Financial Contracts and Bargaining

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Financial Contracts and Bargaining

Matan Tsur

November, 2012

Abstract

An entrepreneur considers whether to invest in a project, the profit of which depends

on the outcome of bargaining with prospective buyers. I study how the financial contract,

in the form of a security, between the outside investor and the entrepreneur influences

the entrepreneurs strategic bargaining position with the buyers. In the case of a single

prospective buyer, I consider the securities that the entrepreneur designs to maximize

her payoff. I characterize the entrepreneurs maximal payoff and construct a security

that takes a simple form and implements it uniquely. In the case of many prospective

buyers, I consider a debt security and characterize the equilibria payoffs. Debt significantly

strengthens the entrepreneurs bargaining position when there are few buyers, but the

bargaining advantage diminishes with the number of buyers. The more buyers there are,

the lower the share of the surplus extracted by the entrepreneur. When bargaining offers

are frequent, the advantage may vanish, in which case debt financing is equivalent to self-

financing. Finally, I identify conditions under which outside financing can increase the

entrepreneurs incentive to invest and overcome the hold-up problem, leading to socially

efficient investment.

I thank Alessandro Lizzeri and Tomasz Sadzik for their guidance and comments. Helpful comments givenby Guillaume Frechette, Douglas Gale, Ariel Rubinstein, Emanuel Vespa, Tom Cunningham, Michael Richter,Alistair Wilson, Daniel Martin, Isabel Trevino, Yunmi Kong, Nicholas Kozeniauskas, Severine Toussaert aregratefully acknowledged.

New York University, Department of Economics, 19 W. 4th Street, New York, NY.

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1 Introduction

An entrepreneurs return from an investment often depends on the outcome of bargaining with

suppliers, buyers or potential partners. In this paper, I study the interaction between financing

and bargaining. I consider an entrepreneur that seeks financing from an outside investor, then

produces a good and bargains with prospective buyers. My aim is to understand how thefinancial contract with the outside investor influences bargaining. In particular, I focus on the

implications for contract design and for the entrepreneurs incentive to invest.

The financial contract consists of a security which determines how the revenue is split between

the outside investor and the entrepreneur. Since the revenue is determined by the prices nego-

tiated with the prospective buyers, the security also influences the bargaining outcomes. There

are no frictions between the entrepreneur and the outside investor: There is complete informa-

tion, a single investor and there are no agency costs. The paper makes several contributions.

First, I consider the securities that the entrepreneur designs to maximize her payoffwhen there

is a single prospective buyer. I identify the upper bound on the entrepreneurs payoff and

construct a security that uniquely implements this payoff. Outside financing enables the en-

trepreneur to extract a larger share of the surplus as compared to self-financing. Second, I show

that when the entrepreneur sells goods in different markets, a bargaining externality between

buyers dilutes the effect of outside financing. I characterize the equilibrium payoffs under a

debt security. Debt allows the entrepreneur to extract a larger share of the surplus compared

to self financing, but the advantage is attenuated. The more buyers there are, the lower the

share extracted. Third, when the entrepreneur also chooses the investment level, I identify

conditions under which outside financing can overcome the hold-up problem, thus leading to

socially efficient investment.

In the taxonomy of Harris and Raviv (1991), the present model falls under the category of

models in which the main force determining the financial contract is the effect on the in-

put/output markets. The pioneering papers are Brander and Lewis (1986) and Fershtman and

Judd (1987), and within the large literature that followed, several papers have stressed the link

between financing and bargaining (e.g., Bronars and Deere (1991); Perotti and Spier (1993);

Sarig (1998)). In contrast, in this paper, bargaining is non-cooperative, and the bargaining

outcome is determined through strategic interaction. The contribution of my paper is in the

analysis of the strategic channel through which financing influences bargaining.

To explain my results in further detail, consider, first, a case with a single potential buyer. The

outside investor invests in return for a portion of the future revenue, which is determined by

the security. The entrepreneur designs a security to maximize her payoff under the constraintthat the outside investor agrees to finance the project with the security. I first characterize the

upper bound on the entrepreneurs payoff. I then show that the entrepreneur can guarantee

the maximal payoffby designing a security which implements a bargaining game with a unique

equilibrium. The degree to which the entrepreneur could be made liable plays an important role

in determining this security. Under full liability, a debt security implements the entrepreneurs

maximal payoff uniquely. But when liability is limited, the bargaining game with debt has ad-

ditional outcomes with lower prices. The reason is that under limited liability, the entrepreneur

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is indifferent between low prices that do not cover the debt, and as a result would have no

incentive to bargain over a price increase. Thus, under limited liability, the bargaining game

with debt has many outcomes in which the investor incurs losses, and in some cases she will

not agree to finance the project through debt.

When liability is limited I show that a simple security, consisting of an options scheme, can

guarantee the entrepreneurs maximal payoff. Since a security determines the entrepreneurs

share for each price negotiated, a security will increase the entrepreneurs payoff by increasing

her share. The option scheme is constructed to incrementally increase the entrepreneurs share

at different prices. It can be shown that when the incremental increases are sufficiently large,

the option scheme can eliminate the low price equilibria, thereby uniquely implementing the

entrepreneurs maximal payoff. To illustrate the basic idea, note that in an equilibrium of

the bargaining game, the buyer offers a price that makes the entrepreneur indifferent between

accepting and rejecting. A security that locally increases the entrepreneurs share at the contin-

uation price, increases the price that the buyer must offer to make the entrepreneur indifferent.

With debt, for instance, the entrepreneurs payoff is constant when the price is low. Providing

the entrepreneur with some equity increases the entrepreneurs payoff at every price, which

eliminates some of the low price equilibria. But globally increasing the entrepreneurs payoff,

as does equity, may not be sufficient. The security must provide incentives that are increasing

in the price bargained. The security constructed creates these local incentive by providing the

entrepreneur with options to increase her equity share that she only chooses to exercise when

prices are sufficiently high.

When the entrepreneur produces multiple goods that are sold in different markets, the en-

trepreneur may have to bargain with more than one buyer. In this case an additional factor

influences bargaining, which is the focus of the second part of the paper. There is no compe-

tition between the buyers and there are no direct externalities: Each buyers value does notdepend on whether other buyers acquire their goods. The entrepreneur separately and simul-

taneously bargains with each buyer as follows: The entrepreneur makes offers simultaneously

and privately to each buyer. The buyers then simultaneously accept or reject. Any buyer

who accepts gets his good and pays the agreed amount. The game then continues with the

remaining subset of buyers. These buyers simultaneously and privately make offers to the seller

who chooses which, if any, to accept. The game continues indefinitely. When the project is self

financed, an agreement with one buyer does not affect the other and the game has a unique

equilibrium that is identical to Rubinsteins outcome (Rubinstein (1982)). However, a financial

obligation to the outside investor that is tied to the total revenue, links between the bargain-

ing games. The reason is that an agreement with one buyer reduces the outstanding financial

obligation in subsequent periods, thereby changing the entrepreneurs bargaining posture. To

focus the analysis on the bargaining externality, I fix the security to be debt (with unlimited

liability): The outside investor has a senior claim to the revenue, and the entrepreneur gets the

residual after the debt is paid out.

To illustrate the bargaining externality, consider a production company that makes a movie

and sells the distribution rights. The previous case corresponds to a situation with a single

global distributor, whereas now there are two separate distributors: a U.S. distributor and a

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European distributor. In a bilateral bargaining game with debt (with unlimited liability), the

entrepreneurs bargaining posture increases with the debt level, and so does the price payed

by the buyer. Therefore, when the U.S. distributor expects the European distributor to pay a

high price, the debt level in the subsequent period is low, and waiting is more favorable. But

waiting is more costly if the European distributor pays a lower price. Hence, the highest price

that each buyer is willing to pay decreases with the price in the other transaction. It is notclear whether this externality works in favor of the producer or the distributors. Put differently,

holding the total value and cost constant, is the producers payoff higher when she bargains

with a single global distributor as opposed to two separate distributors? How does the number

of prospective buyers affect the advantage of debt-financing compared to self-financing?

I characterize the equilibrium payoffs under a standard restriction of the off-equilibrium be-

liefs, with an arbitrary number of buyers. Even though there are multiple equilibria, the

entrepreneurs revenue is unique, and the model makes sharp predictions. In equilibrium, the

bargaining externality between buyers counterbalances the entrepreneurs bargaining advantage

gained through outside financing. The entrepreneur still extracts a larger share of the surplus

when the project is financed with debt as compared to self-financing, but the advantage is

attenuated. The more buyers there are, the smaller is the share extracted by the entrepreneur.

