in case of success is unobservable when they sign the contract, they will learn more about it at the end of the …rst period. If this return is veri…able, the entrepreneur and the private equity fund sign an initial long-term contract contingent on the project's return (contingent …nancing contract). They show that the private equity fund must get a constant share of the outcome only when the project is pro…table. However, when the return is not veri…able, they sign a start-up contract which is negotiated in the expansion stage. They show that the venture capitalist should get no compensation for his initial investment in the lower tail and high compensation in the upper tail of the distribution of returns. They point out that this sharing rule can be approximated by the use of warrants.

Second, another branch of the literature focuses on the various exit vehicles in private equity investments but few papers, mostly empirical studies, focus on the topic of buyouts particularly the write-o¤ route. To our knowledge, most of the theoretical papers study the exit in the venture capital. Most of these papers argue that the project is abandoned when the quality is bad or mediocre in the sense it is only able to return the initial investment.

For instance, Schwienbacher (2002) analyzes the relationship between the level of innovation of the project and the exit decision. He shows that going public is more pro…table than trade sale when the project is very innovative. In an IPO strategy, the entrepreneur remains in the …rm, keeps its control and can get private bene…t. Consequently, she is tempted to distort the innovation strategy so that the IPO looks the preferred exit route. [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 abandoned quickly via write-o¤s. [START_REF] Schmidt | Exit strategies of Buyout Investment-An empirical Analysis[END_REF] focus on buyout exit strategies in Europe and the United States. They consider a sample of 666 buyouts between 1990 and 2005. They analyze the determinants in ‡uencing the choice of the exit option. Their results show strong support for signaling e¤ect. If the return is very poor, the LBO fund writes o¤ the project 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 the most pro…table projects are taken public through an IPO. Nikoskelian and Wright (2005) 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] Das | The private equity discount: An empirical Examination of the Exit of Venture Backed Companies[END_REF] analyze the options of exit of venture and LBO funds in the US market. They estimate the probability of various exit options and they conclude that the probability of an exit via sales is the highest. Groh and Gottschalg (2008) point out that the US buyouts investments clearly outperform the market benchmark. [START_REF] Ick | Private equity Returns: Is here really a Bene…t of low Co-movement 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. [START_REF] Cumming | Venture Capital Exits in Canada and the United States[END_REF] show in an empirical analysis in Canada and USA that IPO are the most pro…table followed by secondary sales, buybacks and write-o¤s for the less pro…table projects.

In all these papers, the exit decision depends only on the performance of the project, but they do not consider the characteristics of the company's market: the private equity fund invests in the project in the hope that he will get a high return in a short period of time and he exits as soon as possible in order to invest his money in a new deal. The LBO fund looks continuously for "LBO stars". This issue is still pending.

Results

The model allows to derive the following results: First, if there is no competitive project, the entrepreneur and the LBO fund provide the …rst best e¤orts only when the project is not very risky and the debt's payments are decreasing with the project's revenues. The presence of the bank induces the entrepreneur and the LBO fund to make high e¤orts because of the threat of liquidation. This result is in line with those of Meckling and [START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure[END_REF] and [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]. In this case, the whole surplus value of the project is retained by the entrepreneur. If the project is very risky, they make the second best e¤orts and the social value of the project is not optimal. Whether the project is very risky or not, the optimal …nancial contracts should reward the entrepreneur and the LBO fund only in the good state of the nature. The bank receives no payment in success states of the world. The optimal debt contract exhibits the features of a "live or die" contract.

When the payments of the bank are non-decreasing with the project's outcome, the bank payments are …xed: the success payments are equal to the liquidation values whether the project succeeds or fails. Besides, the e¤orts are not optimal. The agents provide higher e¤orts when the project is …nanced only with equity than when the entrepreneur asks for advice and for money from the LBO fund and the bank. The debt has no e¤ects on the agent's incentives.

Second, the threat of exit induces the entrepreneur and the LBO fund to provide the high e¤orts when all the agents contribute …nancially into the acquisition. These e¤orts are not optimal.

Third, under the write-o¤ threat, the LBO fund issues more equity but the bank's investment depends on the write-o¤ cost. If the cost of violation contracts is high, then the threat of write-o¤ becomes incredible even if the LBO fund exits, the latter will pay high compensation to the bank and to the entrepreneur. This is why the bank accepts to lend more debt than when the fund keeps the entrepreneur's project. When the violation cost is low, the bank invests less money into the project.

Fourth, if the compensation cost is …xed by the legislature, the optimal sharing rule depends on the quality of the competitive buyout: if it is not very pro…table, it is shared between the bank and the entrepreneur. Otherwise, the whole amount is pledged to the entrepreneur which boosts her incentives to provide high e¤ort in the …rst period even when the LBO fund will probably abandon the project. When this cost is to be determined in the optimal …nancial contracts, it is speci…ed such that the entrepreneur captures the entire amount of compensation whether the new project is very pro…table or not. The amount of compensation is signi…cantly: it is equal to the half of the revenue of the competitive project.

Fifth, optimal sharing rule does not depend on …nancial capital structure of the project but it depends on the incentives to provide e¤orts. The optimal contracts should take into account the countervailing e¤ects of inducing one agent to work, and re ‡ect both incentives.

The paper is structured as follows. The model and the assumptions are presented in section 2. The optimal …nancial contracts are characterized in section 3. The section 4 analyzes the optimal …nancial contracts when the project is …nanced only through equity. We consider the threat of write-o¤ of the LBO fund and study its in ‡uences on the agents' incentives in section 5. Empirical implications are presented in section 6. The section 7 concludes the paper. All proofs are presented in the appendix.

The model

Consider a dynamic agency model with two periods: the starting stage and the productive stage. There are three agents: the entrepreneur E, the LBO fund A and the bank B. The entrepreneur wants to acquire a company. She asks for money and advice from the LBO fund. He issues the amount of equity i and he gets in exchange a part of the revenues of the acquired …rm.

The acquisition of the …rm requires a …xed initial investment K at time 0, K is signi…cantly high. Let W , i and I denote respectively the capital issued by E, A and B such that K = W + i + I. At the end of the period t, the entrepreneur (respectively the LBO fund) gets the part of the bene…t t (respectively 1 t ) such that 0

t 1, t = 1; 2:
First, the entrepreneur and the LBO fund sign the holding contract in order to establish the holding company2 . Then, in order to issue further funds, the new company and the bank sign the debt contract.

The entrepreneur and the LBO fund provide respectively the unobservable e¤orts e and a. These e¤orts enhance the productivity of the project. The entrepreneur's e¤ort is related to her technical skills and/or to her past experience as a manager. The LBO fund provides a managerial and/or control e¤ort to run properly the project. 3

The revenues

The project yields the revenue X1 ( X2 ) at the end of the starting stage (the productive stage) such that: Xt = X t with the probability p t = min fe t +a t ; 1g t 1 C with the probability (1 p t ) = max f0; 1 (e t +a t )g C is the liquidation value of the project at the end of the …rst period. 1 (0 1) is the depreciation rate. The value of the equipment and machines decreases due to the usage, the passage of time and the outdated technologies. We assume that X 1 C and X 2

C.

e t and a t 2 [0, 1] are the e¤orts provided at the stage t, t = 1, 2. These e¤orts are perfect substitutes. If one agent provides unitary e¤ort, the project will succeed with probability 1: p t (0; 1) = p t (1; 0) = 1, t = 1, 2.

The entrepreneur and the LBO fund choose their e¤orts simultaneously. When e¤orts are unobservable, they may be tempted to not provide su¢ cient e¤orts. The entrepreneur will rely on the e¤ort of the LBO fund in hopes that the latter will provide unitary e¤ort. This situation creates an asymmetric information problem, speci…cally it induces a double sided moral hazard problem.

E¤orts are costly. The cost functions are given by: c E (e t ) = 2 e 2 t and c A (a t ) = 2 a 2 t , t = 1; 2

where > 0 is assumed to be signi…cantly high. E¤orts have equal costs when the entrepreneur and the LBO fund make the same levels of e¤orts. Given the fact that they are perfect substitutes, these e¤orts have the same impact on the performance of the project. Besides, we assume C 2 :

The sequence of events

The sequence of events is presented in …gure 1:

At date t = 0, the entrepreneur E and the LBO fund A sign the holding contract. Then, they may ask for additional funds from the bank B. They sign the debt contract.

At date t = 1, the agents E and A provide simultaneously and respectively the e¤orts e 1 and a 1 .

-If X1 = X 1 , the bank perceives the payment D 1 . The entrepreneur and the LBO fund get respectively (1 1 ) (X 1 D 1 ) and 1 (X 1 D 1 ). The LBO fund may write the entrepreneur's project o¤ even if it succeeded, to invest in a competitive buyout which gives him net revenue R 0.

If there is exit, the entrepreneur's project is stopped at date 1. The LBO fund must pay a compensation cost L to the other partners because he violates the contracts. The revenue of the competitive project must be superior to the compensation's cost, otherwise, the LBO fund has no reason to abandon the entrepreneur's project: R L. Let (respectively 1

) denotes the share of the compensation paid to the entrepreneur (respectively the bank) such that 0 1. The variable is an endogenous variable to be determined in the optimal …nancial contracts. We suppose that the LBO fund will write the entrepreneur's project o¤ with the probability , 0 1. Otherwise, the entrepreneur's project is continued and the entrepreneur and the LBO fund choose respectively the e¤orts e 2 and a 2 .

-If X1 = C, the entrepreneur's project is supposed to be liquidated and the bank perceives the collateral C.

At date t = 2, the project is completed.

-If X2 = X 2 , the bank, the entrepreneur and the LBO funds get respectively the payments D 2 , 2 (X 2 D 2 ) and ( 12 ) (X 2 D 2 ) with the probability p 2 .

-If X2 = C, the project fails and it is liquidated. The bank is paid the whole failure's revenue C. The riskless interest rate is normalized to 0. All agents are risk neutral and protected by limited liability: they can only share the outcome of the project.

The contracts

The agents sign two …nancial contracts.

a) The debt contract must specify:

The amount of the debt I issued by the bank.

The bank payments at the starting stage and the productive stages D 1 and D 2 .

The bank lends the money only if he gets a positive expected gain:

E( B ) = p 1 fD 1 C + (1 ) [p 2 (D 2 C) + C] + (1 ) Lg +C I 0 (CP B )
(CP B ) is his participation constraint. The banks compete for the right to fund the project, then we expect the debt contract to be the best possible for the entrepreneur. Accordingly, the bank B earns no pro…ts.

