The Effect of External Finance on

the Equilibrium Allocation of Capital*

Heitor AlmeidaNew York Universityhalmeida@stern.nyu.edu

Daniel WolfenzonNew York Universitydwolfenz@stern.nyu.edu

(This Draft: September 15, 2003 )

Abstract

We develop an equilibrium model to understand how the efficiency of capital allocation dependson outside investor protection and Þrms� external Þnancing needs. We show that when capitalallocation is constrained by poor investor protection, an increase in Þrms� external Þnancing needsmay improve allocative efficiency by fostering the reallocation of capital from low productivityprojects to high productivity ones. We also Þnd novel empirical support for this prediction.

Key words: capital allocation, Þnancial development, investor protection, external Þnance, internal capitalmarkets.JEL classiÞcation: G15, G31, D92.

*We are particularly grateful to an anonymous referee for insightful comments. We also wish to thank YakovAmihud, Ulf Axelson, Murillo Campello, Douglas Diamond, Francisco Gomes, Arvind Krishnamurthy, HolgerMueller, Walter Novaes, Raghuram Rajan, Andrei Shleifer, Jeff Wurgler, Bernie Yeung, and conferenceparticipants at the AFA 2002 meetings, Berkeley University Haas School of Business, Conference in thePolitics of Corporate Governance, Stanford Graduate School of Business, University of Maryland, Universityof North Carolina at Chapel Hill, the Texas Finance Festival, the WFA 2003 meetings, and the Þnanceworkshop at New York University for their comments and suggestions. Victoria Ivashina and Sadi Ozelgeprovided excellent research assistance. We also thank Matias Braun for providing us with data.

1 Introduction

Recent research shows that a country�s Þnancial system affects economic growth (King and Levine,

1993a; Levine and Zervos, 1998; Rajan and Zingales, 1998; Demirguc-Kunt and Maksimovic, 1998;

Beck, Levine and Loayza, 2000a). What are the characteristics of a country�s Þnancial structure

that are more conducive to economic development? A number of authors suggest that banks do

a better job than atomistic markets at raising large amounts of capital (Gershenkron, 1962), at

monitoring and at gathering information about Þrms. Others emphasize the potential advantages of

markets versus banks in effectively allocating capital (e.g. Allen, 1993). Levine (1997) argues that

the important dimension of a Þnancial system is the quality of the Þnancial services and not the

identity �market or bank� of the provider. La Porta et al. (1997, 1998) suggest that the efficiency

of the legal system in protecting outside investors is the main variable driving the quality of the

Þnancial system.1

However, there are other characteristics of a country�s Þnancial structure that can affect real

variables. In this paper we contribute to the debate on the real effects of Þnancial structure by

suggesting an additional factor that inßuences economic efficiency: Þrms� external Þnancing needs.2

In particular, we analyze how this factor affects the efficiency of capital allocation.3

We develop a model in which capital allocation is constrained by the extent of legal protection

of outside investors against expropriation by the manager or �insiders�(La Porta et al. 1997, 1998).

When investor protection is low, there is a limit on the fraction of cash ßows that entrepreneurs

can credibly commit to outside investors (limited pledgeability of cash ßows). Because of this

friction, the economy has a limited ability to direct capital to its best users: high productivity

projects cannot pledge a sufficiently high return to attract capital from low productivity projects.

Our main result is that, starting from this situation in which capital reallocation is constrained by

limited pledgeability, higher external Þnancing needs can improve allocative efficiency by increasing1See Allen and Gale (2000) and Demirguc-Kunt and Levine (2001) for a broad discussion of different aspects of

Þnancial structure.2Consistent with our focus on the relation between external Þnance and economic efficiency, Goldsmith (1969)

considers the relation between internal and external Þnance as a deÞning characteristic of Þnancial structure: �Ofparticular importance for the study of Þnancial structure is the distribution of total sources and funds of varioussectors and subsectors by form and by partner. The tool here is the sources-and-uses-of-funds statement...[that]makes possible a determination of the share of internal Þnancing (savings) and external Þnancing (borrowing orissuance of equity securities)� (Goldsmith 1969, p. 29).

3The empirical results in Beck, Levine and Loayza (2000b) suggest that capital allocation is an important channelthrough which Þnance affects economic growth.

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the liquidation of low productivity projects and making more capital available to high productivity

projects.

The need for external Þnance can improve the allocation of capital even when it introduces

inefficiencies at the Þrm level. The reason is that, due to limited pledgeability, outside investors

prefer to liquidate a mediocre project because they can only get a fraction of the return. However,

in the absence of external Þnance, insiders prefer not to liquidate such projects because they get to

keep the full return. To attract sufficient capital when its Þnancing needs are large, the Þrm has to

commit to liquidate. Thus, Þrms that need to raise external Þnance are liquidated more often than

internally Þnanced Þrms (as in Diamond, 1991). This excessive liquidation is suboptimal for a Þrm

in isolation, but it might be socially beneÞcial because the released capital ßows from mediocre to

high productivity projects, improving capital allocation.

An important result of our model is that limited investor protection constrains the efficiency of

capital allocation. Consistent with this prediction, Wurgler (2000) Þnds evidence that measures of

Þnancial development and protection of outside investors are positively related to the efficiency of

capital allocation. However, this result is not particular to our paper. Levine (1991), Bencivenga,

Smith and Starr (1995) and Shleifer and Wolfenzon (2002) also predict that Þnancial frictions, such

as limited pledgeability, reduce allocative efficiency. The novel empirical predictions of our theory

regard the role of external Þnance.

Controlling for the level of investor protection, the external Þnancing needs of Þrms in a given

country should have an independent effect on the efficiency of capital allocation. We show that

as long as the degree of pledgeability is not too low, the resources that are released by the higher

liquidation induced by external Þnanciers will Þnd their way to more productive activities. This

argument suggests that countries where Þrms have higher Þnancing needs will have a more efficient

allocation of capital.

We present new evidence that is consistent with this implication. We build a proxy for the

aggregate need for external Þnance of the manufacturing sector of a country using Rajan and

Zingales�s (1998) industry-level measure of external Þnance dependence, and the shares of the

different industries in total manufacturing sector output (the country�s industrial structure). We

then run cross-country regressions of Wurgler�s (2000) measure of the efficiency of capital allocation

(the elasticity of industry investment to industry value added) on our country-level measure of

2

external Þnance dependence and controls.

We show that the need for external Þnance is positively correlated with the efficiency of capi-

tal allocation, even after controlling for variables such as economic and Þnancial development, or

measures of investor protection. This correlation also seems to be driven by the component of the

investment elasticity that is more directly related to the model�s predictions, the within-year compo-

nent. This component provides a measure of how investment responds to value-added shocks across

different industries in the same country, and thus better captures the idea of capital reallocation

that we model. Finally, we show that the correlation remains signiÞcant after instrumenting our

index of external Þnancing needs with measures of a country�s commodity endowments (Easterly

and Levine, 2003), which we argue are exogenous determinants of a country�s industrial structure,

and thus of the country�s external Þnancing needs. These results are consistent with the model�s

main prediction.

Our model complements earlier equilibrium models that examine the effect of impediments to

Þnancial market operations on the efficiency of capital allocation (e.g. Levine, 1991; Bencivenga,

Smith and Starr, 1995). There are two aspects of our analysis that distinguish our paper. First, we

study the effect of differences in external Þnancing needs on the allocation of capital. Second, we

focus on limited pledgeability of cash ßows as the main source of imperfections, while these earlier

papers have focused on transaction costs of equity market transactions. Our focus on pledgeability

is crucial for our results. While models without limited pledgeability but with transaction costs

on Þnancial transactions also generate an inefficient equilibrium allocation of capital, they would

not predict the same effects of external Þnance that we obtain in this paper. Limited pledgeability

generates a conßict of interests between insiders and outsiders. As explained above, while insiders

prefer to continue mediocre projects outside investors prefer to liquidate them. As the aggregate

level of external Þnance increases and control shifts to outside investors, liquidation increases. Other

frictions to Þnancial market transactions affect equally all of the Þrm�s investors, and thus do not

generate a similar conßict of interest.

Shleifer and Wolfenzon (2002) also analyze the effect of limited pledgeability on the allocation

of capital. Even though our paper uses the same type of friction, they do not consider the effect of

external Þnancing needs on the allocation of capital, and more importantly they do not focus on the

reallocation of capital across ongoing projects of different productivities like we do. Our results on

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external Þnancing needs clearly require a model in which capital can be reallocated among existing

projects of different productivities, and thus the set up in Shleifer and Wolfenzon cannot generate

our results.4

Finally, our paper is related to previous arguments about the beneÞts of external Þnance for Þrm

performance. For example, Jensen (1986) argues that the presence of debt mitigates managerial

agency costs. Our argument differs in three dimensions. First, our model is an equilibrium one

so we can analyze the effect of external Þnance on economy-wide outcomes, and not just on the

particular Þrm seeking external Þnance. Second, the beneÞt of external Þnance that we propose

is a social beneÞt, not a private one. In our framework, the individual Þrm that raises external

Þnance does not beneÞt from it. Rather, other Þrms in the economy beneÞt from the increased

probability of liquidation of the Þrms that raise external Þnance. In contrast, Jensen suggests that

debt beneÞts shareholders of the Þrm itself. Finally, Jensen�s argument applies mostly to Þrms with

dispersed ownership, where managers may be free to use the Þrm�s excess cash ßows to invest in

potentially negative NPV projects. In contrast, in our model external Þnance is socially beneÞcial

even when introduced in Þrms with 100% ownership by the entrepreneur.

In the next section we characterize the effect of limited pledgeability and of changes in exter-

nal Þnancing needs in the equilibrium allocation of capital. Subsection 2.1 contains an intuitive

description of the main effect. Section 3 summarizes the empirical implications of our model, and

presents some evidence consistent with these implications. Section 4 presents our Þnal remarks.

All the proofs are relegated to the Appendix.

2 Limited pledgeability, external Þnance and the equilibrium al-location of capital

In this section we develop our theoretical framework to analyze the effect of limited pledgeability

and Þrms� external Þnancing needs on the equilibrium allocation of capital.

There are two types of agents in the model. First, there is a set J with measure 1 of agents

(�entrepreneurs�), each with one project opportunity (projects are described below). There is also

a different set of agents with no project opportunities (�investors�). All agents are risk-neutral4Our paper is also related to the literature that analyzes the effects of Þnancial intermediaries on the allocation

of capital (e.g. King and Levine, 1993b; Galetovic, 1995; Boyd and Smith, 1999). This literature, however, has alsonot focused on the effects of external Þnancing needs, or on capital reallocation under limited pledgeability.

4

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Entrepreneursenter contractsto raise Z

6

Productivity realized

Projects liquidatedaccording tocontracts

Entrepreneursannounce P�s

Investors allocatecapital

Cash ßowsrealized andpayments

t0 t1 t2

Figure 1: Timing of events

and do not discount the future. There is a single good in this economy (�capital�). At the initial

date t0 the aggregate amount of capital 1 +K (with K > 1) is held by the two types of agents.

