Informed Principals in the Credit Market when Borrowers and Lenders Are Heterogeneous

Transcription

1 ISSN Informed rincials in the Credit Market when Borrowers and Lenders re Heterogeneous Francesca Barigozzi iero Tedeschi Quaderni - Working aer DSE N 1051

2 Informed rincials in the Credit Market when Borrowers and Lenders re Heterogeneous Francesca Barigozzi University of Bologna iero Tedeschi y Università Cattolica del Sacro Cuore - Milano January 2016 bstract Both borrowers and lenders can be socially resonsible (SR). Ethical banks commit to nancing only ethical rojects, which have social ro tability but lower exected revenues than standard rojects. Instead, no credible commitment exists for SR borrowers. The matching between SR borrowers and ethical banks reduces the frictions caused by moral hazard. However, when the tye of the borrowers is not observable, then standard borrowers have incentives to invest in ethical rojects retending to be SR. We show that the searation of borrowers entails costs that are aid by SR entrereneurs but are relatively low because standard lenders o er an outside otion that relaxes the self-selection constraint of the borrowers. Technically, we solve a Contract roosal Game where informed rincials (borrowers) o er di erent menus of contracts to heterogeneous agents (banks). We show that market segmentation imroves e ciency and solves the roblem of multilicity of equilibria in Contract roosal Games. Jel classi cation: D86, G21, G30. Key-words: cororate social resonsibility, ethical banks, motivated borrowers, informed rincials, moral hazard, adverse selection. Deartment of Economics, University of Bologna and CHILD,.zza Scaravilli 2, Bologna (Italy). francesca.barigozzi@unibo.it y Deartment of Economics and Finance Università Cattolica del Sacro Cuore di Milano. iero.tedeschi@unicatt.it 1

3 1 Introduction The emergence of ethical banks is an exression of the growing demand for cororate social resonsibility (CSR) in the banking industry of develoed countries. Quoting Bénabou and Tirole (2010,.2), CSR is about sacri cing ro ts in the social interest. For there to be a sacri ce, the rm must go beyond its legal and contractual obligations, on a voluntary basis. CSR embraces a wide range of behaviors, such as being emloyee friendly, environment friendly, mindful of ethics, resectful of communities where the rm s lants are located, and even investor friendly. In line with the revious de nition, we interret ethical banks as socially resonsible lenders, because they commit to fund only socially relevant rojects, i.e. ethical rojects, which rovide both social and economic advantages, but which deliver lower exected revenue than standard ones. Ethical banks are not the only socially resonsible agents in the credit market of high income countries, we call motivated those borrowers that refer to engage in socially valuable activities as ethical rojects because, by doing so, they receive a non-monetary remium for social resonsibility. Contrary to ethical banks, motivated borrowers do not commit to ethical rojects and will still invest in standard rojects if their exected return is su ciently higher than the one of ethical rojects. 1 Barigozzi and Tedeschi (2015) show that the assortative matching between ethical banks and motivated borrowers allows to reduce the frictions caused by moral hazard. The e ciency gain can be so high that motivated borrowers trading with ethical banks end u receiving better contract conditions than standard borrowers trading with commercial banks. Better rosects translate in larger loans, higher exected returns for the borrower and lower interest rates and are ossible when the remium for social resonsibility is su ciently high. Such a remium accrues motivated entrereneurs ayo s when they undertake an ethical roject nanced by an ethical bank and the roject turns out to be successful. Imortantly, the surlus arising from the matching between agents caring about social issues imlies that socially resonsible banks, desite the fact of sacri cing ro ts on a voluntary basis, can survive in the long run as well as traditional lenders. Some recent emirical studies focusing on ethical banks (Becchetti et al. 2011; Becchetti and Garcia 2011; Cornée and Szafarz 2012) are in line with the revious results: ethical banks authorize larger loans than commercial banks, and borrowers nanced by ethical banks, having controlled for borrowers characteristics, are charged a lower interest rate. In our revious aer we investigate how socially resonsible lenders and motivated borrowers interact with each other when they articiate in a credit market where standard lenders and borrowers also oerate and moral hazard is the unique market failure, meaning that both borrowers and lenders characteristics are common knowledge. s already mentioned, in the revious aer we roved that motivated borrowers can obtain better credit conditions than other borrowers if the remium for social resonsibility is su ciently high. Suose, now, that borrowers motivation is rivate information and no credible commitment is 1 We refer the reader to Barigozzi and Tedeschi (2015) for the descrition of the tyical mission of ethical banks, for real world examles and for a discussion about ethical rojects, ethical banks and motivated borrowers. 2

4 available to entrereneurs so that their ro-social attitude can be falsi ed. In articular, a borrower could retend to be socially resonsible in order to strengthen its bargaining osition. This is recisely what haens in our framework where standard borrowers are the bad tyes willing to take advantage of their rivate information. In our setting, falsifying borrowers CSR is straightforward because entrereneurs retending to be motivated simly have to undertake ethical rojects nanced by ethical banks. This has, however, negative imlications for ethical lenders as, when a standard entrereneur mimics a motivated one, he/she ossibly misbehaves and the lending contract is not ro table anymore for the bank. The situation we describe may occur in a market where borrowers are start-us, or new rms lacking a reutation and credible commitment to CSR does not exist. To sum u, in this aer we analyze the interaction between heterogeneous borrowers and heterogeneous lenders in a credit market where both moral hazard and adverse selection on the borrowers side have bite because banks are not able to observe borrowers behavior nor they can distinguish motivated from standard borrowers. We show that, under moral hazard and adverse selection, the bene t arising from trade between ethical banks and motivated borrowers is artially o set by the information rent aroriated by standard borrowers. In articular, the equilibrium outcome is such that standard borrowers sign with commercial banks the same contract as under moral hazard only; whereas motivated borrowers trading with ethical banks reserve their higher borrowing caacity but loose the bene t of better contract conditions. In di erent words, motivated borrowers ay the cost of searation by acceting contractual terms that do not aeal to a standard entrereneur. Moreover, as under moral hazard only, the market is fully segmented, meaning that standard agents trade among themselves in the market for standard rojects while ethical banks trade with motivated borrowers in the market for ethical rojects. Turning to the modeling strategy, we study a model where borrowers are the informed arty roosing contracts to lenders. Moreover, as it will be better argued in Subsection 4.1, the tye of borrowers a ects the ethical banks ro ts, through the solution of the moral hazard roblem, making our model a common value one. We are thus in a framework with informed rincials and common values and, borrow from the seminal aer by Maskin and Tirole (1992). In Maskin and Tirole (1992), heterogeneous rincials roose one single menu of contracts to uninformed and uniform agents and a key-concet in the aer is the Rothschild-Stliglitz-Wilson (RSW) allocation (with homogeneous agents). Such allocation indicates the menu of incentive comatible contracts that are ro table tye-by-tye and allow searation at the lowest cost. 2 Conversely, in our model heterogeneous rincials (borrowers) roose contracts to uninformed and heterogeneous agents (lenders). The imortant di erence is that, in our Contract roosal Game, informed rincials o er two 2 Maskin and Tirole (1992) name the RSW allocation, where the incentive-comatible contracts are ro table tye-bytye, after the in uential aers by Rothschild and Stiglitz (1976) and Wilson (1977). s we will further discuss in the aer, informed rincials can always guarantee themselves the ayo reached with the RSW allocation which is the unique equilibrium when it is interim e cient. When it is not, then multilicity of equilibria arises in the contract roosal game. 3

5 menus (one for each tye of agents), instead of only one. 3 We show that all contracts in the two menus break-even tye-by-tye and that the menu for ethical lenders contains the RSW allocation with market segmentation (see below). Instead, the menu for standard lenders always contains a ooling contract, since for standard lenders the two tyes of borrowers are equivalent, as it will be clear in what follows. 4 Once each tye of agents (lenders) has acceted the menu designed for it, each informed rincial (borrower) selects the referred otion among the contracts available in the two menus. More seci cally, each borrower icks the overall referred contract from one menu and the null contract from the other menu. We rst show that the resence of heterogeneous lenders and the fact of o ering them two di erent menus lead to a substantial imrovement in the e ciency of the equilibrium allocation and guarantees its uniqueness. Finally, we show that a su cient condition for the unique equilibrium outcome to be interim e cient is that the share of motivated borrowers in the oulation of investors is lower than 50%. s for e ciency, the cost of searation would be higher if ethical banks were the unique tye of lenders in the credit market. Intuitively, in the RSW allocation with market segmentation contained in the menu for ethical banks, borrowers take into account that trading with standard lenders reresents an outsideotion that relaxes the self-selection constraint and hels reventing standard borrowers from mimicking motivated entrereneurs. We thus conclude that the RSW allocation with market segmentation aretodominates the RSW allocation with homogeneous lenders. Moreover, in the model with homogeneous agents, the RSW allocation is always an equilibrium. In our setting with heterogeneous agents, instead, the equilibrium delivers market segmentation and the RSW allocation with market segmentation is not an equilibrium outcome 5. Indeed, only motivated borrowers ick the contract contained in the RSW allocation inside the menu for ethical banks, whereas standard borrowers ick the contract aearing in the menu o ered to standard lenders. Hence, the RSW allocation with market segmentation contains a latent contract. s for uniqueness, the existence of two menus designed for the two tyes of lenders and the fact that cross-subsidies between di erent menus are imossible (meaning that transfers between di erent lenders are not feasible 6 ), imly that the uniqueness issue is fully solved. Only cross-subsidies inside the menu for ethical banks make sense in our setting. To rove uniqueness we thus show that, in the menu for ethical banks, it is imossible to design searating contracts with cross-subsidies that areto dominate the equilibrium allocation with market segmentation. Hence, we conclude that the equilibrium outcome with market segmentation is necessarily unique. Finally, we study the e ciency roerties of the unique 3 Each menu also contains the null contract which works as an exit otion and allow for market segmentation. 4 In articular, the menu o ered to ethical banks contains two self-selecting contracts and the null one. The menu for standard lenders contains, instead, a ooling contract together with the null one. 5 We thus distinguish between the RSW allocation with market segmentation and the equilibrium allocation with market segmentation. The latter contains only one contract from the RSW allocation with market segmentation (the one designed for motivated borrowers). 6 With cross-subsidies among banks, the menu designed for the standard bank would contain a ooling contract letting the bank earn negative ro ts to be covered with ositive ro ts earned by the ethical bank in the other menu. But, in the second stage of the Contract roosal Game, the standard lender would not accet a menu entailing negative ro ts. 4