Holding the total value and cost constant, the production company always prefers to bargain

with fewer distributors. When bargaining offers are frequent, in some cases, the bargaining

externality offsets the effect of debt, and the entrepreneur is indifferent between debt-financing

and self-financing. Whether the advantage of debt is preserved in the limit, depends on the

number of buyers and the ratio of the total value to total cost of investment. The results have

implications that explain why a firm may prefer to concentrate transactions with few buyers,

and why bundling projects is profitable.

The final results show how the financial contract influences the entrepreneurs incentive toinvest when she also chooses the investment level. In this case, a hold-up problem leads to

underinvestment (compared to socially efficient investment) when the entrepreneur self-finances

the project. But financing through an outside investor may strengthen the entrepreneurs

strategic bargaining position with prospective buyers, increasing the marginal benefit from

investment and the incentive to invest. Whether outside financing, alone, can overcome the

hold-up problem and restore the incentive to invest depends on the number of buyers and on

the ratio of total value to total cost of investment. When the financial obligation is debt and

there are few buyers, the entrepreneur finds it optimal to choose the socially efficient investment

level. But when there are many buyers and the ratio of total value to total cost of investment

is high, investment is not socially efficient. Moreover, when bargaining offers are frequent, the

investment level is identical to the one chosen when the entrepreneur self-finances the project,

and the fundamental hold-up problem is unchanged.

The rest of the paper is organized as follows: Section 2 summarizes the relevant literature.

Section 3 presents an example that illustrates the main components in the analysis. Sections

4 and 5 study the single-buyer and multiple-buyers cases. Concluding comments are offered in

section 6 and the Appendix contains proofs and technical derivations.

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2 Literature

My model and results are related to several strands of literature. The point that a firm may

choose a certain financial structure or sign certain contracts to gain a competitive advantage

when interacting with a third party is a central pillar of models pioneered by Fershtman and

Judd (1987); Brander and Lewis (1986); Fershtman, Judd, and Kalai (1991). Several papershave emphasized the link between financing and bargaining; for example, in Bronars and Deere

(1991); Perotti and Spier (1993), the financial obligation influences the size of the pie that

is bargained over. In numerous other papers, bargaining outcomes influence firms decisions

(Shleifer and Vishny (1989); Rajan (1992, 2012); Stole and Zwiebel (1996); Sarig (1998)Shleifer

and Vishny (1989); Rajan (1992, 2012); Stole and Zwiebel (1996); Sarig (1998)). In those

papers, bargaining power is determined either exogenously or endogenously by outside options.

My work differs in that bargaining is non-cooperative, and the effect of financing is endogenously

determined by the strategic effect on bargaining. The strategic channel leads to new insights

regarding the design of securities and how specific features of the project- e.g., number of buyers

and the ratio of total value to total cost- influence financing decisions. Two recent papers ondelegated bargaining by Bester and Sakovics (2001); Cai and Cont (2004) analyze a seller that

delegates the bargaining assignment to an agent. Sakovics and Bester focus on the question of

renegotiation, while Cai and Cont focus on aspects of moral hazard and adverse selection. The

present paper focuses on a financing relationship, thus, the questions, setup and analysis are

quite different.

In contrast to many papers on security design, such as Innes (1990); Demarzo and Duffie (1999);

Repullo and Suarez (2004), in this paper, there are no frictions between the entrepreneur and

outside investor. The only force governing the security chosen is the effect on bargaining. In

DeMarzo, Kremer, and Skrzypacz (2005), securities affect the revenue from an auction.

Another related literature is on multilateral bargaining. Rubinstein and Wolinsky (1985); Gale

(1986) focus on a competitive environment- for example, when the production company bargains

with two U.S. distributors over a single distribution right. In my paper, there is no competition;

the European and U.S. distributors bargain over different rights. In Baron and Ferejohn (1989),

a single pie is to be split between many players. The present work differs in that there are no

externalities: When one buyer gets the good, it does not affect the others value. Horn and

Wolinsky (1988b) model a single firm that bargains with union(s). Their bargaining game

is similar to the one I consider, but the setting is quite different. In the present work, the

financial obligation of the seller is the link between the bargaining games. In several bargaining

papers, an obligation of one player to an outside party enables some commitment that influences

bargaining. Examples are Haller and Holden (1997); Perry and Samuelson (1994), which model

a bilateral bargaining game in which one player represents a constituency.

According to Williamson (2005), the question of vertical integration is the paradigmatic prob-

lem in the economics of governance. A common explanation is that firms vertically integrate to

overcome transaction costs that arise when contracts are incomplete and commitment problems

hinder investment (see Bresnahan and Levin (2012), for a recent survey). In a cross-country

comparative study, Acemoglu, Johnson, and Mitton (2009) empirically test how the interaction

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between contractual costs and financial innovation determines vertical integration. The work

presented here identifies conditions under which outside financing can overcome this commit-

ment problem and restore incentives to invest. It provides predictions that tie together how

outside financing and industry-specific features, such as the number of buyers and the ratio of

total value to total cost of investment, may affect the benefit of vertical integration. It also

identifies conditions in which downstream firms may benefit from horizontal disintegration- e.g.,when the distributor disintegrated into smaller units.

Zingales (2000)emphasizes the somewhat ignored marriage between Corporate Finance and

The Theory of the Firm, noting that financing decisions are deeply intertwined with the

firms structure and organization. The type of financing that a firm obtains may directly

impact different aspects of the inner structure and organization of the firm, while, at the same

time, the structure of the firm affects the type of financing that a firm obtains. This stands in

contrast to the traditional Corporate Finance approach, which takes the firms organization

and structure as given and analyzes how the firm chooses to finance projects. Rajan (2012)

emphasizes the importance of this inter-linkage and analyzes a compelling link regarding how

outside financing enables different transformations of the inner structure and organization of

the firm. My work brings to light a different link: Outside financing impacts the structure of

firms by influencing the benefit of vertical integration and horizontal factorization.

Lack of commitment that distorts incentive to invest (a hold-up problem) has been the main

component in the engine underlying numerous papers. The topics range from organizational de-

sign, property rights and ownership structure (Klein, Crawford, and Alchian (1978); Williamson

(1979, 1998); Grossman and Hart (1986); Hart and Moore (1990); Aghion and Bolton (1992);

Aghion and Tirole (1997); Rajan and Zingales (2001)) to different aspects of financing deci-

sions (Rajan (1992); Shleifer and Vishny (1989); Bronars and Deere (1991)). Several papers

have analyzed how contractual schemes (Noldeke and Schmidt (1995); Che and Hausch (1999);

Edlin and Reichelstein (1996); Aghion, Dewatripont, and Rey (1994)), long-term relationships

(Baker, Gibbons, and Murphy (2002)) or dynamic investment decisions (Che and Sakovics

(2004)) may mitigate or solve the hold-up problem. To my knowledge, none have studied the

specific channel analyzed in this paper.

3 Example

The following example uses a simple two-period bargaining model to illustrate how precom-

mitment to an outside investor influences bargaining and the bargaining externality between

buyers. Consider a case in which the required investment is I and there is a single buyer whose

value is v. Bargaining lasts for two periods as follows: The buyer makes the first offer; the

entrepreneur makes an offer in the next period; if that offer is declined ,the good loses its value.

The discount factor is .

If the project is self-financed, the entrepreneur incurs the investment cost I and bargains with

the buyer. Since the game ends after two periods, the price can easily be determined by

backward induction. In the second period, the entrepreneur makes a take it or leave it offer,

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and, therefore, the buyer pays price v if the game reaches the final stage. The entrepreneurs

continuation value in the first stage is v, which is the lowest price that she is willing to accept.

Therefore the buyer pays v. The entrepreneurs profit is v I, and the buyer gets v v.

Ifv < I, the entrepreneur does not undertake the project. To illustrate how precommitment

to an outside investor may affect the bargaining outcome, consider two financial contracts:

Equity financing: The outside investor invests and gets a share of the revenue. Precom-

mitting to split a fixed fraction with the outside investor does not influence the price. The

production companys continuation value is (1 )v; the distributor offers a price v;

the production company gets (1)v and is indifferent between accepting and rejecting.

The outside investor agrees only if her share covers the cost v I, and the production

company can get, at most, (1)v v I. Thus, an equity share is not preferred to

self-financing.

Standard debt with limited liability: The outside investor invests I and has a senior claim

to the revenue, implying that if the price covers the debt x I, the entrepreneur gets

x I. The continuation value is (v I), and the lowest price that the entrepreneuraccepts is x I (v I). The price is x = v + (1 )I, and the entrepreneurs profit

increases to (vI), while the buyers profit decreases to (1)(vI). The point is that

a delay in negotiations enables the entrepreneur to roll over the debt, which decreases her

cost of delay and increases her bargaining position.

To understand the effect of multiple buyers, consider the case in which the entrepreneur sells

two goods to two buyers. Each buyer values only her good, and the value for each is v2 .

Bargaining lasts for three periods, and the entrepreneur initially offers a price to each buyer.