Each entrepreneur has an endowment of 1 − Z units (with 0 ≤ Z < 1). The remaining K + Z is

distributed among investors.

The timing of events in the model is shown in Figure 1. At date t0 entrepreneurs issue Þnancial

contracts to raise the necessary capital to pay the investment cost of their projects. Information

about the proÞtability of projects arrives at a later date t1, and cash ßows are realized at the Þnal

date t2. In light of the new information, capital can be reallocated among projects before cash

ßows are realized. The efficiency of this �reallocation market� is the main object of our analysis.

Technologies Capital can be stored with no depreciation. In addition, capital can be invested

in two types of technologies that pay off in period t2. We refer to the Þrst type of technologies as

projects. Each project is of inÞnitesimal size. At date t0 a project requires an investment of one

unit of capital. At date t1, a project can be liquidated, in which case the entire unit of capital is

recovered. If a project is not liquidated, it can receive additional capital or can simply be continued

with no change until date t2. At date t2, projects generate cash ßows.

Projects have different productivities. Each project�s type, s, can take three values: H (high

productivity projects), M (medium productivity projects), and L (low productivity projects) with

probabilities pH , pM , and pL (with pH+pM+pL = 1), respectively. The projects� type is not known

at date t0 (not even to the entrepreneur who owns it) but it is revealed to everyone in the economy

at the beginning of date t1. The probability distribution is independent across projects so that, at

date t1, exactly a fraction pH , pM and pL of the projects are of type H, M and L, respectively.

5

For simplicity, we assume drastic decreasing returns to scale. Depending on its type, each project

generates YH , YM or YL (with YH > YM > YL ≡ 0) per unit invested, but only for the Þrst two

units (i.e., the unit invested at t0 and the unit that is potentially invested at t1). Additional units

invested generate no cash ßows.

The second technology (�general technology�) is available to all agents in the economy at date

t1. We deÞne x(ω) as the per-unit pay off of the general technology with ω being the aggregate

amount of capital invested in it.

Assumption 1 The general technology satisÞes:a. ∂[ωx(ω)]

∂ω ≥ 0 c. x(1 +K) > 1b. x0(ω) ≤ 0

We assume that total output is increasing in the amount invested in the general technology

(assumption 1(a)) but that the per-unit payoff is decreasing (assumption 1(b)). Assumption 1(c)

guarantees that no capital is invested in storage at date t1.

To rank the productivity of the projects and the general technology, we assume that:

Assumption 2 YM > x(0)

The most productive technologies are good projects, followed by medium projects. By assump-

tion 2, medium projects are more productive than the general technology regardless of the amount

invested in the latter (since x(0) > x(ω) by assumption 1(b)). Finally, the bad projects are the

least productive.

Capital market at date t0 In the capital market at date t0, entrepreneurs raise Z from outside

investors in exchange for a set of promised payments. To focus on imperfections brought about

by limited pledgeability of payoffs (described below), we assume that initial contracts can be made

fully contingent on dates and type of project. A contract offered by entrepreneur j is deÞned by

a vector {qjs,Dj1s, Dj2s}s=L,H,M where qjs is the probability that the project is liquidated when the

realized productivity is s, Dj1s is the payment to investor at date t1 when the project is liquidated

(since projects generate no cash ßow at date t1, the payment Dj1s can be positive only when there

is liquidation), and Dj2s is the payment to the investor at date t2.

6

Entrepreneurs approach investors and make them a take-it-or-leave-it offer.5 Since initial in-

vestors can always store their capital from date t0 to date t1 and then invest it in the reallocation

market from date t1 to date t2, they accept the contract only if it offers them a payoff of at least

R∗Z, where R∗ is the expected return in the date t1 reallocation market. Entrepreneurs maximize

their payoff subject to the participation constraint of the investor.

Capital market at date t1 At date t1 the type of the project is realized and liquidation occurs

according to the speciÞed probabilities in the contract. After liquidation occurs, date t1 investors

(investors that stored capital from date t0 to date t1, and agents holding the proceeds from liq-

uidated projects) allocate their capital to continuing projects and to the general technology. The

total amount of capital in the hands of date t1 investors is K + T , where T is the capital released

from liquidated projects. That is, T =Rj /∈C dj with C being the set of projects that continue.

Each continuing project j announces P j , the amount it pays at date t2 for the Þrst unit of

capital. Projects have no use for additional units of capital and thus offer to pay zero for those

units. Next, date t1 investors allocate their capital to either the projects or to the general technology.

An allocation in the capital market can be described by rj , the probability that project j ∈ C getscapital and ω, the amount allocated to the general technology. An equilibrium allocation must be

consistent with date t1 investor maximization and satisfy the market clearing condition

ω∗ +Zj∈C

rjdj = K + T. (1)

Limited pledgeability of cash ßows We consider an imperfection at the Þrm level: Þrms

cannot pledge to outside investors the entire cash ßow generated by projects. In particular, for

date t2 cash ßows, we assume that only a fraction λ of the returns of the second unit invested is

pledgeable.6. For the date t1 reallocation market, this assumption implies that, when its realized

productivity is s, a continuing project j can offer at most

Pjs ≡ λYs −Dj2s. (2)

5The assumption that entrepreneurs have all the bargaining power can be justiÞed by the fact that there is excesssupply of capital at date t0.

6The assumption that the cash ßows from the Þrst unit are not pledgeable is made only for simplicity. It willbecome clear our main results do not hinge on this assumption (see footnote 14)

7

We also assume that the liquidation proceeds are fully pledgeable (we justify this assumption

below).

The limited pledgeability assumption can be justiÞed as being a consequence of poor investor

protection, as shown in Shleifer and Wolfenzon (2002). In their model, insiders can expropri-

ate outside investors, but expropriation has costs that limit the optimal amount of expropriation

that the insider undertakes. Higher levels of protection of outside investors (i.e., higher costs of

expropriation) lead to lower expropriation and consequently higher pledgeability. To justify the

full pledgeability of cash ßows in the case of liquidation, we make the natural assumption that

liquidation proceeds are easier to verify than project returns.

Limited pledgeability also arises in other contracting frameworks. For example, limited pledge-

ability is a consequence of the inalienability of human capital (Hart and Moore 1994). Entrepreneurs

cannot contractually commit never to leave the Þrm. This leaves open the possibility that an en-

trepreneur could use the threat of withdrawing his human capital to renegotiate the agreed upon

payments. If the entrepreneur�s human capital is essential to the project, he will get a fraction of

the date t2 cash ßows. A natural assumption is that the entrepreneur�s human capital is not needed

to liquidate the Þrm. This justiÞes the fact that the entire liquidation proceeds are pledgeable, but

only a fraction of the date t2 cash ßows are. Limited pledgeability is also an implication of the

Holmstrom and Tirole (1997) model of moral hazard in project choice. In that model project choice

cannot be speciÞed contractually. As a result, investors must leave a high enough fraction of the

payoff to entrepreneurs to induce them to choose the project with low private beneÞts but high

potential proÞtability. Since liquidation in our framework can be speciÞed contractually, there is

no need to leave anything to the entrepreneur when liquidation occurs. Again, this justiÞes the full

pledgeability of liquidation proceeds.

2.1 Illustration of the main effect

In this section we present an informal argument to illustrate the main mechanism of the model. In

the next section we formally characterize the equilibrium. The reader who is not interested in the

formal derivation of the equilibrium can safely skip section 2.2 after reading the current section,

and proceed directly to section 2.3.

It is useful to think about the date t1 reallocation market in terms of demand and supply of

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funds.7 On the supply side, there is a total of K +T to be allocated with T being the capital from

liquidated projects. This amount is determined by the contract that entrepreneurs enter at date t0

to raise the necessary capital Z. Since continuation of low productivity projects generates no cash,

these projects are always liquidated. The liquidation proceeds are supplied to the market since

the entrepreneur has no use for them. As a result, aggregate liquidation proceeds are at least pL.

Figure 2 depicts the demand and supply schedules for two different supply scenarios. The supply

of capital (vertical line) in panel A is K + pL while total supply in panel B is higher.

The demand for capital arises from good and medium projects and from the general technology.

To raise capital at date t1, Þrms pledge part of the project�s date t2 cash ßows. Due to limited

pledgeability, Þrms with high (medium) productivity projects can offer a maximum of λYH (λYM )

per unit of capital (for the sake of the argument we assume that D2H = D2M = 0) and the general

technology offers x(ω). The aggregate demand schedule (the downward sloping lines in panels A and

B) shows the capital demanded for different levels of market return. If we assume that x(ω) > λYH

for low ω, as we do in Figure 2, some capital must be allocated to the general technology before

the projects get additional capital (the general technology can initially offer a higher return than

good and medium projects). As additional capital is allocated to the general technology, the return

it offers (the required return on capital) decreases. When this return reaches λYH , good Þrms can

start attracting capital. Additional capital is allocated to good Þrms until they all get one unit.

This explains the ßat region of the aggregate demand. The rest of the demand schedule is derived

in a similar way.

In panel A, the market equilibrium rate is R∗ = x(K + pL). In this case, date t1 investors

allocate their capital to the general technology. Even though high productivity Þrms are the best

users of capital, they cannot attract any capital because the return in the market is higher than

the maximum return they can credibly offer (λYH < R∗). Panel A assumes that the supply of

funds is K + pL. That is, only low productivity projects are liquidated. Notice that since good

Þrms do not attract new capital in equilibrium, it would be socially optimal to liquidate medium

productivity projects and supply their capital to the good projects. However, this reallocation does

not happen voluntarily. Firms with medium productivity projects generate cash ßows of YM if they7In this supply and demand framework the existence of an equilibrium is not guaranteed. In the formal charac-

terization of the equilibria, we model the date t1 capital market using the non-cooperative game described above toguarantee the existence of an equilibrium. However, the intuition for both cases is very similar.

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continue. If these projects are liquidated and the unit recovered is invested in the market, they

receive R∗ = x(K + pL). Since x(K + pL) < YM , Þrms with medium productivity projects do not

liquidate but rather continue.

In sum, limited pledgeability distorts capital allocation in two ways. First, good Þrms cannot

attract the capital that is available for allocation. And second, given that good Þrms cannot pledge

enough cash, Þrms with medium productivity projects choose not to liquidate and supply their

capital to the good Þrms.

How can Þrms with medium productivity projects be forced to liquidate their projects? As we

argue in detail in the next section, this is precisely the role of external Þnance at date t0. If the

initial Þnancing requirement is high enough, the only way Þrms are able to repay initial investors is

by having some liquidation at date t1. Moreover, ex-ante, it is less costly to commit to liquidate a

project when its realized productive is medium rather than high. Thus, if liquidation is necessary

for initial Þnancing, the optimal contract will require medium projects to be liquidated Þrst. This

situation is depicted in Panel B. In this panel the supply of capital is higher than in panel A

(K +T > K+ pL) because of increased date t0 Þnancing requirements. In this case, the additional

capital drives down the market return on capital so that good Þrms attract some funds. In other

words, some of the capital from the liquidated medium projects Þnds its way to the good projects,

potentially improving the aggregate payoff.

