Interest ates in Trade Credit Markets Klênio Barbosa Bano BBM klenio@eon.u-rio.br Humberto Moreira EPGE FGV humberto@fgv.br February 10, 2004 Walter Novaes PUC-io novaes@eon.u-rio.br
Abstrat There is evidene that suliers have rivate information about their ustomers redit risk. Yet, interest rates in trade redit markets are usually industry-not-firm seifi. Why? If the demand for intermediate roduts is inelasti, suliers should raise interest rates until they reah their ustomers outside otion, whih, by definition, annot reflet information that is rivy to suliers. In ontrast, a highly elasti demand indues suliers with monooly ower to waive interest, making rivate information one more irrelevant to the trade-redit rate. By haraterizing these two equilibria, we obtain imliations on when trade-redit rates shouldn t vary with rivate information held by suliers. JEL: G30, G32 Key Words: Trade Credit; Invariane of Interest ates.
1 Introdution Trade redit is one of the most imortant soures of short-term external finaning for firms in the G7 ountries (Canada, Frane, Germany, Italy, Jaan, U.K., and the U..). 1 mith (1987), Mian and mith (1992) and Biais and Gollier (1997) argue that suh rominene is due to an informational advantage: The sales effort of suliers makes it easier for them to assess their ustomers redit risk. Aordingly, Petersen and ajan (1997) show that, vis-à-vis banks, suliers extend more redit to firms with urrent losses and ositive growth of sales; a finding that they interret as evidene that suliers have omarative advantage in identifying firms with growth otential. Yet, a sulier s informational advantage is, at first glane, diffiult to reonile with a standard ratie in the trade redit markets. Ng, mith and mith (1999) and Petersen and ajan (1994) show that the terms of trade redit in the U.. are industry-not-firm seifi. In artiular, a ommon term of trade redit harges an effetive interest rate of 44 erent a year by ombining a 30 day maturity with a two erent disount for early ayment within 10 days of the invoie (2-10 net 30 loans). But if suliers are informed lenders, why don t they harge interest rates that reflet variations in the borrowers risk? This aer exlains when and why interest rates in the trade redit markets do not internalize rivate information held by suliers. In a nutshell, suliers should raise trade-redit rates until they reah their ustomers outside otion, if the demand for the suliers goods is inelasti with reset to the finaning osts. By definition, this outside otion e.g., theinterestrateavailable in banking loans annot reflet information that is rivy to suliers. In ontrast, a suffiiently elasti demand indues suliers with monooly ower to waive interest, making their rivate information one more irrelevant to the equilibrium trade-redit rate. Trade redit rates do not vary with the suliers rivate information, therefore, when the demand is inelasti or if suliers with monooly ower fae a demand that is suffiiently elasti with reset to interest rates. To understand the main ideas of the aer, onsider an industry whose firms require finaning to urhase inuts from their suliers. In a fration f of these firms the safe firms the investment in the inut will be aid bak with robability one. In the remaining firms, fration 1 ee, for examle, ajan and Zingales (1995). 1
1 f, the investment in the inut may fail. We all these latter firms risky. To finane the urhase of inuts, firms an borrow from banks or ask for trade redit. As suh, we onsider firms that, albeit ossibly risky, are not reditonstrained. Inthemodel, banks at ometitively (i.e., interest rates imly that the exeted return on a loan equals the ost of funds), but they annot distinguish between safe and risky firms. Hene, banks harge thesameinterestrater B to all firms in the industry. In ontrast to the banks, suliers know whether their ustomers are safe or risky. Thanks to this informational advantage, a sulier may offer interest rates that vary with the firm s tye. Cometition with banks onstrains the suliers hoies of interest rate, though. In artiular, suliers annot extend trade redit at an interest rate that is higher than the ustomer s outside otion, whih, in our model, is the banking rate r B. To be sure, ometition with banks doesn t revent suliers from extending redit at low interest rates. Is it in the suliers interest to underut banks? This won t be the ase if the demand for the inuts is inelasti with reset to the finaning ost. Intuitively, an inelasti demand indues suliers to, regardless of the ustomer s reditworthiness, raise interest to the bankingrate,whihisashighasatradereditrateanbe. Aninelastidemandthusgivesusa natural andidate for an equilibrium trade redit rate that does not vary with suliers rivate information: the banking rate r B. One roblem remains for this andidate to be legitimate, though. Informed suliers may be unwilling to lend to risky firms at an interest rate that is set by uninformed banks that fierely omete with eah other. As it turns out, there is at least one good reason for suliers to lend to risky firms at the banking rate. Frank and Maksimovi (1998) argue that suliers are more effiient than banks in salvaging value from assets of finanially distressed firms. If so, suliers get a higher return than banks when a borrower beomes finanially distressed; an advantage that may make it rofitable for suliers to extend redit to risky firms at the banking rate. What haens if the demand for inuts is inelasti but suliers are not more effiient than banks in lending to risky firms? The equilibrium at the banking rate breaks down. Petersen and ajan (1997) show that suliers extend less redit in industries that kee a high fration of finished goods in inventory; a finding that they interret as evidene that it is easier for suliers to transform reossessed inuts (rather than finished goods) into liquid assets. Aordingly, our 2
