Robert J Gary-Bobo
email: robert.gary-bobo@ensae.fr

Alain Trannoy
email: alain.trannoy@univ-amu.fr.

Optimal Student Loans and Graduate Tax under Moral Hazard and Adverse Selection

Keywords: Student Loans, Graduate Tax, Adverse Selection, Moral Hazard, Risk Aversion

We characterize the set of second-best optimal "menus"of student-loan contracts in a simple economy with risky labour-market outcomes, adverse selection, moral hazard and risk aversion. The model combines student loans with an elementary optimal income-tax problem. The second-best optima provide incomplete insurance because of moral hazard; they typically involve cross-subsidies between students. Generically, optimal loan repayments cannot be decomposed as the sum of an income tax, depending only on earnings, and a loan repayment, depending only on education. Therefore, optimal loan repayments must be income-contingent, or the income tax must comprise a graduate tax. The interaction of adverse selection and moral-hazard, i.e., self-selection constraints and e¤ort incentives, determines an equal treatment property; the expected utilities of di¤erent types of students are equalized at the interim stage, conditional on the event of academic success (i.e., graduation). But individuals are ex ante unequal because of di¤ering probabilities of success, and ex post unequal, because the income tax trades o¤ incentives and insurance (redistribution).

Introduction

The importance of student loans for the accumulation of human capital, economic growth and welfare cannot be overestimated. In the United States, the total amount of outstanding student debt has reached $1 trillion at the end of 2011. In Great Britain, the rise of tuition fees seems to have caused a sharp increase in average student debt 1 . Student loans pose interesting …nancial engineering and regulation problems. There are many discussions on the optimal design of these loans: for instance, the UK and Australia have a form a incomecontingent repayment system, since loan repayments are based on the graduate's earnings, just like income tax, and interest rates are subsidized 2 . In many continental European countries, student loans play a negligible role but, given the severe shortage of public funds, they could go hand in hand with a substantial raise in tuition fees, and become a new source of funds for universities 3 .

There is an important econometric literature on the impact of credit constraints on university or college attendance 4 . For recent quantitative studies of alternative student-loan policies in the US, see, e.g., [START_REF] Ionescu | The Federal Student Loan Program: Quantitative Implications for College Enrollment and Default Rates[END_REF], [START_REF] Lochner Lance | The Nature of Credit Constraints and Human Capital[END_REF]. These questions are hotly debated, yet, to the best of our knowledge, the micro-economic theory of student loans is still underdeveloped 5 . In the following, we propose a simple model of student loans, under the combined e¤ects of risk aversion, moral hazard and adverse selection.

We explore the structure of the set of second-best optimal (or interim incentive-e¢ cient) allocations of credit to risk-averse students in an economy in which individual talents and e¤orts are not observed by the lender, future earnings are subject to risk and incomes can be taxed. The student-loan problem is compounded with an elementary form of the Mirrleesian optimal income-tax problem. 1 The average student debt is predicted to be around 50,000 pounds, on leaving the university, for those starting in 2012 (for details, see http://www.slc.co.uk/statistics). The number of students experiencing di¢ culties to repay their loans reached 8.8% in the United States, after the economic downturn of 2008.

2 New Zealand and Sweden are also providing income-contingent student loans. On these questions, see,

We consider an economy with unobservable, two-dimensional individual types. Students are characterized by an ex ante type and an ex post type. The ex ante type represents cognitive (and other) skills that in ‡uence academic success. In contrast, the ex post type plays a role once the student …nds a job, and can be interpreted as representing some jobmarket skills, or job-market opportunities that are random, but correlated with the quality of education. Students choose the length (or the quality) of their studies in a "menu"; they know their ex ante type before the choice of how much to invest in education. In contrast, the ex post type, which is private information too, is revealed later, after graduation.

Students exert two kinds of e¤ort. While attending college, students exert some study e¤ort that is not observable. This determines an ex ante moral hazard problem. Ex ante e¤ort (or study e¤ort) in ‡uences the probability of graduation. Students can be either successful, or fail to obtain a certi…cate. The distribution of individual earnings is also a¤ected by unobservable e¤ort exerted at work. E¤ort at work may also be a called ex post e¤ort. The government does not observe ex post e¤ort but observes education, success and individual earnings. A successful student with high ex post skills can choose to behave like a student with low ex post skills and save on e¤ort costs. This is the source of an ex post moral hazard problem, with the consequence that the most productive students can decide to earn less if tax rates are too high.

The student's success, education and ex post type jointly determine her(his) potential wage. There are two sources of risk: random academic success and the drawing of ex post types. Thus, individual wages are risky. Because of risk aversion, students would like to be insured against these random shocks.

We characterize the set of second-best Pareto-optima of this model. These optima can be implemented by a combination of structured loans and income taxes.

Our main results are the following. The optimal menu of contracts exhibits incomplete insurance: this is mainly due to moral hazard. Students obtain the maximal amount of income insurance compatible with the exertion of high study e¤ort among the less-talented.

It follows from this that the talented ex ante types bear more risk than strictly necessary to ensure that they choose a high level of study e¤ort. The planner cannot reduce the risk borne by the talented, because that would violate the self-selection constraints.

As a by-product, we …nd that second-best optimal loan repayments are always incomecontingent, even in the presence of an income tax. More precisely, we …nd a non-decomposition theorem: the second-best cannot be implemented by means of the sum of an income tax, that depends only on observed earnings, and a loan repayment, that depends only on the observed quality of education (or observed years of education). It must be that, either the income tax depends on education, or the loan repayments depend on income. Since the budget is by construction balanced (we did not explore subsidies that would be …nanced by means of external sources of funds), the second-best optima typically exhibit cross-subsidies between types: the talented repay more and subsidize the less-talented, but the former are not fully exploited. The second-best solution can be interpreted, either as an income-contingent loan, or as a graduate tax, with a certain degree of progressivity.

In the standard utilitarian case, we …nd that the second-best optimal menu of loan contracts exhibits a form of pooling, called equal treatment: the students'expected utilities, net of repayments, must be equal, conditional on random academic success 6 . In other words, the post-study but pre-work expected utility of net earnings should be independent of the student's unobservable ex ante type. In spite of being treated equally in this particular sense, students are ex ante unequal, since the talented types have a greater probability of success, and they are ex post unequal, since the optimal income tax trades o¤ the provision of insurance (and redistribution) against that of incentives. We show that equal treatment is a way of solving the screening problem in the presence of moral hazard. The equal treatment property results from the interplay of self-selection and e¤ort incentives constraints. Without an adverse selection problem, each type of student would be subjected to the smallest amount of risk needed to guarantee high e¤ort. This takes the form of a gap between the net income earned in case of success and that earned in case of failure. It is typically easier to obtain high e¤ort from the talented types, so the gap between the utilities of the latter in case of success and in case of failure can be chosen to be smaller than what would be appropriate for a less-talented type of individual. But providing insurance to talented types in this way will always violate the self-selection constraints, with the possible result that the talented would not want to study longer than other students. Hence the equal treatment result: interim equalization is the best way of providing insurance to good students, compatible with the revelation of academic talent, and given that complete insurance against failure is precluded.

Another contribution of the present article is to provide a characterization of the set of second-best optima in an economy with wage risk, risk aversion, moral hazard and adverse selection. In other words, we provide a study of the structure of second-best optima in a screening model with the added complication of moral hazard. The equal treatment result described above is not the textbook allocation of insurance under adverse selection, à la

Rothschild-Stiglitz7 . To obtain the familiar separating menu of loan contracts as a secondbest optimum, we show that the weight of the talented types must exceed their natural frequency in the social planner's objective function, which is not the common assumption in public …nance.

Our theory of student loans combines 4 main ingredients: risk aversion, ex ante moral hazard, a screening problem and ex post moral hazard. To be posed in a meaningful way, the student loan problem should embody these ingredients. None of these ingredients can be removed without losing an important aspect of our results. Without risk aversion, there is no demand for insurance and redistribution. Absent the ex ante moral hazard problem, students may be perfectly insured against the consequences of academic failure. It would not be reasonable to assume that study e¤ort is ine¤ective (or that it can be perfectly monitored).

If ex ante types are observable, we lose the equal treatment property. We model a society in which, to a certain extent, individuals choose their individual investment in education, and, to predict a person's career, it matters a great deal, for instance, if this person went to graduate school or has accomplished 2 years of college. This is why we should consider a menu of educational choices. When ex ante types are assumed observable, the self-selection problem is replaced with an education planning problem. We show that when ex ante selfselection constraints are dropped, the social planner will force all talented students to study longer. In the absence of an ex post moral hazard problem, the optimal solution would exhibit an unpalatable form of "exploitation of the talented": the marginal rate of income taxation could reach 100%.