2.2 Characterization of the equilibrium

We solve the model backwards. Since at date t2 no decisions are taken, we start by solving the

equilibrium in the date t1 reallocation market. Then we solve for the optimal date t0 contracts.

The equilibrium in the date t1 reallocation market We solve the equilibrium for any possible

subgame. After liquidation has occurred, a subgame is deÞned by any set C of projects that

continue and any arbitrary contract for each of these projects. For a continuing project j with

realized productivity s, the relevant part of the contract is Pjs = λYs−Dj2s, the maximum amount

it can offer. In what follows we drop the subscript s.

The equilibrium strategy of date t1 investors is to allocate their capital Þrst to the projects

offering the highest return and then to technologies in decreasing order of the offered return. This

10

implies that there is a cutoff value R∗ such that technologies offering strictly more than R∗ get

capital for sure and those offering strictly less than R∗ do not get capital. Technologies offering

exactly R∗ may or may not get capital. Also, investors are indifferent among all technologies

offering the same return. In particular, they are indifferent among all technologies offering exactly

R∗. If, for example, there are more projects offering R∗ than capital available, any allocation rule

to these projects is consistent with date t1 investors maximization behavior. However, in this case,

the only rule for which an equilibrium exists is one where all the projects that could have offered

more (projects with Pj> R∗) receive capital with probability 1.8

Given this allocation rule, it is optimal for all projects with Pj ≥ R∗ to offer exactly P j = R∗.

These projects beneÞt by raising capital at this price because they generate Ys out of this unit of

capital but pay only R∗ ≤ P j < Ys. They have no incentive to decrease the offer, because offersof less than R∗ receive no capital. Projects with P j > R∗ receive capital for sure so they have

no reason to increase their offer. Projects with Pj= R∗ do not necessarily receive capital with

probability 1 but, due to limited pledgeability, they cannot increase their offer. Finally, the storage

technology receives capital up to the point where its return is R∗.

The above discussion can be summarized in the following Lemma.

Lemma 1 There is a level R∗ such that date t1 investors allocate an amount ω∗ = x−1(R∗) to the

general technology and allocate capital to projects according to the following rule:

rj =

0 if P j < R∗

1 if P j > R∗ or, if P j = R∗ and P j > R∗

r if P j = Pj= R∗

(3)

with r ∈ [0, 1] chosen to satisfy the market clearing condition in equation (1). 9 Projects with

Pj ≥ R∗ offer to pay R∗, and projects with P j < R∗ are indifferent among all feasible offers.

Since, in equilibrium, all date t1 investors receive a return ofR∗, we refer to R∗ as the equilibrium

return of the reallocation market.8 If a project with P

j> R∗ receives capital with probability less than 1 when it offers P j = R∗, then it will be

optimal for this project to offer R∗+ ² and receive capital for sure. Clearly, in this case an equilibrium does not existsince ² can always be made smaller.

9The level of R∗ can be found by solving: x−1(R∗) +

Z{j∈C|P j>R∗}

dj ≤ K + T ≤ x−1(R∗) +

Z{j∈C|P j≥R∗}

dj.

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Date t0 contracts At date t0 an entrepreneur offers the contract that maximizes his payoff

subject to the date t0 investor receiving at least R∗Z at date t2. To check that a set of contracts

{qjs,Dj1s,Dj2s}s=L,M,H for all j ∈ J is an equilibrium, we Þrst use Lemma 1 to Þnd the equilibriumreturn and allocation rule (R∗ and r) at date t1 generated by the proposed set of contracts. Then

we check that, for these R∗ and r, no entrepreneur has an incentive to deviate at date t0 and offer

a different contract.

We focus on symmetric equilibria in which all entrepreneurs offer the same contract. We char-

acterize the equilibrium contracts as a function of the degree of limited pledgeability λ and the

Þnancing requirement Z. Since we consider symmetric equilibria, Pjs (and consequently r

j) is the

same for all Þrms that continue and can take three values depending on realized productivity. We

let these three values be PH , PM , and PL, and those of rj be rH , rM and rL. In the text, we focus

on the most interesting cases, those in which the good projects are capital constrained (receive

capital with probability less than 1). The remaining cases are described in the appendix.10

Lemma 2 There is a function λ1(Z) such that the equilibrium contracts offered at date t0 are:

a. For Z ≤ pL, (qL,D1L,D2L) = (1, Z/pL, 0); (qM ,D1M ,D2M ) = (0, 0, 0); (qH , D1H ,D2H) =

(0, 0, 0). That is, entrepreneurs liquidate their projects for sure when productivity is low and

do not liquidate their projects when productivity is medium or high. In this case, T = pL.

b. For pL < Z ≤ pL+pM and λ ≤ λ1, (qL, D1L, D2L) = (1, 1, 0), (qM ,D1M ,D2M ) = (Z−pLpM, 1, 0);

(qH , D1H ,D2H) = (0, 1, 0). That is, entrepreneurs liquidate their projects for sure when pro-

ductivity is low and also liquidate their projects with some probability when productivity is

medium. They never liquidate high productivity projects. In this case, T = Z.

We Þrst derive some conditions on the equilibrium return R∗. By market clearing, ω∗ must

satisfy 0 < ω∗ ≤ 1 +K. It follows that x(1 +K) ≤ R∗ < x(0) and, by assumptions 1 and 2:

YL ≡ 0 < R∗ < YM < YH . (4)

Since the participation constraint of the date t0 investor binds, the entrepreneur�s problem is

equivalent to maximizing the combined payoff to him and the date t0 investor subject to the date10We also assume an upper bound for the Þnancing requirement, Z ≤ pL + pM . We discuss later what happens for

higher Z.

12

t0 investor receiving exactly R∗Z at date t2. Clearly, the entrepreneur beneÞts from liquidating a

low productivity project. Similarly, total payoff is reduced if the entrepreneur liquidates either the

medium or the high productivity project. The contract offered, however, can involve some liquida-

tion of medium or high productivity projects if it is necessary to satisfy the date t0 participation

constraint.

If the capital requirements are low, Z ≤ pL, then liquidation of the low productivity project

is sufficient to satisfy the participation constraint of the initial investor. The maximum date t2

cash ßow the entrepreneur can promise the date t0 investor by liquidating when the project is of

low productivity is R∗pL because liquidation generates one unit of capital at date t1 which can be

invested at a return of R∗. This explains part a).

However, if the capital requirements are high enough (Z > pL), liquidation of medium or high

productivity Þrms might be the only way to pay back the initial investor. Of course, a more efficient

way to compensate the initial investor is by paying him out of the date t2 payoff (D2s > 0) since this

avoids liquidation of H or M projects. However, this is not always possible. When pledgeability

is very low, Þrms cannot get capital in the reallocation market and, since only the returns of the

second unit are pledgeable, it is impossible to offer a fraction of the date t2 payoffs. But the

problem is more severe. The equilibrium has D2H = 0, even when type H projects get capital in

the reallocation market, as long as they get it with probability less than 1. The lower is D2H (i.e.,

the higher is PH), the more likely a project is to raise capital since it can offer a higher return

in the reallocation market (see Lemma 1). If all the projects were offering D2H > 0 and getting

capital with probability less than one, a single Þrm would be better off deviating and offering a

contract with a slightly lower D2H (i.e., a slightly higher PH) so as to be able to outbid all the other

Þrms. Thus when good Þrms are constrained (rH < 1) or equivalently when λ ≤ λ1, competitionfor capital at date t1 drives D2H to 0. By the same logic, D2M = 0 in this range.

Given that D2H and D2M are 0 in the range of λ considered, initial investors must be paid out

entirely out of date t1 liquidation proceeds. Liquidation of medium and high productivity Þrms

reduce the combined payoff. Thus, when liquidation occurs, the entire liquidation proceeds should

be given to the initial investors as this reduces the probability of liquidation. Since the medium

project is less productive, it is better to start liquidating it Þrst. It is only when liquidation of

the low and medium productivity projects is not sufficient to satisfy the investor participation

13

constraint that the good projects start being liquidated. When Z ≤ pL + pM , liquidation of the

low and medium productivity projects is sufficient to pay back the investor.11 Notice that qM is

increasing in Z. The higher the Þnancing requirement, the more frequent the medium project needs

to be liquidated.

All other cases are described in the appendix. For λ > λ1, good projects can pledge a suffi-

ciently high return to receive capital with probability 1. Since good projects are no longer capital

constrained, there is no beneÞt in decreasing D2H to outbid other Þrms. Thus, it becomes possible

to sustain an equilibrium with D2H > 0. For even higher levels of λ it is possible to sustain equi-

libria with D2M > 0. Finally, when the degree of pledgeability is very large, the entrepreneur is

able to pay the investor out of date t2 proceeds only.

Lemma 2 describes the equilibrium contract for each pair (λ, Z). Using these contracts in

Lemma 1 we obtain the equilibrium allocation and return rH , rM and R∗. Since for parts (a) and

(b) of Lemma 2, D2s = 0, the contracts in Lemma 2 imply that T = pL in case (a) and T = Z in

case (b).

Panels A and B of Figure 2 correspond to cases (a) and (b) of Lemma 2. Panel A corresponds

to case (a) of Lemma 2 when pledgeability is low (λYH < x(K+ pL)). In this case all the capital is

allocated to the general technology and R∗ = x(K + pL). Good projects cannot offer a sufficiently

high return to attract capital (λYH < R∗ = x(K + pL)) and, at the same time, medium projects

are continued and their capital is not supplied to high productivity projects (because YM > R∗).

Panel B corresponds to case (b) Lemma 2 with x(K + Z) ≤ λYH < x(K + Z − pH). In this case,λYH = R

∗ and good Þrms are able to attract some capital. However, Z is not large enough to allow

all good Þrms to raise one unit of capital.

2.3 Comparative Statics in λ and Z

We now analyze what happens to the aggregate payoff when pledgeability and the Þnancing re-

quirements change. In the following proposition, the threshold λ1 is the same as in Lemma 2 (this

threshold is such that for λ < λ1 good Þrms are capital constrained).

Proposition 1 The aggregate payoff11When Z > pL+pM , some liquidation of the good project is necessary. In this case, we would have full liquidation

of medium projects (qM = 1) and partial liquidation of good projects (qH = (Z − pL − pM )/qM ) in order to satisfythe Þnancing requirement.

14

� is non-decreasing in the level of pledgeability, λ,

� for Z < pL, it is unaffected by the Þnancing requirement Z,and

� for pL ≤ Z < pL + pM , there exists a λ0 < λ1 such that the aggregate payoff decreases withZ for low λ (λ < λ0), increases with Z for intermediate levels of λ (λ0 ≤ λ < λ1), and thendecreases with (or is unaffected by) Z for high λ ( λ ≥ λ1).

