In kind finance, collateral and cheap trade credit∗

Daniela Fabbri†

University of LausanneAnna Maria C. Menichini‡

University of Salerno and CSEF

February 2006

Abstract

The paper investigates the determinants of trade credit demand and its interactions with theinput combination of the firm, within an incomplete contract setting with uncertainty, two-inputtechnology and collateralised credit contracts. Assuming that the supplier is better able to extractvalue from existing assets and that she has an information advantage relative to other creditors,the paper: (1) predicts that financially unconstrained firms (with unused bank credit lines) taketrade credit because it is in fact cheaper than bank loans; (2) predicts that the reliance on tradecredit may be invariant to the degree of credit rationing, depending on product characteristics;(3) derives predictions on how trade credit demand (supply) varies among firms using (producing)differentiated or standardised inputs, rather than services; (4) provides a rationale for why sellerslend goods to their customers but seldom lend them cash; (5) predicts that a larger use of tradecredit goes together with an input choice biased towards tangible assets; (6) shows that bothfinancing and input choices are sensitive to changes in the degree of creditor protection in a waythat depends on the degree of credit rationing faced by the firm.

Keywords: Trade Credit, Collateral, Financial Constraints, Asset Tangibility, CreditorProtection.

JEL classification: G32, G33, K22, L14.

∗We would like to thank Stefan Ambec, Alberto Bennardo, Mike Burkart, Tore Ellingsen, Marcello D’Amato, MarcoPagano, Salvatore Piccolo, Bruno Parigi, Pascal St-Amour, Peter Simmons, Dezso Szalay, Elu Von Thadden, LucyWhite, the participants to the CEPR-SITE Conference on “Understanding Financial Architecture: On the Economicsand Politics of Corporate Governance” at Stockholm University, the 1st CSEF-IGIER Symposium on Economics andInstitutions in Capri, the Conference in Tribute to Jean-Jacques Laffont in Toulouse, the EEA 2005 congress inAmsterdam and the seminar participants at INRA-University of Grenoble, University of Bologna and University ofPadova for valuable comments and suggestions. Financial support from the Swiss National Science Foundation throughthe NCCR-FINRISK research project is gratefully acknowledged by Daniela Fabbri. Usual disclaimer applies.

†Contact address: HEC-University of Lausanne, Route de Chavannes 33, 1007 Lausanne, Switzerland. Tel.:+41 216923369. Fax:+41 21 6923345.E-mail address: daniela.fabbri@unil.ch

‡Dipartimento di Scienze Economiche e Statistiche, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano(SA), Italy. Tel.: +39 089 962174. Fax. +39 089 963169. E-mail: amenichi@unisa.it

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1 Introduction

Firms raise funding not only from specialised financial intermediaries but also from suppliers of inputs,

most generally by delaying the payment of the same (i.e., by taking trade credit). There is a general

consensus that trade credit is widespread among firms facing obstacles to raise funds in the bank

credit market. Conversely, following the presumption that it is considerably more expensive than bank

loans,1 financially unconstrained firms should not rely on it, thus suggesting that the dependence on

trade credit increases with the degree of credit rationing. Although it is difficult to investigate the

relation between the use of trade credit and the degree of credit rationing, given the lack of clear

measures of financial constraints, the empirical evidence is not always consistent with these common

beliefs. For example, Petersen and Rajan (1997) provide evidence that accounts payable average 4.4%

of sales for small US firms against 11.6% for large firms, which are less likely to be credit-constrained.

Moreover, they document that the presence of obstacles in raising funds in the bank credit market

has no significant impact on the use of trade credit, suggesting that the reliance on trade credit is

not necessarily increasing in credit rationing.2 Similarly, Marotta (2001) documents that trade credit

financed on average 38.1% of the input purchase of non-rationed firms, as opposed to the 37.5% of

rationed ones in the Italian manufacturing sector.3

A feature in the use of trade credit that is common to all firms, thus regardless of the degree of credit

rationing, is that the supplier’s lending activity is closely tied to the value of the input transaction, i.e.

suppliers extend credit by allowing delayed payment for their products, but they seldom lend cash.

Given that, and given that not all the inputs can be purchased on credit, it is reasonable to expect that

the use of trade credit goes together with some bias in the input combination optimally chosen by the

entrepreneur.4 So far, the empirical literature has investigated each of these choices - financing and

technology - separately. On one hand, firms in developing countries have higher proportions of fixed

assets to total assets and fewer intangible assets than firms in developed countries (e.g., Demirguc-

Kunt and Maksimovic, 2001). On the other hand, several papers document cross-country differences

in the use of trade credit (a larger use in less developed countries) and relate them to a different1The evidence is mostly anecdotal (Petersen and Rajan, 1997; Ng, Smith and Smith, 1999; Wilner, 2000). In support

of it, the canonical “2/10 net 30” agreement (a 2% discount for payment within 10 days, with the net price charged forpayment within 30) is invoked, which implies an effective interest rate above 40% for those who do not take the discount.None has measured the actual use of this agreement in the buyer-seller relationship, though.

2Petersen and Rajan (1997) find that firms that have been denied credit in the previous year receive more trade credit,but the coefficient is not statistically significant.

3Marotta (2001) uses data from the Italian survey on manufacturing sector conducted by the bank MediocreditoCentrale in 1994. Credit constrained firms are identified by two questions: “In 1994, has the firm applied for, but notobtained, more bank loans?” and “in 1994, would the firm have accepted tighter terms (higher interest rates or highercollateral requirements) to obtain more bank loan?”

4For example, intangible assets cannot be financed in general by trade credit.

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degree of creditor protection (see among others, Rajan and Zingales, 1995; La Porta et al.. 1998;

Frank and Maksimovic, 2004).Maksimovic, 2004). This latter pattern is ascribed to a different degree

of protection of creditor rights. Although fragmented, the previous cross-country evidence seems to

suggest a common pattern between financing and input choices and possibly a common denominator

in the degree of legal protection.

The previous discussion poses several questions which are hard to reconcile with the existing

theories. First, given that trade credit is considered to be more expensive than bank credit, what

justifies its widespread use by financially unconstrained firms? Second, why is the reliance on trade

credit not always increasing in the degree of credit rationing? Third, why do suppliers lend inputs but

only seldom lend cash? Last, is the use of trade credit related to the firm’s input combination, and

how are the financing and input choices affected by the degree of creditor protection?

Several theories have been proposed to explain the use of trade credit. Among these, the financial

theories, which give the supplier a comparative advantage in lending over other financiers, are those

most closely related to our work. The lending advantage may be due either to the supplier having

some private information about the firm (information advantage), or to a better ability in salvaging

value from existing assets (liquidation advantage). Theories using an information advantage argument

(see among others, Mian and Smith 1992; Biais and Gollier, 1997; Burkart and Ellingsen, 2004)

have rationalised the use of trade credit by financially constrained firms, but have left unexplained

its reliance upon by non-rationed firms. Theories using a liquidation advantage argument (Frank

and Maksimovic, 2004) have shown that also creditworthy firms (with higher probability of success)

take trade credit, but have left unexplained both its use by financially unconstrained firms and the

constant reliance upon it across rationed and non-rationed firms. Moreover, focusing on a single input

technology, neither of the above theories has accounted for cash lending and allowed to capture the

interrelations between financing decisions and input choices.5

Our paper differs from the extant literature in that we interact the information with the liquidation

advantage arguments and let the entrepreneur choose among different assets (multi-input technology).

We model the information advantage as in Burkart and Ellingsen (2004): due to investment

unobservability, the entrepreneur faces a moral hazard problem towards his creditors. However,5The non-financial theories of trade credit justify its use with a variety of marketing considerations and look at its role

in reducing transaction costs. Under uncertain product quality, allowing the buyer to inspect the good before paying forit, provides a warranty about the quality of the product (Lee and Stowe, 1993; Long et al., 1993; Emery and Naiar, 1998).Moreover, when changes in prices are costly or prevented by antitrust laws, it allows to offer effective price discriminationto different types of customers (Brennan et al., 1988). It is furthermore used to facilitate the establishment of long termrelationships with customers (Summers and Wilson, 2002). Last, it allows to minimise transaction costs. Following Ferris(1981), because of the separation between uncertain time of delivery of goods and payment of the same, the buyer maywant to cumulate obligations and pay them at specified dates.

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because the supplier provides an input which is less liquid than cash (thus less easy to divert) and

the input transaction is observable to him but not to third parties, the incentive problem is less

severe with the supplier than with the bank. In addition, we introduce uncertainty and assume

that the entrepreneur can sign a collateralised credit contract, where the collateral consists of the

inputs not exhausted in production. This allows us to introduce the liquidation advantage into our

framework, by assuming that the supplier extracts a higher value from collateral in case of distress.

Last, we assume that, besides choosing between sources of funding, the entrepreneur can choose

between two inputs, labelled tangibles and intangibles, with different features in terms of degree of

liquidity, collateralisability and financing. These modeling choices are crucial to address the four open

questions listed above.

A novel prediction of our analysis, which accounts for the first question, is that firms with unused

bank credit lines (thus financially unconstrained) purchase inputs on credit for a financial motive.

Because of the supplier’s liquidation advantage, trade credit is in fact cheaper than bank credit,

despite banks having a comparative advantage in raising funds on the deposit market. The result

of trade credit as a cheap source of finance is in sharp contrast with the assumption that scholars

typically use in this literature. Although this assumption is crucial to all the incentive theories of

trade credit, there is little evidence, mostly anecdotal, that suppliers charge huge credit terms on their

buyers when delaying payments of products. Moreover, recent evidence (see Marotta, 2001; Miwa and

Ramseyer, 2005) seems to contradict this practice and provides support for our theoretical finding.

Our prediction comes with an interesting side result. The supplier’s liquidation advantage can be

exploited only if she can act as a liquidator, i.e., if she gets priority in repossessing the goods sold in

case of default. Although our model claims that it would be optimal to give seniority on debt claims

to the supplier, the extent to which this is the case in practice strictly depends on the way bankruptcy

codes are designed (i.e, whether they are debtor rather than creditor oriented and the extent to which

the supplier is protected against other creditors). In the paper, we discuss some interesting predictions

that would follow from incorporating in our model extant legal provisions of bankruptcy codes.

While the liquidation advantage is by itself sufficient to explain the demand of trade credit by

financially unconstrained firms, the interaction between liquidation and incentive motives is crucial to

account for a reliance on trade credit not always increasing with financial constraints. In particular,

our analysis shows that financially constrained firms may take trade credit for both reasons. If for

incentive motive, rationed firms finance a larger share of their inputs on credit than the non-rationed

ones, in line with previous literature. Conversely, when the liquidation motive dominates, this share

stays constant across firms with different degrees of credit rationing. This is a unique prediction of

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our model that, unlike extant literature, is also consistent with the empirical evidence. When the

liquidation motive dominates, we even find that, in order to fully exploit the liquidation advantage,

firms are willing to keep their bank credit line unused. It follows that, in contrast to what one might

expect, firms are never rationed on bank financing, even when they are on trade credit.

We develop even further this argument by showing that the relation between trade credit use and

financial constrains depends crucially on the characteristics of the input. We show that the specific

reason - incentive versus liquidation - that drives the trade credit demand by rationed firms depends

on the degree of liquidity and the collateral value of the inputs sold by the supplier. Firms using inputs

with a high degree of liquidity or a high collateral value (e.g., standardised inputs) are more likely

to take trade credit to exploit the liquidation advantage of the supplier. Conversely, the incentive

motive is more likely to dominate among financially constrained firms purchasing illiquid inputs with

a low collateral value (e.g., services). From the previous discussion, it also follows that the relation

between the maximum credit line offered by the supplier and the degree of input liquidity is non-

monotonic. Intuitively, the higher the input liquidity, the more severe the incentive problem for the

entrepreneur, but at the same time the higher the value of the input in case of repossession. Thus,

with an increasing degree of input liquidity, we have first a dominance of the incentive effect (trade

credit line decreasing), and then of the liquidation effect (trade credit line increasing). This result is

in contrast with the existing theoretical literature that predicts the maximum trade credit line to be

always decreasing in input liquidity. Focusing on the use of trade credit by non-rationed firms first,

and then by rationed ones, our analysis also allows to identify separately the demand and the supply of

trade credit and thus to investigate how the two sides of the market vary with product characteristics.

We derive new predictions which are in line with recent empirical evidence provided by Burkart et al.

(2004).

A further contribution of our paper is to identify the reasons why suppliers lend goods but, in

spite of their information advantage, they do not lend cash. In our model, the lack of cash lending

follows from the assumption that the supplier can only observe the input transaction in which she is

involved. Extending, at least partially, this information advantage to the relationships the firm has

with other parties/suppliers, we find that cash lending arises endogenously. As far as we know, there

is evidence on cash lending only among Japanese trading companies (Uesugi and Yamashiro, 2004).

Interestingly, one of the features of these firms seems to be a higher involvement of suppliers in the

firm’s activity, because of an organisational structure that guarantees a continuous flow of information

from the clients to the suppliers of inputs.

The result that suppliers provide credit only by delaying the payment of the inputs they sell

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implies that this credit can only be used to finance specific inputs, in our case the tangible ones. Since

their purchase can be used to relax the credit constraints, a rationed entrepreneur may be willing

to exploit this advantage by altering even more the input combination towards the tangibles. This

feed-back relationship introduces a strong link between financing and input decisions that has been

so far disregarded by the related literature. Exploiting this link, our analysis generates a number of

novel predictions. We show that a more intensive use of trade credit goes together with a technology

biased towards tangible assets. We also show that changes in the creditor rights vulnerability that

alter the incentives to divert resources strongly impact the input combination. In particular, a weaker

protection of creditor rights induces rationed entrepreneurs to rely more on trade credit and to shift

towards technologies more intensive in tangible assets, while it has no effect either on financing decisions

or on input choices if entrepreneurs are non-rationed. The intuition is that when wealth decreases

or creditor rights vulnerability increases, the moral hazard problem becomes more severe because the

entrepreneur’s incentive to divert resources increases. However, different input observability of the two

inputs implies that this incentive problem is less severe for the input that is financed by trade credit,

even if partially. It follows that tangible assets have a lower shadow cost than the intangible ones

and are therefore preferred. Our predictions are in line with the existing fragmented cross-country

evidence previously discussed (Rajan and Zingales, 1995; La Porta et al., 1998; Demirguc-Kunt and

Maksimovic, 2001; Frank and Maksimovic, 2004)

The paper is organised as follows. In Section 2, we describe the model. In Section 3, we analyse

the determinants of trade credit demand and its relation with financial constraints and product

characteristics. In Section 4, we derive the optimal input combination and we investigate how input

choices and financing policies adjust to changes in entrepreneur’s wealth and in the degree of creditor

rights vulnerability. In Section 5, we provide a discussion of the main results and isolate the necessary

conditions for cash lending. In Section 6, we endogenise the cost of funds of the supplier. In section

7, we conclude.