6 equilibrium outcome and nd a su cient condition such that the latter is interim e cient (meaning that, in the hyothetical case where transfers between di erent menus were feasible, no dominating contracts with cross-subsidies would exist). More seci cally, a su cient condition for the equilibrium with market segmentation to be interim e cient is that the share of motivated borrowers in the oulation of otential borrowers is less than 50%. To sum u, our results show that the existence of heterogeneous agents receiving di erent menus and the fact that transfers between the di erent menus are not feasible, imrove e ciency and solve the roblem of multilicity of equilibria in games with informed rincials. Our aer is related to the small literature on CSR in the credit market of high-income countries. Few works, mainly in the business literature, analyze ethical banks and show the relevant role of ethical banking as an indeendent activity (e.g., Green 1989; Lynch, 1991; San-Jose et al., 2009). From the oint of view of the modelling strategy, the setting we study borrows from Tirole (2006). The aer is also related to the models dealing with signaling and informed rincials in the credit market that are suerbly reviewed in the same book (Tirole 2006, chater 6). In those studies, good borrowers try to signal attractive rosects to investors by introducing distortions that are costly to them but would be even costlier to bad borrowers; the informed arty can act as a rincial by roosing the contract (as in our model) or can accet the contract o ered by the uninformed arty. 7 Notable and more recent examles of this kind of literature are Henessy et al. (2010), Cestone et al. (2014) and Bouvard (2014), all dealing with roblems not closely related with ours. Even if the idea that a rm can follow CSR to strengthen its market osition is not new (see the excellent review on the economics of CSR by Kitzmueller and Shimshack 2012), to the best of our knowledge our aer is the rst studying a situation where no available commitment to CSR exists such that a borrower can falsify its interest for social issues. Our aer is organized as follows. Section 2 describes the model set-u and the ayo s of socially resonsible lenders and motivated borrowers. In Section 3, we show how the matching between socially resonsible agents relaxes the incentive comatible constraint of motivated borrowers and we brie y reort results from Barigozzi and Tedeschi (2015) about the characterization of loan agreements o ered in the credit market under moral-hazard only. Section 4 solves the model with both moral-hazard and adverse selection. More seci cally, Subsection 4.1 de nes the RSW allocation with market segmentation, shows the e ciency gain it entails and characterizes the equilibrium outcome when both moral-hazard 7 Leland and yle (1977) is the rst alication of signaling in nance; they show how a risk-averse owner can signal the underlying quality of its rm in an initial ublic o ering (IO) by retaining a substantial undiversi ed stake in the rm. Bhattacharya (1979) investigate signaling through dividends. Myers and Majluf (1984) show that, when the rm has rivate information about the value of its assets, the decision to rise new caital can be used as a signaling devise. mong others, Besanko and Thakor (1987), Bester (1985) and (1987), Chan and Kanatas (1985) study the ossibility of signaling by ledging collateral. In Diamond (1991), a borrower enters into a short-term borrowing contract in order to signal her creditworthiness. Welch (1989), llen and Faulhaber (1989) have extended the Leland and yle s theory by modeling low IO rice as a signal of the quality of the undertaken roject. 5

7 and adverse selection have bite. Subsection 4.2 deals with uniqueness and investigates e ciency. The aer s Conclusion follows. 2 Model Set-u The model is borrowed from Tirole (2006) and has been illustrated in Barigozzi and Tedeschi (2015). Here we brie y resent the model s setu and refer the reader to the mentioned aer for a detailed discussion of the assumtions about motivated borrowers, ethical banks and ethical rojects and for some real world examles. We consider a credit market with a large number of both risk-neutral borrowers (she) and banks (it). Borrowers undertake a roject that requires an investment. Each borrower can aly for at most one loan, and di erent tyes of rojects exist. We call I k the endogenous amount of the investment, where k 2 f0; 1g indicates the tye of roject. When k = 1 the roject is ethical, and when k = 0 the roject is non-ethical or standard. The di erence between the two rojects is seci ed below. ll the borrowers own the same asset. The borrowers do not have su cient caital and/or collateral no matter which roject they are interested in, and hence need to borrow I k. If the roject is undertaken, it generates a cash ow er unit of investment, which is R k > 0 if successful and zero in the case of failure. 8 The total cash ow if the roject is successful is R k I k 0. Ethical rojects reresent all rojects leading to social bene ts beyond ro ts (for examle, rojects that imrove communities and have a ositive imact on the environment). The ro tability of ethical rojects is on average lower than that of standard ones. In articular, standard rojects have a higher return in the event of success, that is: R 0 R 1 > 0. Both tyes of rojects are erfectly observable and have indeendent distributions. To summarize, two sectors exist in the credit market: the market for ethical rojects and the market for standard ones. The latter assures higher exected returns to investors. ll rojects are subject to moral hazard: entrereneurs can behave or misbehave. If they behave, the robability of success is H ; otherwise it is L, with H > L. We de ne H L. n entrereneur who misbehaves will enjoy a rivate bene t whose value is I. Otherwise, the rivate bene t will be nought. We call a 2 f0; 1g the behavior of the entrereneur. In articular, a = 0 if the entrereneur misbehaves, while a = 1 if she behaves. Thus, (1) = H and (0) = L, resectively. The borrowers are rotected by limited liability: hence their income cannot be negative. Given limited liability, the moral hazard roblem is relevant even though both agents are risk neutral. There are also two tyes of banks and entrereneurs, resectively denoted as i 2 f0; 1g and j 2 f0; 1g. For both lenders and borrowers tye 0 denotes the standard agents, whereas tye 1 indicates socially resonsible agents. In case of success, revenues are shared between lenders and borrowers: L k ij and Bk ij are, resectively, 8 This is without generality loss because we roved that the otimal contract is a debt one. 6

8 the incomes of a lender of tye i and that of a borrower of tye j when trading with each other, if the investment is of tye k, hence obtaining: L k ij + Bk ij = Rk I. In the following sections, we characterize the otimal contracts (Bij k ; Ik ij ) that secify the tye of roject, the amount invested, and how revenues are shared between lenders and borrowers in the case of success given the tye of agents trading together. The crucial ingredients of our model are the remium for social resonsibility and the remium for successful interaction : The remium for social resonsibility is a non-ecuniary bene t with monetary value that a motivated borrower obtains when undertaking an ethical roject, no matter the roject s outcome and whatever the tye of lender. The additional remium for successful interaction is the extra remium of social resonsibility accrued by a motivated agent when imlementing an ethical roject nanced by an ethical bank. In fact, the motivated borrower anticiates that, if the ethical bank makes ro ts, given its commitment to investing in ethical rojects, it will use the liquidity to nance other social and solidarity-based rojects, increasing exected utility of the motivated borrower. The entrereneurs ayo can be written as: U k ij = (a) B k ij + ijk + jk + (1 a) I k ij (1) Note that the remium for social resonsibility is ositive only if a motivated borrower invests in an ethical roject (j = k = 1) ; whereas the remium for successful interaction is ositive only if a motivated borrower invests in an ethical roject with an ethical bank (i = j = k = 1) and the roject succeeds. Standard lenders maximize their ro ts. When the moral hazard roblem is taken care of, exected ro ts become: H L k 0j I k 0j + (2) We assume that ethical banks maximize exected ro ts 9, as do standard lenders, but commit to investing only in ethical rojects (therefore k = 1): Because ethical rojects have a lower ro tability than standard ones, commitment to ethical rojects imlies that ethical banks are sacri cing ro ts for the social interest. 10 with ethical banks: Subscrit j in (3) indicates that both standard and motivated borrowers can invest H L 1 1j I 1 1j + : (3) s already mentioned, borrowers are endowed with all the bargaining ower and roose a contract to lenders so that the banks exected ro ts are zero at the equilibrium. 11 The sequence of actions is thus as follows: the borrower o ers the contract, then the lender accets or refuses the roosal. Subsequently, the borrower decides whether to behave or misbehave, uncertainty concerning the roject is solved, and the contract is imlemented. 9 One can show that nothing changes when assuming that ethical banks maximize total revenue from ethical rojects. 10 Commitment to ethical rojects is ossible because borrowers have all the market ower and roose contracts, in the form of a seci c and observable roject for funding, to banks which can accet or reject them. 11 This is equivalent to assuming Bertrand cometition among lenders. 7