Better pledgeability allows projects to raise capital more easily in the reallocation market. Since

this capital would have ended up invested in the general technology, an increase in pledgeability

leads to higher aggregate output.

For Z < pL, only bad projects are liquidated in equilibrium. In this range an increase in

the Þnancing requirement does not affect the probability that medium (qM ) or high projects (qH)

are liquidated. Since the total amount of capital from liquidation proceeds remains unchanged,

aggregate output is not affected.

For Z ≥ pL, increases in the initial Þnancing requirement make it necessary to liquidate mediumproductivity projects more often. Since the return of the medium project is higher than that of the

general technology, this liquidation creates a social inefficiency when the released capital ends up

invested in the general technology. This happens when pledgeability is so low (λ < λ0), that good

Þrms are not able to offer a sufficiently high return to attract the released capital. In such a case

the reallocation market cannot materialize and external Þnance only has costs. If pledgeability is

higher (λ > λ0), good Þrms are able to attract some of the released capital, increasing the aggregate

payoff. At pledgeability levels above λ1, all good Þrms receive one unit in the reallocation market

and so an increase in liquidation of medium projects cannot raise aggregate output.12 13

The reason higher Þnancing requirements improve the reallocation of capital is that, because

of limited pledgeability, medium Þrms do not voluntarily liquidate their projects to invest in high

productivity Þrms unless forced by outside investors. Outside investors require liquidation as this12A similar result holds when Z > pL + pM . In such a case, either good Þrms are liquidated in equilibrium (for

lower λ), or they are not capital constrained (for higher λ). Thus, if all good Þrms continue they cannot be capitalconstrained, leaving no room for external Þnance to improve the allocation of capital.13The fact that external Þnance cannot be beneÞcial when the pledgeability parameter is higher than λ1 is driven

by the fact that our model only has three types of technologies. When all the good Þrms receive one unit of capital,further reallocation is not possible. With a continuum of productivities, external Þnance would always allow capitalto ßow from lower to higher productivity Þrms. The potential beneÞt would only disappear when all Þrms receivethe efficient amount of capital.

15

is the only way they can get their money back.14 Even though this forced liquidation by outside

investors is privately inefficient from the perspective of an individual Þrm, it is beneÞcial for the

economy because the additional capital supplied to the external market Þnds its way to a better

user.

It is important to note that the potential beneÞts of an increase in external Þnance are an

externality that higher liquidation generates to other Þrms in the economy. The Þrm that seeks

outside Þnance does not beneÞt from it, because the market does not compensate this Þrm for the

capital that it releases to the market. In our model, Þrms seek external Þnance only when they

do not have sufficient funds to invest. Therefore, introducing an outside investor that requires

liquidation when the Þrm is of medium productivity is sub-optimal for a Þrm in isolation.

3 Empirical Implications and Tests

In this section we summarize the empirical implications of our model, compare them to the available

empirical evidence, and present new empirical tests of one of our novel implications.

In the model increases in pledgeability allow projects to raise capital more easily in the real-

location market. Since this capital would have ended up invested in the general technology, an

increase in pledgeability leads to improved capital allocation. Levine (1991), Bencivenga, Smith

and Starr (1995), and Shleifer and Wolfenzon (2002) also predict that Þnancial frictions reduce

allocative efficiency. Wurgler (2000)�s Þnding that the efficiency of capital allocation is positively

correlated with investor protection and Þnancial development supports this prediction.

A novel implication of our model is that the external Þnancing needs of Þrms in a given country

should have an independent effect on allocative efficiency. We show that as long as the degree of

pledgeability is not too low, the resources that are released by the higher liquidation induced by

external Þnanciers will Þnd their way to more productive activities. This argument suggests that14 In the model we assume for simplicity that the cash-ßows from the Þrst unit cannot be pledged. Since medium

Þrms receive no additional capital, it is clear why an outside investor prefers to liquidate a medium Þrm rather thanto allow such a Þrm to continue until the Þnal date. However, this result would hold more generally if we allowed theÞrst unit to be pledged. In order to see this, suppose that the same fraction λ of both units could be pledged. In allpossible equilibria at which good Þrms are capital constrained, the return on capital x would be given by:

x(ω) ≥ λYH > λYMThus, since x(ω) > λYM it would still be true that outside investors strictly prefer to liquidate a medium Þrm thatcannot get additional capital in the external market.

16

controlling for investor protection, higher Þnancing needs will be associated with a more efficient

allocation of capital.

One might worry that the external Þnancing requirement of Þrms in different countries is en-

dogenous. In particular, it would be natural to expect a Þrm�s external Þnancing requirement to

be directly related to the level of investor protection. In countries with worse investor protec-

tion, Þrms might end up having lower Þnancing requirements either because they optimally choose

to operate in industries with lower capital needs, or because they retain more internal funds to

meet Þnancing requirements.15 Such possibilities would not invalidate our main argument. Even

when an increase in retained earnings (for example) is an optimal response to an environment of

low investor protection, it does not necessarily follow that such an increase in savings improves

the equilibrium allocation of capital because higher savings might reduce liquidation and hamper

capital reallocation.

Furthermore, to the extent that external Þnancing needs have an exogenous component, one

should be able to uncover the effect of external Þnance on the allocation of capital by relating the

efficiency of capital allocation to the external Þnancing needs of Þrms, after controlling for other

determinants of allocative efficiency. Previous literature has argued that because of technological

reasons, the need for external Þnance at the industry level will be to some extent exogenous, and

can be measured by the actual use of external Þnance in a relatively well developed capital market

(Rajan and Zingales, 1998). To the extent that the industrial structure of different countries also has

an exogenous component (determined for example by historical and geographical considerations),

we predict that a country�s aggregate demand for external Þnance should be positively related to

the efficiency of capital allocation.

With the above caveat in mind, we test the hypothesis that external Þnancing needs increase

the efficiency of capital allocation by regressing Wurgler�s (2000) measure of the allocative efficiency

(the industry elasticity of investment to value added, η) on the aggregate external Þnancing need

of the manufacturing sector (EFN) and controls:

ηc = α+ βEFNc + δControlsc (5)15Dittmar, Smith and Servaes (2002) show evidence that Þrms hold more cash in countries where shareholder

protection is poor. Their interpretation of the evidence, however, is that Þrms in countries with poor investorprotection hold more cash to increase managerial discretion, in conßict with shareholder interests.

17

where the subscript refers to country c.

Our measure of the aggregate external Þnancing need EFNc is

EFNc =X

ωicEFNi

where ωic is industry�s i share in total manufacturing output in country c (data from UNIDO),16

and EFNi is the external Þnance dependence of industry i from Rajan and Zingales (1998). We

show summary statistics on our index of external Þnancing needs in Table 1. We have data for 58

countries.

We use two alternative sets of control variables. First, we control for Þnancial and economic

development. Wurgler uses per capita GDP in 1960 to measure economic development, and he

uses the sum of the ratios of stock market capitalization to GDP and of private credit to GDP

to measure Þnancial development. These ratios are the averages of three years, 1980, 1985 and

1990. Because Wurgler�s measure of Þnancial development (which we call FD) is likely to be

highly correlated with the extent of external Þnancing needs by construction,17 we use alternative

measures of Þnancial development that attempt to capture more directly the efficiency of the

Þnancial system, as opposed to its size, and would thus not be mechanically linked with external

Þnance. SpeciÞcally, we use bank net interest margin to measure the efficiency of the banking

sector and turnover to measure the liquidity of the stock market. Demirguc-Kunt and Levine

(2001) suggest that bank net interest margin (measured as the 1990-1995 average bank interest

income minus interest expense over total assets) captures the inefficiency of the banking sector,

and Demirguc-Kunt and Levine (1996, 2001) suggest using turnover (the 1980-1995 average values

of trades of domestic equities as a share of the value of domestic equities) to capture stock market

liquidity. Because these two variables capture different aspects of Þnancial efficiency, we use them

together as controls when estimating equation 5, as an alternative to the variable FD.18

The second set of control variables that we use are measures of the legal protection of outside

investors. Because the model suggests that external Þnance inßuences the efficiency of capital

allocation beyond the effect of pledgeability, the regressions that control for investor protection

might capture the model�s prediction more directly. We use La Porta et al.�s (1998) measures16We thank Matias Braun for providing this data.17The correlation in our sample is 0.59.18We obtained the turnover and net interest margin data from the CD included in Demirguc-Kunt and Levine

(2001).

18

of shareholder and creditor rights (share rights and creditor rights), and also a measure of the

effectiveness of law enforcement (La Porta et al.�s (1998) rule of law variable). We also use the

index of effective investor rights constructed by Wurgler (2000), which is equal to the sum of

share rights and creditor rights, multiplied by rule of law. This index, which we call RIGHTS,

captures the intuition that it is the interaction between pro-investor laws and their enforcement

that matters for Þnancial variables. Finally, because these measures of investor protection are not

available for some countries in our sample, we use countries� legal origins to proxy for investor

protection in alternative speciÞcations, instead of the investor rights� measures. La Porta et al.

(1998) have shown that legal origin explains cross-country variation in the measures of investor

rights and law enforcement.

One important consideration is that, because our model focuses on the efficiency of capital

reallocation across Þrms, we expect the effect of external Þnancing needs to be especially strong

for the within-year measure of the efficiency of capital allocation. The within-year measure is the

elasticity of industry investment to value-added controlling for year Þxed effects. As explained by

Wurgler (2000), the overall elasticity of investment to value added is also affected by the between-

year component, which measures how overall investment in the manufacturing sector responds to

time series variation in value added for the whole manufacturing sector in the country. Thus,

the within-year component of the elasticity provides a cleaner measure of how capital responds to

value-added shocks across different industries in the same country. This measure is much closer

to the idea of capital reallocation in our model. To the extent that the elasticity of investment

to value-added is positively correlated with external Þnance, we thus expect this correlation to be

driven mostly by the within-year component.19

In Table 2 we check the extent to which the aggregate external Þnancing need of a given country

is correlated with the efficiency of capital allocation, after controlling for economic and Þnancial

development (the Þrst set of controls). In column (1) we replicate Wurgler�s main Þnding that

overall Þnancial development (measured by the size of capital markets and intermediary�s assets)

is positively correlated to the efficiency of capital allocation. Column (2) shows evidence that

EFN is also positively correlated with the efficiency of capital allocation, when controlling for

GDP (column 2). When we include both FD and EFN in the same regression together with19Jeff Wurgler provides data on within and between year components for each country on his website.

19

GDP (column 3), the signiÞcance of both variables is drastically reduced. This result suggests that

because FD and EFN are highly correlated by construction, it is hard to disentangle the effect of

changes in external Þnancing needs from variations in Wurgler�s measure of Þnancial development

FD. In column (4) we use our alternative measures of Þnancial development (turnover and net

interest margin) to try to mitigate this problem. Consistent with our expectations, the effect of

external Þnance on allocative efficiency and its statistical signiÞcance are higher in this alternative

speciÞcation.