The entrepreneur finances the project with debt and the outside investor has senior claims to

the revenue.

If buyer 1 expects buyer 2 to pay x, then the debt in the subsequent period is I0 = I x, and

buyer 1s continuation value (1 )(v2 I0) (1 )(v2 I + x) increases with the expected

price in the other transaction. The entrepreneurs financial obligation creates a link between

the transactions, in that the price each buyer is willing to pay decreases with the expected price

that the other pays. This externality is the additional force that influences prices.

Considering an infinite-horizon bargaining game, I first focus on the single-buyer case and

solve for the optimal financial contract and investment level. The second part focuses on the

multiple-buyers case. I fix the financial contract to be debt with liability and show that the

bargaining model has a generically unique equilibrium with interesting properties.

4 Single Buyer

4.1 The Model

There are three players: an entrepreneur E, an outside investor, and a buyer B. The en-

trepreneur can produce a good for a buyer, and the cost of production is I. The good is worth

v to the buyer.

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There are two stages: In the first stage, the entrepreneur offers the outside investor a financial

contract. If the outside investor declines, the game ends. If the outside investor accepts,

investment is made, the good is produced and the game moves to the second stage. In the

second stage, the entrepreneur and buyer bargain over a price.

Financial Contract: The outside investor invests in the project in return for rights to the future

revenue. The revenue is the price that the buyer pays and the price is verifiable and contractible.

For a revenue of x, a security s(x) is a function that specifies the portion of the revenue that

the outside investor gets. The buyers value is not contractible. A financial contract between

the entrepreneur and the outside investor consists only of a security. I consider securities that

are continuous and weakly increasing. If there is limited liability, then for each revenue level x:

s(x) x.

The entrepreneur offers the outside investor a security s. If the outside investor decides to

invest, she can costlessly monitor the entrepreneur: Any amount invested in the project cannot

be diverted by the entrepreneur. In the second stage, the entrepreneur bargains with the buyer

over the pie v. I denote the bargaining game over a pie v with security s as B (v, s).

Bargaining game: All actions are observable, and the buyer knows the value and the security.

Bargaining is non-cooperative, the bargaining game follows Rubinstein (1982) with a random

initial proposer and specific payoffs. Agents discount time, with a discount factor of < 1

between bargaining rounds.

Payoffs: Agents are risk-neutral. If the financial contract is s, and the good is sold at date t

for a price of x, the buyer gets

uB(x, t) = t(v x);

the entrepreneur receives

uE(x, t) = t(x s(x));

and the outside investor obtains ts(x) I. If an agreement is not reached, the buyer and

entrepreneur get nothing and the outside investor loses the initial investment.

Strategies: The entrepreneurs strategy consists of a security offered and a bargaining strategy.

The buyers strategy is a bargaining strategy for each security offered s. The outside investors

strategy specifies whether to accept or reject any offer. All actions are observable. The solution

concept is SP E.

Discussion of the assumptions:

1. The security is an increasing function of the revenue. The restriction is standard; if the

security were to decrease the entrepreneur would have incentives to artificially inflate the

revenue and decrease the payment to the outside investor (Innes (1990)). One possible

way in which the entrepreneur could do this here, is through an artificial transaction with

a different buyer in which no good is exchanged: the entrepreneur transfers money to that

buyer; the buyer purchases some service from the entrepreneur; the revenue goes straight

to the project. The scheme is profitable when the security does not increase because

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the entrepreneur neither gains nor loses money on the artificial transaction (she gets the

money from back), while the payment made to the outside investor decreases.

2. Bargaining is non-cooperative because the goal is to endogenously derive the strategic

effect of precommitment to the outside investor on the entrepreneurs bargaining position.

3. Bargaining is dynamic while securities are restricted to be static. Debt contracts usuallyspecify interest rates, maturity dates and term payments, which are not accounted for in

the present model, but may influence bargaining outcomes.

(a) A maturity date leads to different bargaining dynamics, and the outcome is pinned

down by backward induction. If bargaining rounds are frequent, a maturity date

may be of a different order of magnitude as compared to the frequency of bargaining

rounds- as a geological time scale is to a calendar time scale- in which case, an

infinite-horizon bargaining model may be a better approximation.

(b) Term payments and interest rates may shift the cost of delay and influence the

bargaining position. In equilibrium, agreement is immediate and the outside investormay not demand such features.

(c) The outcome of a bargaining model that accounts for these features is sensitive to

the underlying assumptions, and small changes in the assumptions can lead to quite

different outcomes.

4. The buyer knows the financial contract. The financial obligations of firms are usually not

observable to buyers. But since the entrepreneur benefits from disclosing the information,

she would do so.

5. After deriving the main results, I further discuss what happens if the outside investor can

design the security or can purchase the project and hire the entrepreneur; I also discussthe implications of allowing the entrepreneur and buyer to sign a contract.

4.2 SPE

The entrepreneur designs a security to maximize her payoff, while making sure that the outside

investor agrees. A security influences both the price negotiated with the buyer and how that

price is split between the entrepreneur and the outside investor. The entrepreneur takes both

into account when designing the security. To better understand the influence of a security

on bargaining, consider, first, a case in which the project is self-financed. The entrepreneurs

payoff is just the price, and the bargaining game has a unique SP E, as shown in Rubinstein

(1982). The SP E strategies are stationary:

The entrepreneur always offers the price xE =v

1+ and accepts an offer x0 only if x0 xE.

The buyer always offers the price xB =v

1+ and accepts an offer x0 only ifv x0 (v xB).

The prices xE and xB are the unique pair that makes each player indifferent between accepting

the others offer and waiting for her own offer.

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Moving to a case in which the project is financed through an outside investor, consider two

different securities: an equity share s1(x) = ax and a debt claim with limited liability s0(x) =

min[x, I]. Each security implements a bargaining game with different SP E outcomes:

If the security is equity, the entrepreneurs payoff is uE(x, t) = t(1 a)x. It is straight-

forward to check that the above strategies are also an SP E in this case. The bargaininggame with equity has a unique SPE; the expected price is v2 , and the entrepreneurs

expected payoff is (1 a) v2 .

If the security is debt with limited liability, the entrepreneur gets nothing if the price x

is less then the debt I, but receives the entire residual x I if the price covers the debt.

When liability is limited, the bargaining game under a debt security has multiple SP E.

The stationary strategies in which each party offers the prices xE = I +1

1+ (v I) and

xB = I +

1+ (v I) are an SP E1. In this case, the entrepreneurs expected payoff is

12(v I). But there also exists an SP E in which the buyer gets the entire value and pays

nothing.2

The utility frontiers under both securities are depicted below:

-x

Payoff E

ayo

Utility frontier: equity

ax(1-a)x

x

v-y

(1-a)y ay

y

v-I

v-I)/2

(v-I)/2Payoff E

Payoff B

I

Utility frontier: debt with limited liability

As offers become frequent (! 1), the stationary SP E price offers converge to a single price.

Comparing the two securities, the price under equity is always v2 , and the entrepreneur clearly

prefers equity to debt when the low price SPE is played. But, if the high price SPE is played,

the entrepreneur prefers debt to equity. The reason is that the high price under a debt security

is greater than the SPE price under equity. The entrepreneur may prefer equity if her share is

sufficiently high, but the lowest share that the outside investor is willing to accept in return for

her investment is a v2 = I. This case is depicted below:

1This is so because each party is indifferent between accepting the others offer and waiting one period.2If the buyer offers a price of zero, and accepts only that price, undercutting the buyers demand by offering

a slightly higher price is not a profitable deviation. The reason is that the entrepreneurs payoff is flat for pricesthat do not cover the debt.

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v/2

Payoff E

ayo

Utility frontier of debt and equity

a(v/2)

I

(v-I)/2

(v-I)/2

The first lemma establishes an upper bound on the entrepreneurs SP E payoff: The en-

trepreneur can do no better than get half the social surplus. Thus, as long as the high price

SP E is played, debt beats any other security.

Lemma. In an SP E of the entire game, the entrepreneurs payoff is at most 12(v I).

A formal proof is given in the Appendix.

To illustrate the idea, not that a security determines the utility frontier. The restriction that

the security increases with the price, implies that the slope of the utility frontier can not be

too flat (it must be steeper than -1), as depicted in the right panel:

-x

Payoff E

Payoff B

Utility frontier general security

s(x)

s(x)

s(x)

x-s(x)

Payoff E

Payoff B

Utility frontier general security

s(x)

s(x)

The intuition underlying the upper bound is that if investment is made, then the bargaining

game has at least one SP E in which the outside investors payoff covers the debt- i.e., the

security and the SP E price x are such that s(x) I.3 Otherwise, the outside investor would

never agree to invest. Thus, there must be an SP E in which the price is within the shaded

domain in the figure below. The proof shows that as long as the security increases, the highest

SP E payoff that the entrepreneur can get in this domain is: 12(v I). The upper bound relies

only on the restriction that the security increase and, therefore, also holds for the case of full

liability.