9 3 Second-best: Loan greements under Moral Hazard The setting with moral hazard only is investigated in Barigozzi and Tedeschi (2015). Here we summarize their main results because they reresent the starting oint of the analysis of moral hazard and adverse selection which is the focus of the resent aer. ssume that the roject and the borrower tye are common knowledge, while borrowers have rivate information on their behavior, which may or may not increase the robability of success of the roject. The otimal contract hence maximizes the borrower s utility under the borrower s incentive comatibility constraint and the lenders articiation constraint (see endix 6.1). IC Bk ij IR Lk ij One can show that standard borrowers otimally sign a contract with a standard bank for a standard roject. Motivated borrowers, instead, at equilibrium can either sign the same contract (because all borrowers are the same when investing in standard rojects) or they can invest in an ethical roject with an ethical bank. The third otion of motivated borrowers, i.e. investing in an ethical roject with a standard bank, is discarded because always dominated. In fact, when trading with an ethical bank, motivated borrowers may also receive the remium for successful interaction,. Moreover, the matching of motivated borrowers with ethical lenders reduces the frictions caused by the agency issue, as we brie y exlain below, allowing for the ossibility of better contract conditions, and this desite the lower ro tability of ethical rojects. To see why ethical banks are more e cient than standard lenders in solving the moral hazard roblem of the motivated borrower, let us consider the incentive comatibility constraint of a motivated borrower trading with an ethical bank: B I1 11 (IC B 11) The left-hand side of the incentive comatibility constraint includes all the gains obtained by the borrower when a roject is successful: an increase in revenues, B 1 11, and an increase in sychological well-being,. The latter is what makes the incentive comatibility constraint in such a case distinctive, in fact this term aears neither in the incentive comatibility constraint of a standard borrower nor in the incentive comatibility constraint of a motivated borrower when investing in an ethical roject with a standard bank. Hence, we conclude that the matching of motivated borrowers and ethical lenders relaxes the incentive constraint of the borrowers. The market structure at equilibrium deends on the motivated borrower s choice of investing in a standard roject with a standard bank or of investing in an ethical roject with an ethical bank. The following roosition from Barigozzi and Tedeschi (2015) describes the equilibrium in the credit market under moral hazard. The analytical exressions for the terms aearing in the two otimal contracts (B 0 00 ) and (B 1 11; I 1 11 ) and the three threshold values ; and for the remium for successful interactions are illustrated in endix 6.1. roosition 1 Second-best (From Barigozzi and Tedeschi 2015). When moral hazard is the unique asymmetric information characterizing the credit market, three threshold values for the remium for 8

10 successful interactions, > 0; exist such that: when, then the credit market is fully segmented and contracts (B 0 00 ) and (B 1 11; I 1 11 ) are signed by standard and motivated borrowers resectively. 1. If, the contracts are such that I 1 11 > I 0 00 and B 1 11 > B 0 00: 2. If, the contracts are such that: I 1 11 > I 0 00 and B 1 11 < B 0 00: 3. If <, the contracts are such that: I 1 11 < I 0 00 and B 1 11 < B 0 00: When 0 <, then ethical banks are not active and there is no market for ethical rojects: all borrowers invest in standard rojects and choose the contract (B 0 00 ). To sum u, only socially motivated borrowers otentially engage in ethical rojects. If they do not, then ethical banks cannot oerate, and the market for ethical rojects does not exist. If, instead, motivated borrowers undertake ethical rojects, then ethical banks are active and the market is fully segmented. That is, standard agents trade among themselves in the market for standard rojects while ethical banks trade with motivated borrowers in the market for ethical rojects. This occurs when the remium for successful interaction is su ciently high ( ). For larger values of the successful interaction remium ( ), not only are ethical banks active, but they even rovide greater funding to motivated borrowers than what is received by standard borrowers from standard banks. Finally, when the remium increases even further ( to motivated borrowers than what standard borrowers can obtain. ), ethical banks surrisingly guarantee greater revenue In the latter scenario, motivated borrowers trading with ethical banks unambiguously receive better contract conditions. 4 Third-best: Loan greements under Moral Hazard and dverse Selection Now lenders cannot observe neither the borrowers behavior nor their motivation, whereas lenders CSR is still common knowledge, together with the fraction of motivated borrowers in the credit market (q). s already mentioned, this setting ts a situation where lenders are banks with well known characteristics, while borrowers are new rms without reutation. s illustrated in roosition 1, this environment is interesting since, when the remium for successful interaction is su ciently high, motivated borrowers trading with ethical banks obtain better contract conditions than standard borrowers trading with standard lenders. Thus, standard borrowers could take advantage of their rivate information by investing in ethical rojects and retending to be motivated. In this latter case ethical banks may obtain negative ro ts, since standard borrowers mimicking motivated ones ossibly misbehave. In di erent words, the second-best contract for motivated borrowers, (B 1 11; I 1 11 ), is not necessarily incentive comatible for 9

11 standard borrowers. In the following, we study the third-best of the model, i.e. we derive the searating allocation allowing to distinguish di erent borrowers, we study the distortion it entails and rove uniqueness of the equilibrium outcome The RSW allocation with market segmentation Recall that borrowers have all the market ower and roose contracts to banks. Since borrowers are the informed arty, we are considering here a case of contract design by an informed rincial and the equilibrium concet is erfect Bayesian Equilibrium. The seminal aer on this toic is Maskin and Tirole (1992), whose results have been adated to the case of cometition in a credit market with two tyes of borrowers and homogeneous lenders in Section 6.4 of Tirole (2006). In that model two tyes of informed borrowers and many homogeneous lenders interact in the market. s mentioned in the Introduction, the RSW allocation corresonds to the searating allocation entailing the lowest cost of searation when two self-selecting contracts that are ro table tye-by-tye are o ered. Moreover, if the RSW allocation is interim e cient (i.e. it is ro table also in exectation) with resect to the lenders riors, then it is the unique erfect Bayesian equilibrium of the game (the formal result is stated in Corollary to Theorem 1 in Maskin and Tirole 1992). We deart from Section 6.4 of Tirole (2006) for two reasons: rst, in our setting moral-hazard is also an issue, second and more imortantly for what follows, we are considering a credit market with heterogeneous borrowers and heterogeneous lenders. This imlies that two menus (one for each tye of lenders) will be o ered in equilibrium. The three-stage timing of actions is adated to our setting with heterogeneous agents and is the following. In the rst stage, borrowers design a menu for standard lenders and a menu for ethical lenders and o er the menus to lenders. Each menu contains three contracts at most: the null contract together with either two searating contracts (one for each tye of borrowers) or a ooling contract. In the second stage, lenders accet the o er if they earn nonnegative exected ro ts from the contracts aearing in the menu designed for them. 13 In the third stage, each borrower icks one contract from each menu: the contract overall referred among all contracts aearing in the two menus and the null contract from the dominated menu. Then borrowers decide whether to behave or misbehave. Finally, uncertainty concerning the roject is solved, and the contract is executed. It is imortant to stress that each menu must be comlete in the sense that it must contain (selfselecting) contracts for the two tyes of borrowers, contingent on the seci c tye of lender. In the second stage lenders accet the menu designed for them if, given their beliefs on the borrowers tye, their exected ro ts will be non-negative. The null contract aears in each menu so that each tye of 12 We will distinguish between the equilibrium menus o ered to ethical and standard banks and the equilibrium outcome/allocation. We will rove that the menu o ered in equilibrium to ethical banks is not unique whereas the equilibrium outcome is unique. 13 Recall that standard lenders otimally fund standard rojects because social resonsibility of motivated borrowers does not a ect the borrowers incentive constraint. Ethical banks, instead, commit to fund only ethical rojects. 10

