Loans, Insurance and Failures in the Credit Market for Students

Transcription

1 Loans, Insurance and Faiures in the Credit Market for Students Eena de Rey and Bertrand Verheyden y February 2008 Preiminary draft. Do not quote without permission. Abstract We present a mode with perfecty competitive banks and risk averse students who have private information on their abiity to earn. We show that the combination of ex-post mora hazard and adverse seection produces credit market faiure when defaut penaties are ow. Intermediate eves of defaut penaties can resut in the existence of an equiibrium that poos together abiity types. This may be e cient or entai over-investment in higher education. Finay, if defaut penaties are arge enough, a separating equiibrium emerges where di erent abiity types are di erenty insured against faiure and participation is aways e cient. The potentia roe of the government is discussed in each case. Keywords: ex-post mora hazard, adverse seection, income contingent oans JEL Cassi cation: Universitat de Girona and FEDEA, Spain y FNRS research feow at University of Namur, Begium 1

2 1 Introduction Pubic intervention in the student oan market is usuay motivated by the genera view that the market fais to provide such oans. In other words, the market is rationed. An iustration of this sentiment can be found in o cia pubications such as Eurydice (1999), which reports that private banks are reuctant to o er student oans primariy because of high defaut rates. This phenomenon is pointed out by Shen & Ziderman (2007). They show, in a vast internationa study of student oan schemes, that the goba percentage of the tota costs of oans schemes that the ending institution can expect to receive back in repayments is consideraby ower than 40 percent. In many countries, banks o er oans when they bene t from highy subsidized interest rates, or because the government guarantees them against the risk of defaut. However, oans o ered by banks are aways "pure" oans: they do not insure borrowers against the risk of a bad outcome, ike dropping out of university, or being unsuccessfu on the abor market. This ack of insurance in private oans contrasts with the new trend observed among oans provided by many governments. Indeed, the repayment of most of pubic oans is now income contingent. 1 Therefore, in addition to credit rationing, ack of insurance is a second faiure of the student oan market. In this paper, we provide a rationae for these two market faiures. As wi be discussed in the next paragraphs, the direct appication to the education nance probem of standard credit market faiure arguments, as those deveoped by Stigitz & Weiss (1981) and de Meza & Webb (1987), is not satisfactory. On the one hand, we caim that the assumptions that ie at the basis of the Stigitz & Weiss (1981) mode are inappropriate for education. On the other hand, athough its main assumption ts much better the education nance probem, the de Meza & Webb (1987) mode obtains a credit market imperfection resut that is characterized by overinvestment, which is aso at odds with the styised facts we want to expain. In the rest of this introduction, we discuss the modes of Stigitz & Weiss (1981) and de Meza & Webb (1987) and make the above mentioned points cear. Further, we present our methodoogy to sove this puzze. In the credit market iterature, theoretica contributions have shown that credit market 1 This has been the case in countries such as Austraia, New Zeaand, Sweden, Canada, the UK, Thaiand, Canada and, very recenty, Spain. See Chapman (2005) and Chapman & Greenaway (2006) for an internationa overview of ICL s. The US Department of Education aso has deveoped an "Income Contingent Repayment Pan", whereby the monthy payments are pegged to the borrower s income, famiy size and tota amount borrowed. 2

3 faiures resut primariy from the asymmetric information regarding the investor s probabiity of success. In Stigitz & Weiss (1981), projects have the same expected return but risks di er in the sense of a Mean Preserving Spread (MPS). Under this setting of MPS and borrowers imited iabiity, the riskier investor is wiing to pay a higher interest than the ess risky investor. In the case of unsatis ed demand, the standard market mechanism reies on an increase of the price to cear the market. In the Stigitz and Weiss setting however, an increase of the price of credit - the interest rate - may fai to reach this objective. Indeed, since ow risk investors wi drop out before high risk investors, the composition of risks changes, and the expected probabiity of success of an investment decreases. It may then be optima for pro t maximizing banks not to raise the interest rate and to ration credit. de Meza & Webb (1987) use a di erent setting based on the First Order Stochastic Dominance (FOSD) concept: investors di er in terms of expected returns, with the abest having the greatest probabiity of success. In contrast to Stigitz & Weiss (1981), the abest investor is more wiing to pay for a oan. Because of this, a market imperfection in de Meza & Webb (1987) impies overinvestment rather than credit rationing. Indeed, if separation of types is impossibe, bad types do not drive out good types as in Stigitz & Weiss (1981), but instead both types are pooed. If the ow abiity agent s investment is ine cient, asymmetric information thus impies that too many projects are nanced. 2 In the economics of education iterature it is usuay assumed that high abiity students face arger expected returns from investing in education. Moreover, their probabiity of success is aso arger. Of the two modes discussed above, the de Meza & Webb (1987) framework, with its First Order Stochastic Dominance concept, thus seems much more appropriate than Stigitz & Weiss (1981) to describe the conditions of the market for student oans. Strikingy, as expained earier, ony the atter can produce credit rationing. Despite its obvious aws in the context of student oans, the setting proposed by Stigitz & Weiss (1981) has been appied to expain the absence of purey private -that do not rey on any kind of pubic intervention- student oans. As pointed out by Barr (2001), both the average and the variance of the rate of return of the education investment vary across professions. The average income of doctors is high, and the variance ow. In contrast, the average income of actors is ow, and the variance, arge. It may then be considered sensibe to adopt the Mean Preserving Spread setting and concude that adverse seection arises because the student has private information about her career choice. Yet, career choices are easiy 2 See Boadway & Keen (2004) for a unifying anaysis of the two modes. 3