Student loans are a very natural application for the theory of incentives or Mechanism Design under hidden actions and hidden types 8 . The general theory of optimum (or equilibrium) contracts under moral hazard, adverse selection and risk aversion is known to be a very hard problem (see [START_REF] Arnott | Moral Hazard and Competitive Insurance Markets[END_REF] for comments and further references to unpublished essays on this question, see also the recent synthesis of [START_REF] Boadway | Optimal Income Taxation with Uncertain Earnings: a Synthesis[END_REF] on optimal taxation with uncertain earnings). Solutions can be exhibited when principal and agent are both risk-neutral 9 . An extension of Rothschild and Stiglitz's insurance market model to moral hazard is proposed in the often quoted, but unpublished manuscript of [START_REF] Chassagnon | Insurance under Moral Hazard and Adverse Selection: The Case of Pure Competition[END_REF]. Our model is close to that of the latter contribution, but Chassagnon and Chiappori did not study the set of Pareto optima 10 .

The question of student loans is certainly not new in economics 11 . A few papers have recently proposed a theoretical study of income-contingent loans, or of the graduate tax, comparing alternative ways of subsidizing higher education, and performed numerical explorations of their welfare properties 12 . A much smaller number of papers come to grips with asymmetric information 13 . Finally, some contributions were devoted to education in an optimal income-taxation model. This question has been studied in static and two-period settings, see, e.g., [START_REF] Anderberg | Optimal Policy and the Risk Properties of Human Capital Reconsidered[END_REF], [START_REF] Bovenberg | Redistribution and Education Subsidies are Siamese Twins[END_REF], De [START_REF] Fraja | The Design of Optimal Education Policies[END_REF], Fleurbaey et al.

Rothschild and Stiglitz's approach to screening in insurance markets has been applied to banking, albeit with adaptations (see, e.g., [START_REF] Bester | Screening vs. Rationing in Credit Markets with Imperfect Information[END_REF]). Classic theories of credit contracts typically treat adverse selection and moral hazard separately (see [START_REF] Freixas | Microeconomics of Banking[END_REF]). The structure of second-best optima in insurance markets with pure adverse selection has been studied by [START_REF] Crocker | The E¢ ciency of Competitive Equilibria in Insurance Markets with Asymmetric Information[END_REF] and [START_REF] Henriet | E¢ ciency of Market Equilibria with Adverse Selection: The Case of Insurance[END_REF]. 9 See, e.g., [START_REF] Picard | On the Design of Incentive Schemes under Moral Hazard and Adverse Selection[END_REF] and [START_REF] Caillaud | Noisy Observations in Adverse Selection Models[END_REF]; see also the discussion in La¤ont and Martimort (2002, chapter 7). In the …eld of optimal regulation theory, a few contributions have dealt with special cases; see, [START_REF] Mcafee | Bidding for Contracts: A Principal-Agent Analysis[END_REF], [START_REF] Baron | Monitoring, Moral Hazard, Asymmetric Information, and Risk Sharing in Procurement Contracting[END_REF], La¤ont and Rochet (1998). 10 Recent work on the Principal-Agent model in the case at hand required advanced mathematical optimisation techniques (see [START_REF] Faynzilberg | On the Generalized Principal-Agent Problem: Decomposition and Existence Results[END_REF]) or used stochastic calculus, as in the asset-pricing, continuous-time …nance literature (see, e.g., Sung Jaeyoung ( 2005)). These intimidating technicalities mainly explain why we study a simple textbook model here, but it conveys, we think, the essential intuitions and ideas (and yet, some of the proofs are not straightforward).

11 See [START_REF] Friedman | Income from Individual Professional Practice[END_REF], [START_REF] Friedman | The Role of Government in Public Education[END_REF], [START_REF] Shell | The Educational Opportunity Bank: An Economic Analysis of Contingent Repayment Loan Programs for Higher Education[END_REF], [START_REF] Nerlove | Some Problems in the Use of Income-contingent Loans for the Finance of Higher Education[END_REF].

12 See, e.g., Hanushek et al. (2004), Garcia-Penalosa and[START_REF] Garcia-Penalosa | E¢ ciency and Equity E¤ects of Subsidies to Higher Education[END_REF], Del [START_REF] Rey | Financing Schemes for Higher Education[END_REF]. 13 De Fraja (2001) studies the interaction of unobservable skills and risk aversion in the allocation of higher education. [START_REF] Cigno | Scholarships or Student Loans? Subsidizing Higher Education in the Presence of Moral Hazard[END_REF] consider a microeconomic model of student loans with pure moral hazard. [START_REF] Chatterjee | Insuring Student Loans against the Financial Risk of College Failure[END_REF] propose a quantitative analysis of a model of student loans with moral hazard but they do not rely on Mechanism Design techniques as we do here. (2002). More recently, and closer to the present contribution, [START_REF] Findeisen | Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans[END_REF] have studied the combination of an income-tax with income-contingent loans in an optimal taxation model with endogenous investment in human capital; they use a more complicated model than us, with a continuum of types, but have recourse to numerical simulations. They reach similar conclusions about the usefulness of income-contingent reimbursement.

In the following, Section 2 describes the model and studies …rst-best optima. Section 3 is devoted to a preliminary analysis of asymmetric information and incentive constraints.

Section 4 presents the main results, characterizing the second-best optimum and discusses implementation by means of income-contingent student loans, the graduate tax, and the income tax. Section 5 studies what happens if some ingredients are removed in turn from the theory. Section 6 provides a complete characterization of the set of second-best optima.

The proofs are in the appendix.

A Simple Model

Basic Assumptions

We consider a population of students with the same von Neumann-Morgenstern utility u(:).

Students have two-dimensional unobservable characteristics: an ex ante type, denoted i and an ex post type, denoted k. To …x ideas, the reader may interpret the ex ante type as representing the individual's cognitive skills, while the ex post type represents job-markets skills and opportunities. Both factors in ‡uence the distribution of a given individual's future wages, albeit in di¤erent manners. The important di¤erence between the two type components i and k is the timing of information revelation. Both components are private information of individuals. The ex ante type i is known when the student decides how much to invest in education, while the ex post type is revealed only later, when the student enters the labor market. Thus, some important individual characteristics will be revealed in the future, when the student comes to grips with a real job. An individual is characterized by the pair (i; k).

We assume that each component can take one of two values only, i, k = 1; 2. We assume that student types are independent draws from the same distribution. The frequency of the ex ante type i in the student population is denoted i and of course, 1 + 2 = 1. 14

The model has three building blocks: the education phase, the job-market phase and the loan contract. We describe each of these blocks in turn (see Fig. 1 for an illustration of the model's timing). There are three consecutive lotteries: type i, success or failure and type k. At the end, the individual chooses ex post e¤ort.

Education

During the education phase, each student …rst learns his(her) type i, then chooses a quantity (or quality) of education q in a set f1; 2g. There are long (or highly-demanding) studies, i.e., q = 2 and short (or less demanding) studies, i.e., q = 1. 15 The cost of education is simply q for quality q; and we assume 2 1 . The chosen education depends on the ex ante type i only, since k is unknown when the decision is made. The educational choice q is assumed observable. Let q i be the education of (or chosen by) type i. The e¢ cient choice of q for type i will be q i = i, due to Assumption 4, presented below in sub-section 2.2. At the end of the schooling period, each student is successful or fails. To …x ideas, success may be interpreted as graduation, i.e., the student is granted a degree or not (but success may have slightly di¤erent interpretations). The individuals of both types have independent probabilities of success, that depend on individual e¤ort.

Each student chooses an ex ante e¤ort denoted e i . The ex ante e¤ort level is exerted before the realization of success. To …x ideas, we can view e i as study e¤ort. For simplicity, we assume that e i may be equal to 0 or to 1. The study-e¤ort cost is c i e i for type i, where c i > 0 is a parameter. Study e¤ort in ‡uences the probability of success in a type-dependent way. The probability of success of a student exerting ex ante e¤ort, e i , is denoted p i (e i ), where p i (e i ) = Pr( success j i and e i ):

To simplify notation, de…ne P i = p i (1), the probability of success under high ex ante e¤ort, and p i = p i (0), the probability of success under low ex ante e¤ort. Type 2, the "talented 14 i is also the marginal probability of i in the distribution of (i; k) 2 f1; 2g 2 . 15 If q represents a quantity, it measures, for instance, years of education, and if it's a quality, it may be a level in a hierarchy of certi…cates, or the quality of the educational institution.

type", is more likely to succeed given the e¤ort level. We assume that success is observable. This is how we model ex ante moral hazard.