3To simplify the intuition, assume, ! 1, and the price offers converge.

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Payoff E

Payo B

Utility frontier

s(x)

It follows that when the project is financed with debt and the high price SP E is played, the

entrepreneur achieves the maximal payoff. But under limited liability, the bargaining game

with debt has many outcomes in which the investor incurs losses, and in those cases the outside

investor will not agree to finance the project through debt. The next proposition shows that the

entrepreneur can overcome this problem by designing a security, consisting of a simple option

scheme, that uniquely implements her maximal payoff. Thus, in every SP E the entrepreneurachieves her maximal payoff.

Proposition. In every SP E of the entire game, the entrepreneurs payoff is 12(v I) and the

investment is made.

A formal proof is given in the Appendix.

Given some value d < v, the proof constructs a security that implements a bargaining game

that has a unique SP E in which the entrepreneur gets 12(v d) and the outside investor gets

d I. The security takes the following simple form (see figure below):

s(x) = min[1x, 2x + d2, ...,nx + dn, d]

n < n1 < ... < 1 < 1 and 0 < d2 < d3... < dn < d

price

s

x

s(x)

xEB

The security is constructed to incrementally and locally increase the entrepreneurs share at

different prices. The key point is that in order to eliminate low price equilibria, the incremental

increases need to be sufficiently large. The intuition is that in an equilibrium of the bargaining

game, the buyer offers a price that makes the entrepreneur indifferent between accepting and

rejecting. A security that locally increases the entrepreneurs share at the continuation price,

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increases the price that the buyer must offer to make the entrepreneur indifferent. This, in

turn, further increases the entrepreneurs continuation value. The proof show that through this

process, when the incremental increases are sufficiently large, the option scheme can eliminate

the low price equilibria. The security creates these local incentives by providing the entrepreneur

with options to increase her equity share that she only exercises when the prices are sufficiently

high. Below is a graphical illustration of the construction.

Start with a debt claim s0(x) = min[x, I]. There exists an SP E in which the entrepreneur and

buyer offer xE I +1

1+ (v I) and xB I +

1+ (v I), and the entrepreneurs expected

payoff is 12(v I). The problem is that there also exists an SP E in which the entrepreneur

gets nothing. The idea is to modify s0 in a way that maintains the good SP E but gets rid of

the bad SP E.

Step 1: Modify s0 to eliminate the degenerate outcome. Consider the security s1(x) :=

min(x, I) where < 1, depicted below.

Security s1 allocates an equity share < 1 to the outside investor and provides the entrepreneur

with an option to buy back all the equity for a price of I. Since the entrepreneur has someequity, her payoff strictly increases with any price which eliminates a SP E in which the buyer

pays nothing. Additionally, if is sufficiently large, then s1(xE) = s1(x

B) = I, and the good

SP E from the previous step also exists in the bargaining game with this security. However,

the bargaining game under this security may have additional SP E. Specifically, the SP E in

the bargaining game without debt, or Rubinsteins SP E, may still exist in this game if the

pair of price offers from Rubinsteins SP E, xRE and xRB, both lie on the first segment. This

happens when xRE v

1+ is less than the break point denoted by x1 I

, as depicted below.

The reason that Rubinsteins SP E also exists in this case is that when s1(xRE) = xE and

s1(xRB) = xB, each player is indifferent between accepting the others offer and waiting one

period for her own off

er:

v xE = (v xB)

(1 )xB = (1 )xE

pricex

s (x)

xEB

1

I

s (x)0

price

x

s (x)

xEB

1I

xB xERR x

1

In case the investment falls short of Rubinsteins price, I < v1+ , for a sufficiently high < 1

we have that the price xRE does not fall on the first segment, xRE > x1, and Rubinsteins SP E

can be eliminated. But in the case that I v1+ , both prices xRE and x

RB always fall on the first

segment (because for any : xRE x1), and Rubinsteins SP E always exists in the bargaining

game with this security. The figure below depicts both cases:

13


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price

s

x

s (x)

xEB

1I

xB xERR I

price

s

x

s (x)

xEB

1I

xB xERR I

Step 2: Modify s1 to eliminate Rubinsteins SP E and maintain the good outcome.

Consider security s2(x) = min[1x,2x + d2, I] depicted below (1 > 2 and d2 > 0). Let

x1 be the first break point, as long as the security breaks before the price xRE ( x

RE < x1)

Rubinsteins SP E is eliminated.

price

s

xR

I*

x1

The crux of the construction is that if the difference in slopes is sufficiently large- specifically, if1112

2- then there does not exist a stationary SPE in which the buyers offer xB lies on the

first segment and the sellers off

erxE lies on the second segment; alternatively,

xB x

1 x

E.The pair of prices is a stationary SP E when each party is indifferent between accepting the

other partys offer to waiting for her own offer. As the difference between the slopes of the first

and second segments increases, so does the compensation that the entrepreneur demands if he

is offered a price on the first segment, x < x1, and expects a price tomorrow on the second

segment, x0 > x1. And if the difference is sufficiently large, no two prices can compensate both

players demands and make each indifferent.

In order to implement the good outcome, the slope of the second segment must be sufficiently

high so that s2(xB) = I, which places a lower bound on 2, as depicted below:

14


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price

s

xR

I*

xE*

xj*

Recapping the construction steps:

To eliminate the degenerate SP E in which the buyer gets everything, we set 1 < 1.

To eliminate Rubinsteins SP E, the first break point is set before the Rubinstein price:

x1 < x

R

E.

To eliminate a stationary SP E that has price offers on both segments, we set a sufficiently

large change of slope: 1112 < 2.

To ensure that s implements the good outcome, 2 cannot be too small.

A sufficient condition for the bargaining game B(v, s) to have a unique SPE is that i) x s(x)

strictly increases, and ii) there exists a unique price pair that satisfies the indifference condition:

v xE = (v xB) (1)

xB s(xB) = (xE s(xE)) (2)

(see Osborne and Rubinstein (1990)).

If there is a pair of prices on the second segment that satisfies the indifference conditions (1)

and (2), we repeat the same process: Break again just before we reach the SP E price. The

security is constructed to eliminate all pairs of prices that satisfy the above condition, aside

from xE, xB , and, therefore, implements a game with a unique SP E.

Loosely speaking, in order to eliminate SP E in which low prices are offered, we increase the

curvature of s by adding segments. However, s must implement the highest possible payoff,

which requires s to be sufficiently steep. Thus, uniqueness imposes some curvature on the

security s, while optimality restricts the degree of curvature. The proof constructs a security

that balances both.

Unique implementation implies that the incremental increases need to be sufficiently large.

Thus, the second order effect, approximated by the second derivative of the security, is the key

component that guarantees a unique outcome.

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4.3 SPE Securities

An SP E does not pin down a unique security; for example, a debt security can be part of an

SP E as long as the good SP E is played in the bargaining game. The only restriction that

SP E imposes is that the security flattens out and s(xB) = s(xE) = I.

4

In contrast to debt, the security constructed above has the robust feature that it implements

the entrepreneurs maximal payoffuniquely. In addition, this security can be implemented with

standard options in the following way:

The security initially allocates 1 shares to the outside investor and provides the en-

trepreneur with a sequence of options: Option i enables the entrepreneur to purchase

1 i shares from the outside investor for a price of di. The entrepreneur can exercise

a single option.

The number of options decreases with the ratio vI

and increases with the discount factor or

frequency of offers:

If vI

> 1 + , a single option is sufficient .

If vI 1 + , the number of options n satisfies, I

v< 1 n+1 (formulas are derived in

the proof). The first break point is fixed x1 =v

1+ , and as ! 1 the breakpoints

converge to each other and s smooths out for all x > x1.

There are other securities that also achieve the entrepreneurs maximal payoff uniquely, but

the one constructed here takes a simple form.

In the case of unlimited liability, the same SP E payoffs arise. The upper bound is the same:

The entrepreneur can do no better than get 12(v I) even if she could be made liable. The

entrepreneur can guarantee this payoffwith debt- s(x) = I. The difference is that when liability

is unlimited, a debt security implements the entrepreneurs maximal payoff uniquely, whereas

under limited liability, the bargaining game has multiple SP E.

4.4 Continuous investment level

The results extend in a straightforward manner to the case in which the entrepreneur chooses

a continuous investment level I, and the buyers value v(I) increases with investment.

To accommodate for this case, assume that the entrepreneur offers a pair (I, s) that consists ofan investment level and a security to the outside investor. The outside investor either agrees

or declines. For any investment level, the entrepreneur offers a security to maximize her payoff.

The previous result implies that the entrepreneur can do no better or no worse than 12(v(I)I).