4 observed. Charging di erent prices for di erent dipomas is, far from "introducing savery by the back door" (in the words of Barr), a common practice in most countries. If the information about career choice cannot be hidden, adverse seection cannot take pace. Common wisdom perseveres, however. Canton & Bom (2004), in another of numerous exampes, aso argue that market faiure in the student oan market is due to adverse seection. In their words: "Students who are more ikey not to be abe to repay their oan, the "bad risks", woud be more interested in appying for a student oan than the "good risks". Banks therefore charge a risk premium on top of the risk-free interest rate. This further discourages high abiity students from appying for a bank oan, as it entais that the high abiity students cross-subsidize the ow abiity students. Adverse seection woud further drive up the risk premium, possiby turning the credit system unsustainabe over time." Recenty, Jacobs & van Wijnbergen (2007) anayze optima nancing of risky higher education, appying expicitey the Stigitz & Weiss (1981) setting. A chaenge of our anaysis is thus to reconcie the appropriate framework to anayze the student oan market with its observed styized facts. In order to do that, we expore the roe of ex-post mora hazard and its interaction with adverse seection. 3 Ex-post mora hazard may occur after the outcome of the investment is reaized, and borrowers may have incentives to defaut strategicay. Such incentives are conditiona on the design of defaut rues, for which some heterogeneity is observed at the internationa eve. For instance, La Porta et a. (1998) estabish a di erence in credit market deveopment depending on whether the country is in uenced by common aw or by French aw tradition. The di erence among the two is precisey the ega protection of investors, stronger in commonaw countries. Ionescu (2007) studies the impications of the change in the bankruptcy rue for student oans in the US, and shows that the choice of rue a ects the defaut behavior of borrowers, who may defaut for strategic reasons under some institutionna arrangements. Sami (1999) attributes high defaut eves in student oans party to "poor management of the oan recovery function". Bertoa et a. (2006) reca that "Equiibrium modes of defaut recognize that a debt coud be repaid if the punishment were su cienty arge. In reaity, punishment is even ess severe than permanent excusion from further consumption smoothing opportunities". Formay, we foow Kehoe & Levine (1993) who provide microfoundations to study the 3 The type of mora hazard we are refering to shoud not be misperceived as ad interim mora hazard, whereby investors may have incentives to reduce their e orts to succeed in their project. See, among others, Vercammen (2002) for an anaysis mixing ad interim mora hazard and adverse seection in credit markets. 4

5 endogenous debt imits that arise when individua borrowers can defaut on debt. In a simpe mode based on de Meza & Webb (1987) and de Meza & Webb (1990), we aso account for student risk aversion, due to the uncertainty of the educationa outcome. We show that the interaction of mora hazard and adverse seection proves fundamenta in expaining credit rationing in the student oan market when defaut penaties are reativey soft. Our mode con rms that a subsidy on interest rates can incite banks to o er oans despite the reative softness of defaut penaties. When defaut penaties are intermediate, we reproduce the styised fact that private student oans do not incude insurance coverage. If penaties for defaut are very arge, or, aternativey, information about the abiity of types is pubicy avaiabe, banks are abe to separate types and provide both credit and insurance when required. The rest of the paper is organized as foows. In Section 2, we present the mode. In Section 3, we show that the interaction between ex-post mora hazard and adverse seection yieds a compete market faiure, where no oans are o ered, if defaut penaties are ow. Then we present the case of intermediate penaties, where the market equiibrium is characterized by pooing contracts. Finay, we discuss the case where defaut penaties are argest, which resuts in a separating equiibrium. Section 4 discusses the roe of subsidies and section 5 concudes. 2 The mode There is a popuation of unskied agents of measure 1. At the beginning of the period, agents decide whether to invest in higher education or not. This investment is risky and has two possibe outcomes = ff; sg, where f stands for faiure and s for success. In case of success, an agent becomes skied and obtains an exogeneous wage w s. In case of faiure, she remains unskied and receives the same wage as an agent who chose not to study, w f. Agents di er in abiity a 2 f; hg, which a ects their probabiity of success: p a with 0 < p < p h < 1. Athough this probabiity is private information, the share of agents of high (h) abiity in the popuation,, is common knowedge. The investment in higher education is costy. We denote these costs, which comprise tuition fees and iving expenditures, by F. Agents need to borrow in order to nance F. If they do not accept any oan contract, they remain unskied and earn with certainty a wage w f. 5

6 The credit market consists of a set of pro t maximizing banks o ering oans of size F, competing à a Bertrand. A student oan contract is a pair of interest rates (r s ; r f ) 2 R 2, where r s and r f are the interest rates charged respectivey in case of success, s, and faiure, f. The contingency of the interest rate to the state of nature aows the oan contract to provide agents with some insurance, by setting r s > r f. Note that this is precisey what pubicy managed income contingent oan programs do. In particuar, it is often the case that r s > r f = 0. At the pure, uninsured, oan contract, r s = r f. Banks may o er more than one contract, or no contract at a. The banks strategy is thus a set, or menu of contracts. When facing the menu of contracts o ered by banks, unskied agents decide whether to accept one of them or refuse a of them. Agents are risk averse and care for consumption over their productive ife Y 2 R +. The utiity function is continuous, stricty increasing and stricty concave and is denoted U() : R +! R +. The expected utiity of an individua who invests in education and has probabiity of success p a is denoted EU(p a ; Y s ; Y f ) = p a U(Y s ) + (1 p a )U(Y f ); where Y s and Y f are consumption eves contingent respectivey on success and faiure. These consumption eves depend both on the accepted oan contract, and on the penaty the borrower endures in case of defaut. Indeed, the atter decide whether to repay the oan or to defaut by comparing the cost of oan repayment to the defaut penaty, which is determined by aw. As Chen (2005), we mode the defaut penaty as the garnishment by the bank of a share g 2 [0; 1] of the wage, w. 4 Indi erence curves of the two types of agent have negative sopes and satisfy the singe-crossing condition. 5 A particuarity of our mode is that banks are subject to ex-post mora hazard from borrowers. Indeed, the atter decide, depending on the terms of the contract and the severity of the defaut penaty, whether to defaut or pay the oan. Whie repayment of the oan or defaut penaties constitute the banks revenues, their costs entai borrowing the funds on the internationa market at the risk-ess interest rate i. 4 In the US, the garnishment rate is set at a maximum of 10%. In Spain, garnishment is not proportiona but evoves with wage, so that high income defauters have a post garnishment income arger than ow income borrowers. 5 For a (Y f ; Y s ) 2 R 2 +, dys dy f j EU(pa;Y f ;Y s)=u = 1 pa U 0 (Y f ) p a U 0 (Y, where U is constant. Since 1 p a s) p a is increasing in p a, dys dy f j EU(ph ;Y f ;Y s)=u > dys dy f j EU(p ;Y f ;Y s)=u. 6