model redits that suliers are more likely to offer standardized rates in industries that kee a low fration of finished goods in the inventory. This imliation will not hold, for instane, if the observed rigidity of trade redit rates in the U.. reflets suliers ability to use the rie of their roduts to adjust for the riskiness of their ustomers. Our model builds on two reent aers: Biais and Gollier (1997) and Burkart and Ellingsen (2002). These artiles exlain why suliers lend to firms that have exhausted their debt aaity with banks. In Biais and Gollier, suliers an identify firms whose redit risks are overestimated by banks. Knowing that a firm s redit line is unduly low, suliers are willing to extend trade redit. In Burkart and Ellingsen, finanially onstrained firms have aess to trade redit beause it imlies a lower risk of misuse of ororate funds than banking loans. In Biais and Gollier s and Burkart and Ellingsen s models, the otimal trade-redit rate varies with the suliers information. Brennan, Maksimovi and Zehner (1988) is another related work. In this aer, a monoolist sells roduts to safe and risky ustomers, disriminating the demand by offering trade redit to the risky ustomers at a subsidized interest rate. As in Biais and Gollier (1997) and Burkart and Ellingsen (2002), the otimal trade-redit rate in Brennan, Maksimovi and Zehner would vary with the suliers rivate information, had there been trade redit to ustomers in different lasses of risk. As in Brennan, Maksimovi and Zehner (1988), a suffiiently elasti demand for inuts makes low finaning osts so imortant to sales that, in our model, it dissuades the suliers from raising the trade-redit rate to their ustomers outside otion. In fat, we shall demonstrate that suliers with monooly ower have inentives to waive interest, if the demand for inuts is suffiiently elasti. A seond equilibrium in whih the trade-redit rate does not internalize the suliers rivate information thus obtains: trade redit at zero interest. In our model, therefore, interest rates in trade redit markets do not internalize rivate information held by suliers, in two instanes. If the demand for inuts is inelasti, in whih ase the trade redit rate mathes the ustomers outside otion, or if the sulier has monooly ower and the demand for inuts is suffiiently elasti, in whih ase the equilibrium trade redit rate is at zero. In addition to exlaining why trade-redit rates shouldn t vary with rivate information held 3
by suliers, our model links the invariane of trade-redit rates to whether firms are redit onstrained. ine waiving interest attrats all tyes of firms, there is no reason for the demand for trade redit at zero interest to onsist mainly of redit onstrained firms. It is easy to see, however, that, at ositive trade-redit rates, rivate information held by suliers matters when ustomers are redit onstrained. In this ase, ometition with banks does not onstrain the suliers hoies of trade-redit rates, imlying that the otimal terms of trade redit deend on any firm-seifi information that suliers may have: elastiity of demand, robability of default, et. Another redition of our model is thus that suliers are more likely to offer standardized rates to ustomers that are not redit onstrained. The remainder of the aer is organized as follows. After resenting the model in setion 2, setion 3 haraterizes the equilibrium in whih the trade redit rate does not vary with rivate information held by suliers. In setion 4, we disuss the emirial imliations and exhibit suffiient onditions for uniqueness of equilibrium. etion 5 then onludes. Proofs of the roositions that are not in the text an be found in the aendix. 2 The Model Consider an eonomy with two dates, t =0and t =1, and an industry with three risk-neutral agents: firms, banks, and a sulier of the firms inuts. At t =0, firms require finaning to urhase inuts. Banks are always willing to finane the urhase of inuts at an interest rate that overs the ost of funds. Firms, however, may have a seond soure of finaning: trade redit. With an exogenous robability x, the sulier has enough funds to finane its ustomers. Uon the urhase of inuts at t =0, rodution takes lae and firms sell the outut at t =1. At this time, firms reay the debt and distribute any remaining ash flow to shareholders. Below we desribe the agents tehnologies and their information strutures. 2.1 Firms There are two tyes of firms, safe and risky, whih are run by value-maximizing managers who know their firms tyes from the onset. The safe firms reresent a fration f of the oulation and have a deterministi rodution funtion. With this safe tehnology, investing I units of 4
the inut at t =0obtains Q(I) at t =1. We assume that Q(I) is an inreasing and stritly onave funtion, with Q(0) = 0 and satisfying the following ondition: there exist investment levels I and I suh that Q 0 (I) >(1 + r)/f and Q 0 (I) <,wherer is the riskless interest rate, f is the roortion of safe firms in the industry, and is the rie of the inut. These harmless assumtions on the marginal rodutivity assure that firms buy a ositive level of inut. isky firms are endowed with a stohasti rodution funtion. With this tehnology, urhasing I units of inut at t =0yields Q(I) at t =1, where: ( Q (I), with robability π Q (I) = δi, with robability 1 π, and δ (0, 1). Note that, with robability π, the risky tehnology is as rofitable as the safe tehnology. But, with robability 1 π, the risky tehnology gets into trouble; the fration 1 δ of urhased inuts is lost and the only return on the investment is an amount δi of inuts that remained unused, yielding a residual value δi. We assume that both Q(I) and Q(I) are verifiable. As suh, firms an write debt ontrats that are ontingent on the realization of oututs. 2.2 Banks In the model, banks an neither distinguish between firms of different tyes nor observe the terms of trade redit. Banks know only the roortion of safe and risky firms, and the amount of inuts I that firms urhase. ine banks oerate in a ometitive market, they will set an interest rate, r B, that yields their oortunity ost. Given risk neutrality, this oortunity ost is the riskless interest rate r. If a risky firm fails, the lender atures the firm s outut, Q (I), whihisthefrationδ of the amount I originally urhased. It is unlikely, nonetheless, that banks an ostlessly transform δi into liquid assets. In fat, one of the key assumtions of our aer is that banks are not as effiient as suliers in transforming inuts into liquid assets. To emhasize this differene between banks and suliers, and to failitate the analysis, we assume that neither the banks nor the firms an resue the unused inuts, δi, if the tehnology fails. 5