We suppose that:

0 D t X t , t = 1, 2: (1)
b) The holding contract which speci…es

The amount of funds i issued by the LBO fund.

The fraction of the bene…t t , t = 1; 2 (respectively 1 t ) of the entrepreneur (respectively the LBO fund).

The compensation cost L to be shared between the entrepreneur and the bank.

The share of compensation (respectively 1 ) paid to the entrepreneur (respectively the bank) if the LBO fund chooses to exit at the date 1.

The LBO fund issues i only if his expected gain is positive. This condition is written:

E( A ) = p 1 f(1 1 ) (X 1 D 1 ) + (R L) + (1 ) [(1 2 ) p 2 (X 2 D 2 ) c A (a 2 )]g c A (a 1 ) i 0 (CP A )
(CP A ) is his participation constraint. Competition between the investment funds implies that this constraint is binding.

Because of the competition among the LBO funds and the banks, both agents A and B are induced to propose contracts maximizing the expected gain of the entrepreneur which is given by:

E( E ) = p 1 f 1 (X 1 D 1 ) + (1 ) [ 2 p 2 (X 2 D 2 ) c E (e 2 )] + Lg c E (e 1 ) W: Substituting W for K i I into E( E ) gives: E( E ) = p 1 f 1 (X 1 D 1 ) + (1 ) [ 2 p 2 (X 2 D 2 ) c E (e 2 )] + Lg c E (e 1 ) K + i + I: (2)

The …rst best solution

Before solving the game, let us compute the social value of the project without double sided moral hazard and exit problems. The social value of the project is given by:

V = p 1 [X 1 (1 ) C + p 2 (X 2 C) c E (e 2 ) c A (a 2 )] +C K c E (e 1 ) c A (a 1 ):
Given the …rst order conditions of V , the …rst best e¤orts are written:

e F B 1 = a F B 1 = 1 X 1 (1 )C + 1 (X 2 C) 2 (3) e F B 2 = a F B 2 = 1 (X 2 C) : (4)
These e¤orts induce the following success probabilities:

p F B 1 = 2 X 1 (1 )C + 1 (X 2 C) 2 and p F B 2 = 2 (X 2 C) : (5)
In the following, we assume that:

X 1 < 2 + (1 )C and X 2 < 2 + C : (6)
to ensure that p

F B t < 1, t = 1, 2.
The equations (3) and (4) show that the entrepreneur and the LBO e¤orts must make strictly positive e¤orts during the starting and the productive stages. Note that e¤orts of the …rst period depend on revenues of both stages. But, those of the second period depend only on the revenues of the productive stage. They increase with the revenues of the project and decrease with the cost parameter and/or the amount of the collateral C.

Moreover, it is straightforward to see that the e¤orts of the second period are decreasing when increases. In the starting stage, the impact of on the e¤orts of the …rst period is less intuitive:

If is high enough 2X2 2C , 1 , e F B 1
and a F B 1 are increasing with . Note that when the depreciation rate is low, the failure revenue C is high. In contrast, when the depreciation rate is high 0 , 2X2

2C

, a small failure revenue will induce the entrepreneur and the LBO fund to provide low levels of e¤orts e F B 1 and a F B 1 .

Furthermore, the entrepreneur and the LBO fund provide equal levels of e¤orts which is not surprising because the e¤orts have equal impacts on the success probabilities and the agents have the same cost function.

Consequently, the optimal social value of the project is given by:

V F B = 1 X 1 (1 )C + 1 (X 2 C) 2 2 + C K: (7)
We assume that V F B is strictly positive. There are many ways to implement the …rst best solution: the identities of the agents providing advice and …nancial investments are irrelevant as long as the LBO fund and the bank are not wealth-constrained and there are no transaction costs.

The e¤orts in equilibrium

In order to maximize their expected gain, each agent will take into account the levels of e¤ort chosen by the other agent. These strategies are described by their reaction functions.

We solve for optimal …nancial contracts using a dynamic programming approach (the backward induction process). We consider the subgame that begins at the end of the productive stage: …rst, we determine the reaction functions of the second period. Then, we substitute them into the expected gain of each agent so that we can deduce those of the …rst period.

The e¤orts of the second period in equilibrium

Each agent chooses the level of e¤ort that maximizes his expected gain of the second period. The LBO fund provides the e¤ort a 2 solution of:

a 2 2 arg max a2 E( 2 A X1 = X 1 )
where

E( 2 A X1 = X 1 ) = p 2 (1 2 ) (X 2 D 2 ) c A (a 2
) is his expected gain of the second period.

The entrepreneur e¤ort is given by:

e 2 2 arg max e2 E( 2 E X1 = X 1 )
where

E( 2 E X1 = X 1 ) = p 2 2 (X 2 D 2 ) c E (e 2 )
is her expected gain of the second period.

Lemma 1

The e¤ orts of the second period of the entrepreneur and the LBO fund e 2 and a 2 are given respectively by:

e 2 = 1 2 (X 2 D 2 ) and a 2 = 1 (1 2 ) (X 2 D 2 ) (8)
The success probability is then given by:

p 2 = 1 (X 2 D 2 ) (9)
If the fraction of bene…t of one agent increases, his e¤ort will increase: e 2 (respectively a 2 ) is an increasing function of 2 (respectively 1

2 ). When the di¤erence between the success revenue and the bank's payment is large, the levels of e¤orts are high. Note that these e¤orts do not depend on the write-o¤ threat.

Given the fact that the revenue of the second period is …xed, if the optimal …nancial contracts give powerful incentives to one agent, they will reduce the incentives of the other agent. The optimal …nancial contracts must boost simultaneously the two agents'incentives.

The e¤orts of the …rst period in equilibrium

The entrepreneur and the LBO fund provide respectively the e¤orts e 1 and a 1 given by:

e 1 2 arg max e1 E ( E j e 2 , a 2 ) and a 1 2 arg max a1 E( A j e 2 , a 2 )
We substitute the optimal e¤orts deduced in the previous paragraph into their expected gains. The …rst order conditions of E ( E j e 2 , a 2 ) and E ( A j e 2 , a 2 ) enable us to deduce the following lemma: Lemma 2 The e¤ orts of the …rst period e 1 and a 1 are written:

e 1 = 1 1 (X 1 D 1 ) + 1 2 2 (2 2 ) (1 ) (X 2 D 2 ) 2 + L (10)
a 1 = 1 (1 1 ) (X 1 D 1 ) + (R L) + 1 2 1 2 2 (1 ) (X 2 D 2 ) 2 (11)
The success probability of the …rst period:

p 1 = 1 X 1 D 1 + [R (1 ) L] + 1 2 1 + 2 2 2 2 2 (1 ) (X 2 D 2 ) 2 (12)
The more the exit probability is high, the more the entrepreneur and the LBO fund will be induced to provide e¤orts in the starting stage: given high, the e¤orts (10) and (11) are increasing with . If the project fails the LBO fund will not get his money back. The entrepreneur is better o¤ when the project succeeds: if his project is abandoned, he will be paid L, otherwise, he gets nothing. Moreover, the e¤ort of the entrepreneur (respectively the LBO fund) is increasing (respectively decreasing) with the compensation cost. Given high fraction of compensation to the entrepreneur will induce her to provide more e¤ort in the starting stage. Notice that the net revenue of the competitive buyout R has an impact only on the fund's e¤ort which is very intuitive.

As in the previous lemma, we …nd that the entrepreneur's e¤ort e 1 (respectively the LBO's e¤ort a 1 ) increases with his part of bene…t 1 and/or 2 (respectively 1

1 and/or 1 2 ). Besides, these e¤orts decrease with the debt payments D t , t = 1, 2.

The lemmas 1 and 2 show that the e¤orts of the entrepreneur and the LBO fund are independents: they play a strictly dominant strategies.

The probability p 1 does not depend on 1 and p 2 does not depend neither on 1 nor on 2 . This result is closely related to the assumptions of the model because the e¤orts are perfect substitutes and the agents have the same function of cost. The entrepreneur's objective is to maximize her expected gain given the participation constraints of the LBO fund and the bank and the incentive constraints: max i, I; t ; D t ; et; at, t=1, 2

E ( E ) = p 1 f 1 (X 1 D 1 ) + (1 ) [ 2 p 2 (X 2 D 2 ) c E (e 2 )] + Lg c E (e 1 ) K + i + I. s.t. (CP A ) , (CP B ) , (8)
, [START_REF] Chan | Learning, Corporate control, and Performance Requirements in Venture Capital Contracts[END_REF] and [START_REF] Cornelli | Stage Financing and the Role of Convertible Securities[END_REF] with the following conditions:

(1) and 0 t 1, t = 1; 2:

3 The optimal …nancial contracts without the write-o¤ option

In this section, we consider …rst the case where the LBO fund cannot abandon the entrepreneur's project because there is no competitive buyout looking for advice and for money.

If = 0, the equations (10) and (11) are therefore written:

e 1 = 1 1 (X 1 D 1 ) + 1 2 2 (2 2 ) (X 2 D 2 ) 2 (13)
a 1 = 1 (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 (14)
Then, the success probability of the starting stage is given:

p 1 = 1 X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 (15)
The e¤orts of the entrepreneur and the LBO fund and the success probability of the productive stage do not change: they are given respectively by [START_REF] Cestone | Anti-Competitive Financial Contracting: the Design of Financial Claims[END_REF] and [START_REF] Chang | Payout Policy, Capital Structure, and Compensation when Managers Value Control[END_REF].