7 The timing of the game is the foowing: 1. Nature draws the type of an unskied agent. He wi be of high abiity (h) with probabiity, otherwise her abiity is ow (). 2. Banks o er a menu of student oan contracts to the agent. 3. The agent observes the menu of contracts and decides, given her abiity, whether to accept one of the oan contracts or refuse a of them and remain unskied. If banks o er no contract, the agent remains unskied and the game ends. 4. If the agent accepts one contract, the investment in higher education materiaizes and, accounting for the agent s abiity, nature reaizes the outcome ( 2 ff; sg) of the investment. 5. The agent pays the oan or defauts, in which case banks recover the oan up to the ega imit gw. 2.1 Equiibrium The equiibrium concept is Subgame Perfect Nash Equiibrium (SPNE) in pure strategies. 6 As described in the timing of the game, a strategy pro e gathers three strategies: banks o er of the menu of contracts, agents acceptance of one of the contracts or refusa of a of them, and, nay, once the outcome is reaized and in case agents have subscribed to one contract, agents compiance with the contract or defaut. To be an SPNE, a strategy pro e must be such that 1. At stage 5, borrowers maximize utiity by defauting if R F > gw. 2. At stage 3, an unskied agent accepts the contract that provides her with the highest eve of utiity, provided the atter is higher than the one obtained by remaining unskied. Otherwise, she refuses a contracts. 6 Even though there are two types and information is asymmetric, the equiibrium concept does not need to rey on Bayesian expectations. Indeed, the uninformed payers - banks - do not need to formuate beiefs about which type wi take a contract. Because they pay rst, the contracts they design aow them to anticipate with certainty what type(s) they are going to face for each contract o ered. For further discussion of this issue, see Mas-Coe (1995) Chapter 13. 7

8 3. At stage 2, banks o er a menu of student oan contracts that maximize expected pro ts. Because of Bertrand competition, the highest vaue for expected pro ts is zero, so that at equiibrium, every contract (r f ; r s ) in the menu must be such that E(q; r f ; r s ) = q min(r s F; gw s ) + (1 q) min(r f F; gw f ) IF = 0; where q 2 [0; 1] is the expected probabiity of success of the agents for whom the contract is intended. The menu of contracts wi be empty at equiibrium if a possibe oan contracts provide the bank with stricty negative pro ts. If the menu is composed of two contracts, and banks anticipate that each of them wi be seected by a di erent type of agent, the equiibrium is separating and q = p h for the contract seected by high abiity agents, whie q = p for the contract seected by ow abiity agents. Finay, the menu may be a singeton, and two scenarios emerge. Either banks anticipate that both types wi accept the contract, and q = p p h + (1 ) (the equiibrium invoves pooing both types). Or, aternativey, banks anticipate that ony one type (with high abiity) wi accept it and q = p h (in a separating equiibrium where ow types refuse the contract). At equiibrium, then, student consumption eves in outcome 2 ff; sg are Y = max fw R F; (1 g)w g ; (1) for R 1 + r. Conversey, banks pro ts under outcome 2 ff; sg write = min fr F; gw g IF; (2) where I 1 + i. 2.2 Graphica anaysis In order to anayze under which conditions the various types of equiibria wi emerge, it may prove convenient to represent a payers strategies in the space of consumption eves 8

9 Figure 1: Zero-pro t oci Z(q), defaut proof space DP (g) and indi erence curve I of agents in case of faiure and success (Y f ; Y s ), as iustrated in Figure 1. Such a space can be divided into two subspaces reative to the two strategies that agents can pay at stage 5: repay or defaut. Let us de ne in this space the set of aocations such that defaut does not occur: De nition 1 The defaut-proof space, DP (g) is the set of consumption bundes (Y f ; Y s ) such that for a 2 ff; sg, w R F (1 g)w. Inside DP (g), Y = w R F > (1 g)w for a 2 ff; sg, whie outside DP (g), there exists at east one outcome 2 ff; sg such that Y = (1 g)w > w R F. This impies that, in the space of consumption eves, one can estabish a one to one reation between oan contracts (r f ; r s ) and consumption eves (Y f ; Y s ) ony inside DP (g). In other words, banks can crediby obtain interest rates (r f ; r s ) inside DP (g). Outside DP (g), a contract is not respected. Then, banks are egay aowed to garnish gw and consumption is in fact Y = (1 g)w. Such consumption bundes are ocated on the boundaries of DP (g). Let us now represent, in the space (Y f ; Y s ), the set of oan contracts that provide, for a given expected probabiity of success q, zero expected pro ts. Since Y + = w IF, 9