2 Setup and model assumptions

A risk-neutral entrepreneur needs funding to carry out an investment project which requires the

purchase of two inputs, a tangible (e.g., physical capital) and intangible (e.g., human capital) one,

denoted by qk and qL, respectively. The amount of purchased inputs that is invested, denoted by

Ik, IL, and observable to the entrepreneur only, is transformed into a verifiable state contingent

output yσ, with σ ∈ {G, B} and yG > yB = 0. The good state (σ = G) occurs with probability

p. Uncertainty affects production through demand (i.e., production is demand-driven). In times of

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high demand, all inputs are used in production and return output yG according to the increasing and

strictly concave production function fG(Ik, IL) > 0. In times of low demand, no output is produced

(yB = fB(Ik, IL) = 0, for any Ik, IL) and the firm is worth only the scrap value of unused inputs, which

can therefore be pledged as a collateral to the financiers.6 Last, inputs are substitutes in production,

but each one is essential.

The entrepreneur is a price taker in the market for inputs and in the market for the final good.

The output price is normalised to 1. The price of the tangible and intangible inputs is given by ρ and

w, respectively.

The input purchase qk and qL can be financed with the entrepreneur’s own wealth (A), which is

observable to all parties, and with the credit raised from competitive banks (LB > 0) and suppliers

(LS > 0). We place an upper bound on the entrepreneur’s wealth assuming that it is never so high to

finance the first best investment: some credit has therefore to be raised from banks and/or suppliers.

Banks and suppliers play different roles. Banks only lend cash. Suppliers sell inputs, but they can

also act as financiers, lending inputs and cash. Of the two inputs provided by the suppliers, we assume

that only the supplier of the tangible input (physical capital) can finance its purchase. Each unit of

intangible input instead is fully paid for in cash. Thus, if we interpret intangible assets as high-skilled

employees, working in research and development units inside the firm, this assumption captures the

idea that there cannot be any labour credit, i.e. employees do not finance the hiring of labour units.

Banks and suppliers also differ in the type of information they have. Suppliers of each input

can costlessly observe the amount of that input which is purchased by the entrepreneur, but not the

amount that is purchased from the other supplier. Thus, the supplier of tangible inputs observes

which is the amount qk purchased by the entrepreneur, and similarly for the supplier of the intangible

input.7 It follows that the bank, which supplies no input but only lends cash, cannot observe whether

any input transaction takes place, i.e. whether the loan made is used for input purchase.8

Of the inputs qk and qL purchased, the entrepreneur can choose to invest an amount Ik ≤ qk, and

IL ≤ qL. Neither the bank, nor the supplier can observe the actual amount invested in the venture.9

Under the assumption of imperfect creditor rights protection, investment unobservability introduces6This implies that ex-post diversion is not feasible. However, allowing for this case does not alter our qualitative

results, so long as there is a minimal share of the assets which cannot be hidden (e.g., the premises of the firm, or heavymachinery).

7We assume that the cost of observing the purchase of inputs by parties other than their direct suppliers is too highto make observation worthwhile.

8We assume that the bank cannot condition the contract on qk or on a share of that. Assuming full unobservabilityof the tangible input from the bank is not necessary to get our results. We could also assume that the bank can partiallyobserve qk. What matters is that the supplier has a higher degree of observability, which is reasonable, given that sheprovides the input.

9If they could, they would write a contract which is conditional on the realised investment.

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a problem of moral hazard on the entrepreneur, since the funding, either in cash or in kind, raised

from the creditors might not be invested in the venture but diverted to private uses, which do not

maximise the lenders’ expected return. In case of diversion, creditors get paid only to the extent that

project returns are available. If all resources are diverted, by limited liability nothing is repaid. The

profitability of the diversion activity depends on the vulnerability of creditor rights (φ) and on the

characteristics of the inputs, in particular on their degree of liquidity. The more vulnerable creditor

rights are (the higher φ), the higher the return from diversion. In particular, one unit of diverted cash

gives the entrepreneur φ < 1 units of private benefit. To make the problem interesting, we assume

that φ is so high that a zero-wealth firm can never carry out the first best investment. This amounts

to introducing the following assumption:

Assumption 1 : φ > φ.10

Inputs are less liquid than cash and thus less easily divertable. One unit of input qk costs ρ and

gives ρφβk units of private benefit, where βk < 1 is the degree of capital input liquidity.11 Similarly, one

unit of input qL costs w and gives wφβL units of private benefit if diverted, where βL < 1 is the degree

of intangible input liquidity.12 Diversion of cash or inputs implies therefore a loss on the entrepreneur

which is increasing in the degree of creditor’s legal protection. Differential input observability and

differences in input characteristics imply therefore that the entrepreneur’s moral hazard problem is

less severe if funding is raised from suppliers providing an illiquid input.

Inputs are also important if they can be repossessed in the event of firm’s default. We assume

that only tangible assets (qk) can be pledged to financiers, while intangibles have zero collateral value.

Interpreting qL as high-skilled labour, this implies that financiers cannot extract any positive return

from workers after default has occurred.13

Another crucial assumption concerns the different abilities of the financiers to liquidate the tangible

inputs. We assume that the supplier has a better knowledge of the resale market, which gives her a

more efficient liquidation technology relative to the bank. This implies that in case of default, the

scrap value of tangible assets is higher if these are pledged as a collateral to the supplier rather than

to the bank, i.e. βSC ≥ βBC, where C is the collateral liquidation value (C = βkρIk) and βS ≥ βB.10According to the scenario that arises, φ can take two values, φ1 and φ2. We define them in the appendix.11Interpreting capital input diversion as resale, βk < 1 can be justified considering that often the sale of goods is

bundled with services which can only be provided by the dealer, and thus not by the entrepreneur in second-hand sale.12Interpreting qL as high-skilled labour, the assumption that 0 < βL < 1 implies that it is possible to extract private

benefits from workers by assigning them to tasks different from those they were employed for. In many countries, however,such practices are inhibited by the employment protection legislation. Notice that we do not make any assumptionregarding the relative degree of liquidity of the two inputs as it turns out to be immaterial for the analysis.

13A similar argument holds if we consider intangibles assets as patents or goodwill.

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Finally, we assume that the cost of raising one unit of funds on the market is higher for the

supplier than for the bank (rB < rS). This is without loss of generality for several reasons. First,

it is consistent with the role of specialised financial intermediaries of banks. Moreover, the supplier

is likely to be herself credit constrained and have a higher cost of funds than banks. Finally, even

assuming rB = rS , the supplier might internalise her information and liquidation advantage over other

financiers by charging a higher price for the funding provided.14

Contracts. The entrepreneur-bank contract specifies{LB, RB

σ (yσ, LB) , γ}

, where LB is the loan

from the bank to the entrepreneur and RBσ is the state-contingent repayment obligation, depending

on the output realised and on the size of the loan, and γ, which is the share of the collateral obtained

in case of default.

The contract between the entrepreneur and the supplier of the tangible input specifies

{qk, LS , RSσ (yσ, LS) , 1 − γ}, where LS is the credit given by the supplier to the entrepreneur, RS

σ

is the state contingent repayment obligation, depending on the size of the loan, the output realised

and the input purchase qk, and 1− γ, which is the share of the collateral obtained in case of default.

Unlike the bank, the supplier can condition the contract on the input purchase.

Because the intangible input is fully repaid when it is hired, the relevant contract between the

entrepreneur and the supplier of the intangible input specifies only the amount of the input purchased

qL.

Each party is protected by limited liability.

Time-line

1. Competitive banks and suppliers make contract offers, given A;

2. the entrepreneur accepts or rejects;

3. conditional on acceptance, the investment/diversion decisions are taken (q∗k, q∗L; and I∗k , I∗L are

chosen).

4. the uncertainty resolves;

5. the payoff realises and repayments are made.

14In Section 6, we develop more thoroughly this argument showing that the analysis carried out under the assumptionthat rB < rS can be seen as a reduced form of a model in which rB = rS and the supplier has bargaining power.

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E accepts or rejects

If E accepts, investment -diversion decision

CompetitiveB and S make contract offers

Uncertainty resolves

Pay-offs realise and repayments made

t

3 Determinants of bank and trade credit

In our model there are firms, banks and suppliers of the tangible and intangible inputs. Firms carry

out production, which is financed with internal funds and with the funding provided by banks -in cash-

or by suppliers of the tangible input -in cash or in kind.15 Since banks have a comparative advantage

in raising funds on the market (rB < rS), entrepreneurs would prefer bank financing to trade credit.

However, supplier’s credit has two advantages relative to bank’s financing. First, the supplier is better

able to liquidate the inputs if repossessed from a defaulting firm, which may reduce the actual cost of

trade credit. Second, being less liquid than cash, lending inputs rather than cash reduces the scope

for diversion, thus making the entrepreneur moral hazard problem towards the supplier less severe

than that faced with the bank. There are therefore two motivations underlying trade credit demand:

a liquidation motive (i.e. to exploit the supplier liquidation technology) and an incentive motive (i.e.

to relax financial constraints created by moral hazard problems). In this section, we discuss under

which conditions each of the two motives becomes relevant and how they interact.

Firms maximise profits, which can be split into two components: the return from production (EP )

and the return from diversion (D). The expected return from production is given by:

EP = p[fG (Ik, IL)−RB

G −RSG

]+ (1− p)

[fB (Ik, IL)−RB

B −RSB

]where RB

σ , RSσ ≥ 0 are the repayments due to bank and supplier respectively when state σ = {G, B}

occurs. Because low state output is zero and because of limited liability on all players, the repayments

that banks and suppliers get in the bad state are both zero (RBB = RS

B = 0).16

The return from diversion is equal to:

D = φ {βkρ (qk − Ik) + βLw (qL − IL) + [A + LB + LS − wqL − ρqk]}15A remark is warranted here regarding the terminology. We will henceforth use the term trade credit to identify the

credit, either in cash or in kind, provided by the supplier. To be rigorous, however, the term trade credit should be usedonly to define in-kind finance and not include any cash lending. We will show that under our assumptions cash lendingby the supplier never arises in equilibrium and the supplier only lends inputs, which reconciles with our terminology. Wewill come back to this point in Section 5.7.

16Banks and suppliers can still get a repayment in the bad state by getting a share of the scrap value of unused inputsβSC.

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where the first two terms in round brackets denote the return from input diversion and the term in

square brackets denotes the return from “residual” cash diversion (the difference being the amount of

cash not spent in input purchase).

We first show that the entrepreneur never purchases intangibles for diversion purposes, i.e.

qL = IL = 0. This is because: a) the transaction in intangible inputs is a cash transaction, which

implies that it cannot be used to raise credit from its suppliers; b) it is unobservable to any creditor,

which implies that no contract can be conditioned on the amount of intangibles that is actually

purchased. These two considerations, along with the lower degree of liquidity of intangibles relative to

cash, imply that the entrepreneur prefers to divert cash rather than buy intangibles and then divert

them. Hence qL = 0 and thus IL = 0. A similar argument can be used to show that the entrepreneur

does purchase tangibles for diversion purposes, even though they are less liquid than cash. This is

because the supplier is willing to extend credit to the firm and the contract can be conditioned on the

input transaction taking place. Thus, a potentially diverting entrepreneur will purchase the desired

qk from the supplier and keep the residual resources in cash. This gives the following return from

diversion:

D = φ {βkρ (qk − Ik) + [A + LB + (LS − ρqk)]}

The inefficient diversion technology and the uncertain investment return at the time of the

investment/diversion decision imply that partial diversion is never optimal.17 Thus, either all funds

(and inputs) are used for investment (D = 0), or they are diverted, which implies that neither share

of the purchased inputs is invested: Ik = 0. The return from diversion reduces to:

D = φ {βkρqk + [A + LB + (LS − ρqk)]}

The entrepreneur’s problem is therefore given by programme PG :

maxLB ,LS ,Ik,IL,RB

G,RSG,γ

EP + φ (βkρqk + A + LB − (ρqk − LS)) (1)

17Two cases are conceivable here: (i) invest in the venture an amount sufficient to repay the loan in full (it has sufficientrevenues to repay LBrB); (ii) invest in the venture an amount not sufficient to repay the loan in full. (i) Suppose theentrepreneur has to decide whether to invest all resources or to invest almost all of them and divert the marginal unit. Byprofit maximisation Ef ′k = ρrB , Ef ′L = wrB . Is it worthwhile to divert the marginal unit of Ik? Invested in productionthis gives ρrB , but if diverted it gives ρφβk, i.e. the value of the input scaled by φ. Thus, the entrepreneur always prefersto invest this unit, and even more so for the last but one units. Thus once he decides to invest an amount sufficientto repay the loan, he better invests all the resources available. (ii) Suppose the entrepreneur invests in the venture anamount not sufficient to repay the loan in full (pfG (Iu

k , IuL) + (1− p) βBρIu

k < LBrB , where Iu is a level of investmentlower than the minimum required to repay the loan in full). Because output is observable, any return from productionwill be claimed by the creditors and the entrepreneur will get a return only from the residual resources not invested anddiverted. This implies that there is no point in investing any amount insufficient to repay the loan in full. In this caseit is better to divert all resources.

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s.t. EP ≥ φ (A + LB) (2)

EP ≥ φ {βkρqk + (A + LB − (ρqk − LS))} (3)

pRBG + (1− p) γβBC ≥ LBrB (4)

pRSG + (1− p) (1− γ) βSC ≥ LSrS (5)

A + LB + LS ≥ wIL + ρIk (6)

RSG ≥ (1− γ) βSC (7)

where EP = p[fG (Ik, IL)−RB

G −RSG

]is the expected profits from production; LS is the amount of

trade credit raised from the supplier costing rS ; βS and βB are the collateral values of the unused

inputs for the supplier and for the bank respectively, with βS > βB, while γ is the share of collateral

accruing to the bank and (1− γ) the one accruing to the supplier.