Thus, she chooses an investment level to maximize half of the social surplus, and the investment

is efficient.

4The upper bound is not reached if s (st) increases; see remark in the proof.

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4.5 Points for discussion

I discuss below what happens if some of the assumptions are relaxed:

1. The model does not allow for any communication between the entrepreneur and buyer

before investment is made. The main result establishes that, on its own, a financial con-tract between the entrepreneur and outside investor leads to socially efficient investment.

Therefore, relaxing this assumptions and allowing the entrepreneur and buyer to contract

does not improve social efficiency. However, relaxing this assumption may lead to differ-

ent conclusions regarding the role of financial contracts. For instance, a hold-up problem

does not arise if the entrepreneur and buyer can contract on the value. But when the

the value is contractible, Che and Hausch (1999) show that if i) the entrepreneur makes

the initial investment, ii) inefficient outcomes are always renegotiated, and iii) investment

is cooperative (as is the case here) and parties are risk-neutral, then a contract between

the entrepreneur and buyer cannot lead investment levels higher than those that arise

without a contract. Therefore, even if we allow the entrepreneur and outside investor to

contract, a financial contract with an outside investor is necessary to achieve efficiency

when parties cannot commit not to renegotiate inefficient outcomes.

2. In any SP E, the expected price is v+I2 . The entrepreneur could offer a security that

increases the price, but she does not benefit from this because it requires giving up a

larger portion to the outside investor. It follows from the above construction that there

exists a security that implements a bargaining game with a unique SP E in which the

buyer pays her entire value v and all the revenue goes to the outside investor. Therefore,

if the outside investor were to design the security, she could extract the entire surplus,

leaving the entrepreneur and buyer with nothing. Likewise, if the outside investor could

purchase the project and hire the entrepreneur, then the entrepreneur could extract theentire surplus.

5 Multiple buyers, multiple goods

The previous section finds that when there is a single buyer, the entrepreneur can do no better

than finance the project using debt with unlimited liability. This section considers a project in

which the entrepreneur bargains with many buyers. The difference is that, now, an agreement

with one buyer may change the financial obligation in subsequent periods, which adds an

additional factor that influences the bargaining game. To simplify the analysis, I considerdebt-financing (with unlimited liability).

The project. The entrepreneur/seller bargains with multiple buyers. If the seller decides to

undertake the project, she produces n > 1 goods for n buyers. There is no competition, each

buyer demands a different good: Buyer i0s value of good i is vi and is zero for all other goods.

Values are symmetric: vi = vj v. The cost of the project is I. The number of goods and the

values are fixed and do not depend on the investment level. The seller decides whether or not

to undertake the project; the project is potentially profitable: nv > I.

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Bargaining procedure. Bargaining follows a simultaneous procedure.5 The seller makes

offers simultaneously and privately to all buyers. The buyers then privately and simultaneously

accept or reject. Any buyer who accepts gets his good and pays the agreed amount. The

game then continues with the remaining subset of buyers. These buyers simultaneously and

privately make offers to the seller who chooses which, if any, to accept. The game then continues

indefinitely. The discount factor between bargaining rounds is .

5.1 Debt Financing

The entrepreneur finances the project through an outside investor and the financial contract is

debt. The outside investor has a senior claim to the revenue: Any revenue goes first to cover

the debt. For example, if the debt is $100 and a single buyer purchases her good and pays $70,

then the seller gets nothing that period, and the debt in the subsequent period is $30. The

price each buyers pays is determined via bargaining.

The bargaining game with debt: The bargaining game is a dynamic game with two state

variables: the number of buyers nt and the debt level dt. If an agreement is reached with a

buyer, the buyer pays, receives the good and leaves the game. If an agreement is reached at

period t with k buyers over prices x1,...,xk, then the revenue in that period is XPk

i=1 xi.

If the revenue does not cover the debt- i.e., X dt- the seller gets nothing in that period and

the debt in the subsequent period is dt+1 = dt X. If the revenue covers the debt, the debt is

payed out, dt+1 = 0 and the seller gets the residual X dt in that period. In the subsequent

period, the seller bargains with the remaining nt+1 nt k buyers and a debt level of dt+1.

Transactions are publicly observable, and the number of buyers and the debt level in each period

are common knowledge. Let B(n,d,v) be the simultaneous bargaining game with n buyers, an

initial debt level of d and buyers values v. The solution concept is sequential equilibrium.To highlight the main issues, consider a case with two buyers and 1. If buyer 1 expects

buyer 2 to pay x < d, the debt in the subsequent period is d0 = dx, and her continuation value12(v d

0) 12(v d + x) increases with x (see previous section). Thus, each buyer takes into

account the outcome of the other transaction. To see how this externality influences the sellers

revenue compared to the case in which there is a single buyer, consider a case in which the total

value of the project is V = 200 and the debt is d = 100. When there is a single buyer, the seller

gets 12 (V d) = 50 and the price is12 (V+ d) = 150. The seller cannot extract a revenue of 150

when she bargains with two buyers, each of whom has a value of 100. If buyer 1 expects the

other to pay 75, for example, her continuation value is 12(v d0) = 37.5, and the highest price

she is willing to pay is 62.5. At the same time, if buyer 2 expects buyer 1 to pay 62.5, the mostshe would be willing to pay is: 6834 (this price solves v x =

12(v d + 62.5)). The highest

price the seller can offer and have both buyers accept is x = 66 23 ; since v x =12(v d + x),

this price makes each buyer indifferent between accepting and rejecting. The seller gets 3313 ,

which is less than what she gets when there is a single buyer.

The next proposition places a standard restriction on the off-equilibrium and shows that even

5The procedure can be thought of as n Rubinstein bargaining games played simultaneously and independently.

18


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though there are multiple pure strategy sequential equilibria, the sellers payoff is unique. In

this particular example, it is 33 13 .

Before stating the proposition, some more assumptions are needed. As shown in the previous

section, under limited liability, there exist degenerate equilibria in which the seller is indifferent

between accepting and rejecting low prices. To rule out such outcomes, I assume unlimited

liability as follows. In any sub-game with m buyers and a debt level d0, the seller is solvent as

long as the debt level is less then the total surplus- i.e. d0 < mv. Since nv > d, the game always

starts with a solvent seller. If, in some period, the seller is insolvent, then the seller pays the

debt and the game continues without debt.

I also place a restriction on the off-equilibrium beliefs. Following a deviation to an off equilib-

rium offer by the seller, since offers are private, each buyer has conjectures about offers being

made to other players. Thus, off equilibrium beliefs can have a large effect on the continua-

tion game and this may create multiplicity. A buyer has passive beliefs if, when she receives

an off-equilibrium price, she believes that other buyers received their equilibrium price. The

need for such refinements is common when agreements among pairs of players affect payoffs

for other players (for instance in vertical contracting when there are several downstream firms,

e.g., McAfee and Schwartz (1994); Horn and Wolinsky (1988a)).

The main result shows that all pure-strategy sequential equilibria with passive beliefs are gener-

ically unique in the sense that they all give rise to the same aggregate price or total revenue.

Proposition. In all sequential equilibria with pure strategies and passive beliefs of the game

B(n ,d ,v), agreement is immediate and the aggregate price is the same. The sellers profit is

vs(n ,d ,v) =

8 1 + , investment is not efficient, and the seller chooses a

quality level v that is less than the social efficient level. But, otherwise, n1n

V

C 1 + and

the seller chooses the efficient investment level v.

The reason is that as long as the sellers payoff is proportional to the social surplus, the seller

chooses an investment level to maximize the social surplus. But when n1n

V

C> 1 + , her

payoff

from the optimal investment is not proportional to the social surplus, and investment isnot efficient. In this case, as bargaining offers become frequent, the seller chooses the value to

maximize V2 c(v, n). Therefore, the investment level is identical to when the project is self-

financed and financing with debt cannot improve on what can be achieved under self-financing.

When the seller can also choose quantity, the sellers problem is to choose n, v:

maxn,v

vs(n,d,v)

s.t. d = c(n, v)

v 0, n N.

This case adds an additional source of distortion. Similar to a monopolist that accounts for

the effect of quantity on prices, when the seller produces an additional unit, the price of all the

other units changes. In this case, the investment level depends on the cost function. In general,

incentives to invest are distorted when marginal costs increase (assume that n is continuous).

But if the marginal cost in quantity is constant, then the investment level depends on the

fundamentals, and in some cases, investment may be efficient.

For example, if c(n, v) = n + c(v), where c(.) is convex, assume that the social optimum is

n = N. Then, if N1N

V

C 1 + , investment is efficient; otherwise it is not.

6 Conclusion

This paper focuses on how the form of the security used to finance a project affects the en-

trepreneurs bargaining posture with prospective buyers. The analysis of this dimension pro-

vides a novel perspective on security design and on the advantage of outside financing. In the

case of a single prospective buyer and limited liability, neither debt nor equity can guarantee the

7Assume that a solution exists: @c(v,n)@v

> 0 and@2c(v,n)@2v

> 0.