10 E(q; r f ; r s ) can be rewritten as E(q; Y f ; Y s ) = q(w s Y s ) + (1 q)(w f Y f ) IF: (3) Equation (3) aows us to de ne the zero pro t ocus in terms of combinations of consumption bundes in case of faiure and success (Y f ; Y s ). De nition 2 Z(q) is the set of consumption bundes (Y f ; Y s ) in DP (g) such that, for a probabiity of success q, banks make zero expected pro ts: Y s = w s w f + w f IF 1 q Y f : (4) q q Figure 1 depicts, in the (Y f ; Y s ) space, the defaut-proof space and the zero pro t oci: Z(p ) (when contracts are accepted ony by ow abiity agents), Z(p p ) (for contracts that poo together high and ow abiity agents) and Z(p h ) (for contracts that separate high abiity agents). Ceary, the sope of a zero pro t ocus is given by (1 q)=q. Thus, since p < p p < p h, Z(p h ) is the attest of these oci, foowed by Z(p p ) and, nay, Z(p ), the steepest one. On the other hand, zero pro t oci cross at (Y f ; Y s ) = (w f IF; w s IF ). Note that if g is such that w f IF > (1 g)w f (i.e., gw f > IF ), the point (w f IF; w s IF ) actuay beongs to DP (g). Pro ts are then zero when agents pay the risk-ess interest rate independenty of the type of agents the contract addresses. In fact, there is no potentia for a market faiure in this case, since defaut woud never occur: as gw f > IF, a agents are better o paying the risk-ess interest rate than defauting, whether they are successfu or not. We henceforth concentrate on vaues of g for which market faiure can take pace (gw f < IF ). This impies that the intersection between the zero pro t oci is ocated outside of the defaut-proof space, as depicted in Figure 1. Sti in this gure, F I is the certainty or fu insurance ine, characterized by the set of consumption bundes (Y f ; Y s ) such that Y f = Y s. The straight ine G is the set of consumption bundes (Y f ; Y s ) such that Y s When g = 1, Y = max fw = (w s =w f ) Y f, it represents the expansion path of DP (g). R F; 0g, so that the defaut proof space is R 2 +. As g decreases (penaties become softer) the defaut-proof space shrinks, its origin moving aong G towards (w f ; w s ). Finay, point O in Figure 1 represents the outside option of refusing a contracts and remaining unskied (w f ; w f ). 10

11 De nition 3 I is set of consumption bundes (Y f ; Y s ) such that EU(p ; Y f ; Y s ) = U(w f ), i.e., the ow abiity agent s indi erence curve for a utiity eve obtained at the outside option. Note that the position of O has impications on the e ciency of the investment. To see why, et us rst de ne e ciency. When a market does not exist, it does not necessariy impy that there there is market faiure. Market faiure ony takes pace when the market shoud exist and it fais to. It is then important to estabish who shoud have access to credit. Note that the investment in higher education is e cient for an agent with probabiity of success p a if and ony if its expected bene t is arger than its funding cost: p a (w s w f ) > IF. Since we ony consider two types of agent, e ciency may either impy that a invest in higher education or that ony those who have high abiity do. Both assumptions are egitimate. On the one hand, we may assume ony the most abe shoud attend higher education. On the other, it may be the case that after competing secondary education, a potentia students have the minimum required abiity. Yet, they may sti di er in abiity to study and, thus, in wiingness to pay for education. We may want to account for this fact. For convenience in the presentation of resuts, we make the rst of the possibe assumptions in this respect. Yet, given that the poicy impications of our work depend on which of these assumptions is made, we wi consider aong the anaysis the consequences of its non fu ment, which, as wi be made cearer, does not quaitativey a ect our resuts. Assumption A1 Ine ciency of the investment in higher education of ow abiity agents. Formay, A1 writes p (w s w f ) < IF and impies that O is above the zero pro t ocus when contracts are targeted to ow abiity students. To see this, simpy compare the outcome w f with the certain outcome Y f = Y s that provides zero pro ts when ony ow abiity individuas accept the contract. From (4) this certain outcome is Y f = p w s +(1 p )w f IF < w f by A1. We make two further assumptions imiting the degree of risk aversion of the agents: Assumption A2 Limited risk aversion of ow abiity agents. Assumption A3 Limited risk aversion of high abiity agents. 11

12 Assumptions A2 and A3 are formay stated in the appendix. Graphicay, a imited degree of risk aversion is de ned as the intersection between Z(p p ) and I ying to the eft of the ine G. This assumption imits the curvature of I and, hence, the degree of risk aversion of a ow abiity individua. Simiary, we say that the degree of risk aversion of the high abiity type is imited when the tangency between the indi erence curve of a high abiity type and Z(p p ) ies to the eft of G. We wi discuss the consequences of reaxing these assumptions. 3 Low defaut penaties When defaut penaties are su cienty ow, the best strategy at the ast stage is for agents to defaut. Yet, since penaties are ow, banks revenues yied negative pro ts, so they wi not o er any contract. With higher defaut penaties, banks can o er pro tabe contracts provided that ony high abiity individuas, with higher probabiity of success, take them. However, given A2, a contract appeaing to high abiity students is aso appeaing to ow abiity students. And, since the bank cannot distinguish them, it wi tend to end indiscriminatey, and therefore incur osses. Consequenty, it is the interaction between ex-post mora hazard and adverse seection that induces banks not to o er any oan contract in this case. Let g p be the minimum g such that the intersection between Z(p p ) and G beongs to DP (g). We formay de ne ow defaut rues as cases where g < g p. Proposition 1 If defaut penaties are ow, that is g < g p, banks wi not o er oans to students at equiibrium. Proof. First, note that for any g 2 [0; 1], any contract (r f ; r s ) o ered by a bank wi, if it is accepted, provide an agent with a consumption bunde (Y f ; Y s ) that is at east as good as a bunde where she woud defaut in both outcomes, ((1 g)w f ; (1 g)w s ). Since g < g p < g U p, a ow abiity agent wi accept any defaut-proof contract a bank woud o er, because her expected utiity is arger than with no education. Graphicay, a contracts in the defaut proof area ie above I. Consequenty, any contract that corresponds to a point on Z(p h ) - that yieds zero pro ts if it is designed ony for high abiity agents - wi induce negative pro ts, since ow 12