In columns (5) to (12) we separately examine the two components of the elasticity of invest-

ment to value-added, the within and between components. As explained above, the within-year

component captures better the intuition of our theory, and thus we expect EFN to have a stronger

effect on this component. In contrast, as discussed by Wurgler (2000) overall Þnancial development

affects both the between and within components. We replicate Wurgler�s result in columns (5) and

(9). In columns (6), (7) and (8) we show that EFN is not signiÞcantly related to the between-year

component, even when we do not control for FD. In contrast, in columns (10) to (12) we show that

EFN is signiÞcantly correlated with the within-year component of the elasticity, as suggested by

our theory. This is true even when including both EFN and FD in the same regression (column

11). Again, signiÞcance increases when using turnover and net interest margin instead of FD

(column 12). Thus, consistent with the intuition behind our theory, external Þnancing needs seem

to affect mostly the within-year component of the elasticity of investment to value added.

In Table 3 we use our alternative set of control variables that capture the extent of the legal

protection of outside investors. Again, we use the overall elasticity of investment to value-added

(columns 1 to 3), and also the between (columns 4 to 6) and within-year components (columns 7

to 9). The results are consistent with those reported in Table 2. External Þnance seems to affect

mostly the within-year component of the elasticity of investment to value added. This conclusion

also holds when controlling for measures of investor protection and legal origin, as shown in columns

7 to 9. Notice that statistical signiÞcance increases when using legal origin instead of measures of

investor protection, partly due to the increase in the number of observations in the regressions

(from 40 to 58 countries).

As discussed above, an important caveat in interpreting these regression results is the potential

endogeneity of a country�s external Þnancing need. For example, external Þnancing needs might be

20

determined by a country�s overall level of Þnancial development and protection of outside investors.

In countries with low Þnancial development and/or poor protection of outside investors, Þrms might

optimally choose to operate in industries with lower capital needs, which might therefore grow faster

and dominate the economy. While the regressions in Tables 2 and 3 provide some evidence that

external Þnancing needs have an effect on allocative efficiency over and above what can be explained

by Þnancial development and investor protection, we also attempt to provide additional evidence

of an independent effect of external Þnance by pursuing two alternative approaches.

First, we regress EFN on measures of both Þnancial development and investor protection,

and collect the predicted values and the residuals of EFN . Presumably, the residuals of this

regression contain information on determinants of external Þnancing needs that are unassociated

with both Þnancial development and investor protection. We then regress the within-year elasticity

of investment to value-added on the predicted values and residuals.20 In order to maximize the

number of observations we use legal origin instead of the direct measures of investor protection.

We use turnover and net interest margin to measure Þnancial development, but the results are

qualitatively identical when we use Wurgler�s measure of Þnancial development. For brevity, we

report the results in the text. We Þnd the following coefficients and t-statistics:21

ηwithinc = −0.054(−0.40)

+ 1.359(2.47)

(EFpredicted) + 1.251(2.45)

(EFresiduals)

where the coefficients on both EFpredicted and EFresiduals are signiÞcant at a 5% conÞdence level.

The coefficient on EFresiduals suggests that the determinants of EFN that are unassociated with

both legal origin and Þnancial development have a signiÞcant effect on the efficiency of capital

allocation.

A more direct approach to tackle the endogeneity issue is to search for exogenous determinants

of EFN that can be used as instruments for EFN in the type of regressions reported in Tables 2 and

3. Recent literature has argued that pre-determined variables such as geographic endowments and

a country�s natural proclivity for the production of certain commodities is associated with economic

development, perhaps through their effect on long-lasting institutions (Easterly and Levine, 2003).

In the context of this paper, a particularly appealing hypothesis is that a country�s industrial20We thank an anonymous referee for suggesting this test.21The coefficients on the Þrst stage regression show that EFN is lower in countries with French legal origin, with

less liquid markets and with more inefficient banking systems.

21

structure (and thus its external Þnancing needs) is partly determined by that country�s commodity

endowments. To illustrate this point, consider the case of Chile. Chile has very low external

Þnance dependence (0.120, which is in the 5th percentile), because Chilean industry is heavily

concentrated in non-ferrous metals (Copper). This concentration is clearly historically determined

by Chile�s natural Copper endowment, and thus the low external Þnancing dependence of Chilean

manufacturing industry has an arguably exogenous determinant.

To perform a more systematic analysis, we collected data on our sample countries� production

of a given set of leading commodities. We follow Easterly and Levine�s (2003) approach, and

construct dummies for whether a country produced ANY amount of the commodity in 1998-1999.22

The set of commodities is the same one used in Easterly and Levine, and include bananas, coffee,

copper, maize, millet, oil, rice, rubber, silver, sugarcane, and wheat. Because their data only

includes countries that are former colonies, we went to the original sources to collect data for all

the countries in our sample. We obtained data for the production of crops from the Food and

Agricultural Organization statistics. Data for metals comes from the World Mineral Statistics

yearbook and for oil from the World Mineral Statistics yearbook. Similarly to Easterly and Levine,

we use all commodities together in the regressions, and do not attempt to choose a speciÞc subset

of commodities.

We use the commodity dummies as instruments for EFN and estimate equation 5 using a

two-stage least squares procedure. We use the within-year elasticity as the measure of allocative

efficiency. Easterly and Levine (2003) show that commodity endowments help explain cross-country

differences in economic development (through their effect on institutions). Thus, we include GDP

among the set of control variables to control for the possibility that commodity dummies affect the

efficiency of capital allocation through their effect on economic development, and not through their

effect on external Þnancing needs. Given that the model predicts an effect of external Þnancing

needs on allocative efficiency over and above the effect of investor protection, we also include the

country�s legal origin among the set of controls to account for cross-country variations in investor22Easterly and Levine (2003) argue that while the produced quantity of a given commodity is endogenously deter-

mined, whether ANY amount of the commodity is produced reßects exogenous characteristics.

22

protection.23 Therefore, we estimate the following model:

Second stage : ηwithin = α+ βEFN + γGDP + δ(Legal Origin)+ u

First Stage : EFN = a+ b(Commodity Dummies)+ cGDP + d(Legal Origin)+ v

The results (which we report in the text below) suggest that the cross-country variation in

external Þnancing needs that is explained by the commodity dummies is signiÞcantly related to

the efficiency of capital allocation. First of all, the Þrst stage F-test that the coefficient vector b is

equal to zero suggests that the commodity dummies do explain cross-country variation in external

Þnancing needs. The F-statistic is equal to 2.46, with a p-value of 0.018.24 Second, the coefficient

β on EFN in the second-stage regression is equal to 1.877, with a t-statistic of 2.06 and a p-value

of 0.044. The positive and signiÞcant β is evidence that exogenous variation in external Þnancing

needs is indeed related to the within-year elasticity of investment to value-added.

In sum, the results on this section provide some support for the prediction that external Þnance

and the efficiency of capital allocation are positively related. The positive correlation between exter-

nal Þnancing needs and Wurgler�s (2000) measure of allocative efficiency (the elasticity of industry

investment to industry value-added) remains after controlling for several possible determinants of

this elasticity, including overall Þnancial development and measures of investor protection. The

correlation also seems to be driven by the component of the elasticity that is more directly related

to the model�s predictions (the within-year component), and it remains signiÞcant after accounting

for the endogeneity of external Þnance.

4 Conclusion

We develop an equilibrium model to understand how the efficiency of capital allocation depends on

outside investor protection and Þrms� external Þnancing needs. Because of poor investor protection,

Þrms may not be able to raise Þnance for their good projects while, at the same time, other Þrms

fail to liquidate mediocre projects to supply their capital to the market. We show that the presence

of outside investors improves the allocation of capital by increasing the liquidation of mediocre

projects.23We chose not to use the more direct measures of investor protection because of the decrease in the number of

observations.24We use heteroskedasticity-consistent standard errors to draw all inferences.

23

We present some empirical evidence that is consistent with one of the novel implications of the

model, namely that the aggregate demand for external Þnance is positively correlated with the

efficiency of capital allocation. We interpret these results as suggestive that the speciÞc mechanism

in our model is inßuencing the speed and efficiency with which different systems accomplish cap-

ital reallocation across different sectors. However, further empirical research is required to Þrmly

establish this conclusion. For example, while we report some evidence that exogenous variation in

external Þnancing needs increases Wurgler�s (2000) measure of allocative efficiency, we cannot rule

out the possibility that the instruments that we use for external Þnance (commodity endowments)

inßuence allocative efficiency through other mechanisms that we do not control for. It would also

be desirable to experiment with alternative measures of the efficiency of capital allocation, and

if possible explore alternative data sets that allow the researcher to control more effectively for

important variables such as investor protection.

On the other hand, we do believe that the mechanism we describe in this paper can help explain

cross-country differences in the equilibrium efficiency of investment. Our main argument is that

the extent to which Þrms rely on external Þnance affects Þnancial arrangements (the allocation

of control in corporations between external and internal investors) and this, in turn, inßuences

economic efficiency. Thus our paper proposes a new channel through which Þnancial arrangements

affect economic efficiency. While our model is by nature a static one, this is mostly for theoretical

simplicity. Extensions of our theory could thus try to integrate our arguments into dynamic theories

of comparative advantage, and of the endogenous evolution of Þnancial arrangements. A more

dynamic version of our model might also allow us to determine whether the effects we model

in this paper can help explain cross-country differences in income growth, and overall economic

development.

24

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25

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26

Table 1: Summary Statistics - Index of External Finance Dependence (EFN)

This table displays summary statistics for the index of external Þnance depen-dence for countries. The index of external Þnance dependence is computed asa weighted average of the inudustry-level measures of external Þnance dependcetaken from Rajan and Zingales (1998). The weights are the industrial shares intotal manufacturing output, data from UNIDO, Industrial Statistics Database.

Country Index of EFN Country Index of EFN

Australia 0.261 Korea 0.281Austria 0.265 Kuwait 0.138Bangladesh 0.160 Macao 0.243Barbados 0.199 Malawi 0.207Belgium 0.193 Malaysia 0.274Bolivia 0.144 Malta 0.252Canada 0.273 Mexico 0.239Chile 0.120 Morocco 0.219Colombia 0.208 Netherlands 0.293Cyprus 0.150 New Zealand 0.256Denmark 0.301 Nigeria 0.248Ecuador 0.246 Norway 0.262Egypt 0.208 Pakistan 0.157El Salvador 0.184 Panama 0.179Ethiopia 0.131 Peru 0.175Fiji 0.201 Singapore 0.417Finland 0.268 Spain 0.254France 0.300 Swaziland 0.172Germany 0.327 Sweden 0.307Greece 0.229 Tanzania 0.183Guatemala 0.218 Trinidad 0.157Hong Kong 0.377 Turkey 0.177India 0.284 UK 0.310Indonesia 0.191 US 0.346Iran 0.276 Uruguay 0.153Ireland 0.316 Venezuela 0.151Israel 0.387 Zimbabwe 0.186Italy 0.307Japan 0.367Jordan 0.068 Mean 0.211Kenya 0.223 STD 0.079

Table 2:

Efficiency of Capital Allocation as a Function of Financial Development and External Finance Dependece

The dependent variable is the measure of the efficiency of capital allocation calculated by Wurgler (2000), the elasticity of industry investmentto value-added. In columns (1) to (4) we use the overall elasticity of investment to value added. In columns (5) to (8) we use the between-year elasticity of investment to value-added, and in columns (9) to (12) we use the within-year elasticity to measure the efficiency of capitalallocation. The independent variables are the aggregate external Þnancing need summarized in Table 1 (EFN), the summary measure ofÞnancial development used by Wurgler (2000), FD, which is equal to the log of one plus the average sum of stock market capitalization andprivate credit to GDP (1980-1990 averages), the 1960 value of log per capita GDP from Wurgler (2000), GDP , the ratio of stock market turnoverto GDP (Turnover) from Demirguc-Kunt and Levine (2001), and the net interest margin from Demirguc-Kunt and Levine (2001). Robustt-stats (in parentheses).