12 borrower is able to trade with just one tye of lender and market segmentation can occur. Before roceeding, the following clari cations are useful. When dealing with ethical lenders we are considering a setting with common values. In fact, the borrower s tye a ects the exected ayo of the ethical bank throughout the incentive comatibility constraint of the borrower that is tye deendent (see constraint IC B 11 in the revious Section). In addition, if the incentive constraint is not satis ed, then the borrower misbehaves and the ethical lender s exected ro ts become negative. On the contrary, the tye of the borrower does not a ect the exected ayo of commercial banks so that the setting is about rivate values for those lenders. To solve the contract roosal game we roceed as follows. We show that, in third-best, the secondbest contracts are incentive comatible and thus are still o ered when <. We then turn to the case where and we observe that the second-best contract (B 0 00 ) is the (ooling) contract characterizing the menu o ered to standard banks. Then, we derive the menu of contracts that borrowers o er to ethical lenders. To do so, we rst de ne the RSW allocation ignoring market segmentation. Subsequently, we de ne the RSW allocation with market segmentation and we show that the latter dominates the former. 14 This allows us to conclude that the RSW allocation with market segmentation is the allocation which characterizes the menu o ered to ethical lenders. The equilibrium outcome will be given by the contracts selected by the two tyes of borrowers among the contracts aearing in the two equilibrium menus. When borrowers trade with standard lenders, searation has no meaning because all borrowers are the same. Moreover, borrowers otimally o er the second-best contract (B 0 00 ) : Remark 1 In third-best, (i) borrowers o er to standard lenders a menu with the second-best (ooling) contract and the null contract: (B 0 00 ); (0; 0) : (ii) In equilibrium, both borrowers tyes can always guarantee themselves the ayo associated with the second-best contract (B 0 00 ). From the revious Remark and from roosition 1, it straightforwardly follows that, when < ; borrowers roose contract (B 0 00 ) to standard lenders, and no searation is ossible because ethical banks are not active. When instead, for ; borrowers are confronted with ethical and standard banks, searation with market segmentation is ossible if the self-selection constraints of the two tyes of borrowers are satis ed. Obviously, if second-best contracts (B 0 00 ) and (B 1 11; I 1 11 ) verify such constraints, then those contracts will be roosed in third-best as well. From roosition 1 we know that motivated borrowers refer contract (B 1 11; I 1 11 ) to contract (B 0 00; I 0 00 ) when ; so that contract (B 0 00 ) is envy free. Moreover, standard borrowers refer contract 14 s it will be clari ed below, in the RSW allocation with market segmentation, borrowers take into account that trading with standard lenders reresents an outside-otion which relaxes the self-selection constraint reventing standard borrowers from mimicking motivated entrereneurs. 11

13 (B 0 00 ) to contract (B 1 11; I 1 11 ) when < since, with the former, they receive a higher exected utility than with the latter. In fact, for ; B 0 00 > B 1 11 holds so that: H B 0 00 > H B 1 11 : (4) The revious reasoning imlies that, when <, the credit market is fully segmented and the second-best contracts (B 0 00 ) and (B 1 11; I 1 11 ) satisfy the borrowers self-selection constraints. Here, the searation of borrowers tyes occurs without additional agency costs, i.e. the unique friction is the one caused by moral hazard. Remark 2 In third-best, when < ; the second-best allocation illustrated in roosition 1 is still otimal and is imlemented through the two menus: (B10; 1 I10 1 ); (B11; 1 I11 1 ); (0; 0) for ethical banks and (B 0 00; I00 0 ); (0; 0) for standard banks, where (B10; 1 I10 1 ) is the best contract that standard borrowers can o er to ethical banks. 15. In the third stage, when borrowers choose a contract from each menu, motivated borrowers ick (B 1 11; I 1 11 ) from the menu for ethical banks and the null contract from the other one; standard borrowers ick (B 0 00 ) from the menu for standard banks and the null contract from the other one. Interestingly, the menu for ethical banks contains a latent contract, (B 1 10; I 1 10 ); that will never be chosen in equilibrium because it is dominated by (B 0 00 ). We consider now the most interesting case where and the oosite of inequality (4) holds because B 0 00 < B 1 11: Now both borrowers tyes refer contract (B 1 11; I 1 11 ) so that adverse selection has bite and the standard borrowers are the bad tyes who can take advantage of their rivate information. Below we describe the RSW allocation with homogeneous agents (See Tirole 2006, section 6.4) in our setting. It corresonds to the searating contracts aearing in the menu that borrowers o er to ethical banks when they ignore that the ooling contract (B 0 00 ) o ered in the menu designed for standard banks is the best available otion for standard borrowers. De nition 1 The RSW allocation with homogeneous lenders corresonds to the searating contracts (B 1 10; I 1 10 ) and ( ^B 1 11; ^I 1 11) which do not take into account the menu o ered to standard lenders. 15 The analytical exression for B10 1 is the following: B10 1 = 1 H R 1 < B 0 00 and is obtained solving the rogram of a standard borrower contracting for an ethical roject with an ethical bank. corresonds to rogram (8) in endix 6.1 with R 1 relacing R 0. Obviously, contract (B10 1; I1 10 ) is dominated by (B0 00 ; I00 0 ) because of standard rojects higher returns. It 12

14 Contract ( ^B 1 11; ^I 1 11) solves the following rogram: max B 1 11 ;I1 11 H B H + s:t: B I11 1 IC11 B H R 1 1 I11 1 H B IR 11 L H B10 1 L B I11 1 SS hom 0=1 (RSW hom) where the incentive constraint of motivated borrowers IC B 11 is satis ed, together with the articiation constraint of ethical banks IR11 L and where B 1 10; aearing in the standard borrowers self-selection constraint ; is the best contract that standard borrowers can o er to ethical banks. SS hom 0=1 Notice that, in the right-hand side of the self-selection constraint SS hom 0=1 ; standard borrowers mimicking motivated ones misbehave (a = 0 so that the robability of a successful investment is just L ). Indeed, when standard borrowers ick the contract designed for motivated tyes, their incentive constraint is not necessarily satis ed because they do not receive the remium for successful interaction. The RSW allocation ignoring market segmentation assures that standard borrowers refer contract (B 1 10; I 1 10 ) to ( ^B 1 11; ^I 1 11) when, by icking ( ^B 1 11; ^I 1 11); they misbehaves (that is when ( ^B 1 11; ^I 1 11) rovides the highest ayo to standard borrowers). lso note that the Weak Monotonic ro t assumtion (see Tirole 2006, age 267), requiring that contract (B 1 10; I 1 10 ) is ro table also when signed by the good (ethical) borrowers, is trivially veri ed in our setting. In Maskin and Tirole (1992) and Tirole (2006, section 6.4), borrowers can obtain at least the ayo associated with a RSW allocation equivalent to the one described above. Conversely, in our setting with heterogeneous banks, we stated in Remark 1 that borrowers can always obtain the ayo associated with (B 0 00 ): The latter contract dominates (B 1 10; I 1 10 ) for standard borrowers, because standard rojects have higher exected returns than ethical ones. Clearly, then, searating contracts that dominate the RSW allocation de ned above and that are still ro table tye-by-tye do exist. Let us now de ne the RSW allocation with market segmentation. De nition 2 The RSW allocation with market segmentation corresonds to the searating allocation (B 1 10; I 1 10 ) and (B 1 11 ; I 1 11 ) such that motivated borrowers anticiate that trading with standard banks is the best-otion of standard borrowers. Contract (B 1 11 ; I 1 11 ) solves the following rogram: max B 1 11 ;I1 11 H B H + s:t: B I11 1 IC11 B H R 1 1 I11 1 H B IR 11 L H B00 0 L B I11 1 SS het 0=1 (RSW het) where the incentive constraint of the motivated borrowers IC B 11 is satis ed, together with the articiation constraint of ethical banks IR L 11, and where B 0 00; aearing in the standard borrowers self-selection 13

15 constraint SS het 0=1 ; is the exected ayment that standard borrowers roose to standard banks in the second-best contract. gain, (B 1 10; I 1 10 ) is the best contract that standard borrowers can o er to ethical banks. Being B00 0 > B10; 1 the self-selection constraint SS het 0=1 is easier to satisfy than SS hom 0=1 : Then, contract (B 1 11 ; I 1 11 ) assures to motivated borrowers a higher ayo than ( ^B 1 11; ^I 1 11) because it is derived from a less-constrained rogram. In di erent words: the RSW allocation with market segmentation strictly dominates the RSW allocation of De nition Imortantly, borrowers anticiate here that the best available otion for standard borrowers is contract (B 0 00 ) and that the contract targeted to standard borrowers and aearing in the menu for ethical banks will never be chosen in equilibrium (recall that any contract signed by ethical banks and standard borrowers necessarily accrues lower ro ts than (B 0 00 ) to the latter). Nevertheless, notice once again that the menu o ered to ethical banks must be comlete (that is, it must contain the null contract and self-selecting contracts for the two tyes of borrowers) and feasible for the bank (that is, the bank must earn nonnegative exected ro ts from the menu), otherwise the bank will not accet the menu in the second stage. Indeed, we could substitute (B 1 10; I 1 10 ) with any other contract that signed by standard or by motivated borrowers assures non-negative ro ts to ethical banks. We are now able to describe the equilibrium with market segmentation assuring the lowest ayo to the borrowers. Suose that, in the rst stage, borrowers o er the menu (B 1 10; I 1 10 ); (B 1 11 ; I 1 11 ); (0; 0) to ethical banks and the menu (B 0 00 ); (0; 0) to standard lenders. In the second stage, each tye of lender will accet the o er because the menus contain contracts that are feasible (tye-by-tye) and thus assure non-negative ro ts to the targeted bank, no matter the bank s beliefs about borrowers tyes. In the third stage, borrowers will ick their referred contract from each menu: standard borrowers will chose (B 0 00 ); while motivated ones will choose (B 1 11 ; I 1 11 ). Hence, we derived the lower bound for the ayo that borrowers can reach in equilibrium in our setting with heterogeneous lenders (see roosition 5 of Maskin and Tirole 1992). To sum u: Lemma 1 (i) The RSW allocation which takes into account market segmentation strictly dominates the RSW allocation that ignores market segmentation. (ii) The equilibrium outcome is such that standard borrowers sign the second-best contract (B 0 00 ) with standard banks, whereas motivated borrowers sign with ethical banks the third-best contract (B 1 11 ; I 1 11 ): Such equilibrium outcome guarantees the lowest ayo that both borrowers can reach in the credit market with standard and ethical banks. gain, notice that the menu o ered to ethical banks in equilibrium is not unique because of the latent contract (B 1 10; I 1 10 ): s mentioned before, the latter can be substituted with any other contract that signed by standard or by motivated borrowers assures non-negative ro ts to ethical banks. 16 In the end of endix 6.3, we also show that the RSW allocation in De nition 2 is areto suerior to the allocation derived from a rogram similar to RSW het but where the standard borrowers incentive constraint is satis ed so that standard borrowers behave in SS0=1 het. 14