The participation constraints of the LBO fund and the bank enable us to write:

E( A ) = p 1 f(1 1 ) (X 1 D 1 ) + (1 2 ) p 2 (X 2 D 2 ) c A (a 2 )g c A (a 1 ) i 0 (16)
E( B ) = p 1 fD 1 (1 ) C + p 2 (D 2 C)g + C I 0 (17)
Given the previous results, we are able to solve the entrepreneur's program. Her aim is to maximize her expected gain under the incentive and the participation constraints. The program to be solved is the following:

max t ; i; I; Dt; et; at, t=1, 2 E( E ) = p 1 [ 1 (X 1 D 1 )+ 2 p 2 (X 2 D 2 ) c E (e 2 )] c E (e 1 ) K + i + I s.t. (8)
, ( 13), ( 14), ( 16) and, [START_REF] Groh | The Risk-Adjusted Performance of US Buyouts[END_REF] with the following conditions:

(1) and 0

t 1, t = 1, 2:
Because of the competition among the LBO funds and the banks, the participation constraints (CP B ) and (CP A ) are binding. So we can write:

i = p 1 f(1 1 )(X 1 D 1 )+p 2 (1 2 )(X 2 D 2 ) c A (a 2 )g c A (a 1 ); (18)
I = p 1 fD 1 (1 ) C + p 2 (D 2 C)g + C: (19)
We substitute [START_REF] Hirao | Learning and incentive Problems in Repeated Partnerships[END_REF] and (19) into the objective function of the entrepreneur's program. Consequently, the optimal …nancial contracts induce her to maximize the expected social value of the project under the incentive constraints such that:

max t ; D t ; e t ; a t , t=1, 2 V = p 1 [X 1 (1 ) C + p 2 (X 2 C) c E (e 2 ) c A (a 2 )] +C K c E (e 1 ) c A (a 1 ) s.t. (8)
, ( 13) and ( 14)

with the following conditions:

(1) and 0 t 1, t = 1; 2:

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

Proposition 1 The optimal …nancial contracts are given by: If the project is not very risky, in the sense X 1 2 (1 ) C and X 2 2 C:

D 1 = 2(1 )C X 1 + 1 (X 2 C) 2 D 2 = 2 C X 2 I = C 2 X 1 (1 ) C + 1 (X 2 C) 2 2 i = 3 2 X 1 (1 ) C + 1 (X 2 C) 2 2
If the project is very risky, in the sense

X 1 > 2 (1 ) C and X 2 > 2 C: D 1 = D 2 = 0 I = C 1 (1 ) C + 1 X 2 C X 1 + 3 4 (X 2 ) 2 i = 3 8 X 1 + 3 4 (X 2 ) 2 2
The amount of equity issued by the entrepreneur is given by: W = K i I :

Whether the project is highly risky or not, the entrepreneur and the LBO fund have equal payo¤ s:

t = 1 2 , t = 1, 2. See appendix A.
This proposition states that whether the project is very risky or not, the acquisition must be funded jointly by the entrepreneur, the LBO fund and the bank: all agents have to invest strictly positive amount of money in the entrepreneur's project. Besides, the entrepreneur and the LBO fund have equal bene…t's shares. If the optimal contracts attribute high share to one of them, they will induce him to provide high e¤ort but simultaneously, they will reduce the e¤ort of the other one. Hence, the optimal …nancial contracts have to induce both parties to provide optimal e¤orts. Note that the debt's payments are decreasing with the outcome of the project in the starting and the productive stages. Consequently, the entrepreneur and the LBO fund are tempted to announce a success to the bank whatever happens. In case of failure, they are better o¤ if they sell the project's assets and pay D t to the bank and share the di¤erence between them.

When the project is not very risky, the entrepreneur and the LBO fund provide the …rst best e¤orts. The levels of these e¤orts are not very high since the di¤erence between the revenues of success and failure is small, it is easy to induce them to provide the …rst best e¤orts. The presence of the bank, more speci…cally the threat of liquidation induces them to make e¢ cient investment decisions. This is no longer true when the di¤erences between the revenues of failure and success are large. The e¤orts provided by the entrepreneur and the LBO fund are lower than the optimal e¤orts (5), despite the fact that we use a powerful incentive scheme where the optimal debt contract exhibits the features of a "live or die" contract: the entrepreneur and the LBO fund share equally the revenues of success and pledge the entire revenue of failure to the bank. So, they provide the following second best e¤orts:

e 1 = a 1 = 1 2 h X 1 + 3 4 (X 2 ) 2 i and e 2 = a 2 = 1 2 X 2 (20)
These e¤orts induce the success of the project with the probabilities p 1 and p 2 given respectively by: p

1 = 1 h X 1 + 3 4 (X 2 ) 2 i and p 2 = 1 X 2 (21)
The expected gain of the entrepreneur is therefore given by:

E ( E ) = 3 8 X 1 + 3 4 (X 2 ) 2 2
W .

The entrepreneur does not capture the social value of the project.

Whether the project is very risky or not, given high, the amount of debt I is larger than the amount of equity i issued by the LBO fund. This is consistent with the excessive use of debt noticed in the buyout acquisitions.

The LBO fund issues high equity when the project is not very risky. Consequently, the bank invests higher amount of debt when it is very risky. This may be explained as follows: more the collateral is large, more the bank will be induced to lend money. Moreover, the amount of debt I (respectively equity i) is decreasing (respectively increasing) with the revenues of success of the two stages.

The bank' s payments are non-decreasing with the project' s revenues

Consider now that the entrepreneur and the LBO fund cannot lie about the project's result to the bank: the …nancial contracts are speci…ed such that the bank's payments are nondecreasing with the revenues of the project. We add the following conditions D 1 C and D 2 C to the entrepreneur's program.

Proposition 2

The optimal …nancial contracts are given by:

D c 1 = C and D c 2 = C (22)
I c = C 1 + X 1 C + 3 4 (X 2 C) 2 (23)
and c t = 1 2 , t = 1, 2 (24)
i c = 3 8 X 1 C + 3 4 (X 2 C) 2 2 (25)
W c = K i c I c (26)
See appendix B.

Note that the bank's payments do not depend on the quality of the project. In this case, the entrepreneur and the LBO fund provide the following e¤orts:

e c 1 = a c 1 = 1 2 X 1 C + 3 4 (X 2 C) 2 (27)
e c 2 = a c 2 = 1 2 (X 2 C) (28)
These e¤orts are not optimal and they are inferior to the ones given by [START_REF] Ick | Private equity Returns: Is here really a Bene…t of low Co-movement with Public Equity Markets?[END_REF]. Considering that the payments of the bank are non-decreasing reduces the agents'incentive to provide e¤orts.

The e¤orts [START_REF] Jensen | Corporate Control and the Politics of Finance[END_REF] induce the success of the project with the probabilities:

p c 1 = 1 h X 1 C + 3 4 (X 2 C) 2 i and p c 2 = 1 (X 2 C) (29)
In contrast with the previous propositions, the amount of debt is higher than the collateral of the starting stage C. The clause of non-decreasing payments ensures to the bank to get higher success payments D t if the project is not very risky, and a strictly positive payment if the project is very risky. Given high, the amount of additional debt is not very large.

4 What is achieved without debt?

Hereafter, we consider that the bank does not contribute …nancially into the project. The optimal holding contract must determine endogenously the fractions of bene…t given to the entrepreneur and to the LBO fund at the end of each stage.

The entrepreneur and the LBO fund choose the levels of e¤orts ẽ2 and ã2 maximizing their expected gains given by:

E( 2 E = X1 = X 1 ) = 2 [p 2 (X 2 C) + C] c E (e 2 ) E( 2 A = X1 = X 1 ) = (1 2 ) [p 2 (X 2 C) + C] c A (a 2 )
According to the …rst order conditions of E( 2 E = X1 = X 1 ) and E( 2 A = X1 = X 1 ), their reaction functions in the productive stage are written:

ẽ2 = 1 2 (X 2 C) and ã2 = 1 (1 2 ) (X 2 C) (30)
Consequently, the success probability of the second period is given by

p2 = 1 (X 2 C) (31)
We substitute ẽ2 , ã2 and p2 for their expressions given by (30) and (31) into the expected gains of the entrepreneur and the LBO fund:

E( E =ẽ 2 ; ã2 ) = p 1 f 1 (X 1 C) + 2 [p 2 (X 2 C) + C] c E (e 2 )g + 1 C c E (e 1 ) W E( A =ẽ 2 ; ã2 ) = p 1 f(1 1 ) (X 1 C) + (1 2 ) [p 2 (X 2 C) + C] c A (a 2 )g +(1 1 )C c A (a 1 ) K + W
The …rst order conditions of E( E =ẽ 2 ; ã2 ) and E( A =ẽ 2 ; ã2 ) enable us to deduce their reaction functions in the starting stage:

ẽ1 = 1 1 (X 1 C)+ 2 C+ 1 2 2 (2 2 ) (X 2 C) 2 (32)
ã1 = 1 (1 1 )(X 1 C) + (1 2 ) C + 1 2 (1 2 2 ) (X 2 C) 2 (33)
Accordingly, the success probability of the …rst period is given by:

p1 = 1 X 1 (1 ) C+ 1 2 (1 + 2 2 2 2 2 ) (X 2 C) 2 (34)
The optimal …nancial contracts induce the entrepreneur to maximize the social value of the project under the incentive constraints (30), ( 32) and (33) such that: ) , [START_REF] Leland | Informational asymmetries, Financial structure, and the Financial Intermediation[END_REF] and [START_REF] Myers | Determinants of corporate Borrowing[END_REF] with the following conditions:

max t ; i; et; at; t=1;2 V = p 1 [X 1 (1 ) C + p 2 (X 2 C) c E (e 2 ) c A (a 2 )] +C K c E (e 1 ) c A (a 1 ) s.t. ( 30
0 t 1, t = 1, 2
The properties of the optimal holding contract are presented in the proposition 3:

Proposition 3 When the LBO fund issues i = K W , the entrepreneur and the LBO fund must get equal revenues in both periods:

~ t = 1 2 , t = 1, 2.
The entrepreneur and the LBO fund issue the following amounts of equity:

W = K 1 2 C 3 8 X 1 (1 ) C + 3 4 (X 2 C) 2 { = 1 2 C + 3 8 X 1 (1 ) C + 3 4 (X 2 C) 2 See appendix B.
This proposition states that the shares of bene…t given to the entrepreneur and to the LBO fund do not depend on project's revenues: whether the result of the project in the …rst period is a success or failure, they must perceive equal parts of the bene…t.

Given the incentive constraints (30), ( 32) and (33), the optimal e¤orts are written:

ẽ1 = ã1 = 1 2 h X 1 (1 ) C + 3 4 (X 2 C) 2 i and ẽ2 = ã2 = 1 2 (X 2 C) (35)
These e¤orts are inferior to the …rst best e¤orts and even to the e¤orts (20) provided by the entrepreneur and the LBO fund if the project is highly risky. When the debt's payments are non-decreasing with the outcome of the project, in the productive stage, they provide the same levels of e¤orts whether the project is …nanced with debt and equity or just with equity. But, in the starting stage, ẽ1 and ã1 are superior to [START_REF] Jensen | Corporate Control and the Politics of Finance[END_REF]. We conclude that the presence of a passive …nancier such as the bank induces the entrepreneur and the LBO fund to make low e¤orts which reduces the performance of the project. This implies that the e¤orts are the highest when the project is …nanced solely through equity.