Dependent variable: η overall Dependent variable: η between-year Dependent variable: η within-year

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

EFN 1.354∗∗∗ 0.763 0.996∗ 1.086 0.320 2.073 1.145∗∗∗ 0.673∗ 1.152∗∗(2.89) (1.60) (1.92) (1.47) (0.33) (1.08) (2.90) (1.96) (2.15)

FD 0.323∗∗∗ 0.217∗ 0.419∗∗ 0.425∗ 0.213∗∗ 0.010(3.09) (1.77) (2.15) (1.72) (2.59) (0.98)

GDP 0.154∗∗∗ 0.159∗∗∗ 0.143∗∗∗ 0.161∗∗∗ 0.090 0.107 0.064 0.128 0.053∗ 0.046 0.045 0.022(5.07) (5.78) (4.42) (4.54) (0.88) (1.16) (0.61) (0.96) (1.76) (1.47) (1.39) (0.53)

Turnover 0.154 -0.157 -0.020(1.05) (-0.32) (-0.11)

Net interest margin -1.179 5.340 -0.472(-0.89) (0.74) (-0.35)

Intercept 0.162∗∗∗ 0.014 0.051 0.104 0.432∗∗∗ 0.397∗ 0.388 0.128 0.091∗ -0.052 0.001 0.001(3.01) (0.14) (0.53) (0.75) (2.98) (1.69) (1.62) (0.96) (1.78) (-0.60) (0.02) (0.01)

Observations 61 58 57 50 53 57 51 50 61 58 57 50

R2 0.52 0.50 0.52 0.49 0.15 0.10 0.14 0.10 0.21 0.23 0.21 0.18

Table Notes: ***,**,* indicate statistical signiÞcance at 1%, 5%, and 10% (two-tail) test levels, respectively.

Table 3:

Efficiency of Capital Allocation as a Function of Measures of Investor Protection and Legal Origin

The dependent variable is the measure of the efficiency of capital allocation calculated by Wurgler (2000), the elasticity of industry investmentto value-added. In columns (1) to (3) we use the overall elasticity of investment to value added. In columns (4) to (6) we use the between-yearelasticity of investment to value-added, and in columns (7) to (9) we use the within-year elasticity as the measure of the efficiency of capitalallocation. The independent variables are the aggregate external Þnancing need summarized in Table 1 (EFN), the number of importantshareholder (Share Rights) and creditor (Creditor rights) rights in the country�s legal code and a measure of the rule of law from La Porta etal. (1988), the index of effective investor rights (RIGHTS) constructed by Wurgler (2000), which is the product of Ruleoflaw and and the sumof Share Rights and Creditor rights, and measures of the country�s legal origin from La Porta et al. (1998). Robust t-stats (in parentheses).

Dependent variable: η overall Dependent variable: η between-year Dependent variable: η within-year

(1) (2) (3) (4) (5) (6) (7) (8) (9)

EFN 0.723 1.153 2.077∗∗∗ -0.732 -0.005 1.710∗ 0.902∗∗ 0.686∗ 1.246∗∗∗(1.29) (1.55) (4.07) (-0.47) (-0.00) (1.69) (2.05) (1.75) (3.02)

Share rights -0.016 -0.057 0.027(-0.72) (-0.98) (1.23)

Creditor rights -0.021 0.024 0.010(-0.84) (0.28) (0.39)

Rule of law 0.093∗∗∗ 0.137∗∗∗ 0.012(3.96) (3.58) (0.58)

RIGHTS 0.035 0.051 0.031∗(1.56) (1.29) (1.70)

French origin 0.114 0.068 -0.009(1.60) (0.33) (-0.15)

German origin 0.304∗∗∗ 0.274 0.070(3.30) (1.50) (0.81)

Scandinavian origin 0.244∗∗∗ 0.388∗∗ 0.030(2.98) (2.23) (0.30)

Intercept 0.069 0.120 -0.126 0.624∗∗ 0.720 0.283 -0.093 0.002 -0.042(0.45) (0.79) (-0.96) (2.22) (1.64) (0.78) (-0.66) (0.02) (-0.42)

Observations 40 40 58 40 40 58 40 40 58

R2 0.48 0.24 0.40 0.11 0.02 0.10 0.20 0.22 0.21

Table Notes: ***,**,* indicate statistical signiÞcance at 1%, 5%, and 10% (two-tail) test levels, respectively.

Figure 2

6

-

R

Capital

Panel A

x(0)

λYH

R∗

K + pL

6

-

R

Capital

Panel B

x(0)

R∗ = λYH

K + T

AppendixProof of Lemma 1

The allocation rule is optimal as it implies that investors allocate capital to the technologies offering the highestreturn. We also explain in the text that, given the allocation rule, the announcements P j are optimal. Finally, r isdeÞned so that the market clears. Plugging the equilibrium allocation in the market clearing condition

x−1(R∗) +Z{j∈C:P j>R∗}

dj + r

Z{j∈C:P j=R∗}

dj = K + T. (6)

We only need to prove that r ∈ [0, 1]. The value of R∗ is found by solving the equation in footnote 9 which can berearranged as

0 ≤ K + T − x−1(R∗) −Z

{j∈C|P j>R∗}

dj ≤Z

{j∈C|P j=R∗}

dj. (7)

IfZ

{j∈C|P j=R∗}

dj = 0 then equation 7 reduces to the market clearing condition in equation 6 regardless of the value of

r. IfZ

{j∈C|P j=R∗}

dj > 0 then by equation 7, it must be that r ∈ [0, 1].¥

We state the following Lemma that covers all cases for Z < pL + pM . The version of Lemma 2 described in themain text is a special case.

Lemma 2 There are functions λk(Z) for k = 1, 2, 3, 4, 5 such that equilibrium contracts offered by entrepreneurs atdate t0 are:

a. For Z ≤ pL, (qL, D1L,D2L) = (1, Z/pL, 0); (qM ,D1M ,D2M ) = (0, 0, 0); (qH ,D1H , D2H) = (0, 0, 0)

b. For pL < Z ≤ pL + pH YHYM

− pH , (qL,D1L,D2L) = (1, 1, 0), D1M = D1H = 1, qH = 0 and

1. For λ ≤ λ1, qM = Z−pLpM

, D2M = D2H = 0

2. For λ1 < λ ≤ λ2, qM =Z− pHD2HbR −pL

pM, D2M = 0, and D2H = λYH − bR, where bR is deÞned by

x(K + Z − λYHbR pH) = bR3. For λ2 < λ, qM = 0, D2M = 0, D2H =

bR(Z−pL)pH

, where

bR =

x(K + pL − pH) if λYM < x(K + pL − pH)λYM if x(K + pL − pH) ≤ λYM ≤ x(K + pL − pH − pM )

x(K + pL − pH − pM ) if λYM > x(K + pL − pH − pM )

c. For pL + pHYHYM

< Z ≤ pL + pM , (qL,D1L,D2L) = (1, 1, 0), D1M = D1H = 1, qH = 0 and

1. For λ ≤ λ1, the contract is identical to that in case b.12. For λ1 < λ ≤ λ3, the contract is identical to that in case b.2

3. For λ3 < λ ≤ λ4, qM =Z− pHD2HbR −pL

pM, D2M = 0, and D2H = λYH − bR, where bR = λYM

4. For λ4 < λ ≤ λ5, qM =Z− λbR (pHYH+pMYM )−pL+pM+pH

pM (2− λbRYM ), D2M = λYM − bR, and D2H = λYH − bR, wherebR satisÞes x(K + 2(pL + pMqM ( bR))− 1) = bR

5. For λ > λ5, qM = 0, and D2M and D2H can be anything that satisÞes pMD2M + pHD2H + pL bR = Z bRand D2s ≤ λYs − bR where bR is deÞned by bR = x(K + pL − pH − pM )

ProofWe focus on symmetric equilibria. We denote the proposed equilibrium contract by {qs,D1s,D2s}s=L,M,H (no

superscript j). To verify that {qs,D1s,D2s}s=L,M,H is an equilibrium we proceed as follows. First, using Lemma

1

1 , we Þnd the equilibrium return, R∗, and allocation rule, r, in the date t1 reallocation market for the proposedcontract. Then given the equilibrium R∗ and r we check that no entrepreneur has an incentive to deviate.

First, we Þnd R∗ and r. Since all entrepreneurs offer the same contract then Pjs = P s and rjs = rs for all

entrepreneurs j. In addition, the total amount of liquidation T =P

s psqs. The following table is derived assumingthat PH ≥ PM (which will always be the case) and using Lemma 1 to

Case Range R∗ rH rM1 x(K + T ) > PH x(K + T ) 0 0

2 x(K + T ) ≤ PH ≤ x(K + T − pH) PHK+T−x−1(PH)

pH0

3PH > x(K + T − pH) andPM < x(K + T − pH) x(K + T − pH) 1 0

4 x(K + T − pH) ≤ PM ≤ x(K + T − pH − pM ) PM 1 K+T−pH−x−1(PM )pM

5 PM > x(K + T − pH − pM ) x(K + T − pH − pM ) 1 1

Table 1: R∗, rH and rM as a function of T, PH and PM .

It is always the case that rL = 0 because since YL = 0, a low productivity entrepreneur cannot pledgeanything in the reallocation market. Second, given R∗ and r, we Þnd the optimal contract for entrepreneur j,{qjs,Dj

1s,Dj2s}s=L,M,H (with superscript j) that maximizes:

U jE =P

spsnqjs(1−Dj

1s)R∗ + (1− qjs)[Ys + rjs(Ys −Dj

2s −R∗)]o

(8)

subject to the participation constraint of the date t0 investor

U jI =P

spsnqjsD

j1sR

∗ + (1− qjs)rjsDj2s

o≥ ZR∗ (9)

and pledgeability constraints, where rjs is the probability that entrepreneur j raises one unit of capital with the contract{qjs,Dj

1s,Dj2s}s=L,M,H when the productivity is s. Finally, if {qjs,Dj

1s,Dj2s}s=L,M,H = {qs,D1s,D2s}s=L,M,H then

{qs,D1s,D2s}s=L,M,H is an equilibrium.