Formally, we assume, Assumption 1. 0 < p i < P i < 1, i = 1; 2, and P 2 > P 1 ; p 2 > p 1 .

To sum up, types i and e¤orts e i are not observable, but the public authority observes the individual's wage w, the quality of education q i , which is recorded, and the event of success or failure.

Job market

We now describe the job-market phase. In case of failure, the student gets a basic job with the basic wage w. Success, the second chance move, will allow the student to …nd a job with a salary greater than the basic wage 16 .

In case of success, once the student graduated from university, a third chance move determines the student's ex post type k, or job-market skills. Depending on the realization of these skills, students will be able to occupy either a top job or a middle-range job. The probability of ex post type k, given success and education q, is de…ned as follows, Pr(k = q j q , success ) = 1 ;

and obviously, Pr(k 6 = q j q , success ) = . We assume that < 1=2. This may be interpreted as follows: with a small probability , an individual choosing q in f1; 2g will be endowed with ex post characteristics k 6 = q. In other words, a successful type i = 2 who chose q = 2 "normally" meets expectations and becomes a type (i; k) = (2; 2), with high realizations of both types, and can occupy a top job, but some individuals, namely, the types (i; k) = (2; 1), lack the necessary characteristics, in spite of being high quality graduates.

These students will …nd a middle-range job. In the same fashion, a successful type i = 1 with education q = 1 would "normally" become a type (i; k) = (1; 1) and obtain a middle-range job, but some of these students, namely, types (i; k) = (1; 2), "escape their fate" because they possess the necessary skills or characteristics to occupy the top jobs.

To sum up, the quality of education increases the probability of drawing a high job-

market type k = 2.
This assumption is reasonable. The increased probability may be due to knowledge, training, to peer e¤ects, or constitutes a return to network building during college years. It may also be due to some complementarity between job-market skills and higher education, these skills being in a sense catalyzed by education.

During the job-market phase, successful students choose e¤ort at work, or ex post e¤ort. The ex post e¤ort of a type (i; k) is denoted " ik , and the " ik may be equal to 0 or to 1. For simplicity, we assume that ex post e¤ort plays no role in case of failure. High e¤ort at work is required from individuals occupying top jobs and low e¤ort is the norm for middle-range jobs17 . The wage earned by a successful individual depends on education q and e¤ort at work " ik only. In other words, wages depend on type i through q only (academic talent is crystallized in q when q i = i), and type k determines e¤ort at work only through its impact on disutility (as indicated below).

Let !(q; ") be a function measuring the wage of a graduate of quality q when ex post e¤ort " is exerted. We set, !(q; 1) = W q ;

!(q; 0) = w q ;

where W q corresponds to a top job and w q to a middle-range job, q = 1; 2. This means that a successful type (i; k) with education q = i can earn a top salary W q , but only at the cost of high ex post e¤ort, " ik = 1; she obtains a middle-range salary w q otherwise. The above speci…cation is a simple way of representing the fact that, to a certain extent, a type i = 1

with education q = 1 can make up for a low expected productivity w 1 by means of high ex post e¤ort and obtain W 1 > w 1 . Conversely, a type i = 2 with education q = 2 can decide to exert low e¤ort and will in that case content herself with a middle-range salary w 2 < W 2 .

To sum up, there are 5 di¤erent wage levels: the base wage w, and w 1 , w 2 , W 1 , W 2 .

We now state a natural assumption on !(q; ").

Assumption 2.

(a) !(q; ") w for all q and ";

(b) !(2; 0) > !(1; 0);

(c) !(2; 1) !(2; 0) !(1; 1) !(1; 0) > 0;

(d) !(1; 1) !(2; 0).

Assumption 2b guarantees w 2 > w 1 . Assumption 2c implies W q > w q . The return on ex post e¤ort is higher when education is longer. As a consequence of Assumptions 2b and 2c, we also have,

W 2 W 1 w 2 w 1 > 0; (1)
and by 2d, the smallest W q is larger than the highest w q , i.e., W 1 w 2 . 18

The disutility of e¤ort-at-work is equal to ik " ik in state (i; k). Ex post types di¤er in this disutility. The total cost of e¤ort is additively separable and equal to c i e i + ik " ik . In a nutshell, the disutility of e¤ort at work of a type (i; 1) is assumed to be huge. We choose the simplest structure for the ik terms; precisely,

22 = 12 = b > 0; 11 = 21 = B > 0:
The meaning of these assumptions is that a type (i; 2) incurs a disutility b to get a top job, and a type (i; 1) incurs disutility B > b while trying to mimic a type k = 2. We assume that B is very large, in such a way that no individual endowed with a low ex post type will ever try to exert high e¤ort and earn the highest wage.

The public authority does not observe k and " ik , and will not be able to infer k and " ik from the observation of wages in all cases. The tax authority will face an informational problem, because a successful person with observed education q = q i and observed wage w q may be, either a type (i; 1), or a type (i; 2) that chose to behave as a type (i; 1) ex post. By assumption, the latter deviation is not observable. It follows that if the government taxes the income of types (2; 2) too much, these individuals will choose to occupy middle-range jobs (the same is true of types (1; 2) who can also occupy top jobs). This speci…c structure is 18 Assumption 2d is not crucial.

a way of combining a student-loan model with an elementary form of the optimal income-tax problem in a simple and relatively tractable manner, as will be seen below.

Loans

We now turn to the menu of student loans and the distribution of income. Consider a public (or publicly regulated) lending system and assume that the public lending authority distributes all loans. By assumption, a student loan covers the cost of education19 . So, the amount of a loan to an individual choosing education q is q . Students initially choose in a menu of loan contracts. Reimbursement is contingent on earnings and on the quality of education, both observed by the government. Let (R q1 ; R q2 ; r q ) denote the repayment pro…le of a loan of size q . A student choosing education q must repay, (i), r q in case of failure (and earnings are w),

(ii), R q1 if earnings are equal to w q , and …nally,

(iii), R q2 , if earnings are W q .
There are 6 possible repayment levels. This framework creates a Mirrleesian taxation problem, because the most lucky types can decide to reduce their e¤ort ex post if their income is taxed too heavily through repayments R 22 and R 12 .

A standard bank loan would be of the form (R q1 ; R q2 ; r q ) = (R q ; R q ; 0) with R q1 = R q2 = R q = (1 + ) q , assuming that denotes the interest rate and that the student does not reimburse the loan in case of failure: the ordinary bank loan is just a particular case in a much larger set of admissible contracts.

By de…nition, in this economy, an allocation is an array f(e i ; q i ; r i ; R ik ; " ik )g i;k=1;2 . A menu of contracts is an array f(q; R q1 ; R q2 ; r q )g q=1;2 . Given the above assumptions, we now write the resource constraint when (q; R q1 ; R q2 ; r q ) is chosen by type i = q only, i = 1; 2.

(i) With probability p i (e i )(1 ), the public banker collects the amount

X i = " ii R i2 + (1 " ii )R i1
from type i students;

(ii), with probability p i (e i ) , the public banker collects the amount

Y i = " ij R i2 + (1 " ij )R i1 ;
with j 6 = i, from type i students; and

(iii), with probability 1 p i (e i ), the public banker collects the amount r i from type i students.

Using the notation just de…ned, the resource constraint can be written,

X i i fp i (e i )[(1 )X i + Y i ] + (1 p i (e i ))r i i g 0; (RC)
with i = q. We assume that the lending system is self-…nanced, but of course, it would be easy to add an external source of funds: that would simply add a constant in RC. 20Finally, we formally assume that students are risk averse.

Assumption 3. The utility function u(:) is strictly increasing, strictly convave and continuously di¤erentiable.

First-Best Optimality

Let V i denote the ex post utility of a successful student with education q i = i, when the ex post type is k = i. Let v i denote the ex post utility of a student with education q i = i, when k 6 = i. The interim expected utility of a successful student who chose education q i = i is by de…nition,

U i = (v i ij " ij ) + (1 )(V i ii " ii ); (2)
where

v i = u[!(q i ; " ij ) Y i ]; V i = u[!(q i ; " ii ) X i ]:
with i 6 = j. The utility of an unsuccessful type i is denoted u i , and u i = u(w r i ).