The sharing rule does not depend on …nancial capital structure of the project, but it depends on the incentives to e¤orts. As explained in the previous sections, the agents have the same cost function and their e¤orts have equal impacts on the success probabilities, the optimal …nancial contracts must boost simultaneously the agent's incentives to provide the required e¤orts. If one of them has the highest part of the payo¤ so that he makes more e¤ort, the other one will make less e¤ort. [START_REF] Biais | Optimal leverage and aggregate Investment[END_REF] The optimal …nancial contracts under the write-o¤ threat

Consider now that the LBO fund may leave the project if there is a success at the end of the starting stage to invest in a new acquisition with a strictly positive probability of exit: > 0. Hereafter, we consider that the bank's payments are non-decreasing.

The competition among the LBO funds and the banks induce A and B to o¤er the entrepreneur the best contracts possible such that the participation constraints (P C A ) and (P C B ) are binding which enables us to write

I = p 1 fD 1 C + (1 ) [p 2 (D 2 C) + C] + (1 ) Lg +C ( 36
) i = p 1 f(1 1 ) (X 1 D 1 ) + (R L) + (1 ) [(1 2 ) p 2 (X 2 D 2 ) c A (a 2 )]g c A (a 1 ) (37)
Substituting [START_REF] Phan | Organizational restructuring and economic performance in leveraged buyouts: an ex post study[END_REF] and (37) into the objective function of the entrepreneur gives the following program:

max t ; D t ; e t ; a t , t=1, 2 E ( E ) = p 1 fX 1 [1 (1 ) ] C + R + (1 ) [p 2 (X 2 C) c E (e 2 ) c A (a 2 )]g +C K c E (e 1 ) c A (a 1 ) (38)
s.t. (8), [START_REF] Chan | Learning, Corporate control, and Performance Requirements in Venture Capital Contracts[END_REF] and [START_REF] Cornelli | Stage Financing and the Role of Convertible Securities[END_REF] with the following conditions:

(1), D H and 0 t 1, t = 1; 2: (39)

The …nancial structure when the information is perfect

If there is no double moral hazard problem, the expected gain of the entrepreneur is given by [START_REF] Sahlman | Aspects of Financial Contracting in Venture Capital[END_REF]. It is surprising that E ( E ) does not depend neither on nor on L but it depends on R. Hereafter, we show that if the compensation cost is exogenous, the entrepreneur's share of compensation is an increasing function of R. But, if it is endogenous, L increases with R and the whole amount will be paid to the entrepreneur.

When the information is perfect, the e¤orts that maximize the expected gain of the entrepreneur are given by the …rst order conditions of E ( E ). They are written

e P I 1 = a P I 1 = 1 X 1 [1 (1 ) ] C + R + 1 (1 ) (X 2 C) 2 (40)
e P I 2 = a P I 2 = 1 (X 2 C) (41)
When the information is perfect, both agents provide the …rst best e¤orts in the productive stage.

If the net revenue of the competitive buyout is high (R > C), they provide high e¤orts in the starting stage: (40) are superior to the …rst best. In the opposite, the entrepreneur and the LBO fund are induced to provide low e¤orts because the LBO fund is not tempted to exit: if he does, his expected gain -after the payment of the compensation cost-is low.

Then, the project succeeds with the probabilities

p P I 1 = 2 X 1 [1 (1 ) ] C + R + 1 (1 ) (X 2 C) 2 (42)
p P I 2 = 2 (X 2 C) (43)
The expected gain of the entrepreneur is

E P I E = C K + 1 X 1 [1 (1 ) ] C + R + 1 (1 ) (X 2 C) 2 2 ( 44
)
This gain is reached if the e¤orts (8), ( 10) and (11) satisfy the conditions (40) and (41). In this case, the …nancial capital structure are given by:

= R L 1 and t = 1 2 , t = 1, 2 D 2 = 2 C X 2 D 1 = 2 [1 (1 ) ] C X 1 2 L + 1 (1 ) (X 2 C) 2
These contracts are signed only if the project is not very risky in the sense X 2 < 2 C and ) ] C 2 L. Moreover, we assume L R 2L so that 0 1. Note that the debt's payments are decreasing with the revenues of the project.

X 1 < 2 [1 ( 1
If the e¤orts are observable, the LBO fund and the bank issue the following amounts of money

i P I = 3 2 n X 1 [1 (1 ) ] C + R + 1 (1 ) (X 2 C) 2 o 2 (45)
I P I = C 2 n X 1 [1 (1 ) ] C + R + 1 (1 ) (X 2 C) 2 o 2 (46)
W P I = K i P I I P I (47)

5.2

The write-o¤ exit and the agents'incentives

L is exogenous

Hereafter, we assume that L given by the legislature and focus on the particular case where the payments of the debt do not decrease when the project's revenues increase. The following proposition characterizes the optimal …nancial contracts.

Proposition 4 When the LBO fund has the option to write o¤ the entrepreneur's project at the end of the starting stage, the optimal …nancial contracts are given by: If R 2L, the competitive buyout is not very pro…table and

^ = R L 1 { = 3 8 (l) 2 Î = C + 1 l f(1 ) C (R 2L)g
If R > 2L, the competitive buyout is very pro…table and

^ = 1 { = 3 8 l X 1 C + 2 3 (R + L) + 3 4 (1 ) (X 2 C) 2 Î = C + 1 (1 ) C X 1 C + R + 3 4 (1 ) (X 2 C) 2 where l = X 1 C + 2 (R L) + 3 4 (1 ) (X 2 C) 2 .
The entrepreneur issues the amount of equity:

Ŵ = K { Î.
Whether the competitive buyout is very pro…table or not, the bank's payments of success are equal to the liquidation values of the project D1 = C and D2 = C. The entrepreneur and the LBO fund have equal payo¤ s: ^ t = 1 2 , t = 1, 2. The proof of the proposition 4 is presented in the appendix D.

These contracts exhibit the features of the optimal …nancial contracts discussed in proposition 2: the payments of the bank in case of success are equal to the liquidation values of the project. The optimal sharing rule of the compensation cost depends on the quality of the competitive buyout and on the compensation cost.

When the competitive buyout is not very pro…table, the compensation cost is paid to the entrepreneur and to the bank such that: * If the net revenue of the competitive buyout is high in the sense 3 2 L < R 2L, the entrepreneur perceives higher compensation than the bank. * In the opposite, the bank gets the highest compensation if L R 3 2 L.

The e¤orts of the productive stage do not change: they are given by (28). These e¤orts do not depend on the write-o¤ threat. But, in the starting stage, the entrepreneur and the LBO fund provide the following e¤orts

ê1 = â1 = 1 2 l ( 48
)
The threat of write-o¤ induces the entrepreneur and the LBO fund to provide lower e¤orts than in the …rst best in the two stages. The increase of the write-o¤ probability induces them to increase their e¤orts. If the violation cost is high, the exit becomes costly to the LBO fund. Consequently, the threat of write-o¤ is non credible so they make less e¤orts in the starting stage. In this case, the share of compensation paid to the entrepreneur (respectively the bank) decreases (respectively increases) if this cost becomes expensive and increases (respectively decreases) if the revenue of the competitive buyout is high.

When R > L, the alternative project becomes more attractive for the LBO fund but he must provide high e¤ort in order to succeed the entrepreneur's project so that he gets his money back. Despite the fact that the LBO fund will abandon the project, the entrepreneur prefers working hard in order to reach the success so that she gets a positive payment.

The project succeeds with the probabilities

p1 = 1 l and p2 = 1 (X 2 C) (49)
In the presence of the write-o¤ threat, the LBO funds issues higher equity than in the case where he cannot abandon the entrepreneur's project. In the opposite, the bank's investment depends on the write-o¤ cost: if this cost is signi…cantly high, in the sense L 2

h R+ C 2 , R
i , the LBO fund is not tempted to exit otherwise, the banks will perceive a high compensation. This is why he accepts to lend more money to the entrepreneur and the LBO fund. When the amount of compensation is low, we need more restrictive conditions in order to compare Î and I c .

The expected gain of the entrepreneur is therefore given by

E( E ) = 3 8 (l) 2 W = { W
When the competitive buyout is very pro…table, the e¤orts of the productive stage are still given by [START_REF] Kaplan | Leveraged Buyouts and Private Equity[END_REF]. But, in the starting stage, the entrepreneur provides higher e¤ort than the LBO fund such that

ê1 = 1 2 X 1 C + 2 L + 3 4 (1 ) (X 2 C) 2 > â1 = 1 2 l (50)
The higher the revenue of the competitive buyout, the more the entrepreneur will be induced to make e¤ort: the LBO fund will probably leave the entrepreneur's project because the compensation cost becomes not signi…cant and she will get the whole amount of compensation. So she is better o¤ in case of success. This is result is not very intuitive: despite the fact that the bank has equal payments in cases of success and failure, he gets a compensation's share only if the competitive project is not very pro…table. But, notice that in this case the LBO fund issues a low amount of equity. So in order to induce the bank to lend them more money, he must give him a strictly positive compensation's share in case of write-o¤. This is no longer true if the competitive project is very pro…table, the LBO fund issues more equity than before; there is no need of additional funds from the bank. To induce the entrepreneur to invest e¢ cient e¤ort, he must get the entire amount of compensation. In this case, the entrepreneur's e¤ort is higher than the LBO's e¤ort despite the fact that e and a are perfect substitutes and both agents have the same functions of cost. The sharing rule of the compensation depends on the …nancial structure and the agents'incentives.

At the opposite, whether the new acquisition is very pro…table or not, the LBO fund provides the same level of e¤ort.

Then, the project succeeds at the end of the …rst period with the probability :

p1 = 1 X 1 C + R + 3 4 (1 ) (X 2 C) 2
higher than 1 l. This is an interesting fact because the LBO fund may use the write o¤ decision as a powerful incentive given to the entrepreneur to make him working hard. This result is consistent with what is noticed in LBO acquisitions: if the project does not meet his short term objectives, the LBO fund threatens to leave the project. Note that when the competitive project is very pro…table, in contrast with the bank, the LBO fund issues lower equity than when it is not very pro…table.