13 Figure 2: Credit rationing equiibrium types woud accept it too. A pooing contract wi aso yied negative pro ts since g < g p, which impies that everyone defauts, and E(p p ; (1 g)w f ; (1 g)w s ) < 0. Finay, banks cannot o er a menu of separating contracts. Since p p > p, E(p ; (1 g)w f ; (1 g)w s ) < E(p p ; (1 g)w f ; (1 g)w s ) < 0, banks cannot design a contract for ow abiity agents that woud yied non negative pro ts. Summing up, when g < g p, a contracts impy negative pro ts, so that banks do not o er any. At this point, having proved the existence of market faiure in the provision of students oans, it is worth inquiring what can be done to make the market exist. One rst possibiity is to increase defaut penaties. In the next section, we wi show how arger defaut penaties can impy the existence of a student oan market. Another possibiity, compatibe with ower defaut penaties, is to simpy provide the information about the abiity of the individuas. This can easiy be done by impementing an adequate system of nationa examinations or using the information gathered by universities. According to our anaysis, if this information was avaiabe and of good quaity, subsidies and guarantees to banks woud be unnecessary. Further, under A1, not ony woud the market soution be e cient in terms of participation 13

14 (with ony high abiity individuas getting a oan) but aso, they coud be o ered as much insurance as desired. Aternativey, if we reax A1, we can take the market to exist just the same. Yet, the soution is now ine cent, since ony the oans to high abiity students are pro tabe and ow abiity students are eft out of the market. As wi be made cearer in Section 3.1, in this case, it wi be worth increasing defaut penaties in order for the market equiibrium to be of the pooing type. Summing up, we can have a market with ower g but oans wi ony be made avaiabe to higher abiity types, or we can increase defaut penaties and then banks wi o er contracts that wi poo together a agent types. Let us further investigate the atter case. 3.1 Intermediate defaut penaties As defaut penaties increase further, more contract types become avaiabe (DP (g) moves down). Lemma 1, in the Appendix, shows that, among feasibe pooing contracts, banks choose those who impy minimum insurance (i.e. those which ie further away from the certainty or fu insurance ine). Figure 3 shows why the most insuring pooing contract in the shaded defaut proof area is not an equiibrium: there is aways a pro tabe deviation consisting in o ering a contract that ony high abiity types woud accept, thus yieding stricty positive expected pro ts. Proposition 2 formaizes the statement that with intermediate defaut penaties, a market equiibrium with a pooing contract emerges. The equiibrium contract is such that agents receive the owest possibe eve of consumption in case of faiure: Y f = (1 g)w f. This may be interpreted as if agents defauted in case of faiure, su ering the garnishment of a proportion g of their wage, or as if the bank anticipating this charged R f F gw f, consequenty adapting the interest rate in case of success to make zero pro ts. In any event, agents are insured, either by the bank, if the interest paid is ower than it woud in case of success, or by the ega system, if the agent defauts and the bank cannot expropiate her. The fact that payments depend on the resuting state of nature, simpy impies that oans are income contingent. Yet, by Lemma 1, the pooing equiibrium contract is the east-insuring of a pooing contracts avaiabe to banks. Before stating Proposition 2 it is worth noting that, if agents are too risk averse, they might prefer to remain unskied and a pooing equiibrium may not exist. Lemma 2 shows 14

15 Figure 3: No insurance at the pooing equiibrium that Assumptions A2 and A3, which imit the degree of risk aversion of agents, are precisey the necessary conditions under which a pooing equiibrium may exist. Proposition 2 If defaut penaties are intermediate, banks o er a unique pooing contract at equiibrium. This contract is the east-insurance pooing contract of a avaiabe. Proof. First note that since g > g p, Z(p p ) is non-empty. Whie Lemma 1 aows us to focus ony on the non-insuring bunde among a bundes in Z(p p ), Lemma 2 has made two important points. Let g U p be the minimum g such that I \ Z(p p ) 2 DP (g) and et g U p h be the minumum g such that the tangency between the indi erence curve of a high abiity type with Z(p p ) beongs to the defaut proof space. On the one hand, A2 impies that for g < g U p, ow types prefer the non-insuring, pooing contract to remaining unskied. On the other, A3 ensures that for g < g U p, the high abiity individua s indi erence curve that goes h through the pooing, east-insuring bunde - is atter than the zero pro t ocus under pooing of types. Consequenty, no agent has a pro tabe deviation. Banks, whose expected pro ts are zero, cannot nd a separating deviation from the pooing non-insuring contract. Low abiity agents prefer this contract to the aternative of remaining unskied, and so do high 15

16 Figure 4: Pooing equiibrium types. Can there be another type of equiibrium? On the one hand, as shown just above, the "no market" outcome is dominated by the east-insuring pooing outcome. On the other hand, a separating equiibrium does not exist. As shown in the Appendix, if g U h = min g such that I \ Z(p h ) 2 n DP (g), Assumptions o A2 and A3 impy the foowing ordering of threshods of g: g U h > min g U p ; g U p > g. But, as just de ned, when g < g h U h, I \ Z(p h ) =2 DP (g): g is too ow to enabe banks to o er a contract that ony a high abiity agent woud accept. Under Assumption 1, the pooing, east-insurance contract, is not e cient: it attracts ow abiity students who shoud not invest in education. If A1 is reaxed, we can attain the e cient soution in terms of participation. We ony need arge enough defaut penaties. Yet, wefare can be increased by providing arger eves of insurance and in this sense the market sti fais. 16