The ex ante expected utility of a type i student, net of expected e¤ort costs, is simply

p i (e i )U i + (1 p i (e i ))u i c i e i ; (3)
where i = 1; 2.

If we assume that welfare is higher when all types i choose education q = i, i = 1; 2, the …rst-best utilitarian optimum can be obtained as a solution of the following problem,

Maximize X i i [p i (e i )U i + (1 p i (e i ))u i c i e i ] (4)
with respect to f(e i ; r i ; R ik ; " ik )g i;k=1;2 , subject to the resource constraint RC, and each e¤ort level is chosen in the set f0; 1g. In the above formulation of the …rst-best problem, it is understood that type i is assigned to education q i = i, for all i. This assumption is justi…ed below.

We call the solution of this problem a standard utilitarian optimum because the expected utility of each type is weighted by its frequency in the population. There are of course other optima that can be generated by varying the weight of type i in the social objective. These optima are studied in Section 6 below.

To determine the …rst-best e¤ort vector (e i ; " i1 ; " i2 ), we need to compute the optimal allocation of utility in 2 6 = 64 cases, that is, consider in turn each possible vector of e¤orts and compare the value of welfare for each of these combinations. But the only really interesting case is when (e 1 ; e 2 ) = (1; 1), i.e., high study e¤ort is required from both types, and

(" i1 ; " i2 ) = (0; 1), i.e., ex post e¤ort is high if and only if job-market skills are high.

It can be shown that e i = 1 is optimal for all i if the e¤ort costs c 1 , c 2 are small enough and if the di¤erence P i p i is large enough, that is, if e¤ort is su¢ ciently e¤ective in increasing the probabilities of success. We assume that this is indeed the case. In addition, if b is not too large, while B is very large, and if W q is su¢ ciently large as compared to w q , it will always be optimal to let the types (i; 2) choose high e¤ort ex post. In other words, under these assumptions, it is socially e¢ cient to require that all individuals endowed with k = 2 occupy high-pay jobs. It follows that we have,

V 2 = u(W 2 R 22 ); v 2 = u(w 2 R 21 ); V 1 = u(w 1 R 11 ); v 1 = u(W 1 R 12 ): (5)
Given the optimal choice of e¤orts, the expected wage Ew q of a student with education q is de…ned as follows,

Ew 1 = (1 )w 1 + W 1 ; Ew 2 = (1 )W 2 + w 2 :
The social surplus S i is de…ned as follows:

S i = P i Ew i + (1 P i )w i :
S i gives the expected social bene…t of education for a type i student choosing q = i and the "right"amounts of e¤ort, ex ante and ex post. We assume the following, Assumption 4.

(a) S i w for all i;

(b) P 2 [Ew 2 Ew 1 ] 2 1 P 1 [Ew 2 Ew 1 ].
Thus, we assume that an education of quality q i = i is pro…table, on average, for type i, as compared to the no-education alternative, with value w. Assumption 4a will be satis…ed, in essence, if !(q; 0) is high enough as compared to w, and if q is not too large.

Assumption 4b says that assigning type i to education q = i generates more surplus than choosing q 6 = i, given the probabilities of success P i . 21 This assumption eliminates some uninteresting corner solutions. Now, de…ne the inverse utility function,

z(x) = u 1 (x): (6)
By de…nition, z(u) is the minimal amount of resources needed to provide utility u. Using z(:), we obtain

R 22 = W 2 z(V 2 ); R 21 = w 2 z(v 2 ); R 12 = W 1 z(v 1 ); R 11 = w 1 z(V 1 ); (7)
and

r i = w z(u i ), i = 1; 2.
With this interpretation in mind, it is easy to see that the expected amount of resources needed to provide expected utility P i U i + (1 P i )u i to a type i with education q = i is given by the expression,

E(z i ) = P i [(1 )z(V i ) + z(v i )] + (1 P i )z(u i ): (8)
With the help of these de…nitions, the …rst-best optimality problem can be rewritten as follows. Eliminating R ii , R ij and r i from the objective and the resource constraint RC, and recalling that P i = p i (1), we obtain an equivalent expression for the …rst-best utilitarian optimum problem,

Maximize X i i [P i U i + (1 P i )u i c i ] (9)
with respect to (V i ; v i ; u i ) i=1;2 , subject to the resource constraint,

X i i fS i E(z i )g 0: (RC)
Given that the values of (e ; " ) are now given, the cost of e¤ort terms are just additive constants that play no role in the above maximization problem. The …rst-best problem becomes easy to solve.

We can state the following result.

Proposition 1. Under Assumptions 1-4, …rst-best Pareto optimality implies full insurance, that is, for all i,

V i = v i = u i ;
and the standard utilitarian …rst-best optimum exhibits full equality, i.e.,

V 1 = V 2 , v 1 = v 2 and u 1 = u 2 :
In addition, …rst-best e¢ ciency implies that the resource constraint is binding.

Proposition 1 is a standard consequence of risk aversion. Students are fully insured against labour market risk, but also against the risk of being of type 1.

This proposition describes an extremely idealized situation in which any degree of redistribution is possible, and politically acceptable.

We can also show the following.

Corollary 1. Under Assumptions 1-4, (i), there does not exist an optimum with r i 0 for all i;

(ii), if, in a …rst-best Pareto optimum, individuals obtain weakly more in case of success than if they were not educated, i.e., if W 2 R 22 w, and w 1 R 11 w then, …rst-best e¢ ciency implies r i 0 for all i, i.e., the students who fail receive insurance payments.

Corollary 1 shows that the public banker should also be an insurer. If we do not permit negative repayments (i.e., if the banker cannot be an insurer), it would now be easy to show that optimality implies r i = 0: we …nd a contingent reimbursement loan, in the ordinary sense that no repayment is required in case of failure 22 . In the analysis below, negative repayments are permitted.

Asymmetric Information and Incentive Constraints

Let us now study the case in which types are not observed by public authorities. By de…nition, second-best optimal (or interim e¢ cient) allocations maximize a weighted sum of the students'expected utilities, subject to resource-feasibility and incentive-compatibility constraints. Students self-select in a menu of contracts proposed by the public authorities. The allocation determines ex post utility values (V i ; v i ; u i ) and a quality of education q i for each type i. Although in principle, second-best e¤ort levels could be di¤erent from their …rst-best counterparts, we now assume that second-best e¤ort levels are equal to the …rst-best e¤ort levels. More precisely, we assume that the second-best optimal study e¤ort is high for all types, that is, e i = 1 for all i, and that second-best ex post e¤ort is high (" ik = 1) if and only if k = 2. This is at the same time a reasonable assumption and the only interesting case here, given that e¤ort variables are discrete. Other cases, in which some or all of the types exert zero e¤ort, could be studied in a very similar way. Again, high study e¤orts will be optimal if the ratios c i =(P i p i ) and the disutility cost b of ex post e¤ort are not too large, for otherwise, the social cost of providing incentives could be higher than the bene…ts of e¤ort in terms of aggregate surplus. The social bene…ts of e¤ort are clearly the increased probabilities of success and the increased productivity of agents on the labour market. In this model, the social cost of providing incentives is due to the addition of a dose of inequality and risk, imposed by incentive constraints, on top of the direct disutility of e¤ort itself.

Incentives

We …rst consider the ex post incentives. Students know their ex ante types. They will reveal their ex ante types by choosing the quality of their education: type i students choose education q i = i. This result will be obtained if the menu of loans f(q; R q1 ; R q2 ; r q ) q=1;2 g is ex ante incentive compatible. In the ex post stage, the students with education level q = 1 and ex post type k = 2 will not choose a job with a middle-range salary ex post if and only if,

v 1 b V 1 : (ICX 1 )
Similarly, a student with education level q = 2 and type k = 2 will not be tempted to behave as a type k = 1 ex post if and only if,

V 2 b v 2 : (ICX 2 )
We will see below that these constraints must be binding at a second-best optimum. The other constraints, that is, V 1 v 1 B and v 2 V 2 B will always be satis…ed since B is su¢ ciently large, as assumed above.

Given values of (V i ; v i ) satisfying ICX 1 and ICX 2 , we now have a generalized Principal-Agent problem in the sense of [START_REF] Myerson | Optimal Coordination Mechanisms in Generalized Principal-Agent Models[END_REF]. 23 We apply the extended revelation principle.