2.3 The sulier To fous the analysis on the trade redit market, we follow Brennan, Maksimovi and Zehner (1988) and assume that suliers annot use the rie of the inut to disriminate the demand. As Petersen and ajan (1997) oint out, this hyothesis an be justified by anti-trust laws. Yet, we do not want to restrit the attention to industries in whih a sulier has monooly ower in the market for inut. Desite assuming that eah firm buys inuts from a single sulier, the inut market may be ontestable, that is, the otential entry of alternative suliers may drive inut ries down to marginal ost. Hene, we assume that the sulier faes a onstant marginal ost andanexogenousrie of the inut. The mark-u,, measures the degree of monooly ower in the market for inut. For =1, the market for inuts is ometitive, while > 1 imlies that the sulier enjoys some monooly ower. In this latter ase, the sulier faes a two-stage roblem; hoosing the otimal trade-redit rate for any given inut rie in the first stage, and then looking for the otimal inut rie in the seond stage. For the urose of our work, we an restrit attention to the first stage roblem, haraterizing the otimal trade-redit rate as a funtion of the mark-u. A ommon view in the trade-redit literature is that suliers have omarative advantage over banks in finaning urhases of inuts. Biais and Gollier (1997) argue, for instane, that an ongoing sales effort makes it easier for suliers to evaluate their ustomers redit risk; an argument that Petersen and ajan (1997) find evidene for. Aordingly, we assume that, unlike the banks, the sulier knows whether a firm is risky or safe. Ability to evaluate risk of redit is not the only reason for the existene of trade redit, though. Petersen and ajan (1997) also find evidene that suliers are more effiient than banks in transforming ollateral into liquid assets. To model this advantage, we follow Frank and Maksimovi (1998) and assume that, unlike the banks, the sulier an ostlessly resell inuts that they ature from bankruted firms. Hene, when a risky investment of I units of inut fails, the sulier atures the unused inuts, δi, assuring some return on their trade redit. We assume, however, that the sulier loses when a risky ustomer is bankruted. That is, the resent value of the resued inuts δi is lower than the suliers ost of roduing the inut, I, whih imlies that < ; an inequality that is trivially satisfied if the inut market δ is ometitive. 6
But, as Mian and mith (1992) show, some suliers do not have aess to funds that an be used to rovide trade redit. We model this otential onstraint as follows. With a robability x in the interval (0, 1), our sulier has aess to funds at the same ost of banks, r. In this event, the sulier an extend trade redit. With robability 1 x, however, the sulier has no aess to funds, ruling out trade redit. Firms will then have to seure bank loans to urhase inuts. The sulier s stohasti ost of funds assures an ative role for banking redit, desite thesulier sotentialadvantageasalender. 2.4 The game in the extensive form Figure 1 desribes the extensive form of the game. Nature ats first, determining at date t = 1 thetyeofthefirm (safe or risky) and whether the sulier an rovide trade redit. At t =0, the sulier and the firms learn the firms tyes and whether trade redit is available. If the sulier annot extend trade redit, an event with robability 1 x, firms borrow from banks before urhasing I B from the sulier. If, instead, trade redit is available, the banks and the sulier make simultaneous offers to finane urhases of inuts. ine banks do not know the firm s tye, they offer the same interest rate r B to both tyes of firms, while the informed sulier may tailor the interest rate to the firm s tye, offering rt to risky firms and rt to safe firms. These interest rates determine firms returns on the urhase of inuts, induing risky firms to invest IT and safe firms to invest IT. At time t =1,ashflows are generated aording to the rodution funtions Q(I) and Q(I). And firms ay bak their loans whenever ossible. hareholders then ature any ash flow left after the debt is aid. The game has two tyes of equilibria. In the first one, the sulier lends to at most one tye of firm. In the seond tye of equilibrium, suliers lend to both tyes of firms, whenever ossible. The first tye of equilibrium is not interesting for the uroses of our work. If borrowers are all in the same lass of risk, there is no soe for the sulier to vary the trade-redit rate with the ustomers reditworthiness. As suh, our fous in on the equilibrium in whih, whenever ossible, the sulier lends to both tyes. After haraterizing this equilibrium in setion 3, we exhibit onditions for it to be unique in setion 4. 7
3 Equilibrium with Invariane of Interest ates 3.1 Banking redit In the equilibrium that we look for, the sulier extends trade redit to both tyes of firms, whenever ossible. When trade redit is not available, an event with robability 1 x, firms finane urhases of inuts by borrowing from banks. Let us then start our analysis by deriving the demand for inuts of a safe firm that borrows from banks at an interest rate r B. By assumtion, safe firms an always reay loans that are used to finane inuts. 2 As a result, a safe firm s otimal investment in inuts solves: max I Q(I) (1 + r B ) I. (1) (1 + r) Program (1) looks for the investment that maximizes the resent value of a safe firm s rofit. By investing I at t =0,asafefirm obtains Q(I) at t =1with robability one, aying (1 + r B ) I in rinial lus interest to the bank (also at t =1). ine the investment of a safe firm is riskless, we disount its ayoff at the riskless interest rate r. Given the interest rate r B,thefirst order ondition, whih is also suffiient, yields the demand for inuts of the safe firm, IB (r B ),by setting the marginal rodutivity of investing in the inut equal to the ost of finaning: Q 0 (I B)=( B ). (2) Consider now a risky firm that borrows I to urhase inuts. With robability π, the investment will yield the same return Q (I) of the safe firms. With robability 1 π, however, the investment will fail, leaving only δi units of inuts at t =1. egardless of the lender s ability to transform the residual inuts into liquid assets, a failure of the risky tehnology imlies that the firm loses all rights on the residual inuts. Given the assumtion of risk neutrality, the demand for inuts of a risky firm, IB (r B ), maximizes the resent value of the exeted ayoffs, using the riskless interest rate as the disount rate, that is, 2 For any finite interest rate, our assumtions on the marginal rodutivity of investment (see setion 2.1) assure that a small urhase of inuts will more than offset the osts of serviing the debt, leaving a ositive rofit for the safe firm. Hene, an otimal hoie of inuts must imly a ositive rofit as well. In the absene of unertainty, a ositive rofit imlies that any debt will be reaid with robability one. 8