Constraint (2) is the incentive compatibility condition towards the bank, which prevents the

entrepreneur from diverting internal funds (A) and the credit raised from the bank (LB); (3) is

the incentive constraint towards the supplier, preventing the entrepreneur from diverting the input qk

purchased from the supplier (with return βkρqk) plus any spare cash left after the inputs have been

acquired with bank credit and internal funds (A + LB − (ρqk − LS)); constraints (4) and (5) are the

bank and the supplier’s zero profit conditions respectively; (6) is the resource constraint; last, (7) is

the non-decreasing repayments condition, ensuring that repayments to the supplier are non-decreasing

in revenues.18

The liquidation motive Looking at the participation constraints of the two financiers (4) and

(5), we can introduce the first motivation for taking trade credit, the liquidation motive. Because

βS > βB, pledging the collateral to the supplier relaxes the supplier participation constraint more

than the bank’s, reduces the total repayments due by the entrepreneur in the good state and thus

increases total surplus. Moreover, rB < rS implies that the entrepreneur is better off by asking the

supplier to act as a liquidator rather than a financier, but the non-decreasing repayment condition

prevents this. Thus, constraint (7) is binding and, using equation (5), sets the amount of trade credit

demanded for liquidation motives as:

LS−LM =1rS

(1− γ) βSC. (8)

18This is standard in the literature (see Innes, 1990). In our setup, it is justified on the ground that, without thisconstraint, a sufficiently rich entrepreneur would use the supplier only to liquidate the unused inputs in the event ofdefault and never to finance the input purchase, given that rB < rS . In exchange for the proceeds from liquidation, thesupplier would transfer an equal amount to the entrepreneur in the bad state, thereby acting as a pure liquidator. Beinginterested in the supplier’s dual role of liquidator and financier, we do not allow for such contract and require repaymentsto be non-negative and non-decreasing in revenues.

12

The entrepreneur has an incentive to take such amount of trade credit if its cost, discounted for the

saving in repayments obtained by pledging the collateral to the supplier, is lower than the cost of bank

credit, i.e. if the following condition holds:

Assumption 2 rB ≥ rS

{βS

pβS + (1− p) βB

}.

Crucial for this condition to be satisfied is thus the size of the bank’s intermediation advantage as

compared to the size the supplier’s liquidation advantage. If banks and suppliers are equally good in

liquidating the assets (βS = βB), assumption 2 is never satisfied: the advantage in the intermediation

activity makes bank credit always cheaper than trade credit. When βS is sufficiently higher than βB,

the liquidation advantage may offset the intermediation advantage, thereby making the actual price

of trade credit fall short of the price of bank credit.

Under the previous assumption, we derive the following result:

Proposition 1 At equilibrium, γ = 0, i.e. the right to repossess and liquidate the collateral goes to

the supplier.

Proof. In the appendix.

Proposition 1 claims that, as long as the liquidation advantage of the supplier is sufficiently large

(Assumption 2 is satisfied), it is optimal to pledge all the assets to the supplier and set γ = 0. It

follows that, using (8), the demand for trade credit is precisely equal to the discounted value of the

collateral assets:

LS−LM =βSβk

rSρIk (9)

Each unit of trade credit below this amount is cheaper than bank credit, since the supplier is exploiting

the liquidation revenues accruing in the bad state to lower the repayment asked in the good one. In

particular, using constraint (5) and (7), the price of one unit of trade credit is RSG = rS , while the price

of one unit of bank credit, using γ = 0 and (4), is RBG = rB/p. Under assumption 2, the first is lower

than the second. For an amount exceeding the one set by equation (9), an extra unit of trade credit

is more expensive than bank credit, since there is no more collateral to be pledged. Such amount

represents therefore the maximum and the actual amount of trade credit taken for liquidation motive.

The incentive motive Besides extracting more value from existing assets, trade credit can

also serve another scope: that of mitigating the entrepreneur’s moral hazard problems towards his

financiers. Whether this second motivation arises or not depends on which incentive problem (bank’s

or supplier’s) becomes relevant first. Comparing constraints (2) and (3), we find that two alternative

13

regimes may arise, depending on how large is the demand for trade credit for liquidation motive when

no incentive problems are in place. This in turn only depends on the parameters of the model, in

particular the ones defining the scrap value of each unit of collateral. If this is sufficiently low, i.e.,βkβS

rS≤ (1− βk), then the entrepreneur’s moral hazard problem is more severe with the bank than

with the supplier. This implies that, at the level of wealth at which the incentive constraint towards the

bank becomes binding, the one towards the supplier is still slack. The entrepreneur can therefore use

the credit line offered by the supplier to relax financial constraints and keep the investment constant.

When this happens, we argue that trade credit is taken for incentive motive, and we are in a regime

called dominating incentive motive. Given the amount of bank credit taken, suppliers are not willing

to meet any entrepreneur’s request, though. There exists a maximum trade credit line obtained when

both incentive constraints bind, namely:

LS−IMmax = (1− βk) ρIk (10)

Any extra unit of trade credit provided above this amount would induce the entrepreneur to divert all

the resources.

In the complementary parameter spaceβkβS

rS> (1− βk) , we are in the regime called dominating

liquidation motive. In this case, at the level of wealth at which the incentive constraint towards the

supplier becomes binding, the one towards the bank is still slack. Thus, trade credit cannot be (is

never?) used to relax financial constraints, but only to exploit the supplier’s liquidation advantage,

for any level of wealth.

In the next two subsections, we describe the pattern of trade (and bank) credit for varying levels

of wealth in each of the two regimes. We focus our analysis on levels of wealth for which entrepreneurs

need both trade and bank credit to finance the first best investment. Notice that since the opportunity

cost of internal wealth is lower than the cost of any type of external finance, entrepreneurs first exhaust

their own resources and then rely on external financing.

3.1 Dominating incentive motive

This subsection considers the case in which (1− βk) > βkβS/rS , i.e. firms using inputs with low

degrees of liquidity and collateralisability and paying high trade credit prices (βk, βS low, rS high).

Proposition 2 There exist three critical levels of wealth, A1 < A2 < A3 such that:

(i) for A ≥ A3 entrepreneurs finance the first best investment (IFBk , IFB

L ) and take trade credit for

liquidation motive (LS =βkβS

rSρIFB

k ) and bank credit as a residual. They solve programme PG,

with (2) and (3) slack and (7) binding;

14

(ii) for A2 ≤ A < A3, entrepreneurs fully exploit the bank credit line (they are credit constrained),

invest Ik(A) ∈[I∗k , IFB

k

)and IL(A) ∈

[I∗L, IFB

L

)and take trade credit still for liquidation motive

(LS =βkβS

rSρIk (A)), solving programme PG, with (3) slack and (7) still binding;

(iii) for A1 ≤ A < A2, entrepreneurs invest I∗k , I∗L, and take trade credit for incentive motive

(βkβS

rSρI∗k < LS ≤ (1− βk) ρI∗k), solving programme PG, with (3) and (7) slack;

(iv) for A < A1, entrepreneurs invest Ik (A) < I∗k and IL (A) < I∗L; they are constrained on both credit

lines, and take trade credit for incentive motive (LS = (1 − βk)ρIk (A), solving programme PG,

with (7) slack.

Proof. In the appendix.

The content of Proposition 2 is represented in Figures 1 and 2, which display the use of trade credit

for different levels of wealth. Sufficiently rich entrepreneurs (A ≥ A3) finance the first best investment,

take a constant amount of trade credit to exploit the better liquidation technology of the supplier and

ask for some bank credit as a residual. Entrepreneurs prefer trade credit to bank financing because the

first is cheaper than the second. Thus, they fully exploit the liquidation advantage taking an amount

of trade credit equal to the discounted value of the collateralised assets. As wealth approaches A3, the

amount of trade credit stays constant, while the amount of bank credit increases, so as to compensate

for the lack of internal wealth. For A2 ≤ A < A3, entrepreneurs become constrained on bank credit,

but they do not compensate this rationing with the available trade credit line. The reason is that the

cost of any extra unit of trade credit for amounts above the scrap value of the asset is higher than the

cost of bank credit. Thus, as wealth decreases, they are forced to reduce the investment level below

the first best, as well as the absolute level of trade credit and bank finance, but they keep the share of

inputs financed by trade credit constant. Only for wealth at or below A2, when the shadow cost of BC

is as high as the marginal cost of trade credit rS , they start demanding trade credit also for incentive

motives, i.e., to relax financial constraints and keep the investment constant. Thus, both the absolute

amount of trade credit and the share of tangible inputs it finances increase up to the maximum defined

in (10), at which also the incentive constraint towards the supplier binds (A = A1). For A < A1, the

entrepreneur is constrained on both credit lines. The absolute amount of trade credit decreases, but

the share of input it finances stays constant and equal to (1− βk). It follows that the share of inputs

financed by trade credit is decreasing in wealth and thus higher for financially constrained firms, as

reported in Figure 2. Notice also that this share is below unity for any A, meaning that the supplier

finances the purchase of tangible inputs and never lends cash.

15

wealthA2 A3

unconstrained

TC for LMTC for LM

constrained on TC and BC

A1

TC for IM

γ=0

increasing repayments contractγ=0

flat contract

constrained on BC

TC for IM

Figure 1: Liquidation and incentive motives

wealthA1 A2 A3

LS/ρIk

S

Sk

r

ββ

kβ−1

Figure 2: Share on inputs purchased on account

16

The reasons underlying the demand for trade credit, liquidation versus incentive motive, also affect

the properties of the financial contract between the entrepreneur and the financiers, as described in

corollary 1.

Corollary 1 The bank gets a contract with repayments increasing in cash flows for any level of wealth,

while the supplier gets a contract with flat repayments across states when A ≥ A2, and a contract with

repayments increasing in cash flows when A < A2.

Proof. By Proposition 1, the supplier always gets full priority in the liquidation of the collateral.

Moreover, when A ≥ A2, the non decreasing repayments condition (7) is binding. This implies that

the supplier gets the same return across states (equal to the scrap value of unused inputs), namely

βSβkρIk/rS , and provides an amount of credit also equal to the collateral value. Conversely, when

A < A2, entrepreneurs demand trade credit also for incentive reasons. In such circumstances, the value

of unused scrap inputs is not sufficient to repay the supplier for the loan given and any extra unit of

trade credit can be provided only if a higher repayment is promised in good states. She gets therefore

an increasing repayment contract, with an extra return for any unit of trade credit taken above the

collateral value. As for the bank, because the collateral is always repossessed by the supplier, it gets

no return in the bad state but only in the good state, and thus a contract with repayments increasing

in cash flows for any A.

According to Corollary 1, the properties of the financial contract between the entrepreneur and the

supplier depend on the reasons behind the demand for trade credit. The supplier gets a flat contract

when trade credit is demanded for liquidation motive and an increasing repayment contract when the

incentive motive becomes the driver.

3.2 Dominating liquidation motive

In this section, we consider the complementary parameter space (1− βk) < βkβS/rS , i.e. firms using

inputs with high degrees of liquidity and collateralisability and paying low trade credit prices (βk, βS

high, rS low).

Proposition 3 There exists a critical level of wealth, A1, such that:

(i) if A ≥ A1 entrepreneurs finance the first best investment (IFBk , IFB

L ) taking trade credit for

liquidation motive (LS =βSβk

rSρIFB

k ) and bank credit as a residual. The optimal contract solves

programme PG, with (2) and (3) slack and (7) binding;

17

wealthÂ1

unconstrained

TC for LM

γ=0flat contract

unconstrained on BC constrained on TC

TC for LM

Figure 3: Liquidation motive and unused bank credit line

(ii) if A < A1, entrepreneurs invest Ik (A) < IFBk and IL (A) < IFB

L taking trade credit for liquidation

motive (LS =βSβk

rSρIk (A)) and bank credit. The optimal contract solves programme PG, with

(2) slack and (3) and (7) binding.

Proof. In the appendix

The content of Proposition 3 is represented in Figures 3 and 4. As in the previous case, wealthy

firms (those with A > A1) finance the first best investment by using an amount of trade credit equal

to the scrap value of the collateral assets and bank credit as a residual. The reliance on bank credit

increases as wealth goes down to compensate for the lack of internal resources, while the demand for

trade credit stays constant.19 At A = A1, the entrepreneur becomes constrained on trade credit but

is still unconstrained on bank credit. Moreover, as wealth decreases, this credit line remains unused,

so as to preserve the incentive constraint towards the supplier. The entrepreneur therefore is forced

to reduce the level of investment, but he keeps constant the share of inputs financed by trade credit.

As in the previous regime, this share is below unity for any A, which implies that there is no cash

lending.

Proposition 3 leads to the following corollary:

Corollary 2 Entrepreneurs with A < A1 exhaust the trade credit line, but they never fully exploit the

bank credit line.20

19Recall that trade credit is cheaper than bank credit for an amount not higher than the liquidation value of thecollateral. For larger amounts the liquidation advantage is lost and its price is no less than the price of bank credit.

20Notice that in the intermediate case, where 1− βk = βkβS/rS , there exists a threshold level of wealth A, such thatentrepreneurs with A < A1 are unconstrained on both credit lines, while those with A ≥ A1, do not exhaust either creditline. In either case, trade credit is only used for liquidation motives. Incentive reasons never arise.

18

wealthÂ1

LS/ρIk

S

Sk

r

ββ

kβ−1

Figure 4: Share of inputs purchased on account

Proof. The formal proof develops the following three arguments. Because the entrepreneur is

constrained on trade credit, it is impossible to raise extra credit, be it in cash or in kind, as this violates

the incentive constraint towards the supplier. Raising extra credit from the bank is therefore only

feasible if the entrepreneur substitutes trade credit with bank loans, so as to preserve his incentives.

However, this is suboptimal since the first is cheaper than the second.

Suppose that at A < A1, the entrepreneur takes an extra amount of cash. He can use this to

purchase inputs to be invested in production. Because constraint 3 is binding, the marginal return

from investment is lower than the return from diversion.21 Thus, it is better to divert the marginal

unit rather than investing it. However, we have shown that partial diversion is never optimal: either

he diverts all resources or nothing. By diverting all the credit raised in inputs and cash, the maximum

return he gets is given by φ[βkρIk (A) + wIL (A) + (EP −DB)

], where (EP − DB) is the extra

credit.22 This is higher than the return he would get from production without taking the extra credit

from the bank, namely φ(βkρIk (A) + wIL (A)

). The bank anticipates the entrepreneur’s incentive

to take the entire credit line for diversion and does not provide the extra credit demanded, unless

the entrepreneur grants her a share of the collateral γ = 1 − 1−βkβkβS

rS in case of default. This lowers

21Formally, the derivative of the incentive constraint (3) when (7) is binding is negative:

pρ

∂fG(·,·)∂Ik

− 1rS

[(rS − βSβk) rB + prSβSβk] < φβk

pw

∂fG(·,·)∂IL

− rB < φ

otherwise it would be possible to increase Ik and IL without violating the constraint.22The term (EP −DB) is the unused bank credit line, i.e. the difference between the left and the right hand side of

the incentive constraint towards the bank (2), which is slack.

19

the credit line provided by the supplier to (1− βk) ρIk (A) and restores the incentives for production.

However, this is suboptimal since any increase in γ decreases the entrepreneur’s profits (by Assumption

2, the entrepreneur’s profits are decreasing in γ). Thus the entrepreneur prefers not to take the extra

credit line provided by the bank and give full priority to the supplier in case of default (γ = 0).