The constraints bearing on ex ante utilities and e¤orts are revelation and obedience constraints: the students should simultaneously self-select by choosing the right contract in the menu of loans and exert the right amount of e¤ort ex ante. The interim expected utility (i.e., the expected utility knowing success), can now simply be written as follows,

U 1 = (1 )V 1 + (v 1 b); (EU 1 ) U 2 = (1 )(V 2 b) + v 2 : (EU 2 )
Since we assume that high ex ante e¤ort is e¢ cient, incentive constraints can easily be written. First, the ex ante self-selection constraint IC i says that type i should not be tempted to mimic type j ex ante while exerting high e¤ort,

P i U i + (1 P i )u i P i U j + (1 P i )u j ; (IC i )
for all i = 1; 2 and j 6 = i. The cost of ex ante e¤ort being c i on both sides, it cancels from IC i . 24 Second, the ex ante moral hazard constraint M H i says that type i should prefer to exert high study e¤ort over low study e¤ort ex ante,

P i U i + (1 P i )u i c i p i U i + (1 p i )u i (M H i )
(Recall that p i = p i (0) and P i = p i (1)). In addition, the incentive constraint IC i says that type i prefers high study e¤ort to low study e¤ort and mimicking type j ex ante, that is,

P i U i + (1 P i )u i c i p i U j + (1 p i )u j ; (IC i ) with j 6 = i.
By de…nition, the second-best optimality problem is the following:

maximize X i i [P i U i + (1 P i )u i c i ]
subject to RC, IC i , M H i , IC i , ICX i and EU i .

Ex Post Moral Hazard

When the second-best optimality problem is posed as above, it is immediate that the variables (V i ; v i ) appear only in ICX 1 , ICX 2 and in the resource constraint RC. We can therefore decompose the problem as follows. Fix the value of U i . Then, ex post utility levels (V i ; v i ) must be chosen in such a way that they minimize the expected cost (1 )z(V i ) + z(v i ) subject to ICX i , i = 1; 2 and constraints EU i . This can be proved easily, using standard techniques. Intuitively, this minimization operation is just …nding the least costly way of providing incentives, given the ex post e¤ort-incentive constraints. In other words, the social planner should choose the minimal level of risk compatible with ICX i . It follows that both ICX i constraints must be binding at the second-best optimum. We can state this result formally.

Lemma 1. The ex post incentive constraints ICX i , i = 1; 2, must be binding at any secondbest optimum, that is,

v 1 = V 1 + b; and v 2 = V 2 b: (10)
Given this result and constraint EU , the (V i ; v i ) variables can be completely eliminated from the welfare maximization problem. We …nd convenient expressions for ex ante utility, namely,

U 1 = V 1 ; and U 2 = V 2 b: (11)
Eliminating (V i ; v i ) from RC, we obtain a modi…ed resource constraint, in which V i and v i are functions of U i , as follows.

V 1 = U 1 ; V 2 = U 2 + b; v 1 = U 1 + b; v 2 = U 2 : (12)
As a consequence, the resource constraint and all the incentive constraints are expressed in terms of (U i ; u i ) only.

Ex Ante Moral Hazard

We now study the role played by ex ante moral hazard or study-e¤ort incentives. It is not di¢ cult to see that M H i can be rewritten as,

(P i p i )(U i u i ) c i , or U i u i K i (M H i )
where by de…nition,

K i = c i P i p i : (13)
Moral hazard will thus force a gap between the reward of success and that of failure. It is natural to assume that type i = 2 is more e¢ cient than type i = 1 while exerting e¤ort 25 .

Formally, we assume the following.

Assumption 5.

K 1 K 2 0.
Can we now ignore at least one of the two moral hazard constraints? The answer is yes, under IC.

Lemma 2. Under Assumption 5, if IC 1 , IC 2 and M H 1 hold, then M H 2 is satis…ed.

Adverse Selection

We now study the implications of the ex ante incentive constraints. Since IC i can be rewritten P i (U i U j ) (1 P i )(u j u i ), we get the string of inequalities,

P 2 1 P 2 (U 2 U 1 ) u 1 u 2 P 1 1 P 1 (U 2 U 1 ): (IC)
Using IC, we then derive the following useful result.

Lemma 3. IC constraints imply the conditions, The proofs of all Lemmas are in the appendix. Being equipped with these preliminary results, we can now solve the optimal student-loan problem under asymmetric information.

U 2 u 2 U 1 u 1 ; (D) U 2 U 1 ; and u 1 u 2 : (D 0 ) (a) If

The Graduate Tax Problem

The second-best optimality problem described above can be called the graduate tax problem, since we can easily derive the optimal loan repayment schedule from the optimal solution, and a graduate tax can be used to implement this schedule. We then inquire whether the optimal graduate tax can be decomposed as the sum of an income tax, depending only on earnings, and a loan repayment, that depends only on education. The answer to this question is no, in general. To implement the optimal graduate tax, we need a loan repayment schedule depending on education and income, that is, an income-contingent repayment schedule, or else, we need an income tax that depends also on education, i.e., a graduate tax.

A further study of …rst-order conditions for optimality yields the following result, that may be called the equal treatment result.

Proposition 2. (Equal treatment as a second best under moral hazard and adverse selection.) Under Assumptions 1-5, for all < 1=2, the second-best optimal, standard utilitarian solution has the following properties:

U 1 = U 2 = U; u 1 = u 2 = u (equal treatment); U = u + K 1 (incomplete insurance);
RC, M H 1 , IC 1 , IC 2 , ICX 1 and ICX 2 are all binding. If, in addition, K 1 > K 2 , then, M H 2 , IC 1 and IC 2 hold as strict inequalities.

Proposition 2 shows that the second-best optimal solution exhibits equal treatment, in the limited sense that U 1 = U 2 and u 1 = u 2 , and incomplete insurance, since

U i = u i + K 1 .
The equal treatment property means that the post-study but pre-work expected utilities are equalized. In particular, the interim expected utilities U i of both ex ante types i are the same when students are successful. They are also the same conditional on failure.

The intuition for the equal treatment result is relatively easy to convey. Suppose for a moment that the planner observes the ex ante types i. If moral hazard is the only problem in the economy, the planner should impose the minimal amount of risk compatible with high ex ante e¤ort on each type, that is, U i u i = K i for all i. 26 The talented types being more e¢ cient, we have K 1 > K 2 and therefore, U 1 u 1 > U 2 u 2 . This contradicts condition D, which is a consequence of IC, i.e., of self-selection constraints, as shown by Lemma 3. If we now re-introduce the IC constraints, it is intuitively clear that the best is to choose the smallest admissible U 2 u 2 , that is, to set "equal di¤erences"U 2 u 2 = U 1 u 1 = K 1 . Under IC, equal di¤erences in turn imply u 1 = u 2 , as can easily be checked27 . So, we derived the equal treatment property. This shows why equal treatment is the result of the interplay of moral hazard and adverse selection.

However, inequalities will be created ex post, when the ex post types k are drawn and revealed. Indeed, the optimal solution typically involves ex post inequality, since

U 1 = U 2 implies v 1 = V 2 > V 1 = v 2 (

as explained by Proposition 3 and Corollary 3 below).

The solution also entails ex ante inequality. To see this, consider the ex ante expected utility of type i, namely,

Eu i = P i U i + (1 P i )u i .
The second-best optimum entails,

(Eu 2 c 2 ) (Eu 1 c 1 ) = (P 2 P 1 )K 1 + (c 1 c 2 ) > 0;
since c 2 c 1 . The talented types are strictly better o¤ ex ante. Both types obtain the same expected payment in the event of success as well as in the event of failure, but (i), insurance is incomplete: U i u i = K 1 , and (ii), the talented types have a higher probability of success,

P 2 > P 1 .
The optimal second-best allocation is trivially not …rst-best e¢ cient, since …rst-best e¢ ciency requires full insurance. The solution potentially entails a limited form of exploitation of the talented, by means of cross-subsidies between types, since the less-talented are also producing less surplus per capita. In the case described by Proposition 2, this subsidy from the talented is a price paid to solve the incentive problem. Indeed, both IC i constraints are binding and incentive compatibility holds since U 1 = U 2 . It is only when the social welfare function su¢ ciently favors the talented that the incentive problem is solved by means of screening, imposing a higher level of risk (and return) on the most productive agents (this is shown in Section 6 below).

To sum up, the optimal solution involves, (i), ex ante inequality of ex ante types, due to incomplete insurance; (ii), equality of interim expected utilities conditional on success or failure (i.e., equal treatment), and (iii), ex post inequality, conditional on success, due to the operation of the labor market and constraints on income taxation.