π [Q(I) (1 + r B ) I] max. (3) I (1 + r) Like the safe firms, a risky firm s demand for inuts sets the marginal rodutivity of investment equal to the ost of finaning, that is, Q 0 (IB )=( B). It then follows that the demand shedules of safe and risky firms are equal, that is, IB = I B = I B. Indeed, had the demand for loans varied aross firms of different tyes, banks would have been able to infer the tye of a firm that requests a loan. Banks and suliers would then end u with the same information struture, and our model would not be fit to exlain why interest rates in the trade redit markets do not seem to reflet suliers rivate information about their ustomers. Of ourse, a request of a bank loan may onvey information even if safe and risky firms have idential demands for inuts. For instane, in an equilibrium in whih the sulier finanes only safe firms, banks should exet that most of their loans go to risky firms. (Banks should not exet all firmstoberiskybeauselakoftradereditmightleadsafefirms to look for banking redit.) In the equilibrium that we look for, though, the sulier finanes both tyes of firms, whenever ossible. Thus, banks know that lak of funds for trade redit is the only reason for firms asking for bank loans. Aordingly, requests of loans do not onvey information, and banks do not udate their riors about firms tyes. Provided that requests for loans do not onvey information, we an easily derive the equilibrium interest rate r B. ine the tehnologies of both tyes of firms are ommon knowledge, the banks know that safe firms will ay rinial lus interest with robability one, while risky firmswillhonorthedebtontratwithrobability π. (iskyfirms do not ay anything with robability 1 π.) Therefore, the banks will ollet rinial lus interest at t =1if the firm is safe, robability f, orifthefirm is risky but the tehnology does not fail, robability π (1 f). In other words, the robability that a borrower ays a bank at t =1is f + π (1 f). Andthe exeted return of a bank that lends at a rate r B is (1 + r B )(f + π (1 f)). Cometition among banks drives the exeted returns on banking loans to their oortunity ost, whih, under the assumtion of risk neutrality, is the riskless interest rate r. Assuh, the interest rate that assures banks their oortunity ost is r B = 1. (4) f + π (1 f) 9
Having haraterized the equilibrium banking rate and the demand for inuts of firms that borrow from banks, our next task is to introdue trade redit. Two questions then naturally arise. Is it otimal for the sulier to finane urhases of inuts? What is the otimal interest rate in the trade redit market? Answering these questions requires solving for the investment deision of a firm that has the otion of using trade redit to finane urhases of inuts. As it turns out, trade redit does not fundamentally hange firms investment deisions. Whether a firm borrows from banks or from the sulier, all that matters is the ost of finaning. It then follows that the investment deisions of safe and risk firms are still haraterized by, resetively, rograms (1) and (3), one we substitute the minimum ost of finaning for the banking rate. Thus, for both tyes of firms, the demand for inuts, I, is imliitly defined by the equality of the marginal rodutivity of investment and the ost of finaning. Formally, Q 0 (I(s)) = (1 + s), (5) where s is the lowest between the banking rate and the trade redit rate. Equation (5) determines the shae of the demand for inuts. A straightforward aliation of the imliit funtion theorem shows that the demand for inuts dereases with the ost of finaning. Furthermore, the demand is stritly onave if Q 000 (I) < 0. 3 Equied with the demand for inuts, the next two setions haraterize the sulier s otimal strategies, starting with the otimal terms of trade redit of a sulier that faes a safe ustomer. 3.2 The suly of trade redit to safe firms In our model, a sale of inuts is not neessarily linked to trade redit. Firms an borrow from banks to finane urhases of inuts and, rather than extending trade redit, suliers an let banks finane the firms. These outside otions imose restritions on the otimal terms of trade redit. For instane, no safe firm will aet terms of trade redit that ask for an interest rate that is larger than the banking rate r B. As suh, a first restrition for the otimal trade-redit 3 To show that I ( ) is dereasing and onave (under Q 000 ( ) < 0) ontheostoffinaning, aly the imliit funtion to Q 0 (I) =(1+s) to obtain I 0 = Q 00 (I), whih is negative beause, by assumtion, Q00 (I) < 0. Assuming Q 000 < 0 and differentiating a seond time yields I 00 = Q000 (I) Q 00 (I) I 0 < 0. 2 10
rate is rt r B. 4 In turn, the sulier is not obliged to extend trade redit to sell its roduts. Banks an finane urhases of inuts! To see what tye of restrition this outside otion yields, onsider first the sulier s disounted rofits, Φ T (rt ), for a given trade redit rate rt : Φ T (r T )=( T )I(r T ) I(r T ). (6) At a trade-redit rate rt, the demand for inuts is I rt, imlying that the sulier will finane I rt at time t =0and will ollet (1 + r T )I rt in rinial lus interest at t =1, when the trade redit is due. Disounting this riskless ash inflow to time t =0and subtrating the sulier s ost of rodution yield equation (6) as the disounted rofit. uose now that, rather than extending trade redit, the sulier lets banks finane safe firms at an interest rate r B. The demand for inuts of the safe firm will then be I(r B ), generating arofit for the sulier on the amount of ( )I(r B ). The sulier s outside otion of letting banks finane safe firms then requires that rofits with trade redit, Φ T (r T ), are bigger than or equal to ( )I(r B ). The sulier s roblem is thus max r T ( T )I r T I r T (7) subjet to 0 rt r B, (8) ( T )I rt I r T > ( )I(rB ). (9) The objetive funtion of Program (7) is the disounted rofit of a sulier who extends trade redittoasafefirm. The onstraint (8) rules out interest rates that are larger than the banking rate and imoses a lower bound that revents negative interest rates. The rationale for this lower bound is as follows. A negative trade-redit rate is observably equivalent to a ombination of a zero interest and a disount in the rie of the inut. Hene, ruling out negative trade-redit rates 4 A sulier with monooly ower may feth an interest rate higher than r B by denying inuts to firms that do not use trade redit. The analysis in the aer, therefore, ignores distortions in the trade redit markets that are driven by these tyes of bundling strategies. 11