Note that in both cases, the levels of e¤orts (48) and (50) are inferior to those provided when the e¤orts are observable (in perfect information) but superior to [START_REF] Jensen | Corporate Control and the Politics of Finance[END_REF]; when the debt payments are non-decreasing with the revenues of the project and there is no competitive buyout.

Suppose now that the project is implemented without debt. In this case, the entrepreneur and the LBO fund will share equally the revenues (as in the proposition 3). If the fund leaves the entrepreneur's project, he must pledge the entire amount of compensation to the entrepreneur.

Then, the entrepreneur and the LBO fund provide the following e¤orts:

e 1 = 1 2 X 1 C + 2 L + (1 ) 3 4 (X 2 C) 2 + C (51) a 1 = 1 2 l + 1 2 (1 ) C (52)
In the second period, the e¤orts are given by [START_REF] Nikoskelainen | The impact of Corporate Governance Mechanisms on Value Increase in Leveraged Buyouts[END_REF]. It is easy to check that …nancing the acquisition through the LBO fund induces the agents to provide the highest e¤orts even if there is a write-o¤ threat. The e¤orts (51) and (52) do not depend on the quality of the competitive project.

We conclude that it is optimal to issue equity to …nance the entrepreneur's project. Otherwise, they ask for debt. To induce the entrepreneur to provide high e¤orts, the LBO fund must threat her with the exit before the date 2.

L is endogenous

Hereafter, the compensation cost L is a variable to be determined endogenously in the …nancial contracts.

Proposition 5

The optimal …nancial contracts are written

D 1 = C and D 2 = C I = C + 1 (1 ) C and L = 1 2 R , = 1 and t = 1 2 , t = 1, 2 i = 3 8 2 W = K i I where = X 1 C + R + 3 4 (1 ) (X 2 C) 2 .
The proof of this proposition is presented in the appendix E.

The optimal …nancial contracts have the same properties of those analyzed in the propositions 2, 4 and the lemma 3. The proposition 5 states that the entrepreneur must perceive the whole compensation cost because the bank perceives the same revenue whatever happens.

The e¤orts of the second period does not depend on the type of the variable L: they are given by [START_REF] Kaplan | Leveraged Buyouts and Private Equity[END_REF], but those of the …rst period become

e 1 = a 1 = 1 2 (53)
Notice that these e¤orts are higher than the e¤orts (50) but still lower than the e¤orts provided when the information is perfect.

It is easy to check that the amount of compensation is speci…ed such that the competitive buyout looks not very pro…table to the LBO fund. Hence, the threat of write-o¤ becomes incredible. As explained before, if the project is …nanced through the LBO fund, both agents are induced to provide the highest e¤orts.

Conclusion

The aim of this paper was to study the …nancial structure in LBO acquisition and to take into account the active role of the LBO fund. We focused on the debt's in ‡uence on the agent's incentives in the presence of a double sided moral hazard problem: the entrepreneur and the LBO fund provide unobservable e¤orts which improve the project's performance. We considered a dynamic agency model with two periods: the starting and productive periods.

Under the condition that the debt's payments are decreasing with the project's outcome, we show that if the project is not very risky, the presence of the bank boosts the agents' incentives so that they provide the …rst best e¤orts. This no longer true when the acquired company is very pro…table, we need powerful incentive mechanisms: despite the fact that the debt contract exhibits the features of a "live or die" contract, the entrepreneur and the LBO fund provide the second best e¤orts. This research contributes also to the …nancial literature that links the exit route and the agents'incentives. It analyzes some characteristics of the LBO …nance, namely the write-o¤ exit in LBO. This exit route induces the partners to provide high e¤orts when the project is …nanced through a mixture of debt and equity under speci…c conditions. However, these e¤orts are not optimal: the entrepreneur and the LBO fund provide higher e¤orts if the acquisition is …nanced only with equity.

The current paper studied the characteristics of the standard debt-equity contracts and the pure-equity contracts. However, the use of convertible securities such as the convertible bonds ans the convertible preferred stocks becomes prevelant in the buyout projects. This is surprising because these securities are rarely issued in the presence of a passive …nanciers such as the banks and other outside equity holders. In practice, the LBO fund can convert his debt/stocks into equity in order to increase his controle of the company which reduces the pro…t and controle of the entrepreneur. It would be intresting to examine how the use of these securities in ‡uences the agents'incentives as well as the …nancial structure, namely the debt to equity ratio.

The decision concerning the exit route can induce agency con ‡its between the entrepreneur and the LBO fund. The former prefers the IPO exit such that the acquired company is listed on a perfectly competitive stock market. Consequently, she keeps her position and the controle of the company and can get private bene…ts because of the information asymetry: the shares are sold to a wide spectrum of outside investors. In contrast with the entrepreneur, the LBO fund prefers quick exit routes to get back his money and to invest in a new deal. He takes usually full exit decision of the buyout investments such as sale. Emprically, the probability of an exit via sale was the highest in the buyout stage. To my knowledge no theoritical paper adresses the topic of exit routes in LBO …nance. It would be intresting to answer to the following question: how the …nancial structure in LBO investments solve the agency con ‡icts due to the choice of the exit route?

In future research it would be insightful to study the other exit routes such as IPO, trade sale and buybacks and their in ‡uences on the agents'incentives. Research in this direction is still pending.

Although the investors and …nanciers point out that the strip …nancing and the debt syndication play a major role in mitigating the problems of agency con ‡icts and information asymmetry, there are no academic papers analyzing how they solve for such problems in buyout investments.

A.2 The entrepreneur' s project is very risky

When the project is very risky in the sense X 1 > 2( 1)C and X 2 > 2 C, we have to write the Lagrangian program with all the constraints given by:

L= 1 n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o fX 1 (1 )C + 1 (X 2 D 2 )(X 2 C) 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 o 1 2 n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o 2 1 2 n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o 2 +C K + 1 1 + 2 2 + 3 (1 1 ) + 4 (1 2 ) + 5 D 1 + 6 D 2 + 7 (X 1 D 1 ) + 8 (X 2 D 2 )
Where j , j = 1::8 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions give:

@L @ 1 = 1 (X 1 D 1 ) h (1 2 1 )(X 1 D 1 )+ 1 2 (1 2 2 ) (X 2 D 2 ) 2 i + 1 3 0 (56) 1 0 (57) 1 @
L @ 1 = 0 (58)
@L @ 2 = 1 2 (X 2 D 2 ) 2 f(X 1 C) + (1 1 ) (X 1 D 1 ) + 1 X 2 (X 2 D 2 ) + 1 2 2 (2 3 2 )(X 2 D 2 ) 2 1 2 2 (X 2 D 2 ) 2 f2 (X 1 C) + (X 1 D 1 ) + 2 X 2 (X 2 D 2 ) 1 2 (1 6 2 + 6 2 2 )(X 2 D 2 ) 2 + 2 4 0 (59) 2 0 (60) 2 @L @ 2 = 0 (61) @L @D1 = 1 fX 1 (1 ) C (1 1 ) (X 1 D 1 ) + 1 (X 2 C) (X 2 D 2 ) 1 2 (2 2 2 + 2 2 )(X 2 D 2 ) 2 1 1 n (1 2 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 5 7 0 (62)
D 1 0 (63)
D 1 @L @D 1 = 0 (64) @L @D2 = 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) fX 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 1 2 (X 2 C) (1 2 2 + 2 2 2 )(X 2 D 2 ) n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o + 1 2 2 (2 2 ) (X 2 D 2 ) n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o + 1 2 (1 2 2 ) (X 2 D 2 ) n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 6 8 0 (65)
D 2 0 (66) D 2 @L @D 2 = 0 (67)
The complementary slackness conditions are written:

@L @ 1 = 1 0 , 1 0 , 1 1 = 0 (68) @L @ 2 = 2 0 , 2 0 , 2 2 = 0 (69)
@L @ 3 = 1 1 0 , 3 0 , 3 (1 1 ) = 0 (70) @L @ 4 = 1 2 0 , 4 0 , 4 (1 2 ) = 0 (71) @L @ 5 = D 1 0 , 5 0 , 5 D 1 = 0 (72) @L @ 6 = D 2 0 , 6 0 , 6 D 2 = 0 (73) @L @ 7 = X 1 D 1 0 , 7 0 , 7 (X 1 D 1 ) = 0 (74) @L @ 8 = X 2 D 2 0 , 8 0 , 8 (X 2 D 2 ) = 0 (75)
It is straightforward to see that D 2 6 = X 2 otherwise, the project will fail with probability one in the second period. Hereafter, we will discuss the following cases:

1. D 1 = X 1 , 0 < D 2 < X 2 and 0 < t < 1, t = 1; 2
According to (68), ( 69), ( 70), ( 71), ( 72), ( 73) and (75), the Kuhn-Tucker multipliers

1 = 3 = 2 = 4 = 5 = 6 = 8 = 0. We substitute D 1 = X 1 and 1 = 3 = 2 = 4 = 5 = 6 = 8 =
0 into the …rst order condition (56) such that:

(1 2 2 ) (X 1 C) + 1 X 2 (X 2 D 2 ) + 3 2 2 (1 2 ) (X 2 D 2 ) 2 = 0 (76)
The latter condition is satis…ed only if 2 = 1 2 . Given D 1 = X 1 , (64) is satis…ed only if (62) is binding:

7 = 1 X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) + 1 2 2 2 2 2 + 2 2 1 (1 2 2 ) (X 2 D 2 ) 2 (77)
Substituting 2 = 1 2 in (65) gives:

3 2 X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 4 (X 2 D 2 ) 2 + 3 4 (X 2 D 2 ) (X 2 C) 1 2 (X 2 D 2 ) + 9 16 (X 2 D 2 ) 2 = 0 (78)
Let Y = (X 2 D 2 ), the equation (78) becomes therefore:

21 16 Y 2 9 4 (X 2 C) Y 3 2 [X 1 (1 ) C] = 0
The latter equation has two solutions:

Y 1 = 6 7 (X 2 C) 2 7 q 9 (X 2 C) 2 + 14 [X 1 (1 ) C] < 0 Y 2 = 6 7 (X 2 C) + 2 q 9 (X 2 C) 2 + 14 [X 1 (1 ) C] If Y = Y 2 , we get: D 2 = 1 7 X 2 + 6 7 C 2 7 q 9 (X 2 C) 2 + 14 [X 1 (1 ) C] < 0
This leads to a contradiction with our starting hypothesis.