Corollary 3 The bank gets a contract with repayments increasing in cash flows, while the supplier

gets a contract with constant repayments across states for any level of wealth.

Proof. Because trade credit is taken for liquidation motives and the share of inputs it finances

stays constant across wealth and equal to βkβSrS

, by the same argument used in the proof of Corollary

1, the supplier gets a flat contract while the bank gets an increasing repayment contract.

4 Input choices, financing decisions and creditor protection

While in the previous part, we investigated the evolution of trade credit demand for different levels

of wealth, in this section we focus on the relation between input choices and financing decisions, and

how they react to changes in the degree of creditor protection. We perform our analysis under the

assumption that (1− βk) > βSβkrS

, which corresponds to the case described in Section 3.1 (dominating

financing motive). However, as we will discuss in the end of this section, qualitatively similar results

hold also for the complementary parameter space (dominating liquidation motive).

Proposition 4 If the production function is homogeneous of degree j ≤ 1, both the degree of asset

tangibility ( IkIL

) and the trade credit intensity ( LSLS+LB+A) are non-increasing in wealth.

Proof. In the Appendix.

Proposition 5 If the production function is homogeneous of degree j ≤ 1, an increment in creditor

rights vulnerability increases both the degree of asset tangibility ( IkIL

) and the trade credit intensity

( LSLS+LB+A) for any A < A1 and A2 ≤ A < A3; it has no effect either on Ik

ILor on LS

LS+LB+A for any

A ≥ A3; it increases LSLS+LB+A while it leaves constant Ik

ILfor any A1 ≤ A < A2.

Proof. In the Appendix.

Propositions 4 and 5 share a similar intuition related to the determinants of the relative price

between the two sources of financing and between the two inputs (tangible over intangible). Under the

assumption of homogeneity of degree j ≤ 1 of the production function, the optimal input combination

only depends on the input price ratio, defined by ( PkPL

). Each input price reflects both the financing

cost and the cost of moral hazard. The weight of each of these two components depends not only on the

20

entrepreneur’s wealth, but also on differences between inputs, as well as on differences in information

held by suppliers and banks. In particular, the result that suppliers only finance a share of the input

they sell, without providing any cash lending (see Proposition 2), together with the assumption that

intangibles must be paid on spot, imply that intangibles are fully financed through bank credit (and

internal wealth). The higher degree of liquidity of cash relative to any input also implies that the

moral hazard problem behind the use of bank credit is more severe than the one behind the use of

trade credit. All these argument are crucial to explain the pattern of trade credit intensity and input

combination, as shown in Figures 5 and 6, respectively. Let us start with high levels of wealth. For

any A ≥ A3, the firm is unconstrained on both credit lines. This implies that the relative price ratio

between trade and bank credit does not depend on the level of wealth. It follows that both the trade

credit intensity and the input tangibility stay constant.

When the incentive constraint towards the bank binds (A < A3), reductions in wealth increase

the shadow price of bank credit, and therefore decrease the price ratio between trade credit and bank

credit. This implies that the firm gives up more bank credit than trade credit, thereby increasing

trade credit intensity. Such increment affects both input prices but with different magnitudes: it fully

translates into a higher price of intangibles, given that they are totally financed by bank credit, while

it translates into a higher price of tangibles only by the fraction (1− βkβS/rS), which represents the

share of capital inputs financed by bank credit. It follows that Pk/PL falls, thereby inducing the

entrepreneur to increase input tangibility until A = A2.

When wealth falls below A2, the shadow cost of bank credit equals the cost of trade credit. Thus

the firm is indifferent between either source of financing. However, while the firm is constrained on

bank credit, it is still unconstrained on trade credit and can therefore use this credit line at a constant

price to compensate the decrease in wealth. Thus, trade credit intensity increases, as shown in Figure

5. This extra credit line is used to finance the purchase of the tangible input, thereby liberating

resources to finance the purchase of intangible inputs and leaving the input combination constant

(again, the input price ratio does not depend on A).

Finally, when wealth falls short of A1, entrepreneurs become financially constrained on both credit

lines, the price of both sources of finance increase, but it increases more for bank credit than for

trade credit. This is because the moral hazard problem behind the use of bank credit is more severe

than that behind the use of trade credit. Given that the tangible input is partly financed by trade

credit, while the intangible one is fully financed by bank credit, the input price ratio decreases, thereby

increasing input tangibility.

21

wealthA1 A2 A321 3

i

i

i

SB

SLLA

L++

φ1

φ2

φ2>φ1

Figure 5: Trade credit intensity, wealth and creditor rights protection

wealthA1 A2 A3

Ik/IL

Changes in φ

21 3

i

i i

φ1

φ2

φ2>φ1

Figure 6: Input tangibility, wealth and creditor rights protection

22

Consider now how trade credit intensity and input tangibility react to a increase in creditor rights

vulnerability (dotted lines in Figures 5 and 6). Notice that any increase in φ moves to the right all the

threshold levels of A, given that the incentive constraints become binding at higher levels of wealth.

When A < A1, this change makes the entrepreneur’s moral hazard problem towards both the bank

and the supplier more severe, since the benefits from diversion increase. This effect appears both in

the relative prices of the two sources of finance and in the input price ratio, but its magnitude is bigger

for intangibles (bank credit) than for tangibles (trade credit), since only the fraction βk of tangibles

is divertable. It follows that an increase in creditor rights vulnerability increases both trade credit

intensity and asset tangibility. Therefore, the lines shift up for any A < A1 in both figures. Similar

comments hold for A2 ≤ A < A3, with the difference that in this case it is only the price of bank

credit that increases. When A1 ≤ A < A2, we are again in the case in which the two sources of

finance have the same price, but, although constrained on bank credit, the firm is unconstrained on

trade credit. It can therefore use this credit line to keep investment and input combination constant,

thereby increasing trade credit intensity. A similar situation occurs for A ≥ A3, with the difference

that the firm is now unconstrained on both credit lines. This implies that the moral hazard problem

is not binding and neither trade credit intensity nor asset tangibility vary.

Notice that similar results hold for the complementary parameter space . Recall that in this case

we have only two wealth areas. For any A ≥ A1 both input tangibility and trade credit intensity

are constant since entrepreneurs are unconstrained on both credit lines. For any A < A1, both input

tangibility and trade credit intensity are decreasing in A. Again the reason is that the problem of

moral hazard is more severe for bank credit (intangible asset) than for trade credit (tangible asset).

Similarly, an increase in creditor rights vulnerability leaves unchanged input and financing decisions

of wealthy firms, while it increases both asset tangibility and trade credit intensity when firms are

poorer.

5 Discussion of the results

In this section, we discuss some of our results focusing on their testable implications.

5.1 The liquidation motive of financially unconstrained firms

Our paper is the first contribution acknowledging the existence of a financial motivation to take trade

credit for financially unconstrained firms. Thus it provides a rationale for the evidence showing that

firms which are less likely to be credit constrained do finance a very large share of their inputs on credit

(see for example Petersen and Rajan (1997) for the US economy, Miwa and Ramseyer (2005) for Japan

23

and Marotta (2001) for the Italian manufacturing sector). In this respect our analysis complements

the traditional incentive theories proposed to justify the use of trade credit by constrained firms.

The driver of this result is the suppliers more efficient liquidation technology, which makes trade

credit actually cheaper than bank credit, in spite of the bank’s intermediation advantage. The idea

of trade credit as a cheap source of finance is in sharp contrast with the conventional wisdom that

regards it as costly. Although this assumption is crucial to all the incentive theories of trade credit,

there is little evidence, mostly anecdotal, that suppliers charge huge credit terms on their buyers

when delaying payments of products. Generally the canonical “2/10 net 30” agreement, which would

imply an over 40% implicit interest rate, is invoked in support of this assumption. However, none has

measured the actual frequency of use of this agreement in the buyer-seller relationships. Furthermore,

recent evidence seems to support our prediction. Marotta (2001) documents that trade credit provided

by Italian manufacturing firms is, if ever, only slightly more expensive than bank credit. Similarly,

Miwa and Ramseyer (2005) provide evidence that both small and large sized firms in Japan borrow

from their partners at implicit rates that track the explicit rates banks would charge them.

5.2 Priority rule

The previous findings presume that, in case of bankruptcy, priority should be assigned on efficiency

basis, i.e. to the supplier. However, when there are multiple creditors, the legal system may prevent

the supplier from seizing particular goods, thereby neutralising the liquidation motive behind the

demand for trade credit and thus “contractual seniority.” One way to do this is to design bankruptcy

codes which are debtor rather than creditor oriented, thereby subordinating all creditor rights (and

thus the supplier’s ones) to the firm’s survival. A second more specific way is to establish, within given

bankruptcy codes, priority rules which privilege certain creditors over suppliers. Although a complete

and detailed analysis of this topic is beyond the scope of this paper, in this subsection we discuss these

two aspects of bankruptcy codes, so as to understand in which directions they might alter our results.

Regarding the first aspect, the French bankruptcy law has the stated objective of helping distressed

firms (Biais and Mariotti, 2003) and favouring their reorganisation, with an automatic stay of secured

creditors, which prevents them from removing their collateral during the reorganisation period. The

German bankruptcy law has similar provisions to the French law, with a greater role assigned to

creditors in the reorganisation decision. These two systems can be seen as more debtor oriented, as

opposed to the creditor oriented ones, like the US’ and UK’s. In the US bankruptcy law, managers

have the right to choose between filing for bankruptcy under Chapter 7, which concerns the liquidation

procedure, or under Chapter 11, concerning the reorganisation procedure. Thus, liquidation can occur

24

without prior reorganisation. According to Franks et al. (1996), the majority of US bankruptcies are

actually processed through Chapter 7.23 In Britain there are two bankruptcy regimes: the receivership

and the administration order. Under the first, a creditor holding a general secured interest in the

firm’s assets, known as floating charge,24 may appoint a receiver if the firms defaults, with the right

to sell any assets of the firm to repay the debt owed, except those which are not subject to another

creditor’s lien.25 To prevent the liquidation of the firm’s assets an administration order can be issued

by appointing a bankrupt official with the task of proposing a reorganisation plan to the creditors

committee. However, unlike in the US, an administration order cannot block an already started

receivership procedure, unless the floating charge holder consents.

Regarding the role of specific bankruptcy codes, it is generally true that trade credit is a junior

credit, unless it is secured, in which case the supplier can reclaim in bankruptcy any goods which

have not been transformed into outputs. This poses a limitation on the type of goods which can be

secured, generally not raw materials, intermediate goods or services, but rather durable goods like

plants or heavy machinery. One might therefore expect the demand for trade credit to be driven,

among other things, by the possibility for the seller to create a lien on the good sold, and therefore

by input characteristics. Notice that this prediction is fully in line with our analysis, where we found

that the liquidation motive is stronger whenever the scrap value of the inputs is larger (dominating

liquidation motive). In countries like UK and US, trade creditors do have also specific liquidation

rights. In UK, suppliers can incorporate into the sale contract a Retention of Title clause that allows

them to reclaim in case of bankruptcy any goods supplied on credit, as long as they are distinguishable

from other suppliers’ goods. With such Title they become first in the order of seniority along with the

holders of a fixed charge (Franks and Sussman, 2005). In the US, even when the firm is not under the

bankruptcy procedure, the Uniform Commercial Code gives the seller the right to reclaim the goods

sold to an insolvent buyer, within ten days after the receipt.26

If the effects of such legal provisions are to be incorporated into our model, one can expect

that whether the liquidation motive plays a role depends on both input and bankruptcy codes

characteristics. In particular, it should be more important when the good sold is a durable good which

is not transformed into finished goods and in countries with bankruptcy codes protecting contractual

rights more generally and supplier’s debt claims in particular (UK and US as opposed to France23They report that in the Central District of the California Bankruptcy Court there were 57,752 Chapter 7 cases

pending as compared with only 6,739 Chapter 11 cases as of December 1993.24As opposed to the fixed charge, i.e. a security on a specific asset such as heavy machinery.25For example those subject to a fixed charge.26This deadline does not apply if misrepresentation of solvency has been made to the seller in writing. See Garvin

(1996) for more details.

25

and Germany). It follows that cross-country differences in the design of bankruptcy codes may help

explaining cross-country variability in the use of trade credit.

Notice that this discussion acknowledges the role that legal institutions might have on the demand

for trade credit, in line with previous studies. However, this paper also identifies a new channel, as well

as new testable predictions. In our model, the channel through which legal institutions may affect the

reliance on trade credit is the degree of legal protection granted to the supplier and the economic force

behind the demand for trade credit is the incentive to exploit her liquidation advantage. It follows

that a stronger legal protection of the supplier drives a larger demand for trade credit. Conversely, in

the related literature, legal institutions affect the reliance on trade credit through the legal protection

granted to the banking sector, rather than to the supplier, and the economic driver is the incentive

to relax financial constraints. It follows that higher legal protection reduces credit rationing and thus

drives lower (rather than larger) demand for trade credit.

5.3 Trade credit, financial constraints and input characteristics

The existing theoretical literature has always highlighted the positive relation between trade credit

reliance and financial constraints (Biais and Gollier, 1997; Burkart and Ellingsen, 2004). This contrasts

with the evidence provided by Petersen and Rajan (1997), who find that trade credit demand is

uncorrelated with financial constraints. It is also inconsistent with the evidence provided by Marotta

(2001) showing that in 1994 trade credit financed on average 38.1% of input purchases by non rationed

firms as opposed to 37.5% of rationed ones for the Italian manufacturing sector.

Our model provides the first theoretical foundation for this evidence. We argue that the relation

between financial constraints and trade credit depends on the motivations behind the demand for trade

credit. When the incentive motive arises, the share of inputs financed by trade credit is decreasing in

wealth and thus higher for financially constrained firms than for unconstrained ones (see Figure 2), in

line with the related literature. However, it is not always the case that trade credit is used to relax

financial constraints, even if suppliers have an information advantage over banks and entrepreneurs

have very low internal financing. The liquidation motive might be the only driver of trade credit

demand also for low levels of wealth and the incentive motive never arise. In this case, the share of

inputs financed by trade credit is constant and thus equal for both constrained and unconstrained

firms (see Figure 4). This is a unique prediction of our model which follows from the result that trade

credit is indeed cheaper than bank loans and its demand very high already for wealthy firms.

Our analysis goes even further by claiming that whether it is the liquidation or the incentive motive

that dominates depends on the characteristics of the products sold by the supplier. In particular,

26

our model predicts that the incentive motive is more likely to arise among firms purchasing illiquid

inputs or with low collateral value (i.e., services), which corresponds to the case described in Figure

2. Conversely, the liquidation motive dominates among firms using inputs relatively liquid (i.e.,

standardised goods) or with high collateral value (i.e., differentiated goods). This case corresponds

to Figure 4. By comparing the two figures, we also expect that firms buying illiquid inputs with low

collateral value make significantly more purchases on account the tighter credit constraints become.