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entrepreneurs maximal return, but a simple option scheme can. The option scheme provides

the entrepreneur with increasing local incentives, designed to eliminate all low price equilibria,

thereby uniquely implementing the entrepreneurs maximal payoff. When the project delivers

multiple goods that are sold in separate markets, the entrepreneur multilaterally bargains with

different buyers. I limit the analysis to the case of debt-financing (with unlimited liability).

Debt still allows the entrepreneur to extract a larger portion of the surplus compared to self-financing, and the advantage of debt is preserved in the multilateral setting, but is attenuated.

The more buyers there are, the smaller is the share extracted by the entrepreneur. When bar-

gaining offers are frequent and there are many buyers, the advantage may vanish and debt is

equivalent to self-financing. The results have implications that explain why a firm may prefer

to concentrate transactions with few buyers, and why bundling projects together is profitable.

Additionally, when the entrepreneur chooses the investment level, outside financing influences

the incentive to invest, and is some cases, may overcome the hold-up problem. In the basic

hold-up problem the return from investment is determined via bargaining and costs are sunk.

As a result, the entrepreneurs incentive to invest is reduced. Both parties could benefit from

an agreement in which the buyer pays a higher price and the entrepreneur increases the invest-

ment level, but such an agreement cannot be enforced and is not feasible. When the number

of buyers is low, the entrepreneurs payoff under debt-financing is proportional to the social

surplus, therefore she chooses the socially efficient investment level. Thus, through the effect

on bargaining, outside financing implicitly enables an agreement that cannot explicitly be en-

forced. But when the ratio of total value to total cost of investment (at the socially efficient

investment level) is high and there are many buyers, the entrepreneurs payoff is no longer

proportional to the social surplus and the implicit agreement disintegrates. In this case, the

hold-up problem cannot be alleviated by outside financing.

The results also have implications for the influence that outside financing may have on the inner

structure and organization of firms. Vertical integration is one way in which firms can overcome

commitment problems that reduce the value of investment. Thus, in some cases, the ability

to obtain outside financing reduces the incentive for vertical integration. In a cross country

comparative study, Acemoglu, Johnson, and Mitton (2009) empirically test how the interaction

between contractual costs and financial innovation determines vertical integration. One of their

goals is to empirically distinguish between two opposite effects that financial innovation has on

vertical integration. On the one hand, imperfect credit markets limits the number of small firms

which leads to larger firms that tend to be vertically integrated (Rajan and Zingales (1998)).

But on the other hand, lack of financing may also prevent mergers between firms which leads to

less vertical integration (McMillan and Woodruff (1999)). The effect of financing on bargaining

analyzed here depends on industry-specific features, such as the number of buyers and the ratio

of total value to total cost of investment. These features may distinguish this channel from

others.

On the other side, outside financing may also increase the incentive for buyers to horizontally

disintegrate. For example, a single global distributor can reduce the production companys

bargaining advantage and extract a larger share of the surplus by disintegrating into a European

distributor and a US distributor. As long as the production companys incentive to invest is not

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distorted, distributors can extract a larger share of the surplus by localizing distribution and

disintegrating into smaller units. The ratio of the total value to total cost may limit the extent

of factorization, because when the ratio is high, the production companys incentive to invest

may be distorted when there are too many distributors. But if the ratio is low, disintegration

does not affect the production companys incentive to invest and, theoretically, distributors

extract the entire surplus away from the production company by disintegrating into sufficientlysmall units.

The are several issues, both theoretical and empirical, that are left for future work. In the single

buyer-case, allowing for stochastic values may lead to a tighter characterization of the optimal

security. Also, incorporating the effect of various dynamic features of debt securities, such as

interest rate, term payments and maturity dates, may lead to interesting empirical questions.

7 Appendix

7.1 Optimal security single buyer

Lemma. In any SP E of the entire game, the entrepreneurs payoff is at most 12(v I).

Proof. Let uE(x) := x s(x) be the entrepreneurs utility from an agreement at time t = 0. It

is without the loss of generality to assume that uE(x) is weakly increasing, because any price

offered where it is not, is strictly dominated by a lower price.

Let m be the infimum of the buyers payoff in a sub-game starting with his offer. The buyer

will always offer a price xb v m, and will reject any price xE: v xE < m. Therefore

the highest prices that the entrepreneur and buyer can offer in a SP E are xE v m and

xb v m.

Since uE(x) is weakly increasing in price, the most the entrepreneur can get is UE 12 [uE(v m) + uE(v m

12 [2v (1 + )m (s(v m) + s(v m))]

Since E can get no more then uE(v m) when he makes an offer, in any SP E the buyer

always offers a price x such that uE(x) uE(v m) and therefore:

v m s(v m) [v m s(v m)] ()

(1 )v (1 2)m + s(v m) s(v m)]

since s is increasing, s(v

m)

s(v

m) and we have that:

(1 )v (1 2)m + s(v m) s(v m)]

Which implies that (1 + )m v s(v m).

Substituting this into 2 implies:

UE 1

2[v s(v m)]

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There must be at least one SP E in which the outside investor recoups the investment. Since

s is increasing it follows,1

2[s(v m) + s(v m)] I

because otherwise, the outside investor loses money in every SP E. This implies that that

s(v m) I and together with the previous inequality

UE 1

2(v I)

Remark. If s is strictly increasing the the inequality is strict.

The only investment level that achieves the upper bound is the optimal investment.

Proposition. In every SP E of the entire game the entrepreneurs payoff is 12(v I)

The previous lemma establishes 1

2(v I) as an upper bound on upper bound on the en-

trepreneurs SP E payoff. The proof shows that it is also a lower bound.

Lower bound

Lemma. For any d < v there exists 0 < n < ... < 1 < 1 and 0 < d2 < .. < dn < d such that

if s(x) := min[1x, 2x + d2, ....,nx + dn, d] the game B(v, s) has a unique SP E in which the

entrepreneur gets 12(v d) and the outside investor gets d.

Proof. The proof will be by construction. Define (i, di) inductively:

Algorithm:

0. set1

0, n+1 := max[11n2

, 0] and ifn = 0 then n+1 = 0.

3. given n+1, xn:

dn+1 := (nn+1)xn+dn =

(1 2)(1 n)

2

xn+dn =

(1 )(1 n)

2v+

dn

(1 2)(1 n)

2

Since we set d1 = 0 the values n, dn, xn are determined by 1 denote them by n(1), dn(1), xn(1).

For each 1 denote the security s(x, 1) := min [i(1)x + di(1), d].

Claim 1: for a sufficiently high 1 there exists a SP E in B(v, s(., 1) in which the entrepreneur

gets 12(v d) and the outside investor gets d.

A pair of prices (xE, xb) is a stationary pair if

v xE = (v xb)

xb s(xb) = (xE s(xE))

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As long as x s(x) is strictly increasing, a pair of stationary strategies in which each player

offers the same price (xE, xb) and rejects any offer less then the continuation value is a SP E

when (xE, xb) are a stationary pair

The pair of prices, xE := d +1

1+ (v d) and xb = I

+ 1+ (v d) , are a stationary pair if

s(xj ) = s(xE) = d. Stationary strategies that offer these price give rise to the desired payoffs.

Therefore, I will show that for a sufficiently high 1 there exists x < xj such that s(x, 1) = d.

In words: the security s(x,1) reaches the point d early enough, that is before xj .

(i) As long as n > 0 the kink points are independent of1 and for n 2:

xn = v nv

(1 + ) ( n + 1)

To see this, take 1 < 1, using a recursion on the construction of dn,n:

dn

1 n= (1 )v (1 2) +

dn1

1 n1

applying the recursion and because we set d1 = 0 we get:

dn

1 n=

(1 )v (1 2)

1 + + 2 + 3 + ... + n2

= (1n1)v(1+)(1n1)

by construction:

xn =v

(1 + )+

(1 + )

dn

1 n

and therefore

xn = v nv

(1 + ) ( n + 1)

(ii) Therefore, since xj < v, for sufficiently small there exists 1 and n such that xn > xj

(this implies that n > 0). This follows from the above formula: limn!1

xn = v and from

the construction, 8n < 1 : lim1!1 n(1) = 1. Let n := min[n : xn > x

b ], notice that

xn1 xb < xn, let

0 be the lower bound such that if1 > 0: n > 0. The crucial point

is that n is independent of1, as long as 1 > 0.

From the construction, 8n < 1 : lim1!1 n(1) = 1 , lim1!1 dn(1) = 0, and therefore

since xb > d there exists an 00 such that if 1 >

00 : n(1)xb + dn(1) > d. Choose

1 > max[0, 00] and we have that s(xb , 1) = min[nx

b + dn , d] = d.