16 In the following lemma we characterize contract (B 1 11 ; I 1 11 ) belonging to the equilibrium allocation with market segmentation: Lemma 2 The third-best contract (B 1 11 ; I 1 11 ) is such that B 1 11 > B 0 00 > B 1 11 and I 1 11 > I 1 11 > I 0 00 : roof. See the endix 6.3. Hence, when, the good borrowers must ay the cost of searating from the bad borrowers and sign a contract which is worse than the second-best one because it entails a lower exected ayment and a lower investment. We will comment on the roerties of the equilibrium outcome after roosition 3 in the following Section. 4.2 Uniqueness and e ciency From Corollary to roosition 3 of Maskin and Tirole (1992), we know that the RSW allocation (with homogeneous agents) is interim e cient for a non-emty set of beliefs. From the Corollary to Theorem 1 we also know that, if the RSW allocation is interim e cient, then it is the unique equilibrium. If it is not, then a multilicity of equilibria exists in the form of a continuum of areto suerior searating allocations with cross-subsidies. Imortantly, as we will exlain, cross-subsidies between di erent menus are not feasible in our setting so that we are able to show some interesting new result about uniqueness. We rst rove that the allocation with market segmentation given by (B 1 11 ; I 1 11 ) and (B 0 00 ) is the unique equilibrium outcome of the contract roosal game. Then we derive conditions assuring that, given the banks riors (q; 1 the equilibrium allocation with market segmentation is also interim e cient. In our setting we must distinguish between interim e ciency when considering cross-subsidies between contracts o ered to ethical and to standard banks and when considering cross-subsidies between contracts o ered to ethical bank only. We thus de ne: De nition 3 Interim E ciency. (i) The equilibrium allocation with market segmentation is Unconstrained Interim E cient if no searating contracts ro table in exectation exist that are referred by both borrowers tyes when cross subsidies between ethical and standard banks are considered. The equilibrium allocation with market segmentation is Constrained Interim E cient if no searating contracts ro table in exectation exist that are referred by both borrowers tyes when only cross subsidies between contracts nanced by ethical banks are considered. Notice that Unconstrained Interim E ciency is a more demanding criterion than Constrained Interim E ciency. Intuitively, a larger set of allocations that break-even in exectation is available when crosssubsidies between ethical and standard banks are ossible than when only cross-subsidies inside ethical banks are admitted Uniqueness follows from the observation below. n equilibrium allocation must be incentive comatible and, from Corollary 1 (art (ii)), must weakly areto-dominate the equilibrium allocation given by (B11 1; I1 11 ) and (B0 00 ; I0 00 ). q), (ii) 15

17 Because, in the second stage, banks only accet menus that guarantee nonnegative exected ro ts, we state that: Remark 3 In third-best, at the equilibrium, only cross-subsidies between contracts o ered in the menu designed for ethical banks are ossible. In fact, in the case where cross-subsidies among banks were imlemented, the menu designed for standard banks would contain a ooling contract that leads to negative ro ts to be covered with ositive ro ts earned by ethical banks in the other menu. But the menu for standard lenders would not be acceted in the second stage of the Contract roosal Game. We conclude that each menu must be feasible (in isolation) for the banks. Below we show that, in the menu o ered to ethical banks, feasible contracts with cross-subsidies that areto-dominate the allocation with market segmentation do not exist. roosition 2 Uniqueness. (i) The equilibrium allocation with market segmentation is always Constrained Interim E cient. (ii) The equilibrium allocation with market segmentation is unique. roof. (i) See the endix 6.4. (ii) It directly follows from (i): In order to verify whether areto dominating contracts with cross-subsidies nanced by ethical banks exist (oint (i)), we must consider two rograms with ethical bank as the lender. In the rst rogram (see rogram 15 in endix 2) we derive the exected ro ts of the ethical bank when it signs a contract with the standard borrower which entails the second-best exected ro ts, H B 0 00; lus a transfer T (the latter assures that the contract is areto imroving for standard borrowers). From such a contract the ethical bank earns negative ro ts which must be nanced by a ro table contract (B 1 11; I 1 11 ) that motivated borrowers refer to (B 1 11 ; I 1 11 ): Thus, in the second ste, we verify whether the areto imroving new contract (B 1 11; I 1 11 ) is ro table enough to cover the loss on bad borrowers (see rogram 17 in endix 2). Intuitively areto-imroving contracts do not exist because, given the lower exected return of ethical rojects, meeting exected ro ts H B 0 00 is too costly for the ethical bank. The equilibrium allocation with market segmentation is characterized in the roosition below which incororates results from Remark 2 and Lemma 2. roosition 3 Third-best. Considering the threshold values > 0; when, standard borrowers sign the second-best contract (B 0 00 ) with standard lenders. Motivated borrowers sign with ethical banks the third-best contract (B 1 11 ; I 1 11 ) which is characterized by lower revenue and investment than their second-best contract, but higher investment than the second-best contract of standard borrowers. The credit market is fully segmented. However, it cannot strictly areto-dominate such allocation if the latter is interim e cient and so it must yield the same ayo s to the borrowers in this case. 16

18 When, standard borrowers sign the second-best contract (B 0 00 ) with standard lenders. Motivated borrowers sign the second-best contract (B 1 11; I 1 11 ) with ethical banks. The credit market is fully segmented. When < ; then both borrowers tyes sign the second-best contract (B 0 00 ) with standard banks. Ethical banks are not active and the market for ethical rojects does not exist. When <, neither the market for ethical rojects nor ethical banks exist because all borrowers invest in standard rojects. Whereas, for, ethical banks are active and the market is fully segmented as in the second-best. More seci cally, when < ; the second-best contracts are envy free. Instead, when, standard borrowers are willing to mimic motivated ones to receive better loan conditions and the menu that borrowers o er to ethical banks must satisfy the selfselecting constraint of standard borrowers. s a result, motivated borrowers ay the cost of searation and sign with ethical banks a contract entailing a higher investment I00 0 < I11 1 but a lower exected revenue (B 0 00 > B 1 11 ) than the ones characterizing contracts signed by standard borrowers and standard lenders. 18 This roves that, in third-best, the bene t arising from trading between social resonsible agents is artially o set by the information rent aroriated by standard borrowers. However, the cost of searation would be higher if ethical banks were the unique tye of lenders in the credit market (see Lemma 1) and no market segmentation was ossible. In order to verify whether the equilibrium allocation with market segmentation is also Unconstrained Interim E cient, we consider now cross-subsidies between ethical and standard banks. roosition 4 Unconstrained Interim E ciency. su cient condition such that the equilibrium with market segmentation is Unconstrained Interim E ciency is q < 1 2. roof. See endix 6.5. The roof of roosition 4 is built as follows. In order to verify whether areto dominating contracts with cross-subsidies between ethical and standard banks exist, we roceed again in two stes. First, we characterize the ro t maximizing contract for a standard bank when the latter sign with the standard borrower a contract entailing the second-best exected ro ts, H B 0 00; lus a transfer T to the borrower. Such a transfer now must be aid by ethical banks to standard ones. Cross subsidization between di erent lenders is ossible if the ethical bank earns ositive ro ts on an alternative contract (B 1 11 ; I 1 11 ) (solving rogram 20 in endix 6.5) that motivated borrowers refer to (B 1 11 ; I 1 11 ): Thus, in the second ste, we must verify whether the areto imroving new contract (B 1 11; I 1 11 ) is ro table enough to cover the transfer T to be aid to standard lenders. In aendix 6.5 we show that, for q < 1=2; this is imossible and hence no areto imroving contract with cross-subsidy between di erent lenders exists. The intuition for this result is straightforward: cross-subsidized contracts are feasible if T (1 q) + qe ; where 18 Notice that the same relationshi between equilibrium contracts exists in the second-best for (see roosition 1). 17