amounts to restriting attention to testable imliations on the trade-redit rates. Finally, the inequality (9) assures that rofits with trade redit outweigh rofits onditioned on letting banks finane the safe firm, whih is the sulier s outside otion. If this ondition is not satisfied, it is otimal for the sulier not to extend trade redit to the safe firm. As we showed in setion 3.1, Q 000 (.) < 0 is a suffiient ondition for the demand for inuts to be stritly onave. Under this assumtion and defining the interest-elastiity of demand as (rt )= ( T )I0 (rt ), Proosition 1 haraterizes the otimal trade-redit rate with a safe firm. I(rT ) Proosition 1 - Assume that the investment funtion is stritly onave on the interest rate. Thus, it is otimal for the sulier to extend trade redit to safe firms, and the otimal trade redit-rate, rt, is r T = r B, if (r B ) B B (r B ); 0, if > and D (0) (0); br T (0,r B),if (r B ) > (r B ) and either (0) < (0) or. To get some intuition for Proosition 1, let us omare the sulier s rofit withandwithout trade redit. Without trade redit, banks will finane the safe firm at an interest rate r B, imlying that the sulier will sell for ash I(r B ) unitsoftheinutatarofit of( )I(r B ). In turn, equation (6) shows the sulier s disounted rofit withtraderedit, whihanberewrittenas Φ T (r T )=( )I rt r +( T r )I r T. This equation deomoses the disounted rofits in two arts: the oerational rofits, ( )I rt r,andthefinanial rofits, ( T r )I r T. By extending trade redit at the banking rate r B,theoerationalrofit mathes the total rofits without trade redit, ( )I(r B ), and, in addition, the sulier gets a finanial rofit of( r B r )I (r B) > 0. Hene, it is always otimal for the sulier to extend trade redit to safe firms. The banking rate is not neessarily the otimal trade-redit rate, though. On the one hand, a large trade-redit rate inreases finanial rofits er unit of trade redit, r B r. On the other hand, it redues the demand for inuts, at least artially offsetting the benefits of a large finanial margin. A trade off on the hoie of the trade-redit rate thus exists. Analogously to the analysis of monooly riing, the roof of Proosition 1 shows that there is a ut-off for the elastiity of 12
B demand at r B, below whih the gains of a large margin outweigh the loss of demand, B, making it otimal for the sulier to inrease the trade-redit rate to the uer bound r B. What haens if the elastiity of demand does not satisfy (r B ) B Then the otimal B? traderateislowerthanr B, beause the gains from an inreased demand for inuts more than offset the loss of the finanial margin. The mark-u of the sulier and the elastiity of demand determine how low the otimal trade-redit rate will go. Waiving interest is otimal if the marku is large enough to assure rofits at a zero interest, >, and if, desite the zero-interest, the demand remains suffiiently elasti, (0). If either of these two onditions does not hold, then the otimal trade-redit rate makes marginal rofits equal to zero, lying in the oen interval (0, r B ). In setion 3.4, we will use Proosition 1 to exhibit onditions that assure that the sulier hooses the same interest rate for safe and risky firms. But firstwemustderivethesulyof trade redit to risky firms. 3.3 The suly of trade redit to risky firms Let us now move to the risky firms. In our model, risky firms have the otion of borrowing from banks at the interest rate r B. At suh interest rate, the exeted return of lending to risky firms does not over the banks ost of funds; banks lend to risky firms at r B beause they do not know the firm s tye. The banks inability to distinguish between risky and safe firms may have reerussions in the trade redit markets. In artiular, the banking system works as an outside otion for the risky firms, reventing the sulier from extending tradereditataninterestrate larger than r B. But why should then a sulier lend to risky firmsataninterestratethatbanks would deny redit had they known the firm s tye? uliers have at least two reasons for extending trade redit at an interest rate rt r B. First, lending to risky firms at r B may imose an exeted loss to banks and yet assure an exeted rofit to suliers if their omarative advantage over banks in transforming ollateral into liquid assets, δ, is large enough. eond, as hwartz and Whitomb (1997) and Brennan, Maksimovi and Zehner (1998) oint out, suliers with monooly ower may be willing to offer a subsidized rate to boost rofitable sales. Aordingly, one would exet that if either the sulier s omarative advantage in default, δ,oritsmarku,, is large enough, then extending 13
trade redit should outweigh the sulier s outside otion of letting banks finane the risky firm, that is, π ( T ) I rt (1 + r) I rt ( )I (rb ), (10) The left-hand side of the inequality (10) is the sulier s disounted rofit onditioned on extending trade redit at an interest rate rt. With robability π, theriskyfirm sueeds, aying rinial lus interest, (1 + rt )I rt,attimet =1. With robability 1 π, theriskyfirm fails and all the sulier an do is to resue the unused inuts and resell them for I rt.fromthe risky-neutrality assumtion, the exeted ash flow π(1 + rt )I rt + I r T is disounted at the riskless interest rate to t =0. The disounted rofit then obtains one we subtrat the ost of roduing the inut. Condition (10) requires that this disounted rofit belargerthanthe sulier s rofit without trade redit, whih is ashed at t =0and amounts to ( )I (r B ). 5 Writing ondition (10) as a lower bound on the sulier s advantage in default yields δ µ µ I(r T ) I(r B ) 1 π I(rT ) + (1 + r)i(r B) π(1 + r T )I(r T ) (1 π)i(r T ) δ(r T ). (11) Extending trade redit to a risky firm at an interest rate rt an thus be otimal only if the sulier s advantage in resuing assets of bankruted firms, δ, is larger than or equal to a ut-off, δ(rt ),thatassuresthatrofits math the sulier s outside otion, ( )I(r B). As setion 4 will show, this neessary ondition yields testable imliations of our model. Aordingly, we shall substitute ondition (11) for ondition (10), writing the sulier s roblems as 5 The role of the banking rate as an outside otion thus determines the equilibrium level of interest rates in the trade redit markets, even for suliers that rovide all of their ustomers finaning requirements. Felli and Harris (1996) exlore this role of outside otions in a model of investment deisions in human aital. They show that an emloyee s rodutivity in a rival firm matters, even when an investment in firm-seifi human aital redues the hanes that the emloyee hanges jobs. 14