Conversely, firms buying liquid inputs with high collateral value should finance the same share of their

purchases on account independently of the degree of credit rationing they face.

Finally, our analysis also allows us to isolate the demand side of the use of trade credit by focusing

on rich firms. Looking at the right side of Figures 2 and 4, our model predicts that firms buying

differentiated goods make more purchases on account than firms buying services. Interestingly, this

prediction is in line with the evidence found by Burkart et al. (2004) for firms taking trade credit.

5.4 Unused bank credit line

In Corollary 2, we have shown that - contrary to what one might expect - when the liquidation motive

is dominant, less wealthy firms are constrained on trade credit but not on bank credit. The intuition is

the following: when the liquidation value of the assets is very high, rich firms make massive use of trade

credit to take full advantage of the supplier liquidation technology, and take only a very low amount of

bank credit. As wealth decreases, it is the trade credit line which is exhausted first. Moreover, to keep

taking as much trade credit as possible, while preserving the incentive for production, entrepreneurs

are willing to keep the bank credit line unused even for very low levels of wealth.

This result provides new predictions. In contrast with the financial theories of trade credit, it

suggests that, even assuming a supplier’s information advantage over the bank, trade credit is not

necessarily used to relax financial constraints. According to our analysis, when the goods that can be

purchased on credit have high collateral value, the incentive motive is never a determinant of trade

credit demand, not even for constrained entrepreneurs, as the liquidation motive dominates. It also

follows that the use of trade credit cannot be taken as a signal for credit rationing in the banking

sector (as most of the empirical literature does), since there might be another reason, besides the

incentive motive, to demand trade credit.

5.5 Maximum trade credit line and input liquidity

By comparing the left hand side of Figures 2 and 4, we may conclude that the maximum share of

inputs financed by trade credit varies according to the characteristics of the tangible input, namely

27

1

1

S

S

r

β

k

S

I

L

kβSS

S

r

r

+β

kβ−1

S

Sk

r

ββ

SS

S

r+ββ

Figure 7: Trade credit line and input liquidity for constrained firms

LS/ρIk = max{

(1− βk) , βSβkrS

}. Figure 7 displays its pattern as a function of the degree of input

liquidity (βk), which is clearly non-monotonic, with a V-shape. In particular, there exists a threshold

degree of liquidity, β = rSrS+βS

, such that if βk < β, the maximum share of tangible inputs financed

by trade credit decreases with input liquidity. This situation corresponds to the dashed line in Figure

7. Conversely, if βk ≥ β, this share increases with liquidity, which corresponds to the bold line of

the same picture. The two opposite regimes capture the two motives underlying the reliance on trade

credit by less wealthy firms. When the incentive motive arises, this relation is negative. The reason is

that the supplier’s information advantage becomes more important the more inputs differ from cash,

i.e., the lower their liquidity degree in case of diversion. Conversely, when the liquidation motive is the

driver, trade credit use is increasing in input liquidity, the reason being that the supplier’s advantage

becomes more important the higher the degree of liquidity of the input in case of default.

Besides contrasting with the theoretical analysis in Burkart and Ellingsen (2004), who find that the

pattern is always negative for any value of βk, this non-monotonic relation has interesting empirical

implications. Interpreting the maximum trade credit line extended to rationed firms as the supply of

trade credit, our analysis suggests that firms producing inputs with a low degree of liquidity and a low

collateral value (i.e., services) offer more trade credit than firms producing goods with a high degree of

liquidity and a low collateral value (i.e., standardised goods). In particular, producers of services are

willing to finance almost the entire share of inputs. This situation corresponds to the upper part of

the dashed line in Figure 7. Conversely, producers of standardised goods only finance a small share of

inputs, which corresponds to the upper part of the bold line in Figure 7. Interestingly, this prediction

is consistent with the pattern documented by Burkart et al. (2004) for firms supplying trade credit.

28

5.6 Input tangibility, trade credit intensity and creditor protection

Recent empirical literature documents cross-country differences in the use of trade credit and a negative

correlation with the quality of creditor protection (Rajan and Zingales, 1995; La Porta et al., 1998;

Frank and Maksimovic, 1998; Fisman and Love, 2003). Further evidence shows firms in developing

countries having higher proportions of fixed assets to total assets and fewer intangible assets than firms

in developed countries (e.g., Demirguc-Kunt and Maksimovic, 2001). These findings seem to suggest

a cross-country regularity between financing and input choices - a positive correlation between trade

credit intensity and input tangibility - and identify the degree of creditor vulnerability as a possible

explanation for this pattern.

The paper provides the first theoretical contribution which rationalises, within the same framework,

the two pieces of evidence highlighted above. In particular, it shows that financing and input decisions,

which have been so far investigated separately in the literature, i) are indeed closely related, i.e., they

display a common pattern; ii) they both react to changes in creditor protection; iii) the way they do it

depends on the degree of credit rationing faced by the firm. More precisely, we find that a larger use

of trade credit goes together with an input bias towards tangible assets. This bias becomes stronger

when creditor vulnerability increases, since constrained firms prefer higher tangibility and trade credit

intensity, while financially unconstrained ones keep both financing decisions and input combination

unchanged.

Differences in the abilities between suppliers and banks and differences in the characteristics of

the two inputs are crucial to derive our predictions. In particular, as wealth decreases - or creditor

vulnerability increases - the moral hazard problem behind the use of trade credit becomes less severe

than that behind the use of bank credit, given the information advantage of the supplier. This

introduces a bias not only towards trade credit, but also towards tangible inputs. The reason for that

is strictly related to the result that suppliers do not lend cash and intangibles must be fully paid on

spot.

5.7 Do suppliers ever lend cash?

Propositions 2 and 3 show that, in spite of their information advantage, suppliers extend credit for

their products but they do not lend cash. This is surprising if we consider that credit constrained

entrepreneurs choose an input combination biased towards capital inputs. In this case the marginal

productivity of intangible assets is very high and thus lending money to increase the investment in

these inputs would raise productivity. However the supplier is not willing to provide any cash, since

she anticipates that this will be used for diversion rather than for production.

29

Notwithstanding this result, there is some evidence for Japan that cash lending does arise (Uesugi

and Yamashiro, 2004). This raises two questions: first, is the information advantage concerning the

input sold by the supplier insufficient to induce her to lend cash? And second, which features do

Japanese firms have relative to the others?

In our analysis the lack of cash lending crucially follows from the assumption that the information

advantage is limited only to the purchase of capital inputs. If we assume instead that this advantage

extends also to the other input (for example both creditors can partially but asymmetrically observe

the intangible input purchase), cash lending arises. Denoting by δB and δS the degree of observability

of intangibles by the bank and the supplier respectively, the incentive constraints (2) and (3) are

replaced by:

EP ≥ φ (A + LB − δBwIL)

EP ≥ φ (βkρIk + A + LB + (LS − ρIk − δSwIL))

Using the resource constraint, namely (A + LB + LS = ρIk + wIL), and assuming that both incentive

constraints bind, we can find the maximum credit line offered by the supplier, LS , and by the bank,

LB, as follows:27

LS = (1− βk) ρIk + (δS − δB) wIL

LB = ρIkβk + (1− δS + δB) wIL −A

Thus, if we assume that it is less costly for the supplier to observe the purchase of intangibles, i.e.

δS > δB, we have that she not only provides the inputs and accepts a partial repayment equal to βk

of their value, but also provides an amount of cash to finance the intangibles, equal to (δS − δB) of

their value. This implies that in order to have cash lending, the supplier must have an information

advantage with respect to the bank also on intangible input purchases. The bigger this advantage

(δS − δB), the larger the amount of intangible inputs the supplier is willing to finance by cash.

Interestingly, Uesugi and Yamashiro (2004) show that in Japan cash lending is provided by trading

companies,28 large integrated firms dealing with a variety of commodities and carrying out various

businesses transactions including sometimes all the stages of the production and commercialisation of

a good. Thus, they can supply raw materials to manufacturing firms, but also work as sales agents for

these same firms. Commodity transactions are supported by a variety of financial services, from trade27Notice that cash lending can arise only when there is an incentive motive behind the demand of trade credit. Under

dominating liquidation motive, there would be no scope for borrowing cash.28Examples include Mitsubishi Companies, Mitsui and Toyota Tsusho Corporation.

30

credit to long-term and short-term loans, loan guarantees and investment in equities.29 The supplier

provides therefore many types of services to the same buyer. This organisational structure guarantees

a continuous flow of information that allows the seller to have more control over the behaviour of

its customer. In line with our intuition, the Japanese evidence seems to suggest that, to have cash

lending, it is crucial that the supplier informational advantage extends to various aspects of the firm’s

activity, rather than being confined exclusively to the firm-supplier relationship.

6 Extension: supplier with market power

So far, we have assumed that the cost of raising funds on the deposit market is exogenously higher

for the supplier than for the bank (rS > rB), but not so high to induce the entrepreneur to give up

trade credit, even for high levels of wealth (Assumption 2). In this section, we assume that bank and

supplier have the same cost, equal to r. Under this assumption, we show that the analysis developed

so far can be seen as a reduced form of a model in which the supplier has bargaining power due her

information and liquidation advantages over the bank. Because of this, by taking trade credit the

entrepreneur extracts a higher surplus from the contractual relationship than if he contracted with

the bank only. Let us define ΠS and ΠB as the profit obtainable when the entrepreneur can access

trade credit and when he cannot. The spread in profits is the surplus generated by the presence of the

supplier in the credit relationship. Knowing this, the supplier will ask for a share α of this surplus,

under the threat to withhold her participation from the venture. The parameter α is a measure of

the bargaining power enjoyed by the supplier.30 Under these assumptions, the entrepreneur’s problem29As reported by Uesugi and Yamashiro (2004), their scale is colossal, often totaling more than U10 trillion a year in

sales. They deal with between 20,000 to 30,000 goods, including production, intermediate and consumption goods. Theircommercial dealings include both domestic and foreign transactions and are often efficiently integrated. For example, thetextile transactions of a general trading company include the entire distribution process, from cotton and wool purchasesand imports to fiber, textile and clothes sales and exports. These commercial transactions are supported by a variety ofactivities such as investment, financial and management assistance and information provision. Similar types of businessesare rarely seen in the rest of the world, except for Korea and China.

30Notice that the supplier derives market power exclusively from her role as a financier, rather than as supplier ofgoods. If the goods are paid for in cash, the supplier can no longer exploit her liquidation advantage and her presencecannot be used to mitigate the entrepreneur’s moral hazard problem.

31

becomes the following:

maxIk,IL,LB ,LS ,γ,RS

G,RBG

Π = p(fG (Ik, IL)−RS

G −RBG

)s.t. Π ≥ φ {βkρIk + A + LB + LS − ρIk}

Π ≥ φ (A + LB)

pRSG + (1− p) (1− γ) βSC ≥ LS

[r + α

ΠS −ΠB

ΠB

]pRB

G + (1− p) γβBC ≥ LBr

A + LB + LS = wIL + ρIk

Π ≥ ΠB

There are two differences with respect to the original problem formulation: first, for each unit of trade

credit provided, the supplier asks for a return which is proportional to the surplus generated by her

presence(αΠS−ΠB

ΠB

); second, we introduce an extra constraint on the minimal profits required by the

entrepreneur to find it convenient to ask for trade credit (Π ≥ ΠB). Using this constraint we can derive

the maximum share of surplus α∗ that can be appropriated by the supplier without jeopardising the

incentive of the entrepreneur to ask for trade credit. For any α ≤ α∗, we can derive the effective price

charged by the supplier for each unit of trade credit, rS = r + α(ΠS−ΠB)ΠS

> rB. Such price is bigger

than the bank credit interest rate, but never so high to induce the entrepreneur to take only bank

credit. This implies that rS always satisfies Assumption 2. Notice that rS is now endogenous: changes

in wealth vary the surplus and therefore the interest rate asked by the supplier.

7 Conclusions

The paper uses an incomplete contract approach with uncertainty, two-input technology and

collateralised credit contracts to investigate the determinants of the firm’s financing policy. In

particular, we examine trade credit use and its interactions with input combination, financial

constraints and creditor protection.

As a first contribution, we introduce a financial motivation for taking trade credit that works

both for rationed and for non-rationed firms, namely exploiting the seller’s advantage in liquidating

the inputs in the event of default. This makes trade credit actually cheaper than bank credit and

complements the traditional incentive motive, acknowledged by the related literature, that explains

its use only by rationed firms.

By interacting these two motives, we find that the use of trade credit is not always increasing with

financial constraints: when the liquidation advantage is dominant, its use stays constant across firms

32

with different degrees of credit rationing. We also show that the dominance of the liquidation motive

over the incentive motive depends on input characteristics, like input liquidity or collateral value, or

on institutional factors, like the way commercial and bankruptcy codes are designed. We derive new

predictions on how trade credit demand (supply) varies among firms using (producing) differentiated

or standardised inputs, rather than services, which are in line with recent evidence.

Last, we identify the reasons why traditionally suppliers do not lend cash and derive conditions for

cash lending. As long as suppliers only provide credit (and not cash), we find that a more intensive

use of trade credit goes together with an input combination biased towards tangible assets. It follows

that a change in creditor right vulnerability by affecting the use of trade credit also impact the input

combination.

Our analysis could be extended in several directions. An obvious one would be to empirically test

the predictions on the relation among input choices, financing decisions, credit constraints and legal

institutions. From a theoretical point of view, it would be interesting to explore the effects of having

the supplier conditioning the input provision on the purchase of complementary inputs. In our model

this assumption would imply that the intangible input is partially observable. As we have seen, this

generates cash lending by the supplier and might have implications on the input choice and financing

decisions which are still unexplored. We look forward to investigate these extensions in future research.

33

Appendix

Definition 1 The threshold level of φ below which a zero-wealth firm can carry out the first

best investment is given by φ1 =pfG(IFB

k ,IFBL )−rB(wIFB

L +βkρIFBk )−rS(1−βk)ρIFB

k +(1−p)βSβkρIFBk

βkρIFBk +wIFB

L

, with

IFBk , IFB

L solving the first order conditions of the unconstrained optimisation problem (programmePF defined in Proof of Proposition 2) when 1− βk ≥ βkβS

rS.

Proof of Proposition 1. Since the proof of Proposition 1 overlaps with part of the proof ofProposition 2, to avoid duplications, we include this into the proof of Proposition 2.