19 E 1 11 is ethical banks exected ro ts from contract (B 1 11; I 1 11 ): Obviously, a low ercentage of motivated borrowers in the oulation (or a low value of q) is not comatible with the feasibility of cross-subsidized contracts. Since the condition is su cient but not necessary, larger values of q are still comatible with Unconstrained Interim E ciency. However, when q becomes su ciently larger than 1=2; then the equilibrium allocation with market segmentation is not interim e cient anymore and areto-dominating allocations with cross-subsidies from ethical to standard banks exist. 19 s exlained before, a decentralized economy is not able to imlement such areto-imroving allocations because they imly cross-subsidies between di erent menus, i.e. they require menus that are not feasible in isolation and thus are not chosen by lenders in the second stage of the Contract roosal Game. However, in rincile the government could intervene to restore e ciency by imosing taxes to ethical banks to be used to artially nance commercial banks. Nevertheless, the olitical suort for such a olicy would reasonably be extremely low! In fact, ethical banks are sacri cing ro ts in the social interest by nancing rojects which rovide ositive externalities to eole and communities whereas commercial banks are maximizing their ro ts investing in rojects generating high exected returns and no ositive externalities. 5 Conclusion In our model two di erent credit markets exist: the market for standard rojects and the market for ethical ones. We de ne ethical rojects those rojects with both social and economic ro tability but with a lower exected revenue with resect to standard ones. We model ethical banks as lenders which are able to commit to nancing only ethical rojects so that they are not interested in oerating in the markets for standard rojects. Motivated borrowers obtain a non-monetary bene t (a remium for social resonsibility) when they undertake ethical rojects and also an additional bene t from trading with ethical banks in the case their roject is successful. This imlies that motivated borrowers refer to trade with ethical banks as long as the contract conditions are not too unfavorable with resect to those obtained with standard lenders. We investigate how ethical banks and motivated borrowers interact together when credit markets are cometitive and also standard banks and standard borrowers are active. When moral hazard is the unique market failure, we showed in a revious aer that the matching of ethical lenders with motivated borrowers reduces the frictions caused by moral-hazard and makes motivated borrowers better o : not only they receive the remium for cororate social resonsibility but they may also obtain better contract conditions, in terms of larger loans and higher exected returns, than standard borrowers. However, when no credible commitment to borrowers social resonsibility exists and motivated entrereneurs receive a better rosect, then standard entrereneurs have interest in mimicking social 19 Those allocations entail a ayo for motivated borrowers that is in between the minimum ayo obtained with contract (B11 1; I1 11 ) and the maximum one imlied by (B1 11 ; I1 11 ). 18

20 resonsibility by investing in ethical rojects in order to obtain the dominating contract. This is the setting we study in the resent aer: we solve the Contract roosal Game where informed borrowers o er (ossibly searating) contracts to ethical and standard banks. We contribute to the literature on Informed rincials because, in our setting, agents are heterogeneous and, as a consequence, rincials o er multile menus. We show that (i) in the Contract roosal Game with heterogeneous lenders, the equilibrium allocation entails full market segmentation and is characterized by two contracts icked from di erent menus. (ii) The menu o ered to ethical banks contains the RSW allocation with market segmentation in which motivated borrowers ay the cost of searation from standard entrereneurs and lose their higher exected returns. (iii) The coexistence of standard and ethical banks in the credit market imroves e ciency because the RSW allocation with market segmentation dominates the one with homogeneous lenders. This means that the e ciency-loss caused by adverse selection would be higher in a credit market with homogeneous banks. (iv) The equilibrium allocation with market segmentation is always unique. (v) If the share of motivated borrowers in the oulation of entrereneurs is less than 50%, then the equilibrium allocation with market segmentation is (Unconstrained) Interim E cient. In a nutshell, our model shows that market segmentation imroves e ciency and solves the roblem of multilicity of equilibria in Contract roosal Games. 6 endix 6.1 Second-best contracts The net resent value of both rojects (ethical and non-ethical) is ositive if the borrower behaves and negative otherwise. Hence, the investment cannot be imlemented, in either standard or ethical rojects, if it is not ossible to address the moral hazard roblem: H R > 1 (5) L R + < 1: (6) Moreover, the exected ro t of both standard and socially resonsible lenders must be non-negative. The two lenders articiation constraints IR0j L and IR L 1j, thus corresond to: H L ij I ij or: H RI ij I ij + H B ij (7) s illustrated in Barigozzi and Tedeschi (2015), the roblem of a borrower contracting a loan for a 19

21 standard roject with a standard lender is: max B 0 00 ;I0 00 H B00 0 s:t: B00 0 I0 00 IC00 B0 H R 0 1 I00 0 H B IR00 L0 (8) where IC00 B0 is the incentive comatibility constraint for a standard borrower trading with a standard bank for a standard roject and solution is contract (B 0 00 ) such that: IR L0 00 is the articiation constraint of the standard lender. The I 0 00 = 1 H R 0 B 0 00 = 1 H R 0 = I 0 00 (9) The rogram of a motivated borrower contracting for an ethical roject with an ethical bank is very similar to (8); in articular, R 1 must relace R 0 and the incentive constraint IC B1 11 ; illustrated in Section 3, must relace IC B0 00. The solution is contract (B 1 11; I 1 11 ); such that: I 1 11 = + H 1 H R 1 B11 1 = + H = I 1 H R (10) The three threshold values aearing in roosition 1 are: max 0; 1 2 H H R ooling contract R 0 R 1 I 0 00 I 0 00 H R 0 R 1 H R 1 1 = HB 0 00 H R 1 1 R0 R 1 (11) The ooling contract that motivated borrowers roose to ethical banks, if they are not willing to ay the cost of searation from standard borrowers, can be de ned as follows: De nition 4 Suose that motivated borrowers roose a ooling contract The ooling contract solves: B 1 1j ; I1 1j to ethical banks. max B 1 1j ;I1 1j H B 1 1j + H + s:t: B1j 1 I1j 1 IC10 B H R S1 1 I1j 1 H B1j IRL1 1j (ooling Contract) or it satis es the standard borrowers incentive constraint and the articiation constraint of ethical banks. 20

22 Remember that, when the incentive constraint of standard borrowers holds, a fortiori the incentive constraint of motivated borrowers is satis ed. Moreover, it can be easily shown (see Barigozzi and Tedeschi 2015) that the sum of the two remia for social resonsibility H + aearing in the objective function of (ooling Contract), but not in the incentive comatibility constraint IC B 10, does not a ect the otimal contract. Thus, contract (B 1 10; I 1 10 ) (considered in De nition 1) and B 1 1j ; I1 1j are equivalent. Standard borrowers will never choose the ooling contract B 1 1j ; I1 1j because they strictly refer the second-best contract (B 0 00 ). Moreover, B 1 1j ; I1 1j does not make use of the relaxed incentive constraint (IC B 11) that allows to reduce agency costs for motivated borrowers (who would be the only ones to ossibly choose contract B 1 1j ; I1 1j ). Thus, the searating allocations analyzed in the main text strictly dominate the ooling contract solving rogram ooling Contract. 6.3 roof of Lemma 2 In order to characterize contract B 1 11 ; I 1 11 in the RSW allocation with market segmentation, we have to solve the following roblem of a reresentative motivated borrower: max B 1 11 ;I1 11 H B H + s:t: B I11 1 IC11 B H R 1 1 I11 1 H B IR 11 L H B00 0 L B I11 1 SS het 0=1 (12) Notice that SS0=1 het must be binding, otherwise the second-best rogram, which is not feasible by assumtion, would be reached; in fact, for ; the standard borrower refers the motivated borrower s contract. Hence H B 0 00 = L B I 1 11 The three constraints in rogram 12 can be rewritten as: I11 1 B IC11 B I11 1 HB11 1 I11 1 H B0 00 H R 1 1 L B1 11 IR L 11 SS het 0=1 In the sace B11; 1 I11 1 the boundary of the sets are straight lines. That of SS het 0=1 is negatively sloed while those of the other two are ositively sloed. In articular, the lines de ned by IC11 B and IR11 L have ositive and a negative intercet, resectively. We are going to rove that the three linear constraints are comatible with each other and, in the set of feasible allocations, we will nd out which one maximizes the motivated borrower s ayo. Suose that IC B 11 is binding and hence holds with equality. (which is binding) we obtain: H B 0 00 = L I I 1 11 = L + 1 I (13) Then substituting IC B 11 into SS het 0=1 L = H I 1 11 L :

23 Using the revious equation to derive I 1 11 and substituting the exression for B 0 00, we obtain an exlicit exression for I 1 11 : I11 1 = B H R 0 + L = H L H We substitute the exression for I 1 11 back into IC B 11 to obtain the motivated borrower s income in the oint where IC11 B and SS0=1 het cross each other: 0 B H R 0 = 1 H R 0 H L 1 H We now must check whether the articiation constraint of the lender is satis ed for the values of B 1 11 and I 1 11 just derived. By substituting those values into IR L 11 we obtain: H R 1 1 I 1 11 H B = R 0 R 1 H 1 + H H R + HR 1 1 L 0 + H which must be non-negative. Thus, IR L 11 is satis ed if: H R 1 1 L R 0 R 1 + H H 1 + (14) H H R 0 Recall that we are considering the case where : From (11) we know that: = H R 0 R 1 I 0 00 H R 1 = H R 0 R 1 1 H R H R 0 Because the l.h.s of (14) is increasing in, a su cient condition for IR L 11 to be satis ed is that inequality (14) holds when substituting in the lace of : H R 1 1 L H R 0 R 1 H H R 1 1 H R0 R 1 H R H R 0 1 H R 0 H R 0 R H H R 0 which boils down into H H R 1 1 H L = 22