h i max π ( T ) I h i r rt T I r () T (12) subjet to 0 r T r B, (13) δ δ(r T ). (14) Proosition 2 haraterizes the solution of rogram (12), showing that the otimal traderedit rate deends not only on the sulier s advantage in resuing assets of bankruted firms, δ, but also on the interest-elastiity of the demand for inuts, rt,andthemark-u. Proosition 2 - Assume that the demand for inuts is stritly onave on the interest rate and let rt be the trade-redit rate that, onditioned on extending trade redit, maximizes the sulier s disounted rofits. Thus, it is otimal for the sulier to extend trade redit to risky firms if and only if δ δ(rt ), inwhihaser T is the otimal trade-redit rate and is haraterized by π B r B, if δ δ(r B ), (r B ) π B rt (r B );orδ = δ(r B ) and =1; = 0, if δ δ(0), > 1 and (0) π π (0); br T (0,r B ),ifδ δ(br T ), (r B ) > (r B ) and either (0) < (0) or. πδ Unlike in the ase of a safe firm, Proosition 2 shows that trade redit to risky firms may be otimal and yet the best that the sulier an do is to break even. This will haen if the sulier s advantage in default is just enough to assure the sulier s outside otion, δ = δ(r B ), and, in addition, ometition in the market for inuts, =1, drives the outside otion to zero. In this ase, trade redit imlies an exeted loss for any interest rate lower than r B. The banking rate, therefore, maximizes disounted rofits and solves the sulier s roblem regardless of the elastiity of demand. The trade-off between margin of rofits and volume of transations beomes relevant again, though, if the sulier has monooly ower, > 1, or if its advantage in default makes trade redit a stritly dominant strategy for the sulier, that is, δ>δ(r B ).Inbothases,extending trade redit to risky firms at the banking rate yields ositive rofits. And, as in the ase of a safe firm, the banking rate is r B is indeed the otimal trade-redit rate if and only if the demand 15
π B for inuts is suffiiently inelasti at r B,thatis, (r B ) π B δ If the demand is. not suffiiently inelasti, then it is otimal for the sulier to redue the trade-redit, and it is otimal to waive interest if and only if the sulier has monooly ower, > 1, andthedemand π remains suffiiently at the zero interest rate ( (0) π ).6 Proositions 1 and 2 establish the otimal terms of trade redit to safe and risky firms, giving us all we need for haraterizing an equilibrium in whih the trade-redit rate does not vary with the firm s tye. 3.4 Charaterizing the equilibrium An equilibrium in whih the trade-redit rate does not vary with the firm s tye has two main ingredients. First, the sulier must have inentives to extend trade redit to both tyes of firms, or else we rule out variations of interest rates aross firms from the onset. While the sulier always has inentives to extend trade redit to safe firms, Proosition 2 shows that trade redit to risky firms is otimal if and only if the sulier s advantage in resuing unused inuts, δ, is larger than a ertain ut-off δ(rt ). Intuitively,thisfirst restrition limits the ost that a risky firm s default may imose on the sulier. The seond ingredient for an equilibrium with invariant trade-redit rates is standard: there annot be inentives for the sulier to deviate from the invariant rate, regardless of the firm s tye. Aordingly, the equilibrium trade-redit rate must satisfy the first order onditions of roblems (7) and (12). Proosition 3, below, shows that this restrition imlies that the otimal trade-redit rate with a risky firm is larger than or equal to the otimal rate with a safe firms. More imortantly, the roosition shows that these otimal trade-redit rates an be equal only at the banking rate or at zero, whih are, therefore, the only andidates for an equilibrium trade-redit rate that does not vary with the firm s tye. Proosition 3 - The otimal trade-redit rate with a risky firm is bigger than or equal to the otimal trade-redit rate with a safe firm, with equality only at the banking rate and at the zerointerest rate. In artiular, if it is otimal for the sulier to extend trade redit to risky firms 6 The roof of Proosition 2 shows that, for waiving interest to be otimal, the mark-u must be large enough to assure stritly ositive rofits at a zero trade-redit rate, that is, > πδ. The roof of Proosition 2 also shows, nonetheless, that monooly ower, > 1, suffies for stritly ositive rofits if we also require δ δ(0). 16
at zero interest, then waiving interest is also otimal with a safe firm. And if it is otimal for the sulier to offer trade redit to safe firms at the banking rate, then either it is not otimal to extend trade redit to risky firms or the banking rate is the otimal trade-redit rate with a risky firm. The intuition for Proosition 3 is straightforward. Although a high interest rate assures reditors a high margin of finanial rofits, it redues the demand for loans, artly offsetting the benefits of a high margin. These inentives for lowering interest rates are at their eak when the borrower is a safe firm. In this ase, a redution in the volume of trade redit aounts for the loss of a sure rofit. In ontrast, a redution in the volume of trade redit does not lead to a loss in the states of nature that a risky firm beomes bankruted. As suh, theotimaltrade-redit rate with a risky firm is in general larger than the otimal rate with a safe firm. The exetions are at the banking rate, from whih a further inrease is not ossible beause banks rovide an outside otion for the firms; and at a zero-interest rate, from whih a further derease would imly negative interest rates that are observably equivalent to a zero trade-redit rate with a redution in the rie of inut (rie disrimination). Aordingly, let s start looking for an equilibrium in whih the sulier extends trade redit to both firms at the banking rate. From Proosition 3, if the banking rate maximizes disounted rofits with a safe firm, so it does with a risky firm. In turn, Proosition 1 tells us that a neessary and suffiient ondition for the trade-redit rate r B to maximize disounted rofits with a safe firm is that the elastiity of demand satisfies (r B ) B Our task is thus to show that, for B. admissible arameter values, the sulier s advantage in resuing assets may be large enough to outweigh rofits without trade redit (δ δ(r B )), while assuming that (r B ) B B. The ondition on the elastiity has a lear eonomi ontent. For the sulier to raise the interest rate as muh as ossible, the elastiity annot be too large or else the loss in the volume of transations will more than offset the benefits of a high finanial margin. Moreover, this ondition an be subsumed in the shae of the rodution funtion, Q(I), without imosing further restritions on the other arameter values. Hene, the seond restrition for the equilibrium, δ δ(r B ), is atually the key one for assuring existene of equilibrium at the banking rate. Plugging the baking rate into equation ondition (11) yields δ(r B )=1 πr B r,whihdoes 1 π 17