We now list the ex post utilities associated with the solution, and derive the optimal graduate tax schedule. Given that

U 1 = U 2 , we have V 1 + b = V 2 and therefore V 2 > V 1 (recall that b > 0). Recall that ICX i constraints impose, v 1 = V 1 + b; and v 2 = V 2 b:
We can therefore state the following result.

Proposition 3. In the standard utilitarian case, the second-best optimal allocation has the following properties,

u 1 = u 2 = u; v 2 = V 1 = u + K 1 ; v 1 = V 2 = u + K 1 + b; It follows that, (a) v 1 = V 2 > V 1 = v 2 ; (b) V 1 > u 1 .
The proof of Proposition 3 is very easy, given Proposition 2, and the relationships between U i , V i and v i that it implies. In practice, it follows from this result that the successful "selfmade man", i.e., type (1; 2), earns the same after-tax income than the successful graduate from a top school with the necessary job-market skills, i.e., type (2; 2), since v 1 = V 2 . A top-school student lacking the necessary job-market skills, i.e., type (2; 1), has the same after tax income than the plain student, i.e., type (1; 1), since V 1 = v 2 . These consequences of the optimal graduate tax become explicit when we derive the implications of Propositions 2 and 3 in terms of income-contingent repayment (or education-contingent income tax). The next result shows that the equal treatment property is achieved by means of an unequal ex post treatment of graduates.

Corollary 3. (Optimal graduate tax, repayment schedule)

The optimal graduate tax has the following properties, (a) u 1 = u 2 implies r 1 = r 2 (equal treatment in case of failure);

(b) V 2 > V 1 implies W 2 w 1 > R 22 R 11 (the talented are not fully exploited); (c) v 1 = V 2 implies R 22 > R 12 (self-made (wo)men repay less); (d) V 1 = v 2 implies R 21 > R 11
(top-school students lacking the job-market skills repay more);

In the statement of Corollary 3, points (c) and (d) show that successful individuals who studied longer repay more. Thus, in case of success, past education choices matter:

q i contributes to the determination of repayments. Point (c) is a consequence of v 1 = W 1 R 12 = W 2 R 22 = V 2 implying R 22 R 12 = W 2 W 1 > 0. Similarly, to prove point (d), we have, V 1 = w 1 R 11 = w 2 R 21 = v 2 implying R 21 R 11 = w 2 w 1 > 0.
In case of failure, point (a) shows that the lender "forgets" the choices of q 1 and q 2 : this is mainly due to the fact that ex post e¤ort does not matter for the students who failed.

Nobody can falsely report failure, and the base wage w does not depend on e¤ort or type.

In a more general model, (r 1 ; r 2 ) would be shaped by the tradeo¤ between incentives and insurance, and the planner might be willing to set di¤erent values for r 1 and r 2 . Point (b)

shows that the talented cannot be fully exploited (the marginal income tax rate is smaller than 100%): this is due to the presence of an ex post moral hazard problem, for otherwise, the planner would be tempted to tax the W 2 w 1 increment completely, in order to achieve greater equality.

In the …rst-best world, full insurance can be implemented with the help of a lumpsum tax: transfers would depend on type, success, and therefore, would also depend on education, insofar as ex ante types determine education. When negative repayments are permitted, it follows that income-contingent loans would be useless. Indeed, absent informational problems, the private lending sector could distribute standard loans e¢ ciently, while the government would protect individuals against risk and redistribute the income.

In contrast, in the second-best economy, income-transfer instruments are constrained by limited observation and incentives. The income tax is typically a function of observable income; standard loan repayments are proportional to years of education, with the addition of a bankruptcy clause that provides some form of insurance. These standard tools are not enough to achieve the second-best optimum. In particular, the loan repayments of successful individuals have no reason to be simply expressed as the product of an interest rate times the cost of education, that is, (1 + ) q .28 More ‡exible instruments are needed: appropriately designed income-contingent loans and (or) a graduate tax (with 100% subsidization of tuition fees) can do the required job.

In general, the optimal graduate-tax repayment schedule cannot be implemented by means of the addition of an ordinary income tax, (that would depend only on earnings) and of a standard loan repayment (that would depend only on education q i ). In other words, the optimal graduate tax cannot be expressed as the sum of two pieces: an income tax plus a standard loan repayment.

To be more precise, we would like to choose an income tax schedule

T i = T (w i )
and a loan repayment schedule L i = L(q i ), with q i = i, such that for any given 4 numbers (R 11 ; R 22 ; R 12 ; R 21 ) representing optimal repayments, we have,

R 11 = T 1 + L 1 ; R 22 = T 2 + L 2 ; R 12 = T 2 + L 1 ; R 21 = T 1 + L 2 : (14)
This implies

R 22 R 12 = L 2 L 1 = R 21 R 11 ; R 22 R 21 = T 2 T 1 = R 12 R 11 ;
which happens to be typically impossible. We can now state the non-decomposition result:

the optimal graduate tax is generically non-decomposable.

De…ne the quantity,

= (R 22 R 21 ) (R 12 R 11 ): (15)
The linear system stated above (i.e., ( 14)) implies that if the second-best repayments are decomposable, then = 0. We now show that for a generic (R 11 ; R 22 ; R 12 ; R 21 ), these relations will not be true and 6 = 0. Using the de…nitions of V i and v i , we …nd

= z(v 1 ) z(V 1 ) + z(v 2 ) z(V 2 ) + W 2 W 1 + w 1 w 2 : Using the properties V 1 = v 2 and V 2 = v 1 it is easy to see that, = (W 2 W 1 ) (w 2 w 1 ): (16)
By inequality (1), we have 0. We can have = only in a special case. Generically, we will have > 0. This proves the non-decomposability property.

We have just proved the following result.

Proposition 4. (Non-decomposition Theorem: justi…cation of income-contingent loan repayments) Under Assumptions 1-5, it is generically impossible to decompose the second-best optimal transfers R ik and r i as the sum of an income tax, depending only on earnings, and student-loan repayments, depending only on education. It must be that, either the student-loan repayments are income-contingent, or the income tax is education-contingent.

Application of Occam' s Razor

In the above theory, we have introduced 4 important ingredients: (i), wage risk and risk aversion; (ii), unobservable ex ante types leading to a screening, or self-selection problem;

(iii), unobservable study e¤ort generating ex ante moral hazard; (iv), unobservable ex post types and e¤ort causing an ex post moral hazard problem. Can we remove each of these ingredients from the model, to simplify the theory, and still obtain the key results? What do we …nd if we remove each of the ingredients in turn?

Without risk or risk aversion

If there is no wage risk, the problem posed above becomes rather trivial because, as soon as they would know their ex ante types, students could perfectly predict their future payo¤s.

There are two sources of risk in the model: random success and ex post type draws. In a riskless world, study-e¤ort choices would have deterministic e¤ects and ex post types would play no role.

If, on the other hand, we impose risk neutrality, the model becomes quite degenerate, because there is no private demand for insurance and no social motive for redistribution (i.e., no demand for insurance against the lack of talents, behind the veil of ignorance). Insurance is an essential element of student loan design. It follows that we need some risk and some degree of risk aversion.

Without ex ante moral hazard and ex ante screening

Before discussing what happens in the absence of moral hazard or of a screening problem ex ante, we brie ‡y consider the situation in which ex post moral hazard is the only problem,

i.e., the model in which ICX i inequalities are the only constraints. This can be called the "pure income taxation problem".

If we maximize i i (P i U i + (1 P i )u i ) subject to ICX i and RC, we …nd the following result.

Proposition 5. (Second best with observable ex ante types and observable study e¤ort.)

When the only constraints are RC, and ICX i , i = 1; 2, then, in the standard utilitarian case, the second-best optimal solution has the following properties:

u 1 = u 2 = u; Ez 0 i = z 0 (u);
for all i, where by de…nition,

Ez 0 i = z 0 (v i ) + (1 )z 0 (V i ).
The constraints RC, ICX 1 and ICX 2 are all binding.

Proposition 5 is a relatively direct consequence of the …rst-order conditions for optimality. The solution requires e¢ cient interim insurance given the remaining ex post moral hazard problem 29 . To achieve this goal, the marginal social cost of providing the basic 29 Remark that when = 0, the marginal social cost boils down to Ez 0 i = z 0 (U i ) and the condition

Ez 0 i = z 0 (u) boils down to z 0 (U i ) = z 0 (u), or U i = u.
welfare level u, i.e., z 0 (u), must be equated with Ez 0 i , the expected marginal cost of providing the interim expected welfare level U i = (1 )V i + v i . As a consequence, we have

z 0 (v 1 ) > z 0 (u 1 ) > z 0 (V 1 ) and z 0 (v 2 ) < z 0 (u 2 ) < z 0 (V 2 ) or equivalently, v 1 > u 1 > V 1 and V 2 > u 2 > v 2 .
These properties are unpalatable, because a student who failed ends up being better o¤ than an academically successful student with an unlucky draw of job market opportunities.