24 or H H R 1 1 = 1 + H H R 1 0 which is certainly true, being the revious exression the denominator of (10). Hence, the constraint IR L 11 is satis ed imlying that the two constraints, IR L 11 and IC B 11 are comatible with each other. That is, IR L 11 (taken with equality) crosses SS het 0=1 at a lower investment level, I1 11, and (more imortantly) at a bigger borrower s revenue, B 1 11, with resect to IC B 11 (again taken with equality). This means that the solution where IR L 11 and SS het 0=1 cross each other is characterized by the highest B1 11, in the intersection of all constraints. s can be checked in the following gure, the feasible allocations are inside the three straight lines reresenting the three constraints and the motivated borrower s exected revenue is maximized in the intersection of IR L 11 and SS B 0. Insert Figure 1 here The oint where IR11 L crosses SS0=1 het is characterized by the system HR 1 1 H L 5 4 I1 11 B = 4 H B with solutions: I 1 11 = B 1 11 = 2 H B0 00 L L ( H R 1 1) + H H R 1 1 H B L ( H R 1 1) + H Substituting the value of B 0 00 we obtain: 11 = L H R H H R 0 R 1 L ( H R 1 1) + H B 1 I 1 11 = L H R H + L H R 0 R 1 L ( H R 1 1) + H 1 H R 0 1 H R 0 Note that B 1 11 and I 1 11 do not deend on and, by comarison with exressions in endix 6.1, they are such that B 1 11 < B 0 00 and I 1 11 > I 0 00 : Moreover, for and B 1 11 > B 0 00, we have that B 1 11 > B 0 00 > B 1 11 : Finally, we showed before that the third-best contract is at the intersection between SS0=1 het and IRL 11. The second-best contract is instead at the intersection between IC11 B and, again, IR11. L Since IR11 L and SS0=1 het are ositively and negatively sloed (see Figure 1), resectively, it must be true that the level of investment in the third best is lower than in the second best, I11 1 < I11 1. We roved that, at the solution of rogram 12, the incentive constraint IC B 11 is slack. One may then wonder whether the solution to a rogram where the incentive constraint of the standard (instead of the motivated) borrower is satis ed and where, in the self-selection constraint SS 0=1 ; the mimicker behaves 23

25 (instead of misbehaving), is areto-suerior to the allocation we just derived. To see that, let us consider the following rogram: max B 1 11 ;I1 11 H B H + s:t: B11 1 I11 1 IC11 B H R 1 1 I11 1 H B IR11 L H B00 0 H B11 1 SS 0=1 The answer is not. Indeed, the revious rogram is mathematically equivalent to the ooling Contract analyzed in endix 6.2, which delivers the areto-dominated ooling solution B 1 1j ; I1 1j : 6.4 roof of roosition 2 The equilibrium allocation with market segmentation is Constrained Interim E cient if a air of areto dominating contracts nanced by ethical banks and feasible in exectation does not exist. First ste. We will derive the exected ro ts of an ethical bank as a function of T; where T is the standard borrowers rent above their exected utility H B 0 00; in a candidate equilibrium with cross subsidy: C 1 10 indicates that the new contract max H R 1 1 I 1 B10 1 (T );I1 10 (T ) 10 (T ) H B10 1 (T ) + st: B 1 10 (T ) I 1 10 (T ) (IC 1 10) H B 1 10 (T ) H 1 H R 0 the exected ayo obtained with (B00; 0 I00 0 ); where B00 0 = + T ( C 1 10) (15) B 1 10 (T ) ; I 1 10(T ) imlies, for standard borrowers, a gain of T over 1 H R 0 The articiation constraint C 1 10 of the standard borrower must be binding. If not, the standard lender could decrease B 1 10 (T ) and increase ro ts. By substituting C 1 10 taken with equality into the objective function, the rogram becomes: : max H R 1 1 I 1 10 (T ) + 1 H R 0 1 H R 0 T st: B 1 10 (T ) I 1 10 (T ) (IC 1 10) The objective function is thus increasing in I Hence, also the incentive comatibility constraint IC 1 10 has to be binding. s a consequence, we can study rogram (15) with both constraints binding. From the C 1 10: and substituting B 1 10 (T ) into IC 1 10 we nd: B10 1 (T ) = T + 1 H R 0 I 1 10 (T ) = B1 10 (T ) = H + 1 H R 0 H T 24

26 Now we can relace the exressions for B 1 10 (T ) and I 1 10 (T ) into the objective function of the ethical bank to derive its (maximized) exected ayo as a function of the transfer T : L1 (T ) = T 1 H R 1 H R 0 R 1 (16) H 1 H R 0 Thus, L1 (T ) < 0 8T and L1 (T ) = H (R 0 R 1) 1 H R 0 < 0 if T = 0. ro ts L1 (T ) are decreasing in T: In words, ethical banks always earn negative ro ts (also if T = 0) when o ering to standard borrowers an exected ayo H B 1 10 (T ) H B T: In fact R 0 > R 1 and the best contract that ethical banks can o er to standard borrowers is always dominated (i.e. B 1 10 < B 0 00 holds), indeendently from the magnitude of the transfer T: This imlies that ethical banks need very high ro ts from motivated borrowers to cover losses from contract B 1 10 (T ) ; I 1 10(T ) : Second ste. The transfer T can be o ered to standard borrowers only if ethical banks earn ositive ro ts on motivated borrowers. We have to solve a rogram where motivated borrowers o er a contract to ethical banks subject to the incentive comatibility constraint of motivated borrowers IC B 11, the selfselection constraint SS T 0=1 and the articiation constraint for ethical banks f IR L 11. max H B H + B11 1 ;I1 11 ;T s:t: B I1 11 IC B 11 T H 1 H R 1 H R 1 1 I11 1 H B q + H(R 0 R 1 ) 1 H R 0 H B T L B I 1 11 (1 q) 0 L fir 11 SS0=1 T (17) In f IR L 11, ethical banks must earn ositive ro ts from the fraction q of motivated borrowers in order to cover the exenditure of roviding T to the 1 ro ts L1 (T ) just derived before in (16) has been substituted in f IR L 11. q standard borrowers. The exression for the exected Let us focus on the three constraints of the revious rogram when T = 0 : B I1 11 ( H R 1 1) I 1 11 H B q H B 0 00 L B I 1 11 IC11 B H(R 0 R 1 ) L (1 q) 0 fir 1 H R 0 11 SS0=1 het The system taken with equality is: I 1 11 = I 1 11 = B H(R 0 R 1 ) q( H R 1 1) 1 H R 0 1 H R 0 I 1 11 = H IC11 B L fir H R H H R 1 1 B1 11 L B SS het 0=1 (18) By comaring system (18) with system (13) analyzed in the roof of Lemma (2), we observe that the only di erence is in f IR L 11, which now has the same sloe but a higher intercet. In articular the intercet 25

27 is now larger of the term H(R 0 R 1 ) q( H R 1 1) 1 H R 0 > 0: This imlies that the line deicted by f IR L 11 shifts on the left with resect to the one deicted by IR L 11 and illustrated in Figure 1. We need now to understand whether f IR L 11 crosses IC B 11 on the left or on the right of the intercet between IC11 B and SS0=1 het : To do so let us rst calculate BSS IC ; that is B1 11 in the intercet between IC11 B and SS het that is: 0=1 : B = H H BIC SS = 1 H R 0 1 H R 0 H 1 H R 0 B11 1 L IR Let us now calculate B f IC ; that is B11 1 in the intercet between IC11 B and IR f L 11: B H R 0 R 1 q 1 H R 1 B = + 11 q ( H R 1 1) 1 1 H H R 0 H R 1 1 that is: IR B f IC = H R H H R H R 1 q H R 0 R 1 1 q ( H R 1 1) 1 H R 0 We can easily check that B SS IC between IC11 B and SS0=1 het : In fact: B SS IC IR B f 1 H R 0 IC = > B f IR IC The revious inequality takes into account that because imlying that f IR L 11 crosses IC B 11 on the left of the intercet H R H 1 H R 1 q + 2 H R0 R 1 > 0 q H 1 H R 0 1 H R 1 H R H 1 H R 1 > 0 H R H 1 H R 1 = ( H L ) H R H 1 H R 1 = which comes from the ine ciency of misbehaving. L H + L R 1 > L L + L R 1 = L 1 L R 1 > 0 ll this roves that, when T = 0; the constraints of system 18 can be deicted as in Figure 2 below. In articular, IC11 B and IR f L 11 cross each other before reaching SS0=1 het. Then the highest ayo for motivated 26