Proof of Proposition 2. Proposition 2 is proved in steps: we first prove that a) Ik (AA<A1) <

I∗k < Ik (AA≤A<A3) < IFBk , IL (AA<A1) < I∗L < IL (AA≤A<A3) < IFB

L ; then that b) the critical levelsA1, A2 and A3, exist and are unique.

To establish part a), consider the constrained maximisation problem defined by PG . Because(1− βk) > βSβk/rS , wealthy entrepreneurs ask for trade credit only for liquidation motives. Thus,

constraint (7) is binding and trade credit is given by LS =1rS

(1− γ) βSC. Substituting LS out in

the objective function and in the constraints, we get:

maxIk,IL,LB ,γ

EP = pfG (Ik, IL)− LBrB − p (1− γ) βSC + (1− p) γβBC (11)

s.t. EP ≥ φ (A + LB) (12)

EP ≥ φ

{A + LB +

(1− γ) βS

rSC − (1− βk) ρqk

}(13)

A + LB +(1− γ) βS

rSC = wIL + ρIk (14)

Solving the resource constraint for LB, we can rewrite the general programme PG under the bindingconstraint (7), thus getting programme PF :

maxIk,IL,γ

EPF (15)

EPF ≥ φ

(wIL + ρIk −

1− γ

rSβSC

)(16)

EPF ≥ φ (wIL + βkρIk) (17)

where EPF = p [fG (Ik, IL)− (1− γ) βSC] −(

wIL + ρIk −A− 1− γ

rSβSC

)rB + (1− p) γβBC. The

first constraint is the entrepreneur incentive constraint towards the bank, while the second is theincentive constraint towards the supplier. Setting up the Lagrangean, we get:

ΛF = EPF + λ1

[EPF − φ

(wIL + ρIk −

1− γ

rSβSC

)]+ λ2 [EPF − φ (βkρIk + wIL)]

where λ1 and λ2 are the multipliers of the firm’s incentive constraints. The FOC’s are:

∂ΛF

∂Ik:{

pρ

∂fG∂Ik

− rB − βk

[βS (1− γ)

(p− rB

rS

)− (1− p) γβB

]}(1 + λ1 + λ2) =

=[λ1

(1− 1−γ

rSβSβk

)+ λ2βk

]φ (18)

∂ΛF

∂IL:(

pw

∂fG∂IL

− rB

)(1 + λ1 + λ2) = φ (λ1 + λ2) (19)

34

∂ΛF

∂γ:(p (βS − βB)− 1

rS(rBβS − rSβB)

)(1 + λ1 + λ2)− λ1

βSβkrS

φ ≤ 0 (20)

∂ΛF

∂λ1: EPF − φ

[wIL +

(1− βSβk

rS(1− γ)

)ρIk

]≥ 0 (21)

∂ΛF

∂λ2: EPF − φ (βkρqk + wIL) ≥ 0 (22)

Notice that∂ΛF

∂γ≤ 0 given Assumption 2. Therefore, we can solve the problem setting γ = 0 for any

level of wealth. This proves Proposition 1.Suppose wealth is sufficiently high (A ≥ A2) that the incentive constraint towards the supplier

is slack (λ2 = 0). Since γ = 0, the firm pledges all of the collateral to the supplier in case ofdefault. The supplier gets flat repayments across states for the funding provided and is repaid in fullupon liquidation, while the bank gets an increasing repayment contract. The optimal Ik, IL solve thefollowing FOC’s:

pρ

∂fG∂Ik

= rB + βk

(pβS − βS

rSrB

)+ λ1

1+λ1φ(1− βSβk

rS

)(23)

pw

∂fG∂IL

= rB + λ11+λ1

φ (24)

These can also be written as:rS

rS−βSβk

pρ

∂fG∂Ik

− pβkβSrSrS−βSβk

= pw

∂fG∂IL

(25)

We can distinguish between two cases: A ≥ A3 and A2 ≤ A < A3.

A ≥ A3 : Both incentive constraint are slack and IFBk , IFB

L solve (23) and (24) with λ1 = 0. Theoptimal financial contract has the following properties:

RSG = βSβkρIFB

k

LS =1rS

βSβkρIFBk

LB = wIFBL +

1rS

(rS − βSβk) ρIFBk −A

RBG =

rB

p

(wIFB

L +1rS

(rS − βSβk) ρIFBk −A

)A2≤ A < A3 : The incentive constraint towards the bank is binding and Ik (A) ∈ [I∗k , IFB

k ),IL (A) ∈ [I∗L, IFB

L ), where Ik (A) , IL (A) solve (23) and (24) with λ1 > 0. We prove now that, under theassumption that factors are substitutes, (23) and (24) imply that I∗k < IFB

k , and I∗L < IFBL . Consider

the FOC’s (23) and (24) and notice that, relative to the first best (λ1 = 0), there is now an increasein the cost of both factors. In order to derive the implications of such rise the levels of Ik and IL, wetotally differentiate (23) and (24), and we get:

∂2fs (·, ·)∂I2

k

∂2fs (·, ·)∂Ik∂IL

∂2fs (·, ·)∂Ik∂IL

∂2fs (·, ·)∂I2

L

[ dIk

dIL

]=[

dPk

dPL

]. (M1)

35

where dPk > 0, dPL > 0 are the changes in the cost of factors induced by a change in one of theirdeterminants. Inverting (M1), we can solve for the vector of unknowns :

[dIk

dIL

]=

1H

∂2f (·, ·)

∂I2L

−∂2f (·, ·)∂IL∂Ik

−∂2f (·, ·)∂Ik∂IL

∂2f (·, ·)∂I2

k

[ dPk

dPL

](M2)

where H is the determinant of the Hessian, which is positive assuming the Hessian to be negative

semidefinite. Thus, if factors are substitutes, i.e.∂2f(·, ·)∂k∂L

> 0, then

dIk =1H

(∂2f (·, ·)

∂IL2

dPk −∂2f (·, ·)∂IL∂Ik

dPL

)< 0

dIL =1H

(−∂2f (·, ·)

∂Ik∂ILdPk +

∂2f (·, ·)∂I2

k

dPL

)< 0

which implies that both factors are under-invested.31

The optimal financial contract has the following properties:

RSG = βSβkρIk (A) ,

LS =1rS

βSβkρIk (A) ,

LB = wIL (A) +1rS

(rS − βSβk) ρIk (A)−A,

RBG =

rB

p

(wIL (A) +

1rS

(rS − βSβk) ρIk (A)−A

)where Ik (A) , IL (A) solve (23) and (24) with λ1 > 0.

For levels of wealth lower than A2, the shadow price of bank credit becomes so high that theentrepreneur finds it convenient to take trade credit not only for the liquidation motive, but also forthe financing motive. This is feasible since the trade credit taken for liquidation motive does notexhaust the maximum credit line offered by the supplier (recall that we are in the case in which(1− βk) > βkβS

rS). However, to persuade the supplier to increase the financing, the entrepreneur

has to offer her a contract with repayments increasing in cash flows. This implies that the optimalcontract solves programme PG with constraint (7) slack. We can therefore rewrite the programme inthe following way:

maxIk,IL,LB

EPI (26)

s.t. EPI ≥ φ (A + LB) (27)

EPI ≥ φ (βkρIk + wIL) (28)

where the constraints are the usual ones and

EPI = pfG (Ik, IL)− LBrB − (ρIk + wIL −A− LB) rS + (1− p) βSβkρIk.

31Notice that this result holds also for the case in which factors are complements, provided the Hessian has a dominantdiagonal.

36

Setting up the Lagrangean, ΛI = EPI + λ3 (EPI − φ (A + LB)) + λ4 (EPI − φ (βkρIk + wIL)), andworking out the FOC’s

∂ΛI

∂Ik:(

p

ρ

∂fG

∂Ik− rS + βSβk (1− p)

)(1 + λ3 + λ4) = λ4φβk (29)

∂ΛI

∂IL:(

p

w

∂fG

∂IL− rS

)(1 + λ3 + λ4) = λ4φ (30)

∂ΛI

∂LB: (rS − rB) (1 + λ3 + λ4) = λ3φ (31)

∂ΛI

∂λ3: EPI − φ (A + LB) ≥ 0 (32)

∂ΛI

∂λ3: EPI − φ (βkρIk + wIL) ≥ 0 (33)

and solving for λ3 =rS − rB

φ− rS + rB(1 + λ4) , we can derive the following FOC’s:

p

ρ

∂fG

∂Ik= rS − βSβk (1− p) +

λ4

1 + λ4βk (φ− rS + rB) (34)

p

w

∂fG

∂IL= rS +

λ4

1 + λ4(φ− rS + rB) (35)

where λ4 is the multiplier of the incentive constraint towards the supplier (28).We can distinguish again between two cases: A1 ≤ A < A2 and A < A1.

A1≤ A < A2 : The incentive constraint towards the supplier is still slack and λ4 = 0. This impliesthat the entrepreneur can keep investing I∗k , I∗L even for decreasing levels of wealth until this constraintbecomes binding. The properties of the optimal contract are defined by:

LS ∈(

βSβkrS

ρI∗k , (1− βk) ρI∗k

]LB ∈

[(wI∗L + βkρI∗k −A) , wI∗L + ρI∗k

(1− βkβS

rS

)−A

)RS

G ∈((

βSβkrS

− (1− p) βSβk

)ρI∗kp , [(1− βk) rS − (1− p) βSβk]

ρI∗kp

]RB

G ∈[

rBp (wI∗L + βkρI∗k −A) ,

rB

p

[wI∗L + ρI∗k

(1− βkβS

rS

)−A

])where I∗k , I∗L solve (34) and (35) with λ4 = 0. Notice that while the level of the two inputs is constantin the above interval, the repayment due to the bank and the supplier in the two states vary withwealth.

A < A1 : The trade credit incentive constraint becomes binding (λ4 > 0) and, under theassumption that factors are substitutes, (34) and (35) imply that Ik (A) < I∗k , and IL (A) < I∗L.32

The contract has the following properties:

LS = (1− βk) ρIk (A)

LB = wIL + βkρIk (A)−A

RSG =

1p

((1− βk) rS − (1− p) βSβk) ρIk (A)

RBG =

1p

(wIL (A) + βkρIk (A)−A) rB

32The proof of this result is analogous to the one obtained for the case in which A2 ≤ A < A3 and hence omitted.

37

where Ik, IL solve (34) and (35) with λ4 binding. Notice that the optimal Ik and Ik vary with wealth.

Part (b) is proved using the following lemma:

Lemma 1 For any rB + φ − rS > 0 and 1 − βk >βSβk

rSthere exists a triple of threshold values

Ai (φ, βk, βB, βS) , i = 1, 2, 3, such that:

1. pfG

(IFBk , IFB

L

)− LBrB − pβSβkρIFB

k − φ(A + LB

)= 0 for A = A3;

2. pfG (I∗k , I∗L)− LBrB − LSrS + (1− p) βSβkρI∗k − φ(A + LB

)= 0 for A = A2;

3. pfG (I∗k , I∗L)−LBrB−LSrS +(1− p) βSβkρI∗k−φ(A + LB

)= 0 and pfG (I∗k , I∗L)−LBrB−LSrS +

(1− p) βSβkρI∗k − φ{A + LB + LS − (1− βk) ρqk

}= 0 for A = A1;

4. A3 > A2 > A1 > 0.

Proof. 1. : The threshold A3 (φ, βk, βB, βS) is the minimum wealth that allows the entrepreneurto invest IFB

k , IFBL fully exploiting the bank credit line and taking trade credit only to exploit the

liquidation motive.33 Thus A3 must satisfy:

A3 =1rB

{(φ + rB)

(wIFB

L + ρIFBk − βSβk

rSρIFB

k

)− p

[fG

(IFBk , IFB

L

)− βSβkρIFB

k

]}(36)

To prove that this threshold exists and is unique we need to show (a) that 0 + LB + LS−liq <

ρIFBk + wIFB

L , which follows from Assumption 1 (φ > φ1); (b) and that LB and LS are continuouslyincreasing in A. To establish part (b), it is useful to define the following functions, obtained by takingthe derivatives of constraint (16) wrt Ik and IL respectively:

hk1 = pρ

∂fG(·,·)∂Ik

− (rB + φ)(1− βSβk

rS

)− pβSβk (37)

hL1 = pw

∂fG(·,·)∂IL

− rB − φ (38)

Constraint (16) is only binding if hk1, hL1 < 0, otherwise Ik and IL could be further increased withoutviolating the constraint.

We first prove that∂LB

∂A> 0. Using the resource constraint wIL + ρIk = A + LB + LS and

LS = βSβkrS

ρIk, we deduce that LS = (A+LB−wIL)βSβk

rS−βSβkand Ik = A+LB−wIL

ρ(rS−βSβk) rS . The maximum bankcredit line LB is given by the binding incentive constraint (12). This constraint can therefore be written

as a function of IFBL , LB and A : pfG

((A+LB−wIFB

Lρ(rS−βSβk) rS

), IFB

L

)− LBrB − pβSβkrS

(A+LB−wIFB

LrS−βSβk

)−

φ(A + LB

)= 0. Differentiating it with respect to IFB

L , LB, and A,{pw

∂fG(·,·)∂IL

+ pβSβkrSrS−βSβk

− prS

ρ(rS−βSβk)∂fG(·,·)

∂Ik

}dIL+{

prS

ρ(rS−βSβk)∂fG(·,·)

∂Ik− rB − pβSβkrS

rS−βSβk− φ

}dLB +

{prS

ρ(rS−βSβk)∂fG(·,·)

∂Ik− pβSβkrS

rS−βSβk− φ

}dA = 0

and noting that the multiplier of dIL is null by (25), we can solve fordLB

dA:

dLB

dA= −

pρ(rS−βSβk)

∂fG(·,·)∂Ik

− pβSβkrSrS−βSβk

−φ

pρ(rS−βSβk)

∂fG(·,·)∂Ik

−rB−pβSβkrSrS−βSβk

−φ

33This amounts to say that A3 + LB + LS−liq = A3 + LB +βSβk

rSρIFB

k = ρIFBk + wIFB

L .

38

The denominator is negative whenever the incentive constraint (12) binds, i.e. when hk1 < 034

(otherwise it would be possible to raise the credit limit LB, and thus raise either Ik or IL, withoutviolating the constraint). The sign of the numerator can be inferred by rearranging condition (23) as

follows: prS

ρ(rS−βSβk)

∂fG

∂Ik− pβkβSrS

rS−βSβk−φ = rB− 1

1+λ1φ. Because the right hand side is positive ( φ

1+λ < 1),

the numerator ofdLB

dAis also positive and the whole expression is positive.