Since s(xb , 1) = s(xE,1) = d, the prices (x

b , x

E) are a stationary pair, which concludes the

first claim.

Claim 2 : for 1 > max[0,00] the game B(v, s(x,1)) has a unique SP E

Fix 1 > max[0, 00] and a security s := s(x, 1).

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It follows immediately from Rubinsteins proof (Osborne and Rubinstein (1990)) that B(v, s)

has a unique SP E if i) x s(x) is strictly increasing, and ii) there exists a unique stationary

pair.

Since (i) holds by construction, I show that B(v, s) has a unique stationary pair.

A xE, xb are a stationary pair if:

xE = (1 )v + xb

xb s(xb) = (xE s(xE))

Together these imply:

(1 2)xb s(xb) + s ((1 )v + xb) = (1 )v

Define the function g to be the LHS:

g(x) := (1 2)x s(x) + s ((1 )v + x)

By construction g(xb) = (1 )v , we want to show that this is the unique solution to

g(x) = (1 )v. The boundary conditions are: g(0) < (1 )v < g(v).

Both x and (1 ) + x lie on the first segment as long as x < 1+ v

(because then

(1 ) + x < x1). Denote by x0 the point at which s goes flat: nx

0 + dn = d.

1. for all x s.t. x 1+ v

: g is increases: The reason is that x and x + (1 )v are on

the same segment, therefore s0(x) = s0(x + (1 )v) = 1 < 1 and g0 = (1 1)(1

2) > 0.

2. for all x such that

1+v < x x

0

g decreases. For x x0

we have that xn1 x xnand n is such that n > 0 . Which implies that x and x + (1 )v lie on different segments.

This follows because x + (1 )v xn + (1 )v and as long as n > 1 we have that

xn + (1 )v > xn+1. The inequality is true because:

xn+1 xn = (v

(1 + ) )(1 ) +

(1 + )(

dn+1

(1 n+1)

dn

(1 n)) = (1 )v (1 2)

If n = 1 we have that x + (1 )v > x1.

Therefore s0(x) = n and s0 ((1 )v + x) n+1. Which means that g

0(x) 12n +

2n+1 0.

3. For all x such that x0 < x g increases, because s0(x) = s0(x + (1 )v) = 0

The solution is unique because g( 1+v

) < (1 )v, and for all x > xb : g(x) > (1

)v.

Proof of proposition:

Lemma 1 establishes an upper bound: 12(v I). Lemma 2 establishes a lower bound:12(v I).

The reason is that the entrepreneur can make an offer that guarantees a continuation payoff

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that is arbitrarily close to 12(v I). The offer consists of the security s, that is contracted to

implement a continuation sub-game B(v, s) that has a unique SP E in which the entrepreneur

gets 12(v I ) and the outside investor gets I + I (lemma 2). Since the outside investor

always accepts the offer when > 0 , the lower bound follows. If = 0, the outside investor

breaks even and the entrepreneur can do no better that 12(v I) (lemma 1). Therefore, such

an offer is a SP E of the entire game that achieves the bound.

7.2 Many Buyers

The first lemma establishes that if there is a single buyer left, and the debt is 0 d < v then

there is a unique SP E8

Lemma. The gameB(1, d , v) withv > d has a unique SP E, the proposing party gets 11+ (vd)

and the receiving party gets 1+ (v d)

Proof. The pair of prices xE = d+1

1+(vd) and xj = d+

1+

(vd) are the unique stationary

pair, and since preferences satisfy required assumptions, there is a unique SP E (Osborne and

Rubinstein). The party that makes the initial proposal gets 11+ (v d) and the other party

1+ (v d).

7.2.1 Proof of main proposition

Proposition. In all sequential equilibria with pure strategies and passive beliefs of the game

B(n ,d ,v), agreement is immediate and the aggregate price is the same.

The proof is established by 4 Lemmas. Let be a pure strategy sequential equilibrium with

passive beliefs:

Lemma. The seller makes n offers that are all accepted.

Proof. By induction on n. If n = 1 given by first lemma.

Induction step: assume the seller makes offers and m < n buyers accept.

No agreement with any buyer is not a SP E outcome, because the seller has a deviation.

Let t be the first period in which any offer is accepted. By the induction hypothesis, the game

ends in at most t + 2 periods.

The remaining n m make positive profits. The seller either makes an offer at period t or

period t + 1, by induction all remaining buyers accept the sellers offer, and no-one accepts a

price x = v, (the buyer can always reject the offer, be the only buyer left, and makes positive

profits, assuming full liability).

The seller can offer a slightly higher price to all buyers that reject. Since beliefs are passive,

those buyers expect to get a continuation price that is the same and therefore accept8With a single player the SPE and sequential equilibria are the same

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Lemma. The seller makesn symmetric offers given by: xs(n, d) =

8 d0.

From previous step, we know that the game ends in the next round.

First, if the continuation game B(n k, d0) is low debt game, then we know that continuation

price is v1+ , and the sellers continuation value is: (n k)v

1+ d0.

Consider the deviation by a buyer whose offer is rejected, in which he offers the price x = v1+ +.

If the seller accepts, the continuation game B(n k 1, d0 x) has k 1 buyers, with debt

d0 v1+ . By transition lemma, this game is also low debt, thus the seller still gets the same

price x = vv+ from the remaining k 1 buyers.Thus, if d

0 > x, the sellers continuation value

from accepting is:

((n k 1)v

1 + d0 +

v

1 + + )

which is strictly better. If is sufficiently small, < v1+ (1 ), then the buyer is also better

off: v x > (v v1+ ). A similar argument can be made in the cases in which d0 = 0, or if

d0 x.

Second, if the continuation game B(n k, d0) is a high debt game, the buyers continuation

price is d0+v

1+(nk) and the sellers continuation value is(nk)vd0

1+(nk) .

9To relax notation, since v is fixed, I denote a bargaining game only be the state variables.

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Consider a deviation in which a buyer whose offer is rejected, offers a slightly higher price then

his continuation price: x = d0+v

1+(nk) + .

The seller is strictly better off for any > 0. To see this, there are two cases to consider: the

continuation game with n k 1 buyers and debt of d0 x is high debt game for some (if

its a high debt game for some , its a high debt game for all 0 < ), or for every > 0 the

continuation game is a low debt

Case 1: If B(n k 1, d0 x) is a high debt game for some > 0, it is a high debt game for

all 0 < then the sellers continuation value is: vE(n k 1, d0 xk) = (nk1)vd

0+x1+(nk1)

and

notice that

(nk1)vd0+x1+(nk1)

>(nk)vd0

1+(nk)() x > d

0+v1+(nk) and therefore the seller is strictly better off for

any . Also, for a sufficiently low epsilon the buyer is also strictly better off.

Case 2: If B(n k 1, d0 x) is a low debt game for all > 0 then we have that 8 > 0 :v

1+ (n k 1) d0+v

1+(nk) + !v

1+ (n k 1) d0+v

1+(nk) , and therefore accepting a price

of d0+v

1+(nk)

moves us from a high debt game to a low debt game. By the transition lemma, it

must be that d0+v

1+(nk)> v1+ ()

(1 + )(1 + (n k))((1 + b)d0 (n k 1)v) > 0 ()

(1 + b)d0 (n k 1)v > 0

Also, the sellers continuation value in case he accepts is: (n k 1) v1+ d0 + x which is

greater then what he gets if he rejects if:

(n k 1)v

1 + d0 + x >

(n k)v d0

1 + (n k)

Now, since x = d0+v

1+(nk)+ then we have that the above hold true when:

(1 + )d (n k 1)v > 0

which holds from the previous inequality.

The next lemma pins down the SP E price

Lemma. The seller is indifferent between accepting all offers and rejecting all offersP

xid =

(nxs(n, d) d).

Proof. The proof establishes a sequence of claims:

Notice that from the previous stage, the outcome of the continuation game is uniquely pinned

down by xs. Let vs(m, d0) be the sellers value entering a round in which he makes an offer,

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their are m buyers left and the outstanding debt is d0: vs(m, d0) := mxs(m, d

0) d0

vs(m, d0) =

8 0 then the k + 1 highest offers must all be the same: x1 = x2 = .. = xk+1.

Since k is lowest number of offers that the seller is indifferent between accepting and rejecting,

if j < k the seller strictly prefers accepting k highest offers to j highest offers. Therefore, if

x1 > xk+1 then buyer 1 can decrease the price and still be accepted.