We deÞne λ1 ≡ x(K + Z − pH)/YH , λ2 ≡ x(K+pL−pH)YH

³1 + Z−pL

pH

´, λ3 ≡

x(K+Z−pH YHYM

)

YM, λ4 ≡ x(K + 2(Z −

pHYHYM

+ pH)− 1)/YM and λ5 ≡³Z−pL+pH+pMpHYH+pMYM

´x(K + pL − pM − pH).

Step 1: a) ∂UjE

∂qjL

≥ 0 (with equality only when Dj1L = 1), b)

∂UjE

∂qjs< 0 for s = H,M, and c)

∂UjE

∂Dj2s

≤ 0 for s = H,M(with strictly inequality qjs < 1 and r

js > 0)

PjL = 0 since in the low state the project generates no cash. Thus P

jL < R∗ and, by Lemma 1, rjL = 0. Now,

∂UjE

∂qjL

= (1−Dj1L)R

∗ ≥ 0. Also, for s = H,M , Ys > R∗ and thus ∂UjE

∂qjs= (1−Dj

1s)R∗ − [Ys + rjs(Ys −Dj

2s −R∗)] < 0.Finally, an increase in Dj

2s for s = H,M has two effects on U jE . The direct effect weakly decreases UjE . The indirect

effect works by weakly decreasing rjs (because Pjs = λYs − Dj

2s decreases and, by Lemma 1, rjs weakly decreases),

which in turn weakly decreases U jE .

Step 2 (Proof of part a): If Z ≤ pL then (qjL,Dj1L,D

j2L) = (1, Z/pL, 0) and (q

jM ,D

j1M , D

j2M ) = (q

jH ,D

j1H , D

j2H) =

(0, 0, 0)From step 1, every entrepreneur maximizes the objective function by setting qjL = 1, q

jM = qjH = 0 and Dj

2M =Dj2H = 0. Since qjM = qjH = 0, Dj

1M and Dj1H are irrelevant (we set them to zero). Also, since qjL = 1, Dj

2L isirrelevant (we set it to zero). Finally since qjL = 1, D

j1L = Z/pL the participation constraint is satisÞed. Therefore,

the above is the optimal contract for every entrepreneur j.

We now consider the cases where pL < Z ≤ pL + pM .Step 3: For Z > pL, then no contract dominates one with (qjL, D

j1L,D

j2L) = (1, 1, 0) and D

j1M = Dj

1H = 1. (XXX)Since Z > pL, then by the participation constraint, it must be that max{qjMDj

1M , qjHD

j1H , (1− qjM )rjMDj

2M , (1−qjH)r

jHD

j2H} > 0. First, we show that qjL = 1. Suppose not, i.e., qjL < 1. If Dj

1L < 1 then an increase in qjL increases

U jE while not violating the participation constraint (U jI is also increasing in qjL). Contradiction. If D

j1L = 1 an

increase in qjL increases UjI , relaxing the participation constraint (U

jI > ZR∗). Since Z > pL, at least one of the

following holds: qjsDj1s > 0 ( s = H,M) or (1−qjs)rjsDj

2s > 0 (s = H,M). If the Þrst case holds, then a slight decrease

2

in qjs increases UjE without violating the constraint. Contradiction. If the second case holds, then a slight decrease

in Dj2s increases U

jE (since it must be that q

js < 1 and r

js > 0) without violating the constraint. Contradiction.

Now we prove that Dj1L = 1. Suppose not, i.e., Dj

1L < 1. Consider an increase in Dj1L to D

j1L + ² with ² > 0.

Again, since Z > pL, at least one of the following holds: qjsDj1s > 0 ( s = H,M) or (1 − qjs)rjsDj

2s > 0 (s = H,M).Suppose the Þrst case holds. Since qjs > 0, it must be that U

jI is increasing in q

js or D

j1sR

∗ > rjsDj2s (otherwise setting

qjs = 0 would increase the objective function and relax the constraint). Entrepreneur j can decrease qjs to qjs − δ

where δ = ² pLR∗

ps(Dj1sR

∗−rjsDj2s)

is chosen so as to leave the participation constraint unchanged. Note that δ > 0 since

Dj1sR

∗ > rjsDj2s. The change in the objective function is given by: δps(Ys + r

js(Ys − R∗) − R∗) > 0. Contradiction.

If the second case holds, a similar procedure shows a contradiction.Next we show that Dj

1s = 1 for s = M,H . First, if qjs = 0 then D

j1s is irrelevant (we set it to 1). Suppose that

qjs > 0 and Dj1s < 1. As before, since qjs > 0, Dj

1sR∗ > rjsD

j2s. An entrepreneur can increase D

j1s to D

j1s + ² and

decrease qjs to qjs−δ where δ = ² qjsR

∗

Dj1sR

∗−rjsDj2s+²R

∗ > 0 is chosen so as to leave the participation constraint unchanged.

The change in the objective function is δps(Ys + rjs(Ys −R∗)−R∗) > 0. Contradiction.

We only need to Þnd qjM , qjH , D

j2M and Dj

2H to complete the characterization of the contract. We Þrst show apreliminary and intuitive result that Þrms of type M are liquidated before Þrms of type H :Step 4: If rjMD

j2M ≤ rjHDj

2H and rjM ≤ rjH then qjH > 0 implies qjM = 1.25 This, together with Z ≤ pL+pM , implies

that qH = 0.Suppose not, i.e., rjMD

j2M ≤ rjHD

j2H , r

jM ≤ rjH , q

jH > 0 and qjM < 1. Entrepreneur j can deviate by increasing

qjM to qjM = qjM + ² and decreasing qjH to qjH = qjH − δ with δ = ²pM(R∗−rjMD2M )

pH (R∗−rjHD2H). The choice of δ guarantees

that the participation constraint is unchanged. rjMDj2M ≤ rjHD

j2H implies that pHδ ≥ pM². The change in the

objective function is given by pHδ(1 + rjH)(YH − R∗)− pM²(1 + r

jM )(YM − R∗) > 0. The inequality follows from

YH − R∗ > YM − R∗ > 0 (by equation 4), rH ≥ rM and pHδ ≥ pM ². Contradiction. This step implies that Þrm oftype M is liquidated Þrst and only then is a Þrm of type H liquidated. Since Z ≤ pL + pM , then qH = 0.

In what follows, the conditions of step 4 always hold so qjH = 0. With all these preliminary results we are readyto prove the different cases of the Lemma. But Þrst, we describe the equilibrium R∗, rH and rM .Step 5 (proof of part b.1 and c.1): For pL < Z ≤ pL+pM and λ ≤ λ1 the equilibrium contract has (qM ,D2M ) =(Z−pL

pM, 0) and (qH ,D2H) = (0, 0).

First, we Þnd the equilibrium given the contracts. Assuming all entrepreneurs offers the same contract, totalliquidation is T = pL + qMpM + qHpH = Z. Also PH = λYH and PM = λYM . Now λ ≤ λ1 is equivalent toλYH ≤ x(K + Z − pH). By Table 1, R∗ ≥ λYH .

Second, assuming every entrepreneur is offering the proposed contract we solve for entrepreneur�s j optimaldeviation (by maximizing equation 8 subject to equation 9). Setting Dj

2H > 0 is not optimal. It guarantees thatrjH = 0 and so it does not increase the payoff nor does it relax the participation constraint. Similarly, Dj

2M > 0 isnot optimal. Finally, by the participation constraint qjM = Z−pL

pM. Since (qjs,D

j2s)s=H,M = (qs,D2s)s=H,M , we have,

in fact, an equilibrium.

Step 6 (proof of part b.2 and c.2): For λ1 < λ ≤ min{λ2,λ3} (this region is non-empty), every entrepreneursets (qM , D2M ) = (

Z− pHD2HbR −pLpM

, 0) and (qH ,D2H) = (0,λYH − bR), where the market return bR is the solution to

x(K + Z − λYHbR pH) = bR.First, we compute the equilibrium R∗ given the contracts. Total liquidation is given by T = pL + pMqM =

Z − pHD2HbR = Z − pHλYHbR + pH . Since PH = λYH − D2H = bR = x(K + Z − λYHbR pH) = x(K + T − pH),this casealways falls under case 2 of Table 1 . Thus rM = 0 and rH =

K+T−x−1(PH )pH

= K+T−(K+T−pH )pH

= 1. Also, according

to Table 1, R∗ = PH = bR. The other two variables of interest can be written now asD2H = λYH −R∗ (10)

and,

T = Z − pHD2H

R∗(11)

To analyze the properties of R∗, we deÞne

F (R,λ) = x(K + Z − λYHR

pH)−R. (12)

25 It can be shown that in any equilirbium rMD2M ≤ rHD2H and rjM ≤ rjH . Thus, qH > 0 always implies qM = 1.However, for the rest of the proof, this less general result is sufficient.

3

For any λ, the equilibrium R∗(λ) satisÞes F (R∗(λ),λ) = 0. Next, by the implicit function theorem, ∂R∗

∂λ= − Fλ

FR> 0.

This implies, by equation 11 that ∂T∂λ< 0 and consequently ∂qM

∂λ< 0. Since ∂R∗

∂λ> 0 and ∂T

∂λ< 0, by equation 10, it

must be that ∂D2H∂λ

> 0.Since Table 1 assumes that PH ≥ PM , we need to show that this is true. Since PH = R∗ and D2M = 0, it

suffices to show that R∗ ≥ λYM . The function F (λYM ,λ) = x(K + Z − YHYMpH) − λYM is strictly decreasing in λ

and F (λ3YM ,λ3) = 0. Therefore F (λYM ,λ) > 0 for λ < λ3. But since FR < 0,it must be that R∗(λ) > λYM for allλ < λ3.

Now, we prove that contract is feasible, i.e., that 0 ≤ qM ≤ 1 and D2H ≥ 0. First, we consider qM . SinceF (x(K + pL− pH), λ2) = 0, then R∗(λ2) = x(K + pL− pH). Plugging the value of λ2 and that of R∗ in equations 10and 11, leads to T = pL or qM = 0. Since ∂qM

∂λ< 0, then for λ ≤ λ2, qM > 0. The result that qM ≤ 1 follows from

Z ≤ pL + pM . In the range considered D2H ≥ 0. Since R∗(λ1) = λ1YH then at λ = λ1, D2H = 0. D2H ≥ 0 followsfrom the result shown above that ∂D2H

∂λ> 0.