To complete the proof we need to show thatdLS

dA> 0. The proof is similar to that followed

to show thatdLB

dA> 0. Using the resource constraint wIL + ρIk = A + LB + LS and LS =

βSβk

rSρIk, we deduce that LB = wIL −

(1− rS

βSβk

)LS − A and Ik =

rS

ρβSβkLS . The incentive

constraint towards the bank (12) can therefore be written as a function of IFBL , LS and A :

pfG

((rS

ρβSβkLS

), IFB

L

)−(wIL −

(1− rS

ρβSβk

)LS

)(rB + φ) + ArB − pLSrS = 0. Differentiating the

incentive constraint with respect to IFBL , LS , and A,{

p∂fG(·,·)∂IL

− w (rB + φ)}

dIL + rBdA +{

prSρβSβk

∂fG(·,·)∂Ik

+ (rB + φ)(

βSβk−rSβSβk

)− prS

}dLS = 0

which, using hk1, hL1, we can write as

hL1wdIL + rBdA +rS

βSβkhk1dLS = 0. (39)

Differentiating (25) wrt IL, LS and A, dIL = −rS

ρβSβk

„1ρfkk−

1w

„1−βSβk

rS

«fLk

«1ρfkL−

1w

„1−βSβk

rS

«fLL

dLS , substituting out in

(39) and solving fordLS

dA, we get:

dLS

dA= rB

hL1w

rSρβSβk

(1ρfkk − 1

w

(1− βSβk

rS

)fLk

)1ρfkL − 1

w

(1− βSβk

rS

)fLL

− rSβSβk

hk1

−1

.

Given our assumptions on the production function (fkk, fLL < 0 and fkL > 0), this is certainly positivewhenever the incentive constraint (12) binds, i.e. when hk1, hL1 < 0.

2. : The threshold A2 (φ, βk, βB, βS) is the minimum wealth that allows the entrepreneur to investI∗k < IFB

k , I∗L < IFBL fully exploiting the bank credit line and taking trade credit for the incentive

motive (therefore not fully exploiting the trade credit line). The level A2 must satisfy:

A2 = 1rB

{(φ + rB)

(wI∗L + ρI∗k −

βSβkrS

ρI∗k

)− p [fG (I∗k , I∗L)− βSβkρI∗k ]

}(40)

The proof that this level exists and is unique is analogous to the one for A3 and is omitted.3. : The threshold A1 (φ, βk, βB, βS) is the minimum wealth that allows the entrepreneur to invest

still I∗k , I∗L fully using both credit lines. The level A1 must satisfy:

A1 =1rB

{(φ + rB) (βkρI∗k + wI∗L) + (1− βk) ρI∗krS − pfG (I∗k , I∗L)− (1− p) βSβkρI∗k} . (41)

34It can be deduced by rearranging (37).

39

To prove that A1 exists and is unique we need to show (a) that at zero wealth the amount offunding raised by the bank and the supplier is strictly less than the first best investment, i.e.0 + LB + LS = ρIFB

k + wIFBL and (b) that LB and LS are continuously increasing in A. Part (a)

follows from assumption 1 (φ > φ1). To establish part (b) it is helpful to rewrite the entrepreneur’sproblem as

maxIk,IL

EP1

s.t. EP1 ≥ φ {βkρIk + wIL} (42)

where (42) is the global incentive constraint with EP1 = pfG (Ik, IL) − (βkρIk + wIL −A) rB −(1− βk) ρIkrS + (1− p) βkβSρIk. This gives the following FOC’s:

pρ

∂fG∂Ik

= βk

(rB + φ λ

1+λ

)+ rS(1− βk)− βSβk (1− p) (43)

pw

∂fG∂IL

= rB + λ1+λφ (44)

where λ ≥ λ3 = rS−rBrB+φ−rS

which is the multiplier of the bank credit constraint when the entrepreneuris unconstrained on trade credit. The FOC’s can also be written as:

pρ

∂fG∂Ik

+ βSβk (1− p)− βkpw

∂fG∂IL

= rS (1− βk) . (45)

It is moreover useful to define the following functions, obtained taking the derivative of the globalincentive constraint wrt Ik and IL :

hk2 = pρ

∂fG∂Ik

− βkrB − (1− βk) rS + (1− p) βkβS − φβk (46)

hL2 = pw

∂fG∂IL

− (rB + φ) . (47)

Constraint (42) is only binding if hk2, hL2 < 0, otherwise Ik and IL could be further increased withoutviolating the constraint.

We first prove that∂LB

∂A> 0. Using LS = (1− βk) ρIk in the resource constraint wIL + ρIk =

A + LB + LS , it follows that Ik = A+LB−wILρβk

and LS = (1−βk)βk

(A + LB − wIL) . Substituting LS andIk in the incentive constraint (either 27 or 28, since they are both binding), this can be written as afunction of IL and LB :

pfG

((A+LB−wIL

ρβk

), IL

)−LBrB − ((1− βk) rS − (1− p) βkβS) ρ

(A+LB−wIL

ρβk

)−φ (A + LB) = 0 (48)

Differentiating wrt IL, LB and A, we obtain the following:{p

ρβk

∂fG∂Ik

− rB − φ− 1βk

((1− βk) rS − (1− p) βkβS)}

dLB+

+{− pw

ρβk

∂fG∂Ik

+ p∂fG∂IL

+ wβk

((1− βk) rS − (1− p) βkβS)}

dIL{p

ρβk

∂fG∂Ik

− φ− 1βk

((1− βk) rS − (1− p) βkβS)}

dA = 0

Using (45), the multiplier of dIL is zero, while the multiplier of dLB ishk2

βk. Hence, solving for

dLB

dA

and rearranging, we getdLB

dA= −

pρ

∂fG∂Ik

−rS+(1−p)βkβS+βk(rS−φ)

hk2

40

The sign of this depends on the sign of the numerator, given that the denominator is negative. Using

the FOC on Ik, (43), we deduce that the sign of the numerator is always positive, whencedLB

dA> 0.

To complete the proof we need to show that∂LS

∂A> 0. To prove this (and subsequent comparative

statics results), we use the same procedure as for LB. We rewrite the incentive constraint as a functionof IL and LS , using LB = βk

1−βkLS −A + wIL, as

pfG

((LS

ρ(1−βk)

), IL

)−(

βk1−βk

(rB + φ) + rS − (1−p)βkβS

1−βk

)LS + ArB − (rB + φ) wIL = 0. (49)

Differentiating it wrt IL, LS and A, we get

11−βk

{pρ

∂fG∂Ik

− βk [rB + φ− (1− p) βS ]− rS (1− βk)}

dLS +{

pρ

∂fG∂IL

− w (rB + φ)}

dIL + rBdA = 0

which, using hk2 and hL2, we write as:

11−βk

hk2dLS + hL2dIL + rBdA = 0 (50)

Differentiating (45) wrt IL, LS and A

p(1−βk)

(1ρfkk − βk

w fLk

)dLS + p

(1ρfkL − βk

w fLL

)dIL = 0,

solving for dIL = −1

(1−βk)

„1ρfkk−

βkw fLk

«1ρfkL−

βkw fLL

dLS , and substituting out in (50), we can solve fordLS

dA

dLS

dA= −rB (1− βk)

(hk2 −

1ρfkk−

βkw fLk

1ρfkL−

βkw fLL

hL2

)−1

> 0

4. A3 > A2 > A1 > 0.

To prove that A3 > A2, we have to confront the levels of wealth obtained from the bindingincentive constraint (16) when I = IFB and I = I∗ respectively. This amounts to calculate the effectof a change in Ik or IL on A leaving the incentive constraint unaltered. Totally differentiating theincentive constraint, we get:

p

(∂fG

∂IkdIk +

∂fG

∂ILdIL

)− (rB + φ)

(1− βkβS

rS

)(ρdIk + wdIL)− pβSβkρdIk + rBdA = 0

From this we obtain

dA

dIk= − 1

rB

{pρ

∂fG(·,·)∂Ik

−[(rB + φ)

(1− βSβk

rS

)+ pβSβk

]}dA

dIL= − 1

rB

{pw

∂fG(·,·)∂IL

− (rB + φ)(1− βSβk

rS

)}Whenever the incentive constraint binds, the terms in curly brackets, hk1, hL1, are negative, which

implies thatdA

dIk,

dA

dIL> 0. Thus, as Ik, IL decrease, A decreases, which proves that A3 > A2.

Similarly, to prove that A2 > A1, we compare (16) and (42). In this case, the level of investmentis the same. Thus we only need to compare the parameters. This leads to

A2 −A1 > 1rB

(1− βk − βSβk

rS

)(rB + φ− rS) ρI∗k > 0

which is positive. Hence, A2 > A1.

41

Definition 2 The threshold level of φ below which a zero-wealth firm can carry out the first best

investment is given by φ2 =p[fG(IFB

k ,IFBL )−βSβkρIFB

k ]−“wIFB

L +ρIFBk −βSβk

rSρIFB

k

”rB

βkρIFBk +wIFB

L

, with IFBk , IFB

L solving

the first order conditions of the unconstrained optimisation problem (defined by programme PF) when1− βk < βkβS

rS.

Proof of Proposition 3. This is the case in which (1− βk) <βkβS

rS. The line of the proof is

similar to that followed in Proposition 2. Given that (7) is binding, the relevant maximisation problemfor any level of wealth is the one given by programme PF and the FOC’s are those defined by (18)through (22). Having restricted the parameter space to that identified by Assumption 2, we can set

γ = 0. From the incentive constraints, because 1− βk <βkβS

rS, we deduce that:

φ(wIL + ρIk − βkβS

rSρIk

)< φ (βkρIk + wIL) (51)

which implies that there is a level of wealth, say A1, at which constraint (17) binds while constraint(16) stays slack. This in turn implies that λ1 = 0 for any A while λ2 ≥ 0. Using γ = 0 and λ1 = 0,

the FOC’s become the following:

pρ

∂fG∂Ik

− rB − βk

(pβS − rB

rSβS

)= λ2

1+λ2φβk (52)

pw

∂fG∂IL

− rB = λ21+λ2

φ (53)

which can be rewritten as:

1βk

{pρ

∂fG∂Ik

− rB − βk

(pβS −

rB

rSβS

)}= p

w∂fG∂IL

− rB (54)

We can distinguish between two cases, according to whether A ≥ A1 or A < A1.

A ≥ A1: Both incentive constraints (16) and (17) are slack and IFBk , IFB

L solve (52) and (53) withλ2 = 0. The optimal financial contract has the following properties:

RSG = βSβkρIFB

k ,

LS =1rS

βSβkρIFBk ,

LB = wIFBL +

1rS

(rS − βSβk) ρIFBk −A,

RBG =

rB

p

(wIFB

L +1rS

(rS − βSβk) ρIFBk −A

)Thus, the firm pledges all of the collateral to the supplier in case of default, the supplier gets flatrepayments across states for the funding provided and is repaid in full upon liquidation, while thebank gets an increasing repayment contract.

A < A1: The incentive constraint towards the supplier becomes binding and the one towards thebank stays slack. Ik, IL solve (52) and (53). Under the assumption that factors are substitutes, (52)

42

and (53) imply that Ik < IFBk , and IL < IFB

L .35 The contract has the following properties:

RSG = βSβkρIk,

LS =1rS

βSβkρIk,

LB = wIL +1rS

(rS − βSβk) ρIk −A,

RBG =

rB

p

(wIL +

1rS

(rS − βSβk) ρIk −A

)

Part (b) is proved by using the following lemma:

Lemma 2 For any 1− βk <βSβk

rSthere exists a unique threshold value A1 (φ, βk, βB, βS) such that

pfG

(IFBk , IFB

L

)− LBrB − pβSβkρIFB

k − φ{

A + LB + βSβkrS

ρIFBk − (1− βk) ρIFB

k

}= 0

Proof. The threshold A1 (φ, βk, βB, βS) is the minimum wealth that allows the entrepreneur toinvest IFB

k , IFBL fully exploiting the bank credit line and taking a minimum amount of trade credit

only for the liquidation motive.36 This level must satisfy:

A1 = 1rB

{(φ + rB) wIFB

L +((

1− βSβkrS

)rB + φβk + βSβk

)ρIFB

k − p[fG

(IFBk , IFB

L

)]}To prove that this threshold exists and is unique we need to show (i) that 0+LB +LS < ρIFB

k +wIFBL ,

which follows from assumption 1 (φ > φ1); (ii) and that LB and LS are continuously increasing in A.

To establish part (ii), it is useful to define the following functions, obtained by taking the derivativesof constraint (17) wrt Ik and IL respectively:

hk3 = pρ

∂fG(·,·)∂Ik

− 1rS

[(rS − βSβk) rB + prSβSβk + φrSβk] (55)

hL3 = pw

∂fG(·,·)∂IL

− rB − φ (56)

Constraint (17) is only binding if hk3, hL3 < 0, otherwise Ik and IL could be further increased withoutviolating the constraint.

We first prove that∂LB

∂A> 0. Using the resource constraint wIL + ρIk = A + LB + LS and

LS = βSβkrS

ρIk, we deduce that Ik = A+LB−wILρ(rS−βSβk) rS . The maximum trade credit line LS , given that the

bank credit line is not fully exploited, is given by the binding incentive constraint (17)

pfG (Ik, IL)− wIL (rB + φ) + ArB −{

pβSβk +(1− βSβk

rS

)rB + φβk

}ρIk

This can therefore be written as a function of IFBL , LB and A : pfG

(A+LB−wILρ(rS−βSβk) rS , IL

)−wIL (rB + φ)+

ArB −{(

1− βSβkrS

)rB + pβSβk + φβk

}A+LB−wIL

rS−βSβkrS . Totally differentiating

w{

pw

(1− βSβk

rS

)∂fG∂IL

− pρ

∂fG∂Ik

− φ(1− βk − βSβk

rS

)+ pβkβS

}dIL+ (57){

pρ

∂fG∂Ik

−[(

1− βSβkrS

)rB + pβSβk + φβk

]}dLB +

{pρ

∂fG∂Ik

− [pβSβk + φβk]}

dA = 0

35The proof of this result is analogous to the one obtained for the case in which A2 ≤ A < A3 for Proposition 2 andhence omitted.

36This amounts to say that A3 + LB + LS−liq = A3 + LB +βSβk

rSρIFB

k = ρIFBk + wIFB

L .