2. D 1 = 0, 0 < D 2 < X 2 and 0 < t < 1, t = 1; 2

According to the conditions (68), ( 69), ( 70), ( 71), ( 73), ( 74) and (75), 1 = 2 = 3 = 4 = 6 = 7 = 8 = 0. Consequently, the conditions (56) and (59) are written:

1 X 1 (1 2 1 )(X 1 )+ 1 2 (1 2 2 ) (X 2 D 2 ) 2 = 0 (79) (1 2 2 ) (X 1 C) + 1 X 2 (X 2 D 2 ) + 1 2 2 (1 2 ) (X 2 D 2 ) 2 + (1 1 2 ) X 1 = 0 (80)
(79) and (80) are satis…ed only if t = 1 2 , t = 1, 2. Substituting D 1 = 0, t = 1 2 , t = 1, 2 and 1 = 2 = 3 = 4 = 6 = 7 = 8 = 0 into the equation (65) gives:

3 2 (X 2 D 2 ) 1 2 X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 (X 2 C) 1 2 (X 2 D 2 ) n X 1 + 3 4 (X 2 D 2 ) 2 o = 0 (81)
Let Y = (X 2 D 2 ), (81) becomes:

21 16 Y 3 + 9 4 (X 2 C) Y 2 + Y 1 4 X 1 3 2 (1 ) C + X 1 (X 2 C) = 0
The latter equation has no real solution.

3. D t = 0 and t = 0, t = 1; 2

According to the equations (70), (71), ( 74) and (75), the Kuhn-Tucker multipliers 3 = 4 = 7 = 8 = 0. Substituting D t = 0, t = 0, t = 1; 2 and 3 = 4 = 7 = 8 = 0 into the …rst order condition (56) gives:

1 1 X 1 X 1 + 1 2 (X 2 ) 2 < 0
which does not satisfy (68).

4. D t = 0 and t = 1, t = 1; 2

According to the conditions (68), ( 69), ( 74) and (75), 1 = 2 = 7 = 8 = 0. We substitute D t = 0, t = 1; 2, 1 = 1 and 1 = 2 = 7 = 8 = 0 into (56) gives:

3 = 1 X 1 X 1 + 1 2 (X 2 ) 2 < 0 5. D 1 = X 1 , D 2 = 0 and 0 < t < 1, t = 1; 2
According to the equations (68), (69), (70), (71), ( 72) and (75), the Kuhn-Tucker multipliers 1 = 2 = 3 = 4 = 5 = 8 = 0. Then, the …rst order condition (62) implies:

7 = 1 X 1 (1 ) C + 1 X 2 (X 2 C) + 1 2 [ 2 (2 2 ) + 1 (1 2 2 )] (X 2 ) 2 < 0 (82) 6. 0 < D t < X t and 0 < t < 1, t = 1, 2
According to (68), ( 69), ( 70), ( 71), ( 72), ( 73), ( 74) and (75), j = 0, j = 1::8. We substitute these multipliers into the …rst order conditions (56), ( 59), ( 62) and (65) such that we obtain the following system:

(1 2 1 )(X 1 D 1 )+ 1 2 (1 2 2 ) (X 2 D 2 ) 2 = 0 (83) (X 1 C) + (1 1 ) (X 1 D 1 ) + 1 X 2 (X 2 D 2 ) + 1 2 2 (2 3 2 )(X 2 D 2 ) 2 2 2 (X 1 C) + (X 1 D 1 ) + 2 X 2 (X 2 D 2 ) 1 2 (1 6 2 + 6 2 2 )(X 2 D 2 ) 2 = 0 (84) X 1 (1 ) C (1 1 ) (X 1 D 1 ) 1 2 (2 2 2 + 2 2 )(X 2 D 2 ) 2 + 1 (X 2 C) (X 2 D 2 ) 1 n (1 2 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o = 0 (85) (1 + 2 2 2 2 2 ) (X 2 D 2 ) X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 (X 2 C) (1 2 2 + 2 2 2 )(X 2 D 2 ) n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o + 2 (2 2 ) (X 2 D 2 ) n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o +(1 2 2 ) (X 2 D 2 ) n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o = 0 (86)
Note that the equations (83) and (84) are satis…ed only if t = 1 2 , t = 1, 2. Then, (85) and (86) are written

X 1 (1 ) C 1 2 (X 1 D 1 ) + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 = 0 (87) 3 2 (X 2 D 2 ) X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 4 (X 2 D 2 ) 2 (X 2 C) 5 4 (X 2 D 2 ) n X 1 D 1 + 3 4 (X 2 D 2 ) 2 o = 0 (88)
The equation (87) enables us to deduce:

D 1 = 2 (1 ) C X 1 2 (X 2 C) (X 2 D 2 ) + 5 4 (X 2 D 2 ) 2 (89)
Substituting D 1 for its expression given by (89) into (88) gives:

3 2 (X 2 D 2 ) X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 4 (X 2 D 2 ) 2 + (X 2 C) 5 4 (X 2 D 2 ) 2 [X 1 (1 ) C] + 2 (X 2 C) (X 2 D 2 ) 1 2 (X 2 D 2 ) 2 = 0 (90)
Let Y = (X 2 D 2 ), the equation (90) becomes:

1 4 Y 3 3 2 (X 2 C) Y 2 + n 2 (X 2 C) 2 [X 1 (1 ) C] o Y +2 (X 2 C) [X 1 (1 ) C] = 0
This equation has three solutions:

Y 1 = 2 (X 2 C) ) if Y = Y 1 , D 2 = 2 C X 2 < 0 Y 2 = 2 (X 2 C) + 2 q (X 2 C) 2 + [X 1 (1 ) C] ) if Y = Y 2 , D 2 = 2 C X 2 2 q (X 2 C) 2 + [X 1 (1 ) C] < 0 Y 3 = 2 (X 2 C) 2 q (X 2 C) 2 + [X 1 (1 ) C] < 0. 7. 0 < D 1 < X 1 , D 2 = 0 and 0 < t < 1, t = 1, 2.
According to (68), ( 69), ( 70), (71) (72), ( 74) and (75), j = 0, 8j 2 f1::8g = f6g. Then the condition (56) is written:

(1 2 1 )(X 1 D 1 )+ 1 2 (1 2 2 ) (X 2 ) 2 = 0
It is satis…ed only if t = 1 2 , t = 1, 2. Substituting D 2 = 0, j = 0, j = 1::8= f6g and t = 1 2 into (62) and (65) gives the following system:

X 1 (1 ) C 1 2 (X 1 D 1 ) + 1 2 X 2 3 4 X 2 2 C = 0 (91) 6 3 2 2 X 2 X 1 (1 ) C + 1 X 2 3 8 X 2 C 1 2 (X 1 D 1 ) + 1 2 2 (X 2 2 C) n X 1 D 1 + 3 4 (X 2 ) 2 o (92)
According to (91), if there is success, the bank's payment in the starting stage is written:

D 1 = 2(1 )C X 1 1 X 2 ( 3 4 X 2 2 C) (93)
It is straightforward to see that D 1 < 0 when X 1 2 (1 ) C.

(1 2 1 ) X 1 + 1 2 (1 2 2 ) (X 2 ) 2 = 0 (94) (2 1 ) X 1 C + 1 2 (2 + 2 2 3 2 2 )(X 2 ) 2 2 (3X 1 2C) + 1 2 (3 + 6 2 6 2 2 )(X 2 ) 2 = 0 (95)
The equation (94) and (95) are satis…ed only if t = 1 2 , t = 1, 2: Then, we substitute D t = 0, t = 1 2 , t = 1; 2 and 7 = 8 = 0 into the (62) and (65) such that:

5 1 2 X 1 2 (1 ) C + 1 X 2 3 4 X 2 2 C 6 3 4 2 X 2 X 1 2 (1 ) C + 1 X 2 3 4 X 2 2 C + 1 2 2 (X 2 2 C) h X 1 + 3 4 (X 2 ) 2 i
If 5 and 6 are Kuhn-Tucker multipliers, we should get the following conditions:

X 1 2 (1 ) C 1 X 2 3 4 X 2 2 C (96) X 2 2 C
Given high, the condition X 1 > 2 (1 ) C enables us to meet the inequality (96). Consequently, the entrepreneur and the LBO fund provide the following e¤orts:

e 1 = a 1 = 1 2 X 1 + 3 4 (X 2 ) 2 (97) e 2 = a 2 = 1 2 X 2 (98)
To get the results of the second part of the proposition 1, we substitute (97) and (98) into the participation constraints of the LBO fund and the bank. The optimal …nancial investments made by the LBO fund and the bank are given respectively by:

i = 3 8 X 1 + 3 4 (X 2 ) 2 2 I = C 1 1 1 + 1 X 2 X 1 + 3 4 (X 2 ) 2
We have not presented all the discussed cases but just the most interesting ones, more detailed proof is available under request.

B Proof of the proposition 2

The Lagrangian is written

L= 1 n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o n X 1 (1 )C + 1 (X 2 D 2 )(X 2 C) 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 o 1 2 n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o 2 1 2 n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o 2 +C K + 1 1 + 2 2 + 3 (1 1 ) + 4 (1 2 )+ 5 D 1 + 6 D 2 + 7 (X 1 D 1 ) + 8 (X 2 D 2 ) + 9 (D 1 C) + 10 (D 2 C)
Where j , j = 1::10 are the Kuhn-Tucker multipliers.