Without a screening problem

If there are no ex ante types, there is no ex ante screening problem; students do not selfselect by choosing their level (or quality) of education. In a simple version of the model, there would be only one level of education and all students would face the same probability of success, conditional on study e¤ort. Student loans would then be designed so as to induce e¤ort, that is, to solve the moral hazard problems, both ex ante and ex post, but the graduate tax decomposition problem would be vacuous, because we need at least two types (and two levels of education) to pose this problem in a meaningful way.

Yet, there is another possibility, which is to study a model with two fully observable ex ante types: each type of student would then be assigned to an education level by the authoritarian social planner. To study this case, it is enough to compute second-best optima while ignoring all self-selection constraints, that is, both IC i and IC i constraints. The only constraints taken into consideration are RC, M H i and ICX i . The results are quite di¤erent.

When ex ante types are public information, we …nd the following result.

Proposition 6. (Second best with observable ex ante types and moral hazard.) When the constraints are RC, M H i and ICX i , i = 1; 2, then, in the standard utilitarian case, the second-best optimal solution has the following properties:

U i = u i + K i ; for all i = 1; 2:
RC, M H 1 , M H 2 , ICX 1 and ICX 2 are all binding, and

Ez 0 1 = Ez 0 2 ;
where by de…nition,

Ez 0 i = P i [ z 0 (v i ) + (1 )z 0 (V i )] + (1 P i )z 0 (u i ).
We skip the proof of Proposition 6 here. This proof relies on a discussion of the Kuhn and Tucker conditions for optimality.

In the above statement, it is striking that the equal treatment property is no longer required. To see this, note that M H 1 and M H 2 are simultaneously binding. It follows that, if K 1 6 = K 2 , we cannot have U 1 = U 2 and u 1 = u 2 . In the "normal"case, that is, if

K 1 > K 2 ,
we …nd that IC constraints must be violated. Indeed, condition D (in Lemma 3 above) implies U 2 u 2 U 1 u 1 , or equivalently, K 2 K 1 , a contradiction.

If K 1 > K 2 , since the M H i are binding, we must have u 1 u 2 < U 1 U 2 . It follows that U 1 > U 2 is now a possibility, in particular if u 1 is close to u 2 . Thus, IC 2 may be violated.

In other words, better insurance against a lack of academic talents, in case of failure, would imply that high quality studies are not chosen, because type i = 2 students would all prefer q 1 . As a result, if ex ante types are assumed observable, the social planner would have to force the talented students to study longer.

The solution is indeed strikingly di¤erent from that of Proposition 2 above. Proposition 6 shows that equal treatment is a consequence of IC constraints. Interim equalization is not simply implied by the redistribution motive (risk aversion), it is a way of solving the self-selection problem. Absent self-selection constraints, Pareto-optimality does not impose equal treatment or full equality 30 .

We conclude that with observable ex ante types, we would lose an important and realistic feature of our model, namely, the fact that education is an object of individual choice, which is also used to screen individuals31 . In addition, the ex ante selection problem has a nontrivial in ‡uence on the shape of optimal repayment schedules.

With an ex ante screening problem but without ex ante moral hazard

If we remove study e¤ort from our theory, and if ex ante types are not publicly observable, the only constraints are RC, ICX i and IC i , i = 1; 2. In this case, we know that the best we could possibly achieve is equal treatment in case of failure with e¢ cient interim insurance, that is, u 1 = u 2 = u and Ez 0 i = z 0 (u) for all i (as in Proposition 5). But the self-selection constraints IC i would now impose U 1 = U 2 = U in addition (by Lemma 3c). Given RC, we would then have three equations to determine two variables, namely (U; u). This is typically impossible. It follows that the solution would not simply be the allocation of Proposition 5, with the addition of U = u: we would …nd Ez 0 i 6 = z 0 (u) for at least one i. Another way of looking at the same di¢ culty is to take the equal treatment solution of Proposition 2 and let the cost of study e¤ort go to zero, i.e., c i ! 0 for all i, so that, in the limit, we obtain K 1 = 0, and ex ante moral hazard disappears as an economic problem.

By continuity, the solution is now such that U = u, but we have imposed the constraint U 1 u 1 0, and this constraint is not a consequence of IC alone. In fact, we need a speci…c treatment of this case. The results are disappointing: absent M H i constraints, it can be shown that the structure of the optimal solution depends on the shape of inverse marginal utility z 0 (:).

Proposition 7. (Second best with unobservable ex ante types and observable study e¤ort.)

When the only constraints are RC, IC i and ICX i , i = 1; 2, then, in the standard utilitarian case, the second-best optimal solution may be an equal treatment or a separating allocation.

If is su¢ ciently small, (a) if z 0 (:) is concave, equal treatment is optimal:

u 1 = u 2 = u; U 1 = U 2 = U; Ez 0 2 z 0 (u) Ez 0 1 ; z 0 (u) is a weighted average of Ez 0 2 and Ez 0 1 , z 0 (u) = i i P i Ez 0 i i i P i ;
where, by de…nition, Ez 0 i = z 0 (v i ) + (1 )z 0 (V i );

(b) if z 0 (:) is strictly convex, IC 1 is binding and IC 2 is slack. We have U 2 > U 1 , u 1 > u 2 , and

z 0 (u 1 ) = Ez 0 1 > Ez 0 2 > z 0 (u 2 ):
(c) There is no solution such that IC 2 is binding and IC 1 is slack. The constraints RC and ICX i are always binding.

We skip the proof of this Proposition 32 . The result shows that the second-best optimal allocation depends on speci…c properties of the inverse marginal utility function z 0 = 1=u 0 .

The solution is less clear-cut than when ex ante moral hazard is taken into account. In fact, adding the M H i constraints simpli…es the solution and the proof of the optimum characterization result: the solution is easier to …nd under moral hazard, combined with adverse selection, than under adverse selection alone. The equal treatment solution described by Proposition 2 is also more robust in many ways. It is robust to changes in the values of key parameters and in the shape of the utility function, provided that the latter is strictly concave; it is also robust to changes in the weights of types in the social welfare objective, as shown by Proposition 2'below. Finally, in the student loan problem, it is not reasonable to assume that study e¤ort can be perfectly monitored, or that study e¤ort does not a¤ect the probability of success. We conclude that dropping ex ante moral hazard does not yield an equivalent, but more parsimonious theory.

Without ex post moral hazard

This case is easy to study; the solution can be obtained by letting ! 0: the ex post type k 6 = i is drawn with probability zero. Proposition 2 is still valid when ex post moral hazard does not play a role: the optimum exhibits equal treatment and incomplete insurance. The important di¤erence with the general case is now that V 1 = V 2 . These relations indicate that the talented are completely exploited. We have R 22 R 11 = W 2 w 1 . In other words, the marginal income-tax rate is 100% for earnings above w 1 . The student loan problem is now combined with a …rst-best taxation problem. This solution is clearly unrealistic. The 32 The proof is available upon request.

general conclusion of this discussion is that none of the "ingredients" of our model can be dropped without losing an important feature of the solution.

Non-decomposition results

If we look at the proof of the non-decomposition theorem (i.e., Proposition 4), we see that the proof does not depend on the …ne details of the optimal allocation. In particular, if equal treatment does not hold, i.e., if U 1 6 = U 2 , the non-decomposition theorem is still valid.

The non-decomposition property is still true in the simpli…ed versions of the model studied above. Under the assumptions of Proposition 5, for instance, when ex post moral hazard is the only problem, the non-decomposition property holds.

Proposition 8. (Non-decomposition result when ex post moral hazard is the only problem.)

When the only constraints are RC, and ICX i , i = 1; 2, then, in the standard utilitarian case, the second-best optimal transfers R ij cannot be decomposed as the sum of an income tax, depending only on earnings, and student-loan repayments, depending only on education.

It must be that, either the student-loan repayments are income-contingent, or the income tax is education-contingent.