43

Adding and subtracting(1− βSβk

rS

)rB and using (55) and (56), the multiplier of dIL can also be

written as (1− βSβk

rS

)hL3 − hk3 (58)

After some manipulations,37 this reduces to

w(1− βk − βSβk

rS

)(pw

∂fG∂IL

− rB − φ)

. (59)

Given that 1−βk < βSβkrS

and knowing that hL3 = pw

∂fG∂IL

−rB−φ < 0 whenever the incentive constraint

binds, we deduce that the multiplier of dIL, in particular 58 is positive ((1− βSβk

rS

)hL3 > hk3). Hence,

using (55) and (56), (57) writes as

w(1− βk − βSβk

rS

)hL3dIL + hk3dLB +

{pρ

∂fG∂Ik

− [pβSβk + φβk]}

dA = 0 (60)

Differentiating (54) wrt IL, LS and A, pρ

∂fG∂Ik

−βkpw

∂fG∂IL

= rB (1− βk)+βk

(pβS −

rB

rSβS

), and recalling

that Ik = A+LB−wILρ(rS−βSβk) rS , we get:

rSρ(rS−βSβk)

(1ρfkk − βk

w fLk

)(dA + dLB − wdIL) +

(1ρfkL − βk

w fLL

)dIL = 0.

Solving for dIL, we get dIL = −rS

ρ(rS−βSβk)

„1ρfkk−

βkw fLk

«(dA+dLB)

den(dIL) ,where den (dIL) = 1ρfkL − βk

w fLL −wrS

ρ(rS−βSβk)

(1ρfkk − βk

w fLk

)> 0. Substituting out in (60), we get:hk3 −

„1−βk−

βSβkrS

«wrS

ρ(rS−βSβk)

„1ρfkk−

βkw fLk

«den(dIL) hL3

dLB =

=

„

1−βk−βSβkrS

«wrS

ρ(rS−βSβk)

„1ρfkk−

βkw fLk

«den(dIL) hL3 −

[pρ

∂fG∂Ik

− (pβSβk + φβk)] dA

whence

dLB

dA=

8>>>>>>>><>>>>>>>>:

<0︷ ︸︸ ︷(1− βk − βSβk

rS

)wrS

ρ(rS−βSβk)

<0︷ ︸︸ ︷(1ρfkk − βk

w fLk

)den(dIL)

>0

hL3<0

−[

pρ

∂fG∂Ik

− (pβSβk + φβk)]

︸ ︷︷ ︸>0 by FOC

9>>>>>>>>=>>>>>>>>;8>><>>:hk3−

„1−βk−

βSβkrS

«wrS

ρ(rS−βSβk)

„1ρfkk−

βkw fLk

«den(dIL) hL3

9>>=>>;Using hk3, hL3, fii < 0 and fij > 0 and using the FOC on Ik (18), we deduce that the numerator ofdLB

dAis negative. The sign of

dLB

dAdepends on the sign of the denominator. Using the equality of (58)

with (59), we can write the denominator as

den(

dLBdA

)= hk3 −

wrSρ(rS−βSβk)

„1ρfkk−

βkw fLk

«den(dIL)

((1− βSβk

rS

)hL3 − hk3

).

37By adding and subtracting βk

“pw

∂fG∂IL

− rB

”to

“1− βSβk

rS

”hL3 − hk3 and using (54).

44

This reduces to

den(

dLBdA

)= 1

den(dIL)>0

hk3<0

(1ρfkL − βk

w fLL

)︸ ︷︷ ︸

>0

− wrSρ(rS−βSβk)

(1ρfkk − βk

w fLk

)︸ ︷︷ ︸

<0

(1− βSβk

rS

)hL3<0

︸ ︷︷ ︸<0

< 0

which is unambiguously negative. This completes the proof thatdLB

dA> 0.

To complete the proof we need to show thatdLS

dA> 0. Using the resource constraint wIL + ρIk =

A + LB + LS and LS = βSβkrS

ρIk, we deduce that LB = wIL −(1− rS

βSβk

)LS − A and Ik = rS

ρβSβkLS .

The incentive constraint towards the supplier (13) can therefore be written as a function of IFBL , LS

and A :

pfG

(rS

ρβSβkLS , IFB

L

)− wIFB

L (rB + φ)− LS

βSβk{(rS − βSβk) rB + prSβSβk + φβkrS}+ ArB = 0

Differentiating it with respect to IFBL , LS , and A,{

p∂fG(·,·)∂IL

− w (rB + φ)}

dIL+rBdA+ rSβSβk

{pρ

∂fG(·,·)∂Ik

+ 1rS

[(rS − βSβk) rB + βkrS (pβS + φ)]}

dLS = 0

which, using hk3, hL3, we can write as:

hL3wdIL + rBdA +rS

βSβkhk3dLS = 0 (61)

Differentiating (54) wrt IL, LS and A, and rearranging, we get: dIL =rS

ρβSβk

„1ρfkk−

βkw fLk

«1ρfkL−

βkw fLL

dLS .

Substituting this out in (61) and solving fordLS

dA

dLS

dA= rB

hL3w

rSρβSβk

(1ρfkk − βk

w fLk

)1ρfkL − βk

w fLL

− rSρβSβk

hk3

−1

Given our assumptions on the production function (fkk, fLL < 0 and fkL > 0),dLS

dAis certainly

positive whenever the incentive constraint (12) binds, i.e. when hk3, hL3 < 0.Proof of Proposition 4. Under the assumption that the production function is homogeneous

of degree j ≤ 1, the input combination (Ik/IL) only depends on the input price ratio (Pk/PL). Usingthe proof of Proposition 2, we can write Pk/PL as a function of the parameters of the model. Let usconsider the four wealth areas separately.

When A ≥ A3,PkPL

=ρ

hrB−βkβS

“rBrS

−p”i

wrBand LS

LS+LB+A = βSβkρ[rSw

(IFBL

IFBk

)+ rSρ

]−1. Notice that

trade credit intensity is increasing in asset tangibility. Since ∂(Pk/PL)∂A = 0 and trade credit intensity

(LS/ (A + LB + LS)) depends on wealth only through Ik/IL, both input tangibility and trade creditintensity are independent of A.

45

When A2≤ A < A3:(

PkPL

)A2≤A<A3

=ρ

hrB+

φλ1(1+λ1)

(1−βSrS

)−βSβk(rBrS

−p)i

whrB+

φλ1(1+λ1)

i <(

PkPL

)A>A3

and

LSLS+LB+A = βSβkρ

rSw“

ILIk

”+rSρ

. Notice that trade credit intensity is increasing in asset tangibility. It

follows that ∂(Pk/PL)A2≤A<A3∂A =

h(1−βSβk

rs)−Pkw

PLρ

iw

hrB+

φλ1(1+λ1)

i ∂λ∂A

φ

(1+λ1)2> 0, since ∂λ

∂A ≤ 0 and[(1− βSβk

rs)− Pkw

PLρ

]=

(−βSβkp) ≤ 0. Given that LS/ (A + LB + LS) depends on wealth only through Ik/IL, both inputtangibility and trade credit intensity are decreasing in wealth.

When A1≤ A < A2:(

PkPL

)A1≤A<A2

= ρ[βkrB+(1−βk)rS−βSβk(1−p)]wrS

<(

PkPL

)A2≤A<A3

and

LSLS+LB+A = µ

w

„I∗L

I∗k

«+ρ

, where (βkβS/rS) ≤ µ ≤ (1− βk) and varies with A. Since ∂(Pk/PL)A1≤A<A2∂A = 0

and∂µ

∂A≤ 0,38 while input tangibility is independent of A, trade credit intensity is decreasing in A.

When A < A1:(

PkPL

)A1<A

=ρ[βk(rB+ λφ

1+λ)+(1−βk)rS−βSβk(1−p)]w[rB+ λφ

1+λ ] <(

PkPL

)A1<A<A2

and LSLS+LB+A =

(1−βk)ρ

w“

ILIk

”+ρ

. Notice that trade credit intensity is increasing in asset tangibility. It follows that

∂(Pk/PL)A<A1∂A =

hβk−

wρ

PkPL

iw

hrB+ φλ

(1+λ

i ∂λ∂A

φ

(1+λ1)2> 0, since ∂λ

∂A ≤ 0 and[βk − Pkw

PLρ

]= βSβk(1− p)− (1− βk)rS ≤ 0

when (1 − βk) ≥ βsβkrs

. Since LS/ (A + LB + LS) depends on wealth only through Ik/IL, both inputtangibility and trade credit intensity are decreasing in wealth.

Proof of Proposition 5. Notice that, from the proof of Proposition 4, trade credit intensity isan increasing function of asset tangibility. Let us consider separately the four relevant wealth areas.

When A ≥ A3,∂(Pk/PL)

∂φ = 0. It follows that Ik/IL does not depend on φ. Since φ, affects tradecredit intensity only through the input combination, both asset tangibility and trade credit intensityare independent of φ.

When A2≤ A < A3, the sign of ∂(Pk/PL)∂φ depends on

the sign of[

λ11+λ1

+ φ ∂λ1/∂φ

(1+λ1)2

] [βSrB

rS(βk − 1)− pβkβS

]. Since (∂λ1/∂φ) > 0 and (βk − 1) < 0, the

whole expression is negative. This implies that asset tangibility increases. Since φ affects trade creditintensity only through the input combination, also LS/ (A + LB + LS) is increasing in φ.

When A1≤ A < A2,∂(Pk/PL)

∂φ = 0, which implies that Ik/IL is independent of φ. However, since∂µ∂φ > 0, LS/ (A + LB + LS) is increasing in φ. As shown in Figure 7, when φ increases, the shadowcost of bank credit equals the cost of trade credit at A2 > A2. For decreasing A, the firm substitutesbank credit with trade credit, thereby increasing the absolute level of trade credit from βSβkρI∗k/rS

at A = A2 to (1− βk) ρI∗k at A = A1 > A1.

When A < A1, the sign of ∂(Pk/PL)∂φ

depends on the sign of βk

[λ

1+λ + φ ∂λ/∂φ

(1+λ)2

][βSβk (1− p)− (1− βk) rS ] . Since (∂λ/∂φ) > 0 and the

term in the second square brackets is negative,39 the whole expression is negative. This implies thatasset tangibility increases. Since φ affects trade credit intensity only through the input combination,

38This follows from Proposition 2 and from Figure 2. The intuition is the following: when A = A2, the firm uses ashare of trade credit equal to βSβk/rS and the shadow cost of bank credit equals the cost of trade credit. Since the firmis constrained on bank credit but still unconstrained on trade credit, any reduction in wealth is compensated by a rise intrade credit, keeping investment at I∗k , I∗L. Thus the share of trade trade credit increases until it reaches (1− βk) whenA = A1.

39Given that we are in the case in which (1− βk) > βkβS/rS .

46

also LS/ (A + LB + LS) is increasing in φ.

References

[1] Bias, Bruno and Christian Gollier (1997), “Trade Credit and Credit Rationing,” Review ofFinancial Studies, 10 (4), 903-937.

[2] Brennan, Michael, Vojislav Maksimovic and Josef Zechner (1988), “Vendor Financing,” Journalof Finance, 43 (3), 1127-1141.

[3] Burkart, Mike and Tore Ellingsen (2004), “In-Kind Finance: A Theory of Trade Credit,” AmericanEconomic Review, 94, 569-590.

[4] Burkart, Mike, Tore Ellingsen and Mariassunta Giannetti (2005), “How You Lend is What YouLend? Evidence from Trade Credit,” mimeo.

[5] Demirguc-Kunt, Asli and Vojislav Maksimovic (2001), “Firms as Financial Intermediaries:Evidence from Trade Credit Data,” mimeo.

[6] Emery, Gary and Nandkumar Nayar (1998), “An Optimal Financial Response to VariableDemand,” Review of Finance and Accounting, 10, 269-284.

[7] Ferris, J. Stephen (1981), “A Transaction Theory of Trade Credit Use,” Quarterly Journal ofEconomics, 96, 247-270.

[8] Fisman Raymond, and Inessa Love (2003), “Trade Credit, Financial Intermediary Development,and Industry Growth,” Journal of Finance, 58 (1), 353-374.

[9] Frank, Murray Z. and Vojislav Maksimovic (2004), “Trade Credit, Collateral and AdverseSelection,” mimeo.

[10] Franks, Julian R., Kjell G. Nyborg, and Walter N. Torous (1996), “A Comparison of US, UK,and German Insolvency Codes,” Financial Management, 25, 86-101.

[11] Franks, Julian R. and Oren Sussman (2005), “Financial Distress and Bank Restructuring of Smallto Medim Size UK Companies,” Review of Finance, 9 (1), 65-96.

[12] Garvin, Larry T. (1996), Credit, Information, and Trust in the Law of Sales: The Credit Seller’sRight of Reclamation,” UCLA Law Review, 44 (2), 247-360.

[13] La Porta, Rafael, Florencio Lopez-de-Silanes, Andrei Shleifer and Robert W. Vishny (1998), “Lawand Finance,” Journal of Political Economy, 106, 1113-1155.

[14] Lee, Yul W. and John D. Stowe (1993), “Product Risk, Asymmetric Information, and TradeCredit,” Journal of Financial and Quantitative Analysis, 28, 285–300.

[15] Long, Michael S., Ileen B. Malitz and S. Abraham Ravid (1993), “Trade Credit, QualityGuarantees, and Product Marketability,” Financial Management, 22, 117–127.

[16] Marotta, Giuseppe (2001), “Is Trade Credit More Expensive than Bank Credit Loans? Evidencefrom Italian Firm-Level Data,” mimeo, Universita’ di Modena e Reggio Emilia.

47

[17] Mian, Shehzad L. and Clifford W. Smith (1992), “Accounts Receivable Management Policy:Theory and Evidence,” Journal of Finance, 47 (2), 169-200.

[18] Miwa, Yoshiro and Mark J. Ramseyer (2005), “Trade Credit, Bank Loans, and Monitoring:Evidence from Japan” mimeo, Harvard Law School.

[19] Ng, Chee K., Janet K. Smith and Richard L. Smith (1999), “Evidence on the Determinants ofCredit Terms Used in Interfirm Trade,” Journal of Finance, 54 (3), 1109-1129.

[20] Petersen, Mitchell A. and Raghuram G. Rajan (1997), ”Trade Credit: Theory and Evidence,”Review of Financial Studies, 10 (3), 661-691.

[21] Rajan, Raghuram and Luigi Zingales (1995), “What Do We Know About Capital Structure?Some Evidence from International Data,” Journal of Finance, 50, 1421-1460.

[22] Summers, Barbara and Nicholas Wilson (2002), “Trade Credit Terms Offered by Small Firms:Survey Evidence and Empirical Analysis,” Journal of Business and Finance Accounting, 29,317-335.

[23] White, Michelle J. (1996), “Corporate Bankruptcy: A U.S.-European Comparison,” in Bandharied., Cambridge University Press.

[24] Wilner, Benjamin S. (2000), “The Exploitation of Relationships in Financial Distress: The Caseof Trade Credit,” Journal of Finance, 55 (1), 153-178.

[25] Uesugi, Iichiro and Guy M. Yamashiro (2004), “How Trade Credit Differs from Loans: Evidencefrom Japanese Trading Companies,” mimeo.

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