The Kuhn-Tucker conditions give:

@L @ 1 = 1 (X 1 D 1 ) h (1 2 1 )(X 1 D 1 )+ 1 2 (1 2 2 ) (X 2 D 2 ) 2 i + 1 3 0 (99) 1 0 (100) 1 @L @ 1 = 0 (101) @L @ 2 = 1 2 (X 2 D 2 ) 2 f(X 1 C) + (1 1 ) (X 1 D 1 ) + 1 X 2 (X 2 D 2 ) + 1 2 2 (2 3 2 )(X 2 D 2 ) 2 1 2 2 (X 2 D 2 ) 2 f2 (X 1 C) + (X 1 D 1 ) + 2 X 2 (X 2 D 2 ) 1 2 (1 6 2 + 6 2 2 )(X 2 D 2 ) 2 + 2 4 0 (102) 2 0 (103) 2 @L @ 2 = 0 (104) @L @D1 = 1 fX 1 (1 ) C (1 1 ) (X 1 D 1 ) + 1 (X 2 C) (X 2 D 2 ) 1 2 (2 2 2 + 2 2 )(X 2 D 2 ) 2 1 1 n (1 2 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 5 7 + 9 0 (105)
D 1 0 (106) D 1 @L @D 1 = 0 (107) @L @D2 = 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) fX 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 1 2 (X 2 C) (1 2 2 + 2 2 2 )(X 2 D 2 ) n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o + 1 2 2 (2 2 ) (X 2 D 2 ) n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o + 1 2 (1 2 2 ) (X 2 D 2 ) n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 6 8 + 10 0 (108)
D 2 0 (109) D 2 @L @D 2 = 0 (110)
The complementary slackness conditions are written:

@L @ 1 = 1 0 , 1 0 , 1 1 = 0 (111) @L @ 2 = 2 0 , 2 0 , 2 2 = 0 (112) @L @ 3 = 1 1 0 , 3 0 , 3 (1 1 ) = 0 (113) @L @ 4 = 1 2 0 , 4 0 , 4 (1 2 ) = 0 (114) @L @ 5 = D 1 0 , 5 0 , 5 D 1 = 0 (115) @L @ 6 = D 2 0 , 6 0 , 6 D 2 = 0 (116) @L @ 7 = X 1 D 1 0 , 7 0 , 7 (X 1 D 1 ) = 0 (117) @L @ 8 = X 2 D 2 0 , 8 0 , 8 (X 2 D 2 ) = 0 (118) @L @ 9 = D 1 C 0 , 9 0 , 9 (D 1 C) = 0 (119) @L @ 10 = D 2 C 0 , 10 0 , 10 (D 2 C) = 0 (120)
We rule out the case where D 2 6 = X 2 otherwise, the project will fail with probability one in the second period.

. D 1 = X 1 , 0 < D 2 < X 2 and 0 < t < 1, t = 1; 2
According to (111), (112), ( 113), (114), (115), ( 116), ( 118) and (119), the Kuhn-Tucker multipliers 1 = 2 = 3 = 4 = 5 = 6 = 8 = 9 = 0. We substitute D 1 = X 1 and 1 = 3 = 2 = 4 = 5 = 6 = 8 = 9 = 0 into the …rst order condition (99) such that

(1 2 2 ) (X 1 C) + 1 X 2 (X 2 D 2 ) + 3 2 2 (1 2 ) (X 2 D 2 ) 2 = 0 (121)
The latter condition is satis…ed only if 2 = 1 2 :Given D 1 = X 1 , (107) is properly speci…ed only if (105) is binding. Substituting 2 = 1 2 into the latter condition implies

7 = 1 X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 (122)
7 is a Kuhn-Tucker multiplier only if

X 1 (1 ) C 1 X 2 (X 2 D 2 ) + 5 8 (X 2 D 2 ) 2 (123)
Given high, (123) is never satis…ed.

2. D 1 0, 0 D 2 X 2 and 0 < t < 1, t = 1; 2

If D 1 = 0, the complementary condition (119) cannot be satis…ed because @L @ 9 < 0.

3. 0 D 1 X 1 , D 2 = 0 and 0 t 1, t = 1; 2 If D 2 = 0, (120) is not satis…ed since @L @ 10 < 0. 4. 0 < D t < X t and 0 < t < 1, t = 1, 2
According to (111), (112), ( 113), (114), (115), (116), ( 117) and (118), j = 0, j = 1::8.

We substitute these multipliers into the …rst order conditions (99), (102), ( 105) and (108), we obtain the following system:

(1 2 1 )(X 1 D 1 )+ 1 2 (1 2 2 ) (X 2 D 2 ) 2 = 0 (124) (X 1 C) + (1 1 ) (X 1 D 1 ) + 1 X 2 (X 2 D 2 ) + 1 2 2 (2 3 2 )(X 2 D 2 ) 2 2 2 (X 1 C) + (X 1 D 1 ) + 2 X 2 (X 2 D 2 ) 1 2 (1 6 2 + 6 2 2 )(X 2 D 2 ) 2 = 0 (125) 1 X 1 (1 ) C + 1 (X 2 C) (X 2 D 2 ) (1 1 ) (X 1 D 1 ) 1 2 (2 2 2 + 2 2 )(X 2 D 2 ) 2 1 1 n (1 2 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 9 = 0 (126) 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 1 (X 2 C) (X 2 D 2 ) X 1 (1 ) C 1 2 (1 2 2 + 2 2 2 )(X 2 D 2 ) 2 1 2 (X 2 C) (1 2 2 + 2 2 2 )(X 2 D 2 ) n X 1 D 1 + 1 2 (1 + 2 2 2 2 2 ) (X 2 D 2 ) 2 o + 1 2 2 (2 2 ) (X 2 D 2 ) n 1 (X 1 D 1 )+ 1 2 2 (2 2 ) (X 2 D 2 ) 2 o + 1 2 (1 2 2 ) (X 2 D 2 ) n (1 1 ) (X 1 D 1 ) + 1 2 (1 2 2 ) (X 2 D 2 ) 2 o + 10 = 0 (127)
Note that the equations (124) and (125) are satis…ed only if t = 1 2 , t = 1, 2. Then, (126) and (127) are written

9 = 1 X 1 (1 ) C 1 2 (X 1 D 1 ) + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 (128) 10 = 3 2 2 (X 2 D 2 ) X 1 (1 ) C 1 2 (X 1 D 1 ) + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 1 2 (X 2 C) 1 2 (X 2 D 2 ) n X 1 D 1 + 3 4 (X 2 D 2 ) 2 o (129)
If 9 = 10 = 0, the equation (128) enables us to deduce that:

D 1 = 2 (1 ) C X 1 2 (X 2 C) (X 2 D 2 ) + 5 4 (X 2 D 2 ) 2 (130)
Given , D 1 < C which does not satisfy (119) because @L @ 9 < 0.

If 9 = 0 and 10 > 0, according to (120), D 2 = C and D 1 satis…es (130).

Substituting D 2 = C in (130) gives D 1 = 2 (1 ) C X 1 3 4 (X 2 C) 2 < C (131)
If 9 > 0 and 10 = 0, the condition (119) enables us to deduce that D 1 = C which is substituted into (129) so that we get

3 2 (X 2 D 2 ) X 1 (1 ) C 1 2 (X 1 C) + 1 (X 2 C) (X 2 D 2 ) 5 8 (X 2 D 2 ) 2 (X 2 C) 1 2 (X 2 D 2 ) n X 1 C 1 + 3 4 (X 2 D 2 ) 2 o = 0 (132) Let Y = (X 2 D 2 ), (132) gives 21 16 Y 3 + 9 4 (X 2 C) Y 2 + 1 2 3 [X 1 (1 ) C] 5 2 (X 1 C) Y + (X 1 C 1 ) (X 2 C) = 0 (133)
which has no real solutions.

If 9 > 0 and 10 > 0, given the conditions (119) and (120), we deduce that D 1 = C and D 2 = C which satisfy (128) and (129) :

Consequently, we conclude that

t = 1
2 ; t = 1; 2, D 1 = C and D 2 = C (134)
We substitute (134) into the participation constraints of the entrepreneur and the LBO fund which completes the proof of the proposition 2.

The …rst order conditions are written 22 ) (1 ) (X 2 D 2 ) 2 + 6 8 + 10 0 (147)

@L @ = 1 L fX 1 [1 (1 ) ] C + (R L) 1 (X 1 D 1 ) + 1 (1 ) (X 2 D 2 ) (X 2 C) 1 2 (1 ) 1 + 2 2 (X 2 D 2 ) 2 + 11 12 0 (135) 0 (136) @L @ = 0 (137) @L @ 1 = 1 (X 1 D 1 ) f(1 2 1 ) (X 1 D 1 ) + [R (1 + ) L] + 1 2 (1 2 2 ) (1 ) (X 2 D 2 ) 2 + 1 3 0 (138) 1 0 (139) 1 @L @ 1 = 0 (140) @L @ 2 0 (141) 2 0 (142) 2 @L @ 2 = 0 (143) @L @D1 = 1 X 1 [1 (1 ) ] C + 1 (1 ) (X 2 D 2 ) (X 2 C) (1 1 ) (X 1 D 1 ) + L 1 2 (1 ) 2 2 2 + 2 2 (X 2 D 2 ) 2 1 1 f(1 2 1 ) (X 1 D 1 ) + (R (1 + ) L) + 1 2 (1 2 2 ) (1 ) (X 2 D 2 ) 2 + 5 7 + 9 0 (144)
D 1 0 (145) D 1 @L @D 1 = 0 (146) @L @D2 = 1 2 1 + 2 2 2 2 2 (1 ) (X 2 D 2 ) fX 1 [1 (1 ) ] C + R + 1 2 (1 ) 2(X 2 D 2 ) (X 2 C) 1 2 2 + 2 2 2 (X 2 D 2 ) 2 + 1 2 (1 ) (X 2 C) + 1 2 2 + 2 2 2 (X 2 D 2 ) fX 1 D 1 + [R (1 ) L] + 1 2 1 + 2 2 2 2 2 (1 ) (X 2 D 2 ) 2 + 1 2 1 2 2 (1 ) (X 2 D 2 ) f(1 1 ) (X 1 D 1 ) + (R L) + 1 2 1 2 2 (1 ) (X 2 D 2 ) 2 + 1 2 2 (2 2 ) (1 ) (X 2 D 2 ) f 1 (X 1 D 1 ) + L + 1 2 2 (
D 2 0 (148) D 2 @L @D 2 = 0 (149)
The complementary slackness conditions are written: @L @ 1 = 1 0 , 1 0 , 1 1 = 0 (150) @L @ 2 = 2 0 , 2 0 , 2 2 = 0 (151) We discuss the most important cases as in the appendix B but we will present only the case solving the entrepreneur's program. Consider that 0 < < 1, 0 < D t < X t and 0 < t < 1, t = 1, 2, according to (150), (151), (152), (153), (154), (155), (156), (157), (160) and (161), all the Kuhn-Tucker multipliers are null except 9 and 10 . Substituting all these multipliers into the …rst order condition (138) gives the following equation:

@L @ 3 = 1 1 0 , 3 0 , 3 (1 1 ) = 0 (152)
(1 2 1 ) (X 1 D 1 ) + [R (1 + ) L] + 1 2 (1 2 2 ) (1 ) (X 2 D 2 ) 2 = 0 (162)
Given 0 < D t < X t , this equation is solved only if

= R L 1 and t = 1 2 , t = 1, 2 (163)
Note that we assumed L R 2L to ensure that 0 < < 1 which means that the competitive buyout is not be very pro…table. In this case, substituting (163) into the equations (144) and (147) gives:

9 = 1 2 X 1 2 [1 (1 ) ] C 5 4 (1 ) (X 2 D 2 ) 2 +D 1 + 2 L + 2 (1 ) (X 2 D 2 ) (X 2 C) (164) 10 = 3 4 2 (1 ) (X 2 D 2 ) fX 1 2 [1 (1 ) ] C +D 1 + 2 L 5 4 (1 ) (X 2 D 2 ) 2 + 1 (1 ) (X 2 D 2 ) (X 2 C) + 1 2 2 (1 ) [2 (X 2 C) (X 2 D 2 )] fX 1 D 1 +2 (R L) + 3 4 (1 ) (X 2 D 2 ) 2 (165)
If 9 = 0, the success payment of the bank can be written

D 1 = 2 [1 (1 ) ] C X 1 2 L 2 (1 ) (X 2 D 2 ) (X 2 C) + 5 4 (1 ) (X 2 D 2 ) 2 < C (166)
which leads to a contradiction: @L @ 9 < 0.