28 borrowers is obtained at the intersection between IC B 11 and f IR L 11; where B 1 11 is at the highest value comatible with the three constraints. Insert Figure 2 here Notice that only f IR L 11 deends on T: In articular, for T > 0; f IR L 11 can be re-written as: I 1 11 HB 1 11 H R T H H R 1 H R q q + 1 H R 0 H(R 0 R 1 ) ( H R 1 1) Starting from T = 0 and by increasing T, the intercet rises but the sloe is unchanged. Hence, by increasing T, f IR L 11 shifts even more on the left. s a consequence, the otimal B 1 11 moves left along IC B 11 and therefore decreases. 1 q q 6.5 roof of roosition 4 The equilibrium allocation with market segmentation is Unconstrained Interim E cient if a air of areto dominating contracts with cross-subsidies between ethical and standard banks does not exist. First ste. We will derive the exected ro ts of a standard bank as a function of T; where T is the standard borrowers rent above their exected utility H B 0 00; in an allocation with cross subsidy: max H R 0 1 I 0 B00 0 (T );I0 00 (T ) 00 (T ) H B00 0 (T ) + st: B 0 00 (T ) I 0 00 (T ) (IC 0 00) H B 0 00 (T ) H 1 H R 0 + T ( C 0 00) (19) The right hand side of C 0 00 indicates that the new contract B 0 00 (T ) ; I 0 00(T ) imlies, for standard borrowers, a gain of T over the exected ayo obtained with (B00; 0 I00 0 ); where B00 0 = 1 H R 0 The articiation constraint C 0 00 of the standard borrower must be binding. If not, the standard lender could decrease B 0 00 (T ) and increase ro ts. By substituting C 0 00 taken with equality into the objective function, the rogram becomes: : max H R 0 1 I 0 00 (T ) + 1 H R 0 1 H R 0 T st: B 0 00 (T ) I 0 00 (T ) (IC 0 00) The objective function is thus increasing in I Hence, also the incentive comatibility constraint IC 0 00 has to be binding. s a consequence, we can study rogram (19) with both constraints binding. From the C 0 00: and substituting B 0 00 (T ) into IC 0 00 we nd: B00 0 (T ) = T + 1 H R 0 I 0 00 (T ) = B0 00 (T ) = H + 1 H R 0 H T 27

29 Now we can relace the exressions for B 0 00 (T ) and I 0 00 (T ) into the objective function of the lender to derive its (maximized) exected ayo as a function of the transfer T : 0 1 L0 (T ) = H R 0 T H R 0 H 0 1 T H R 0 H T = 1 H R 0 H Thus, L0 (T ) < 0 if T > 0 and L0 (T ) = 0 if T = 0. s exected, the standard bank earns negative ro ts when o ering the transfer T to the standard borrower. Second ste. The transfer T can be o ered to standard borrowers if ethical banks ay for it. In turn, ethical banks can a ord to ay for T only if they earn ositive ro ts on motivated borrowers. In order to verify whether areto imroving contracts with cross-subsidy between banks are feasible, we have to solve the following rogram where motivated borrowers o er a contract to ethical banks subject to the incentive comatibility constraint IC11 B and the self-selection constraint SS0=1 T : Moreover, now the transfer T must enter the articiation constraint for ethical banks: max H B H + B11 1 ;I1 11 ;T s:t: B I1 11 IC B 11 H R 1 1 I11 1 H B q T H 1 H R 0 (1 q) 0 H B T L B I 1 11 IR11 L SS0=1 T (20) notice that, in IR L 11, ethical banks must earn ositive ro ts from the fraction q of motivated borrowers in order to cover the exenditure of roviding T to the 1 exected ro ts L0 (T ) just derived before has been substituted in IR L 11. q standard borrowers. The exression for the In the main text of Subsection 4.2, the solution to rogram 20 has been called (B 1 11 ; I 1 11 ). We know that the solution when T = 0 (derived in Lemma 2) lies in the intersection between IR 11 L and SS0=1 het SST 0=1. We want to check conditions such that a marginal transfer T; aid to standard borrowers, makes the exected ro ts of motivated borrowers decrease, or such that the derivative is negative. However, since the transfer T enters both constraints IR11 L and SS0=1 T in a linear T =0 T =0 db 1 11 dt way, the derivative db1 11 dt does not deend on the magnitude of T as long as the solution remains in the intersection between R11 L and SS0=1 T : 28

30 By totally di erentiating IR11 L and SS0=1 T we obtain: 2 4 H R H 5 d 4 I1 11 L B H 1 H R 0 (1 q) q 1 Hence: Therefore db1 11 dt db 1 11 dt < 0 if and only if = H(HR 1 1)q ( H L ) 1 H R 0 H ( H +( H R 1 1) L )q q < H ( H R 1 1) ( H L ) 1 H R 0 ( H L ) = 3 5 dt ( H L ) + 1 = q (1 q) and since one can check that ( H L ) 1 H R 0 > H H R 1 1 ; ( H L ) it must be q > 1 2. Thus, q < 1 2 is a su cient condition for db1 11 dt < 0: We can conclude that, when the solution is in the intersection between R11 L and SS0=1 T and q < 1 2 ; the searating allocation with cross-subsidy is not welfare imroving because motivated borrowers are worse o. It remains to see what haens when T increases so much that the solution is no more in the intersection between R L 11 and SS0=1 T : The three constraints aearing in rogram 20 can be rewritten as: I11 1 B IC11 B I11 1 HB 1 11 H R T H R 0 (1 q) H H R 1 1 q IR 11 L I11 1 H B00 0 L B T SS0=1 T s T increases, the line of the SS T 0=1 constraint in Figure 1 moves u, while the line of IRL 11 moves left. Instead, IC11 B does not move. Hence, for T su ciently big, SS0=1 T becomes irrelevant and the otimal contract lies on the intersection between IC11 B and IR11. L However, for a continuity argument, it would still be true that db1 11 dt < 0. In fact IC11 B is unchanged, IR11 L moves left, and we showed before that < 0 for smaller values of T: db 1 11 dt We can conclude that, by increasing the transfer T; the exected ayo of motivated borrowers always decreases and, thus, no ro table air of contracts with cross-subsidies exists. References [1] llen, F. and G.R. Faulhaber (1989), Signalling by underricing in the IO market, Journal of Financial Economics, 23(2)

31 [2] Barigozzi F. and. Tedeschi (2015), Credit Markets with Ethical Banks and Motivated Borrowers, Review of Finance 19(3), [3] Becchetti L. et al. (2011a), "Credit Rationing and Credit View: Emirical Evidence from an Ethicak Bank in Italy", Journal of Money, Credit and Banking 43(6), [4] Becchetti L. et al. (2011b), "Informal collateral and default risk: do Grameen-like banks work in high-income countries?", lied Financial Economics 21, [5] Bénabou R. and J. Tirole (2010), Individual and Cororate Social Resonsibility, Economica 77, [6] Besanko, D. and. Thakor (1987), Collateral and Rationing: Sorting Equilibria in Monoolistic and Cometitive Credit Markets, International Economic Review 28, [7] Bester H. (1985), Screening vs. Rationing in Credit Markets with Imerfect Information, The merican Economic Review 75(4), [8] Bester H. (1987), The role of collateral in credit markets with imerfect information, Euroean Economic Review 31(4), [9] Bhattacharya, S. (1979), Imerfect Information, Dividend olicy, and "The Bird in the Hand" Fallacy, The Bell Journal of Economics 10(1), [10] Bouvard M. (2014), Real otion nancing under asymmetric information, The Review of Financial Studies doi: /rfs/hhs068 [11] Cestone G., J. Lerner and L. White (2014), The Design of Syndicates in Venture Caital, mimeo, Cass Business School. [12] Chan Y-S. and G. Kanatas (1985), symmetric Valuations and the Role of Collateral in Loan greements, Journal of Money, Credit and Banking 17(1), [13] Cornée S. and. Szafarz (2014), Vive la Di érence: Social Banks and Recirocity in the Credit Market, Journal of Business Ethics, 125(3), [14] Diamond, D. (1991), Debt maturity structure and liquidity risk, The Quarterly Journal of Economics 106(3), [15] Green C.F. (1989) Business Ethics in Banking, Journal of Business Ethics, 8(8), [16] Hennessy C.., D. Livdan, and Bruno Miranda (2010), Reeated Signaling and Firm Dynamics, The Review of Financial Studies 23(5), [17] Kitzmueller M. and J. Shimshack (2012), Economic ersectives on Cororate Social Resonsibility, Journal of Economic Literature, 50(1), [18] Leland, H. and yle, H. (1977), Informational symmetries, Financial Structure, and Financial Intermediation, Journal of Finance, 32(2), [19] Lynch J.J. (1991), Ethical Bank: Surviving in an ge of Default, London McMillan. 30

32 [20] Maskin E. and J. Tirole (1992), The rincial-gent Relationshi with an Informed rincial, II: Common Values, Econometrica, vol. 60(1), [21] Myers, S.C. and N.S. Majluf (1984), Cororate nancing and investment decisions when rms have information that investors do not have, Journal of Financial Economics 13(2), [22] Rothschild, M. and J. Stiglitz (1976), Equilibrium in Insurance Markets: an essay on the economics of imerfect information, Quarterly Journal of Economics 90, [23] San José L., J.L. Retolaza and J. Gutierrez (2009), re Ethical Banks Di erent? Comarative nalysis Using the Radical nity Index, Journal of Business Ethics, vol. 100(1), [24] Tirole J. (2006), The theory of Cororate Finance, rinceton and Oxford: rinceton University ress. [25] Welch, I. (1989), Seasoned O erings, Imitation Costs, and the Underricing of Initial ublic O erings, Journal of Finance, 44(2), [26] Wilson C. (1977), Model of Insurance Markets with Incomlete Information, Journal of Economic Theory 16(2),

33 otimal contract / Figure 1. The RSW allocation with market segmentation: the three constraints aearing in rogram RSWhet of Definition 2. otimal contract / Figure 2. Searating contracts with cross-subsidies: the new articiation constraint of ethical banks.

34