17 3.2 Largest defaut penaties Finay, if defaut penaties are argest, banks are abe to separate types by o ering very risky contracts - with a very ow consumption eve in case of faiure - that ony a high abiity agent, with ower probabiity of faiure, woud accept. We say that defaut penaties are argest when g > min g such that I \ Z(p h ) 2 DP (g) and I is the indi erence curve of a ow abiity agent that exhibits a eve of utiity of U(w f ). Proposition 3 If defaut penaties are argest, that is g > g U h, then a separating equiibrium exists. Under A1, ony high abiity agents accept the contract, whie ow abiity agents remain unskied. Proof. First note that since g > g U h > g U p, the pooing, non-insuring contract is undesirabe for ow abiity agents. Pooing cannot thus be an equiibrium. The fact that g > g U h aso impies that there exists a su cienty risky contract, that corresponds to a consumption bunde (Y f ; Y s ) 2 Z(p h ) with Y f (1 g U h )w f. By de nition, a ow abiity agent prefers remaining unskied than accepting this contract, which is not the case of a high abiity agent. Separation is thus an equiibrium since banks expected pro t is zero and cannot deviate in order to increase it. Aso, given the o ered contract, both types of agents maximize utiity. Proposition 3 is iustrated in Figure 5. Note that separation now resuts as a market outcome, without any need to provide banks with the reevant information. Assumption A1 impies that the pooing equiibrium is ine cient. As in de Meza & Webb (1987) the pooing of types invoves overinvestment in education: ow abiity types, who shoud not invest in education because their expected returns do not exceed the costs, do. Conversey, this assumption aso impies that the separating equiibrium, emerging when defaut penaties are the argest, is e cient and is such that ony high abiity individuas invest in education. Yet, unike in the case where g is much ower and the information on abiity eves is made avaiabe (see section?-ow g-), banks cannot provide any desired eve of insurance. In fact, it is precisey because the contract is risky that high abiity types are abe to separate from ow abiity types. Reaxing Assumption A1 makes the pooing equiibrium e cient, since it is then pro tabe for both types to invest in education. More interestingy, when g is argest, the separating equiibrium provides ow abiity agents with the most insuring contract avaiabe (see Figure 5). At the separating equiibrium, banks thus o er menus of two contracts, instead of a singe 17

18 Figure 5: Separating equiibrium one for the high abiity type. Both contracts are income contingent, and o er insurance, but rather di erent eves of it. Low abiity agents wi choose the more insuring of the two contracts whenever it is e cient for them to study. If g is su cienty arge, they wi have fu insurance. 7 It is cear from our anaysis that, due to the presence of asymmetric information, the market wi never provide fu insurance to a agents, which for su cienty arge eves of risk aversion, coud maximize wefare. The government can do such a thing. If the government decided to provide an identica, insured oan to both abiity types, it shoud make sure that banks are not aowed in the market, as they coud ceary o er some contract that woud attract high abiity agents ony and make the pubic system incur oses. 8 It needs to be said that equiibrium does not aways exist. There is a set of intermediate vaues of g, too ow for the separating equiibrium to resut, too arge for the pooing equi- 7 In De Rey and Racionero (??) e cient participation is attained by giving fu insurance to the east abe student who shoud attend higher education. Here we show that the market can o er such an arrangement provided that defaut penaties are arge enough. 8 This point was made by Moshe Justman. 18

19 ibrium, such that this game does not have an equiibrium. Such a range of penaty eves g do not exist if risk aversion is ow, as we have shown that imited risk aversion of the agents guarantees the existence of the pooing equiibrium. Simiary, neither does that range exist if risk aversion is very arge, since then either separation woud be possibe or no market wi take pace at equiibrium. The set of defaut penaty eves g is characterized by Proposition 4, in the Appendix. 4 The roe of a subsidy on the interest rate Many wordwide interventions reated to student oans invove giving subsidies to banks. In this section we expore the e ects of bank subsidies. Suppose that the government bene ts from an exogenous in ow of cash that it uses to subsidize banks costs of borrowing i. Because of Bertrand competition, this ower cost wi immediatey be transferred to the borrower: interest rates wi be ower and aow higher consumption bundes in case of faiure and success. Lower interest rates, on the other hand, make it ess pro tabe to defaut. Thus the existence of the market is compatibe with ower eves of the penaty g when banks are subsidized. In other words, subsidies of this kind can take the economy from a no market equiibrium to a pooing equiibrium. Graphicay, the reduction in I = 1 + i transates into an upward shift of the zero pro t oci, whie their sope remains unatered (see Equation (4)). Given g, the fa in i thus incites banks to o er contracts, as some of those contracts now generate non-negative pro ts despite asymmetric information and ex-post mora hazard. 9 Moreover, subsidizing i wi make the separating equiibrium more easy to obtain. This is iustrated in Figure 6. Interestingy, this resut suggests that governments may not need to organize income contingent oans themseves, they might just strongy subsidize banks, and the competitive equiibrium woud provide insurance for ow abiity agents. 9 As the risk-ess interest rate fas, the zero pro t oci progressivey move upwards, so that a subsidy on i has impications simiar to those of an increase in g. However, xing g in a case where a market exists, the no equiibrium soution does not take pace for any subsidy on i (see the proof of Proposition 3 for an iustration of why an increase of defaut penaties, unike a reduction in the risk-ess interest rate, can under some circumstances yied no equiibrium). 19

20 Figure 6: E ect of a subsidy on i 5 Concusion In this paper, we addressed two questions. Firsty, why do private banks not o er student oans, uness they receive pubic support in terms of interest subsidies or guarantee against risks of defaut? This market faiure has ed many governments to o er student oans themseves. Pubic oans have sometimes a particuarity that di erentiates them from those o ered by private banks, namey that their repayment is made income contingent. Students are then insured against the risk of faiure, in the sense that those who perform bady at university or on the job market have to pay ess than successfu ones. This raises the second question, when private banks o er student oans, why don t they make them income contingent? Appropriate answers to such questions shoud hep formuate poicy recommendations in this market. We show that credit rationing in the student oan market can resut from a combination of ex-post mora hazard and adverse seection. Unike previous attempts based on the inappropriate set of assumptions of the Stigitz & Weiss (1981) mode, we produce the market faiure resut assuming that more abe individuas face arger expected returns when investing 20