Consider now 9 > 0, the condition (158) is satis…ed only if D 1 = C. Substituting D 1 = C in the equation (165)

10 = 3 4 2 (1 ) (X 2 D 2 ) fX 1 [1 2 (1 ) ] C + 2 L 5 4 (1 ) (X 2 D 2 ) 2 + 1 (1 ) (X 2 D 2 ) (X 2 C) + 1 2 2 (1 ) [2 (X 2 C) (X 2 D 2 )] fX 1 C +2 (R L) + 3 4 (1 ) (X 2 D 2 ) 2
We show that 10 > 0. Then, (159) implies that D 2 = C.

To get the other results of the proposition 4, we substitute (163), D 1 = C and D 2 = C into the reaction functions of the entrepreneur and the LBO fund and in the participation constraints of the LBO fund and the bank.

D.2 The competitive project is very pro…table: R > 2L

A straight application of the Kuhn-Tucker theorem gives the following results:

= 1 and t = 1 2 , t = 1, 2 (167)

E The proof of the proposition 5

As in the appendix D, after substituting the e¤orts into the objective function, we solve: max , L, t ; Dt, t=1, 2

E ( E ) = 1 X 1 D 1 + [R (1 ) L] + 1 2 (1 ) 1 + 2 2 2 2 2 (X 2 D 2 ) 2 X 1 [1 (1 ) ] C + R + 1 2 (1 ) 2(X 2 D 2 ) (X 2 C) 1 2 2 + 2 2 2 (X 2 D 2 ) 2 1 2
(1 1 ) (X 1 D 1 ) + (R L) + 1 L @L @L = 0 (171)

@L @ = 1 L fX 1 [1 (1 ) ] C + (R L) 1 (X 1 D 1 ) + 1 (1 ) (X 2 D 2 ) (X 2 C) 1 2 (1 ) 1 + 2 2 (X 2 D 2 ) 2 + 11 12 0 (172)

0

(173) @L @ = 0 (174)

@L @ 1 = 1 (X 1 D 1 ) f(1 2 1 ) (X 1 D 1 ) + [R (1 + ) L] + 1 2 (1 2 2 ) (1 ) (X 2 D 2 ) 2 + 1 3 0 (175) 1 0 (176) 1 @L @ 1 = 0 (177) @L @ 2 = 1 2 (1 ) (1 2 2 ) (X 2 D 2 ) 2 fX 1 [1 (1 ) ] C + R + 1 2 (1 ) 2(X 2 D 2 ) (X 2 C) 1 2 2 + 2 2 2 (X 2 D 2 ) 2 + 1 2 (1 ) (1 2 2 ) (X 2 D 2 ) 2 fX 1 D 1 + [R (1 ) L] + 1 2 (1 ) 1 + 2 2 2 2 2 (X 2 D 2 ) 2 1 2 (1 2 ) (1 ) (X 2 D 2 ) 2 1 (X 1 D 1 ) + 1 2 2 (2 2 ) (1 ) (X 2 D 2 ) 2 + L 1 2 2 (1 ) (X 2 D 2 ) 2 f(1 1 ) (X 1 D 1 ) + (R L) + 1 2 1 2 2 (1 ) (X 2 D 2 ) 2 + 2 4 0 (178) 2 0 (179) 2 @L @ 2 = 0 (180) @L @D1 = 1 X 1 [1 (1 ) ] C + 1 (1 ) (X 2 D 2 ) (X 2 C) (1 1 ) (X 1 D 1 ) + L 1 2 (
D 2 @L @D 2 = 0 (186)
The complementary slackness conditions are written: @L @ 1 = 1 0 , 1 0 , 1 1 = 0 (187) @L @ 2 = 2 0 , 2 0 , = L 0 , 13 0 , 13 L = 0 (199) @L @ 14 = R L 0 , 14 0 , 14 (R L) = 0 (200)

Hereafter, we focus only on the most interesting cases.

Consider that 0 < L < R, 0 < D t < X t and 0 < t < 1, t = 1, 2: According to (187), ( 188), (189), (190), (191), (192), (193), (194), (197), (198), (199)and ( 200), all the Kuhn-Tucker

Figure 1 -

 1 Figure 1-Time line of the game.

8 . 2

 82 D t = 0 and 0 < t < 1, t = 1;The conditions (68), (69), (70), (71), (74) and (75) give 1 = 2 = 3 = 4 = 7 = 8 = 0. Substituting D t = 0, t = 1; 2 and 1 = 2 = 3 = 4 = 7 = 8 = 0 into the conditions (56) and (59) gives:

1

 1

5 = D 1 0 , 5 0 , 5 D 1 6 = D 2 0 , 6 0 , 6 D 2 7 = 8 = 9 =

 5155162662789 X 1 D 1 0 , 7 0 , 7 (X 1 D 1 ) = 0 (156) X 2 D 2 0 , 8 0 , 8 (X 2 D 2 ) = 0 (157) D 1 C 0 , 9 0 , 9 (D 1 C)

D 1 =

 1 C and D 2 = C (168)To get the second part of the proposition 4, we substitute (167) and (168) in the reaction functions of the entrepreneur and the LBO fund. Then, we substitute all in the participation constraints (CP A ) and (CP B ) to deduce the amounts of issued debt and equity.

2 D 2 )t 1 , t = 1

 2211 2 2 + C K with the following conditions (1), D 1 C, D 2 C, 0 L R and 0

5 = D 1 0 , 5 0 , 5 D 1 6 = D 2 0 , 6 0 , 6 D 2 7 = X 1 D 8 = X 2 D 2 0 , 8 0 , 8 (X 2 D 2 9 = D 1 C 0 , 9 0 , 9 (D 1 C

 515516266271822882291991 1 0 , 7 0 , 7 (X 1 D 1 ) = 0

At the end of each period, the entrepreneur sign a new contract with a new venture capital fund. The next period, she is looking for another one; the VC market is supposed to be competitive.

The holding company acquires another 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 uses these assets as collateral for the debt in hopes that the future cash- ‡ows will cover the debt's payments.

The LBO fund provides the same kind of e¤ort as the entrepreneur. He learns these skills from his past investments.

Appendix

A Proof of proposition 1

A.1 The entrepreneur' s project is not very risky

The question raised now is the following: what are the parameters'values that the incentive constraints [START_REF] Cestone | Anti-Competitive Financial Contracting: the Design of Financial Claims[END_REF], [START_REF] Das | The private equity discount: An empirical Examination of the Exit of Venture Backed Companies[END_REF] and [START_REF] Dessi | Start-up …nance, monitoring and collusion[END_REF] must satisfy in order to implement the …rst best solution?

Technically, this is possible if the …rst best e¤orts given by (3) and (4) satisfy the conditions (8), [START_REF] Das | The private equity discount: An empirical Examination of the Exit of Venture Backed Companies[END_REF] and [START_REF] Dessi | Start-up …nance, monitoring and collusion[END_REF]. We deduce that:

Note that these solutions exist under the conditions:

(1

The …rst term of (55) is always satis…ed given high and X 1 C but the second term is ful…lled if X 1 2( 1)C.

We conclude that when the project is not very risky, in the sense X 1 2(1 )C and X 2 2 C, the entrepreneur and the LBO fund provide the …rst best e¤orts in the two periods.

Substituting t = 1 2 and D t , t = 1, 2 into the equations [START_REF] Hirao | Learning and incentive Problems in Repeated Partnerships[END_REF] and [START_REF] Holmstrom | Corporate Governance and Merger Activity in the United States: Making Sense of the 1980s and 1990s[END_REF] gives the optimal …nancial investments of the LBO fund and the bank so that we get the results of the …rst part of the proposition 1:

Given high, check that I is strictly positive and signi…cantly high.

C Proof of the proposition 3

We substitute (30), ( 32) and (33) into the objective function. The entrepreneur's program is written:

The …rst order conditions of V give the following equations system:

This system has two possible solutions but one of them is real and varies from 0 to 1. This solution is given by:

with the following conditions

The Lagrangian is given by

The Lagrangian is given by

The …rst order conditions are written

multipliers are null except 9 , 10 , 11 and 12 . Substituting all these multipliers into the …rst order condition (175) gives the following equation:

We assume L R 2L to ensure that 0 1.

If 0 < < 1, according to (199) and (200), 13 = 14 = 0. As shown in the previous appendix, the equations (181) and (184) imply that:

The discussion of the signs of 9 and 10 leads to the following conclusions: both multipliers are strictly positive and the debt's payments are non-decreasing with the revenues of the project such that D 1 = C and D 2 = C (see appendix D for more details). But, substituting D 1 = C, D 2 = C and 13 = 14 = 0 in the …rst order equation (169) gives

The latter equation is satis…ed only if:

1. the …rst term of the equation (203) is null. It implies that L = 1 2 R which leads to a contradiction: given 0 < < 1, substituting L = R 2 into (202) gives = 1. 2. the second term of this equation is null, the compensation cost is strictly negative:

As a result, the …rst order conditions (181) and (184) give

If 9 = 0, (206) enables us to deduce

< 0.

Given C X 2 and D 2 X 2 , it is easy to check that 10 > 0. Then, given (196), D 2 = C which completes the proof of the proposition 4.