Finally, given the equilibrium, we solve for entrepreneur j�s optimal deviation (qjs,Dj2s)s=H,M . Since R

∗ > λYMfor the range considered then rjM = 0 regardless of Dj

2M . We set Dj2M = 0. By setting Dj

2H = λYH − R∗, theentrepreneur raises one unit with probability 1 (rjH = 1). The entrepreneur does not gain by lowering Dj

2H sincedoing so leaves rjH unchanged and the entrepreneur needs to raise qjM to compensate the date t0 investor. Also,

the entrepreneur does not gain by increasing Dj2H since, in that case, P

jH < R∗ and rjH = 0. This would decrease

his payoff and would require more liquidation to compensate the date t0 investor as well. From the participation

constraint (it holds with equality) it follows that qjM =Z−pHD2HbR −pL

pM. Since (qjs ,D

j2s)s=H,M = (qs,D2s)s=H,M , we

have, in fact, an equilibrium.

Step 7 (proof of part b.3): If pL < Z ≤ pL + pHYHYM

− pH , then for λ > λ2, every entrepreneur sets qM = 0,

D2M = 0, D2H =bR(Z−pL)

pH, where bR =

x(K + pL − pH) if λYM < x(K + pL − pH)

λYMif x(K + pL − pH) ≤ λYM ≤≤ x(K + pL − pH − pM )

x(K + pL − pH − pM ) if λYM > x(K + pL − pH − pM )First, we compute the equilibrium. Since qM = 0 then T = pL. Now, we show that PH ≥ x(K + pL − pH) for

all λ covered in case b.3. Consider V = PH − x(K + pL − pH) = λYH − bR(Z−pL)pH

− x(K + pL − pH). Now, bR(λ2) =x(K+ pL− pH). Because if Z < pL+pH YH

YM− pH (or 1+ Z−pL

pH< YH

YM) then λ2YM = x(K+pL−pH )

YH

³1 + Z−pL

pH

´YM <

x(K + pL − pH) and so bR(λ2) = x(K + pL − pH), and if Z = pL + pH YHYM

− pH (or 1 + Z−pLpH

= YHYM) then λ2YM =

x(K+pL−pH)YH

³1 + Z−pL

pH

´YM = x(K + pL − pH) and again bR(λ2) = λYM = (K + pL − pH). Function V evaluated at

λ2 is λ2YH − x(K+pL−pH)(Z−pL)pH

−x(K+pL−pH) = x(K+pL−pH)³1 + Z−pL

pH

´−x(K+pL−pH)

³1 + Z−pL

pH

´= 0.

Now ∂V∂λ= YH − ∂ bR

∂λ(Z−pL)pH

≥ YH − ∂ bR∂λ

³YHYM

− 1´≥ YH − YM

³YHYM

− 1´= 1, where the Þrst inequality follows from

the deÞnition of case b.3 and the second inequality follows because ∂ bR∂λ≤ YM . Thus, in all the range of λ considered

PH ≥ x(K + pL − pH). Since PH ≥ x(K + pL − pH), PM = λYM , and T = pL then comparing the deÞnition of bRwith R∗ from cases 3,4, and 5 of Table 1, leads to R∗ = bR. In addition, in this case (b.3), rH = 1.

Second, an entrepreneur has no incentives to deviate since this contract allows him not to liquidate the type Mproject. Also by setting D2M = 0, the entrepreneur maximizes his probability of receiving capital when his projectis of type M .

Step 8 (proof of part c.3): If Z − pH YHYM

> pL − pH , then for λ3 ≤ λ ≤ λ4 (this region is non-empty), every

entrepreneur sets (qM ,D2M ) = (Z− pHD2HbR −pL

pM, 0) and (qH , D2H) = (0,λYH − bR), where bR = λYM .

Again, we Þrst compute the equilibrium. The total amount of liquidation is T = pL + pMqM = Z − pHD2HbR =

Z − pHYHYM

+ pH . Since λ ≥ λ3, then PM = λYM ≥ x(K + Z − pH YHYM) = x(K + T − pH) and so this case falls under

case 4 of Table 1. Since we are in case 4, R∗ = λYM , rH = 1 and rM = K+T−pH−x−1(λYM )pM

. Note that rM is strictlyincreasing in λ. At λ = λ3, rM = 0 and at λ = λ4, rM = 1.

We need to check whether the contract is feasible. We show that D2H ≥ 0 and 0 ≤ qM ≤ 1. First, D2H =

λYH − λYM > 0. Now, qM =Z− pHYH

YM+pH−pL

pM> 0 by the deÞnition of this step (Z − pH YH

YM> pL − pH). Also, since

Z − pHYHYM

+ pH − pL ≤ Z − pL ≤ pM (the last inequality follows from our assumption that Z ≤ pL + pM ) thenqM < 1.

Finally, given the equilibrium, we solve for entrepreneur j�s optimal deviation (qjs ,Dj2s)s=H,M . Just as in step

6, there is no proÞtable deviation from Dj2H = λYH − R∗. Also for Dj

2M = 0, rjM ∈ [0, 1] whereas any deviation to

4

Dj2M > 0, would lead to rjM = 0 (since P

jM < λYM = R∗) and is therefore not proÞtable. Given these values of Dj

2H

and Dj2M , q

jM satisÞes the participation constraint with equality. Finally, since (qjs,D

j2s)s=H,M = (qs, D2s)s=H,M , we

have, in fact, an equilibrium.

Step 9 (proof of part b.4): If Z − pH YHYM

> pL − pH , then for λ4 < λ < λ5, every entrepreneur sets (qM ,D2M ) =

(Z− λbR (pHYH+pMYM )−pL+pM+pH

pM (2− λbRYM ),λYM− bR) and (qH ,D2H) = (0,λYH− bR), where bR satisÞes x(K+2(pL+pMqM ( bR))−

1) = bR.The proof of this step is almost identical to step 6. So we do not repeat is here. The equilibrium R∗ = bR. In this

case, rH = rM = 1. Also, qM > 0 for λ < λ5 and qM = 0 for λ = λ5.

Step 10 (proof of part b.5): If Z − pH YHYM

> pL − pH then for λ > λ5, qM = 0, and D2M and D2H can be

anything that satisÞes pMD2M + pHD2H + pL bR = Z bR and D2s ≤ λYs − bR for s = H,M where bR is deÞned bybR = x(K + pL − pH − pM ).First, we Þnd the equilibrium. Since qM = 0 then T = pL. The deÞnition of D2s guarantees that P s ≥ bR.

Then, since P s = bR = x(K + pL − pH − pM ) we are in case 5 of Table 1. From the Table, R∗ = x(K + pL −pH − pM ) = bR. To check whether the contract is feasible we need to show that there are D2H and D2M thatsatisfy 0 ≤ D2s ≤ λYs − bR for s = H, M and pMD2M + pHD2H + pL bR = Z bR. First, λYM − bR > λ5YM − bR =bRpHYH+pMYM

[ZYM + YM (−pL + pH + pM )− pHYH − pMYM ] > 0, where the Þrst equality follows from the deÞnition

of λ5 and the second from the condition that Z − pH YHYM

> pL − pH . Since λYH − bR > λYM − bR > 0, it is possibleto have D2s > 0 for s = H,M . The minimum value that pMD2M + pHD2H + pL bR takes is pL bR < Z bR whenD2H = D2M = 0. The maximum value is pM (λYM− bR)−pH(λYH− bR)+pL bR > λ5(pMYM+pHYH)− bR(pL−pH−pM ) =bR(Z − pL + pH + pM ) − bR(pL − pH − pM ) = bRZ. Thus there are non-negative values for D2H and D2M such thatpMD2M + pHD2H + pL bR = Z bR and D2s ≤ λYs − bR for s = H,M .

With the proposed contract, an entrepreneur does not liquidate when the project is of type M or H. In addition,the entrepreneur always raises capital for sure with these two types of projects. Since R∗ = bR, the contract satisÞesthe investor�s participation constraint. Thus an entrepreneur cannot do better.¥

Proof of Proposition 1For Z < pL + pM , qH = 0 and therefore aggregate payoff can be written as:

Π = pH(1 + rH)YH + pM (1− qM )(1 + rM )YM + ωx(ω),

where ω = K + pL + pMqM − pHrH − pM (1− qM )rM .In region (a), qM = qH = 0 (so T = pL), and D2H = D2M = 0 so PH = λYH and PM = λYM . Using PH ,

PM and T = pL in Table 1, it can be readily seen that rH and rM are weakly increasing in λ. Since ddω(ωx(ω)) =

x(ω) + ωx0(ω) < YM < YH , then ∂Π∂rH

= pH [YH − ddω(ωx(ω))] > 0 and ∂Π

∂rM= pM (1 − qM )[YM − d

dω(ωx(ω))] > 0.

Therefore, in this region, dΠdλ≥ 0. Finally, in this region, the equilibrium is independent of Z, and so dΠ

dZ= 0.

In region b.1 and c.1, qH = 0, qM = (Z − pL)/pM (so T = Z) PH = λYH , PM = λYM . Letting λ0 =

x(K + Z)/YH < λ1and using Table 1 we obtain that rH =

½0 if λ < λ0

[K + Z − x−1(λYH)]/pH if λ0 ≤ λ ≤ λ1 , and

rM = 0. As before, rH is weakly increasing in λ and so dΠdλ

≥ 0. For λ < λ0,dΠdZ

= −YM + ddω(ωx(ω)) < 0. For

λ0 ≤ λ ≤ λ1, dΠdZ = YH − YM > 0.Now we analyze region b.2 and c.2. In the proof of Lemma 2, we showed that, in this region rM = 0, rH = 1

and ∂qM∂λ

< 0. Inside this region, an increase in λ or Z does not affect the equilibrium allocation rH or rM but itdoes affect the amount of liquidation qM . Since

∂qM∂λ

< 0, then dΠdλ= pM [−YM + d

dω(ωx(ω))]∂qM

∂λ> 0. To obtain the

effect of Z on qM , consider Þrst the effect of Z on the equilibrium return R∗. Recall that, in this region, F (R, λ, Z) =x(K + Z − λYH

RpH) − R and R∗ is such that F (R∗,λ, Z) = 0. By the implicit function theorem, ∂R

∗∂Z

= −FZFR

< 0.

This implies that R∗ = x(K + T − pH) so that it must be the case that ∂T∂Z

> 0. Since qM = (T − pL)/pM then∂qM∂Z

> 0 and so dΠdZ< 0.

In region b.3, qM = qH = 0 and so T = pL. Using the fact that T = pL, PM = λYM and PH ≥ X(K + pL − pH)(see proof of Lemma 2) the allocation rH and rM can be derived. In case b.3, rH = 1 and rM is (weakly) increasingin λ and are not affected by Z. Since rH and rM are not affected by Z, dΠ

dZ= 0. Also, higher levels of λ leads to

(weakly) higher rM and so dΠdλ≥ 0.

In region c.3, qM does not change with λ (see proof of Lemma 2), rH = 1 and rM is increasing in λ and sodΠdλ> 0. In this region ∂qM

∂Z= ∂rM

∂Z= 1

pMand dΠ

dZ= −(qM + rM )(YM − d

dω(ωx(ω))) < 0.

Region c.4 is similar to b.2 and region c.5 is similar to c.3.¥

5