not deend on the mark-u. 7 The ut-off δ(r B ) does deend, however, on the banking rate, the robability of default, π, and the riskless interest rate r. Beause δ is the fration of inuts that thesulieranresueuonthedefaultofariskyfirm,itannotbelargerthanone. Thequestion then is whether there are admissible arameter values that allow for δ(r B )=1 πr B r < 1. 1 π Taking into aount that the banking rate is r B = 1, we an rewrite the restrition on f+π(1 f) δ as (1 + r)f δ f +(1 f)π. (15) f 1 f For some δ (0, 1) to satisfy the inequality (15) we must have ()f f+(1 f)π < 1, or equivalently, π. In other words, the robability that a risky firm reays the debt, π, mustbehigh r relative to the fration of safe firms in the industry, f. f It then follows that, π, δ ()f,and (r 1 f r f+(1 f)π B) B imly that there is an B equilibrium in whih the sulier extends trade redit to both tyes of firms at the banking rate at r B. Moreover, this equilibrium is unique, if we restrit attention to equilibria in whih the sulier extends trade redit to both tyes of firms. To see why remember that Proositions 1 shows that (r B ) B is a suffiient ondition for the banking rate r B B to be a strit otimal trade-redit rate with a safe firm; a result that, from Proosition 3, extends to trade redit to a risky firm if we add δ ()f. We have thus established: f+(1 f)π Proosition 4 - uose that the demand for inuts is stritly onave and f π. Thus, 1 f r there exists an equilibrium in whih the sulier finanes both tyes of firms at the banking rate r B = 1 if and only if the interest-elastiity of the demand for inuts at r f+π(1 f) B is smaller B B than or equal to and the fration of unused inuts that the sulier resues in ase a risk firm fails satisfies δ ()f (0, 1). Moreover, the equilibrium is unique in the lass of f+(1 f)π equilibria in whih the sulier extends trade redit to both firms. Consider now our seond andidate for an equilibrium with invariant rates, that is, trade redit to both tyes of firms at a zero interest rate. One again, Proosition 3 is the key to haraterize the equilibrium: If it is otimal to extend trade redit to risky firms at zero 7 The intuition for the irrelevane of the mark-u is that, vis-à-vis banks, rivate information gives some monooly ower to the sulier, making the elastiity of the demand for inuts imortant for the otimal interest rate even if ometition in the market for inuts drives ries down to the marginal ost. 18
interest, then it is also otimal to offer trade redit to safe firms at zero interest. Aordingly, we simly have to show that there are arameter values that satisfy the onditions in Proosition 2 for zero to be the otimal trade-redit rate with a risky firm, that is, δ δ(0), > 1 and π (0) π. We an ertainly assume that some suliers enjoy monooly ower, > 1, andthatthe π elastiity of demand at zero satisfies (0) π. howing that there exists δ (0, 1) suh that δ δ(0) is not so obvious, though. Plugging zero into equation ondition (11) yields δ (0) = ³ I(0) I(r B ) + ()I(r B) πi(0). One more, the ut-off deends on the on the 1 π I(0) (1 π)i(0) banking rate, r B, the robability of default, π, and the riskless interest rate r. Unlike in the equilibrium at the banking rate, though, the ut-off dereases with the mark-u. Intuitively, a large mark-u makes it easier for waiving interest to outweigh rofits without trade redit. Proosition 5 shows that, for a suffiiently large mark-u, there exist admissible arameter values that make δ (0) < 1, allowing for some δ (0, 1) to satisfy δ δ(0). Proosition 5 - uose that the investment funtion is stritly onave on the interest rate and that the mark u,, is suffiiently large. Thus, there exists an equilibrium in whih the sulier extends trade redit to both tyes of firms at a zero interest rate if and only if the interest-elastiity π of the demand for inuts satisfies (0) π and the fration of unused inuts that the sulier resues in ase a risk firm fails satisfies δ ³ I(0) I(()(f+(1 f)π) 1 1) + 1 π I(0) ()I(()(f+(1 f)π) 1 1) πi(0) (1 π)i(0) (0, 1). Moreover, the equilibrium is unique in the lass of equilibria in whih the sulier extends trade redit to both firms. A quik insetion of Proositions 4 and 5 shows two major differenes in the restritions for the two equilibria at an invariant trade-redit rate. For an equilibrium at the banking rate, all we need is that demand for inuts is suffiiently inelasti. For an equilibrium at the zero interestrate,weneedmore. Thedemandmustbesuffiiently elasti and the sulier must enjoy monooly ower. Proositions 4 and 5 thus suggest that the equilibrium at the banking rate is more ervasive. Nonetheless, Proositions 4 and 5 also show the two equilibria with invariant trade-redit rates share a ommon restrition. They both require a minimum level for the sulier s advantage in resuing assets of bankruted firms, δ. Eonomi intuition suggests, though, that a large δ might 19