21 in education. The penaty associated to defaut has shown to be crucia in producing answers to our two motivating questions. In particuar, our mode estabishes that the causes of credit rationing ie with ow defaut penaties and banks inabiity to identify the high abiity students. When e ciency requires that ony high abiity individuas study, governments shoud hep produce accurate information about abiity. When everybody shoud study, increasing defaut penaties seems more appropriate. On the other hand, private oans do not incude more insurance because defaut penaties are not su cienty arge. Note that our second resut about the existence of a pooing, east-insuring contract in the case of intermediate defaut penaties is perfecty compatibe with the high defaut rates observed in reaity. Aso, this equiibrium may invove e cient participation, when a abiity types shoud invest in education. Insurance is however too itte. The government can force pooing contracts with arger eves of insurance. Yet, it is then important to make sure that banks wi not o er arrangements that are more attractive to high abiity types aone, as this wi make the pubic system incur oses. With arger defaut penaties, separation can be obtained, where ow abiity agents no onger try to poo with high abiity agents and prefer to be identi ed as ow abiity agents, but bene t from a high eve of insurance. No government intervention is then required to achieve arge eves of insurance provided that defaut penaties are arge enough. The mode is simpe and eaves out of the scope of the anaysis important aspects of credit markets such as market power, ega costs associated to coecting penaties, other costs associated to defaut or the roe of coatera, among others. Yet, the mode provides a usefu benchmark and can easiy be extended to account for some of these issues. As a way of exampe, the mode can be extended by introducing asymmetries between what defauters pay as penaties and what banks obtain from these penaties. For instance, it coud be costy for banks to sue defauters. If the cost of initiating ega action is higher than defaut penaties when students fai, banks wi ony sue successfu defauters. Unsuccessfu borrowers anticipe this and defaut with impunity. Consequenty, it can be shown that a separating equiibrium can never be obtained. This is quite natura: we have attributed market faiure to insu cient penaties. High ega action costs can ony reinforce the market faiure resuts. Other issues may require ess trivia extensions and we eave them for future research. For exampe, we have expored the impact of the introduction of an initia individua 21

22 endowment which can be used as coatera. The resuts do not change quaitativey in this case, athough the presence of coatera naturay makes banks more incined to o er oans. The anaysis undertaken in terms of g can be reproduced for di erent eves of coatera. Like arger defaut penaties, a arger coatera improves the case for private student oans. If the size of coatera is arge enough, separation of types and insurance take pace at equiibrium. The consideration of di erences in the initia endowment can add an interesting dimension to the probem. We discussed the roe of interest rate subsidies. Our resuts con rm the styised facts that such subsidies provide incentives for banks to o er student oans. The reason is that such subsidies aow banks to o er better contracts that wi not be subject to defaut. Another mechanism used by governments to encourage banks to o er student oans consists in guaranteeing them against the risk of defaut. As informay pointed out by Sami (1999), this raises a new mora hazard probem between the government and the banks, which woud then have ess incentives to appropriatey design the contracts. Having highighted the roe of defaut penaties, one ast reevant question arising from our discussion is why governments do not opt for arger defaut penaties, which woud aow the market to provide insurance to ess abe students without pubic intervention. 22

23 References Barr, N. (2001). The wefare state as piggy bank : information, risk, uncertainty, and the roe of the state. Oxford: Oxford University Press. Bertoa, G., Disney, R., & Grant, C., Eds. (2006). The Economics of Consumer Credit. MIT Press. Bönda, S., Fied, S., & Girouard, N. (2002). Investment in human capita through postcompusory education and training: Seected e ciency and equity aspects. OECD Economics Department, (333). Boadway, R. & Keen, M. (2004). Financing new investments under asymmetric information: A genera approach. (1017). Canton, E. & Bom, A. (2004). Can student oans improve accessibiity to higher education and student performance? an impact study of the case of sofes, mexico. (3425). Chapman, B. (2005). Income contingent oans for higher education: Internationa reform. (491). Chapman, B. & Greenaway, D. (2006). Learning to ive with oans? internationa poicy transfer and the funding of higher education. The Word Economy, 29(8), Chen, H.-j. (2005). Educationa systems, growth and income distribution: a quantitative study. Journa of Deveopment Economics, 76(2), de a Croix, D. & Miche, P. (2007). Education and growth with endogenous debt constraints. Economic Theory, 33(3), de Meza, D. & Webb, D. (1990). Risk, adverse seection and capita market faiure. Economic Journa, 100(399), de Meza, D. & Webb, D. C. (1987). Too much investment: A probem of asymmetric information. The Quartery Journa of Economics, 102(2), Eurydice (1999). Key topics in education. Technica report, European Commission, Brusses. Ho, K. & Lyon, A. B. (1995). Non-eaky buckets: Optima redistributive taxation and agency costs. Journa of Pubic Economics, 58(3),

24 Ionescu, A. F. (2007). Bankruptcy rues, coege enroment, and defaut incentives for student oans. Mimeo. Jacobs, B. & van Wijnbergen, S. J. G. (2007). Capita-market faiure, adverse seection, and equity nancing of higher education. FinanzArchiv: Pubic Finance Anaysis, 63(1), Kehoe, T. J. & Levine, D. K. (1993). Debt-constrained asset markets. Review of Economic Studies, 60(4), La Porta, R., Lopez-de Sianes, F., Sheifer, A., & Vishny, R. (1998). Journa of Poitica Economy, 106(6), Law and nance. Lochner, L. & Monge-Naranjo, A. (2002). Human capita formation with endogenous credit constraints. (8815). Mas-Coe, Andreu, M. D. W. e. J. R. G. (1995). Microeconomic Theory. Oxford University Press. Sami, J. (1999). Student oans in an internationa perspective: The word bank experience. LCSHD Paper Series, 44. Shen, H. & Ziderman, A. (2007). Student oans repayment and recovery - internationa comparisons. Paper presented at the FHE-2007 conference in Portoro. Stigitz, J. E. & Weiss, A. (1981). Credit rationing in markets with imperfect information. American Economic Review, 71(3), Vercammen, J. (2002). Wefare-improving adverse seection in credit markets. Internationa Economic Review, 43(4),