Claim 3: k = 0

Let x be the k + 1 highest offers. The crux of the proof is that is that the seller can not strictly

prefer accepting k + 1 offers to k offers, because then the seller would also (st) prefer accepting

k + 1 to all offers. This places an upper bound on how high the price x can be: if x is too

high the seller would prefer accepting k + 1 offers to k offers, and this upper bound leads to a

contradiction. Formally, it must be that:

vE(n k, d kx) vE(n k 1, d (k + 1)x) (3)

and lemma 1 provides the contradiction:

Lemma1 : if vs(n k, d kx) vs(n k 1, d (k + 1)x) for some k, then it must be:

vs(n, d) vs(n k, d xk)

Lemma1 : if vE(n k, d kx) vE(n k 1, d (k + 1)x) for some k, then it must be:

vE(n, d) vE(n k, d xk)

Proof of lemma 1:

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Proof. To prove lemma 1, assume that x and k are such that vE(n k, d kx) vE(n k

1, d (k + 1)x), to show that vE(n, d) vE(n k, d kx), I break it down to three cases.

Case 1: B(n k, d xk) is high debt

The value vE(n k, d xk) =(nk)v(dxk)

1+(nk)

Case 1a: B(n (k + 1), d x(k + 1)) is also high debt games.

The value vE(n (k + 1), d x(k + 1)) =(nk+1)v(dx(k+1))

1+(n(k+1))

And (1) implies that x d+v1+n .

If B(n, d) is a low debt game, then v(n, d) = n v1+ d and v(n k, d xk) =(nk)v(dxk)

1+(nk)

and as long as x d+v1+n the first is greater then the second. (mathematica)

If B(n, d) is a high debt game, we have that v(n, d) = d+v1+n and v(n k, d xk) =(dxk)+v1+(nk)

and again, as long as x d+v1+n the first is greater then the second.

Case 1b: B(n (k + 1), d x(k + 1)) is a low debt game.

By the transition lemma x v1+ otherwise moving from k to k + 1 buyers would leave us in

a high debt game. Applying the transition lemma again, it must that , B(n, d) is a high debt

(otherwise, since x v1+ accepting k offers moves us to a low debt game).

Since B(n, d) is high debt, it must be that (n 1) v1+ < d

From 1 it must be that vE(n k, d kx) vE(n (k + 1), d (k + 1)x):

(n k)v (d xk)

1 + (n k) (n (k + 1))

v

1 + (d (k + 1)x)

()

x

v + (n k)[(1 + )d + (2 + k n)v)]

(1 + )((1 + (1 + k)(n k)))

Second since B(n (k + 1), dx(k +1)) is low debt game: (nk2) v1+ dx(k + 1) which

implies:

x (1 + )d (n k 2)v

(1 + )(1 + k)

Notice that (1+)d(nk2)v

(1+)(1+k)>

v+(nk)[(1+)d+(2+kn)v)](1+)((1+(1+k)(nk)))

()(1+)d(n1)v

(1+)(1+k)[1+(1+k)(nk)]> 0

() d > v1+ (n 1)

Which is always true since B(n, d) is high debt game.

Case 2: B(nk, dxk) is a low debt game, and dxk > 0. This implies that vE(nk, dxk) =(n k) v1+ (d kx).

If B(n, d) is a low debt game, it must be that x = v1+ . To see this, we can rewrite

vE(n k, d xk) = nv

1 + d + (x

v

1 + )k = vE(n, d) (x

v

1 + )k

Since vE(n, d) vE(n k, d xk) we have that x v

1+ . The transition lemma implies

that B(n (k + 1), d x(k + 1)) is also high debt game (because x v1+ ). But since

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vE(n k, d kx) vE(n (k + 1), d x(k + 1)) it must be that

(n k)v

1 + (d kx) (n (k + 1))

v

1 + (d (k + 1)x) ()

x v

1 +

Which implies that x = v1+ . Now, since x =v

1+and both B(n, d) and B(n k, d xk) are

low debt games we have that:

vE(n, d) = vE(B(n k, d xk) () x =v

1 +

and we are done with this case.

IfB(n, d) is a high debt game, then from transition lemma: x > v1+ , and since B(nk, dxk)

is low debt, the transition lemma implies that B(n (k + 1), d x(k + 1)) is also low debt.

Since, vE(n k, d kx) vE(n (k + 1), d x(k + 1)) it must be that

(n k) v1 + (d kx) (n (k + 1)) v1 +

(d (k + 1)x) ()

x v

1 +

Which is a contradiction.

Case 3: B(n k, d xk) is a low debt game kx d

If accepting k offers moves us to a game without debt. Then for E to not prefer accepting k + 1

offers it must be that x v1+ . The transition lemma implies that B(n, d) is a low debt game,

which means that if E rejects all offers he gets: vE(d, n) = (nv

1+ d) but the most he can

get by accepting k offers is n v1+ d which is strictly less.

7.2.2 Existence

Define the following stationary strategies:

For a sub-game with m buyers and debt d0, the seller offers:

xs(m, d0) =

8 1+ (v d + (m 1)d+v1+m )

Beliefs are passive, the buyers off-equilibria beliefs are that other buyers get the equilib-

rium offers.

Proposition. The above strategy are a sequential equilibrium

Proof. By construction, buyers beliefs are consistent with equilibrium play.

Start with a seller:

1. It is never profitable for the seller to offer a lower price. By induction on B(n, d)

2. The seller can deviate, and offer a higher price to some buyer, or a subset of buyers. Assume

he offer k prices that are accepted, the subsuming sub-game has n k buyers, with a debt of

d0 = n k. There are two options, either B(n, d) is a low debt game or high debt game.

IfB(n, d) is a low debt game then by the transition lemma ifk buyers accept a price of v1+ , the

subsuming game is also a low debt game, and the seller gets a price that is less then v1+ ....show

this....

If B(n, d) is a high debt game then if k buyers accept a price of x = d+v1+n the subsuming sub-

game, B(n k , , d kx) can either be low or high debt. If it is a high debt game, the sellers

value is vE(n k, d xk), and since x =d+v1+n we have that vE(n k, d xk) = vE(n, d). If

B(n k, d kx) is a low debt game, then the sellers continuation price is less then v1+ and

since B(n, d) is a high debt game: d+v1+n >v

1+ , and therefore he is better off offering a price

of x to the n k buyers

3.. The seller can deviate and accept a subset of off

ers. Assume the seller accepts 0< k < n

offers.Again there are two options:

3a. B(n, d) is a low debt game: the buyers offer a price of xb = (v

1+ ) +(1)d

n.

Assume first that accepting k offers ofx move the game to a low debt game i.e. B(nk, dkxb)

is low debt, the sellers total revenue from the n k units tomorrow is (n k) v1+ but he can

sell those units today for a revenue of (n k) v1+ + (n k)(1)d

n, which is more.

Second, if accepting k offers of x moves the game to a high debt game, i.e. B(n k, d xk)

is high debt, in which case his profits are vE(n k, d xbk) = (

(nk)v(dxbk)

1+(nk) ) and if he

accepts all offers he gets (n v1+ d) and it holds true that for xb = (

v1+ ) +

(1)dn

we have

that (nk)v(dxk)

1+(nk) n v

1+ d ()

n[k bkn bn + bn2]v (1 + )d[n(n k) + k(1 )]

(1 + b)n(1 + b(n k)) 0

Since B(n, v) is a low debt game, we have that (1 + )d (n 1)v and therefore we have that

LHS enumerator:

n[kbknbn+bn2]v(1+)d[n(nk)+k(1)] n[kbknbn+bn2]v(n1)v[n(nk)+k(1)] = (1)

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and we are done.

3b. if B(n, d) is a high debt game, the buyers offer a price xb = (d+v1+n ) + (1 )

dn

then accepting a subset of k offers, can either move the game to a low debt one or a high debt

game.

If accepting a subset of prices moves the game to a low debt game, the notice that the continu-

ation price is v1+ which is less then xb and therefore, the sellers total revenue is greater if he

accepts all prices today then rejecting a subset.

If accepting a subset of prices moves the game to a high debt game, the we have that the sellers

continuation value is ((nk)v(dkx

b)

1+(nk) ) and if the seller

accepts all offers he gets: (nvd1+n). And the latter is greater when:

nv d

1 + n

(n k)v (d kxb)

1 + (n k)() xb

d + v

1 + n

But since xb = ( d+v1+n) + (1 ) dn we have that xb d+v1+n () dn < d+v1+n () nv > d

4. The buyers SP E offer is xb = xs(n, d)+(1)dn

. A buyer can deviate and offer a different

price and get rejected, in which case the continuation price is s(n, d), but this is not profitable

because the continuation price is always higher:

xb < xs(n, d) ()d

n< xs(n, d)

To see this: If B(n, d) is low debt, d (n 1) v1+ , and xs(n, d) =v

1+ and therefore the

above holds. If B(n, d) is high debt, d > (n 1) v1+ and xs(n, d) =d+v1+n and we have that

n d+v1+n > d () nv > d. And again this holds.

5. A buyer can deviate and reject the offer, but by construction a buyer is indifferent between

accepting and rejecting given that everyone else rejects.

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