not be neessary if the mark-u and the elastiity of demand are so large that a redution in the ost of finaningtozerosignifiantly enhane exeted rofits. Proosition 6 formalizes this intuition. Proosition 6 - uose that the investment funtion is onave on the interest rate, the marku of the sulier is suffiiently large, and that the interest-elastiity of demand for inuts is f+(1 f)π f (1 f)π greater than 1 π at a zero interest rate. Then there is an equilibrium in whih the sulier extends trade redit to both tyes of firms at a zero interest rate, even if the sulier annot resue any inut when a ustomer beomes finanially distressed (i.e., δ =0). Elliehausen and Wolken (1993), Petersen and ajan (1994) and Ng, mith and mith (1999) all reort that interest rates in trade redit markets are often standardized. And that, in some industries, suliers waive interest when their ustomers reay the loans within 10 days. Waiving interest rates is onsistent with Proosition 4 and 5 if the demand for inuts is elasti with reset to interest rates of loans of very short maturity. Moreover, Proosition 6 redits that industries with strong monooly ower are more likely to waive interest uon an early reayment. 4 Emirial Imliations and Disussion 4.1 Do suliers have inentives to release information truthfully? Consider an equilibrium with invariant interest rates and suose that banks request information on the redit standing of a sulier s ustomer. Is it in the sulier s interest to release the information truthfully? As it turns out, announing that the ustomer is a risky firm is a weakly dominant strategy for a sulier that an extend trade redit. To see why, onsider first the equilibrium in whih the interest rate in the trade redit market is equal to the banking rate. In this equilibrium, the banking rate is an outside otion for the firms that revents the sulier from further inreasing the interest rate. If the sulier an onviningly announe that the ustomer is a risky firm, banks will inrease the interest rate aordingly, letting the sulier inrease the interest rate as well. The higher interest rate moves the sulier loser to its unonstrained otimal. 8 In turn, the banking rate is not relevant to the 8 More formally, let r B betheinterestratethatbankswouldhaveoffered to a known risky firm. Cometition 20
sulier in the equilibrium in whih the sulier waives interests. Hene, for a sulier that an extend trade redit, it is a dominant strategy to announe that a safe ustomer is risky. Consider now a sulier that annot extend trade redit. Here, the inentives to release information are reversed. If the sulier an onvine the banks that its ustomers are safe, the banking rate will derease aordingly, and the ustomer will demand more inuts. If the sulier has some monooly ower, then it is a stritly dominant strategy to announe that risky ustomers are safe. Otherwise, the sulier is indifferent about the banking rate and announing thatriskyustomersaresafeisaweaklydominantstrategy. To be sure, banks an offer some revelation mehanism to suliers. For instane, rofitsharing mehanisms between a bank and a sulier should rovide inentives for the sulier to redibly reveal rivate information. till, we are not aware of any study that douments revelation mehanisms between banks and suliers in standard trade-redit transations. It is oneivable, though, that some sort of revelation mehanism is in lae in rojet loans that are tyially strutured around very omlex ontrats. In these tyes of transations, we do not exet an equilibrium with invariant interest rates. 4.2 Monooly ower and informational advantage o far, we have imosed no onstraints on the struture of the market for inuts. uliers an enjoy some monooly ower or fae ometitive fores that drives the inut rie down to the marginal ost ( =1). The question that we address in this setion is whether the equilibrium with invariant rates survives if ometition in the market for inuts drives to zero not only oerational rofits but also finanial rofits. uose that suliers are all equally informed. In this ase, there is no soe for a sulier to rofit bylendingtoasafefirm. Cometition extends to the trade redit market, driving down among banks imlies that the exeted return of a banking loan at the interest rate rb equals the ost of funds r. Hene, π(1 + rb )=() whih imlies that r B = π 1 >r B. This means that the interest rate in banking loans inreases if a sulier onvines the banks that the ustomer is a risky firm. As a result, the onstraint in the sulier s rogram hanges from 0 rt r B to 0 rt r B. ine r B >r B, the onstraint is relaxed, imlying an inrease in exeted rofits beause a onave investment funtion imlies that the sulier s rofit funtion is onave. 21
theinterestrateofloanstosafefirms to the riskless rate r. Note, though, that the equilibrium interest rate in loans to risky firms will not be equal to the riskless rate. In these loans, the sulier takes into aount the robability 1 π that the debt ontrat will not be honored and that, in default, only δi < I will be olleted. As suh, the interest rate rt that equals the exeted return on the loan to the riskless rate is larger than the riskless rate r. And we onlude that ometition among equally informed suliers breaks down the equilibrium with invariane of interest rates in trade redit markets. It is unlikely, nonetheless, that all suliers of inuts are equally informed about their ustomers. It should be easier to learn rivate information about your best ustomers. Hene, although a threat to buy inuts from an alternative sulier may fore a ometitive rie for the inuts, it should not break down the urrent sulier s informational advantage, whih is all we need for the analysis of our model to hold. 4.3 Invariane of interest rates and effiieny in default From Proosition 4 and 5, the equilibrium with invariant trade-redit rates requires that suliers be more effiient than banks in salvaging assets of finanially distressed firms. As it turns out, all that the equilibrium needs is that suliers are more effiient lenders to risky firms than banks. For instane, a sulier might have no advantage in resuing assets and yet, as Cuñat (2002) oints out, be a more effiient lender due to a threat of stoing the suly of vital intermediate goods. till, a low omarative advantage in salvaging assets of finanially distressed firms makes it more diffiult for the threat of terminating the suly of inuts to be strong enough to assure that trade redit to risky firmsisrofitable. As we argue below, a testable imliation of our model then follows. Petersen and ajan (1997) show that suliers offer larger lines of redit to firms with a low fration of their inventory in finished goods. Petersen and ajan interret their finding as evidene that suliers have a stronger advantage of salvaging assets of firms that hold a low fration of their inventory in finished goods. Intuitively, one firms transform intermediate goods into finished goods, suliers an no longer use their regular sales fore to sell the firms inventory. In this sirit, our model redits that suliers are more likely to offer standardized interest rates in industries with a low fration of finished goods in inventory. This redition will 22