25 Appendix Formaization of the assumptions In order to formaize assumptions A2 and A3, we need to de ne usefu threshods of the penaty g. De nition 4 g U h = min g such that I \ Z(p h ) 2 DP (g). In Figure 7, I \ Z(p h ) is characterized by point A, and the defaut-proof space whose boundary crosses A is DP (g U h ). De nition 5 g U p is the minimum g such that I \ Z(p p ) 2 DP (g). In Figure 7, I \ Z(p p ) is characterized by point B, and the defaut-proof space whose boundary crosses B is DP (g U p ). Ceary, g U h > g U p. Let B = (Yf ; Y s ). Aso, keep in mind that ine G is de ned as the set of consumption bundes (Y f ; Y s ) such that Y s = (w s =w f )Yf. Ceary, by imposing that B = (Yf ; Y s ) to the eft of G, we can imit the curvature of this indi erence curve and, thus, the degree of risk aversion of the ow abiity type. Assumption A2 Limited risk aversion of the ow abiity individua: Y s (w s =w f )Y f. De nition 6 g p is the minimum g such that Z(p p ) \ G 2 DP (g). In Figure 7, Z(p p ) \ G is characterized by point C, and the defaut-proof space whose origin is at C is DP (g p ). Finay, et (Y h f ; Y h s ) (not depicted in Figure 7) be the point where the indi erence curve of the high abiity individua is tangent to the zero pro t ocus under pooing of types, i.e. (Y h f ; Y h s ) = arg max (Yf ;Y s)2z(p p) EU(p h ; Y f ; Y s ). De nition 7 g U p h h is the minimum g such that (Yf ; Y s h ) 2 DP (g). Athough this threshod has not been depicted in Figure 7 in order to keep things as simpe as possibe, we can, as before, imit the degree of risk aversion of the high abiity type by imposing that (Yf h; Y s h ) ies to the eft of G. Assumption A3 Limited risk aversion of the high abiity type: Ys h (w s =w f )Yf h. 25

26 Figure 7: Characterization of the equiibria with respect to g The resuts we present in the next section are obtained under A1, A2 and A3. By A2, g U p > g p. In turn, A3 impies g U p > g h p. Taken together, these assumptions impose the n o foowing ranking on the reevant threshods for g: g U h > min g U p ; g U p > g p. As the next h section wi show, the range of defaut-proof spaces corresponding to each interva impies a di erent equiibrium. Note that Proposition 1 hods if A2 is reaxed, as ong as g < g U p. The reason is that the arger the risk aversion of the ow abiity type (which is imited by A2) the easier it is to o er a contract that is ony pro tabe for the high abiity individua. It is then more di cut that the market does not exist or, put di erenty, we need a ower penaty for this to be the case. In contrast, when the degree of risk aversion of the ow abiity type is ow (and, at the imit, if she is risk neutra) the no market resut is compatibe with higher penaty n eves. o More generay, it can then be stated that no market exists as ong as g < min g p ; g U p. 26

27 Lemma 1 Lemma 1 If a pooing equiibrium exists, it is unique and it is such that the contract o ered by banks is non-insuring. Proof. For a pooing equiibrium to exist, Z(p p ) must be non-empty. Lemma 1 caims that, among a contracts that are pooing equiibrium candidates, or equivaenty, among a corresponding consumption bundes (Y f ; Y s ) 2 Z(p p ), ony the non-insured bunde (1 g)w f ; ppws IF +(1 pp)gw f p p emerges at the pooing equiibrium. To see why, et us consider any (Y p f ; Y s p ) 2 Z(p p ) with (Y p f > (1 g)w f and Ys p < ppws IF +(1 pp)gw f p p, and show that there exists a pro tabe deviation from (Y p f ; Y s p ), so that the atter cannot be an equiibrium. By singe crossing of the two types indi erence curves, there aways exists some other bunde (Yf d; Y s d ) 2 DP (g) such that EU(p ; Yf d; Y s d ) < EU(p ; Y p f ; Y s p ) but EU(p h ; w f ; w f ) > EU(p h ; Y p f ; Y s p ) and such that E(p h ; Yf d; Y s d ) > 0. In other words, if banks o er a pooing contract that impies a consumption bunde (Y p f ; Y s p ), there aways exists a pro tabe deviation, which consists in o ering a contract they know that ony high types woud accept, and that woud yied stricty positive expected pro ts. The dark shade area in Figure 3 represents such pro tabe deviations from (Y p f ; Y s p ). Finay, note that (1 g)w f ; ppws IF +(1 p p pp)gw f is the ony bunde in Z(p p ) such that such a pro tabe deviation does not exist. Lemma 2 Lemma 2 Assumptions A2 and A3 are necessary conditions for the existence of a pooing equiibrium. Proof. Let us assume successivey that A2 and A3 are not satis ed and show that a pooing equiibrium cannot exist in these cases. Let us start by considering the case where A2 is not satis ed, which impies Ys < (w s =w f )Yf, where (Yf ; Y s ) = I \ Z(p p ), where I is the set of consumption bundes (Y f ; Y s ) such that EU(p ; Y f ; Y s ) = U(w f ). By de nition, both (Yf ; Y s ) and (1 g)w f ; ppws IF +(1 pp)gw f p p - the unique pooing equiibrium candidate - ie on Z(p p ). However, by de nition, the non-insured bunde ies on the other side of G, which impies that Yf > (1 g)w f and Y. We need to show that the eve of consumption in case of success of s < ppws IF +(1 pp)gw f p p this bunde, ppws IF +(1 pp)gw f p p, is insu cient for the ow type to accept the contract, so that he 27