Complete Characterization of Second-best Optima

Our goal is now to characterize all second-best Pareto optima in our model. To this end, we will consider weights for ex ante types that may di¤er from their frequencies in the population. Let 1 and 2 be the weights of type i = 1 and type i = 2 in the welfare function. We assume 1 + 2 = 1 without any loss of generality, and i > 0 for all i. A …rst-best optimum can be obtained as a solution of the following problem,

Maximize X i i [p i (e i )U i + (1 p i (e i ))u i c i ] (17)
with respect to f(R i ; R 0 i ; r i ; e i ; " ik )g i;k=1;2 , subject to the resource constraint RC, and e i , " i1 , " i2 are equal to 0 or 1.

Under the assumptions made above, using the inverse utility function z(x) = u 1 (x) and eliminating R ik and r i from the objective and the resource constraint RC, the secondbest optimality problem can be rewritten as follows:

Maximize X i i [P i U i + (1 P i )u i c i ] (18)
with respect to (V i ; v i ; u i ) i=1;2 , subject to the resource constraint RC and all incentive and moral-hazard constraints, exactly as de…ned in Section 3 above. The statement of Proposition 2 is in fact a particular case of a more general result, called Proposition 2', that we now state.

The proof of this proposition is provided in the appendix.

Proposition 2' . (Equal treatment as a second best under adverse selection, ex ante and ex post moral hazard.) Under Assumptions 1-5, for all < 1=2, there exists an open interval ( 2 ; 2 ), including 2 , such that if 2 2 ( 2 ; 2 ), then, the second-best optimal solution has the following properties:

U 1 = U 2 = U; u 1 = u 2 = u (equal treatment); U = u + K 1 (incomplete insurance);
where U, u are functions of . Constraints RC, M H 1 , IC 1 , IC 2 , ICX 1 and ICX 2 are all binding. If, in addition, K 1 > K 2 , then, M H 2 , IC 1 and IC 2 hold as strict inequalities.

Proposition 2'shows that the result of Proposition 2 can be extended: the equal treatment optima are obtained for all weights in a neighborhood of type frequencies .

At this point, several remarks can be made. We …rst …nd a local invariance property:

when is close to , the optimal solution does not depend on .

Corollary 2. The second-best optimal solution of Proposition 2'is independent of , when 2 is small enough, i.e., if 2 < L 2 , we have, @ @ 2 (U 2 ; u 2 ; U 1 ; u 1 ) = 0:

Proof of Corollary 2: The second-best allocation is the solution of a system of four equations with four unknowns: (i), U 1 = U 2 ; (ii), u 1 = u 2 ; (iii), U 1 = u 1 + K 1 ; and (iv), given these constraints, RC pins down u 1 = u. None of these equations involve . Q.E.D.

For the sake of completeness, the remaining question is to …nd the second-best optimal solutions when 2 > 2 . We look for a second-best allocation in which a single IC constraint is binding. By Lemma 5, proved in the appendix, we know that IC 1 must be the binding

[P 2 (1 P )Ez 0 2 P (1 P 2 )z 0 (u 2 )] (P 2 P 1 ) = P 1 Ez 0 1 + (1 P 1 )z 0 (u 1 ); (K)
where P = 1 P 1 + 2 P 2 and Ez 0

i = (1 )z 0 (V i ) + z 0 (v i ).
We have completely characterized the second-best optima under adverse selection, ex ante and ex post moral hazard, in the optimal graduate tax problem. Again, since the Kuhn-Tucker conditions are necessary and su¢ cient, we have shown that when 2 > 2 , the solution is that described by Proposition 9.

Proposition 9 shows that the second-best optimum is a separating allocation à la Rothschild-Stiglitz, i.e., U 2 > U 1 > u 1 > u 2 , when the weight of the talented types is su¢ ciently higher than their frequency in the population. In other words, to get a separating optimum, the social planner must be willing to markedly favor the talented ex ante types.

It is still true that types are ex ante unequal since, using IC i and M H i constraints, we …nd that Eu 2 = u 2 + P 2 (U 2 u 2 ) > u 2 + P 1 (U 2 u 2 ) = u 1 + P 1 (U 1 u 1 ) = Eu 1 :

But the talented types are now less well insured in case of failure, since u 1 > u 2 .

Let us now study the possible decomposition of transfers in separating optima. To this end, compute,

= (R 22 R 21 ) (R 12 R 11 ):
If the repayment schedule can be decomposed, equations ( 16) above must hold, and we must have = 0. Using ( 7) and ( 11)-( 13), i.e., the de…nitions of R ij and the consequences of Lemma 1, we …nd = [z(U 1 + b) z(U 1 )] [z(U 2 + b) z(U 2 )] + (W 2 W 1 ) (w 2 w 1 ):

By Assumption 2, we have (W 2 W 1 ) > (w 2 w 1 ). Since z(:) is increasing and strictly convex, and since by IC, we must have U 2 > U 1 , it must be that

z(U 2 + b) z(U 2 ) > z(U 1 + b) z(U 1 ):
Therefore, the sign of is ambiguous, but has no reason to be exactly zero: nondecomposability is generic. If we draw the model's basic elements from a suitably dispersed distribution, that is, if we choose at random a strictly concave utility function, u(:), productivities w q and W q , and a vector of parameters (w; ; b; i ; i ; p i ; P i ; c i ; i ) i=1;2 satisfying the assumptions stated above, then, quantity will almost surely be nonzero. Indeed, the di¤erence (W 2 W 1 ) (w 2 w 1 ) does not vary with 2 , while U 2 and U 1 do depend on 2 through condition K. This shows intuitively that varies with 2 and can be zero only "by chance"on a negligible subset of the admissible models.

Conclusion

We have studied optimal student-loan contracts in a simple private information economy with unobservable student types. Types di¤er in the probability distributions of individual labour-market outcomes (adverse selection). Future earnings are risky. Students are risk-averse and choose an ex ante e¤ort variable, that is not observed by the lender (moral hazard) but a¤ects the probabilities of academic success. This poses an optimal insurance problem. Students can also reduce their e¤ort ex post and thus reduce their earnings below potential. This poses an additional income taxation problem à la Mirrlees. We completely describe the set of second-best optimal (or interim e¢ cient) incentive-compatible menus of loan contracts. There are two types of optima: the separating and equal treatment allocations. Equal treatment arises when the social weights of types are in the neighborhood of their frequencies in the student population. In this case, the interim expected utility of students of di¤erent types are equalized, conditional on the student's observable success or failure. However, students are ex ante unequal since they di¤er in their probability of success.

In addition, students are ex post unequal since the second-best allocation trades o¤ incentives and insurance-redistribution motives. This type of allocation is di¤erent from the familiar menus of separating contracts in screening models à la Rothschild-Stiglitz. The separating menus, in which the talented students bear more risk than the less-talented ones, appear only if the weight of talented types is su¢ ciently greater than the latter type's frequency. In both cases, the optimal menus of contracts exhibit incomplete insurance, as a consequence of moral hazard; they typically involve cross-subsidies in favor of the less-talented. The less-talented obtain the maximal amount of insurance, compatible with e¤ort incentives.

Optimal student loans are always income-contingent, even in the presence of an income tax.

In other words, the second-best transfers cannot be decomposed as the sum of an income tax, depending only on earnings, and a loan repayment, depending only on education. It must be that the optimal loan-repayments are income contingent, or that the income tax is itself education-contingent. The student-loan contracts can be interpreted as a form of graduate tax. Future work on this topic should try to solve the second-best optimality problem with more than two types, study the case of continuous e¤ort and also the case in which repayments are constrained to be positive.

and the resource constraint must be binding. If, in addition, i = i , we get full equality, i.e., V 1 = V 2 , v 1 = v 2 and u 1 = u 2 . Q.E.D.

Proof of Corollary 1

(i) There does not exist an unconstrained optimum with r i 0 for all i. If such an optimum did exist, then, because of full insurance, we would have !(q i ; " ii ) R ii = !(q i ; " ij ) R ij = w r i w and therefore, i i Ez i w < i i S i , a contradiction, since resources would then be wasted.

(ii) First-best e¢ ciency implies full insurance and full insurance in turn implies !(q i ; " ii ) R ii = w r i . This yields !(q i ; " ii ) R ii w = r i . So, if we require !(q i ; " ii ) R ii w, we must have r i 0. Q.E.D.

Incentives. Proofs of Lemmas

Proof of Lemma 2 :

Adding up the IC i constraints immediately yields (P 2 P 1 )(U 2 U 1 ) (P 2 P 1 )(u 2 u 1 )

and P 2 > P 1 in turn implies U 2 U 1 u 2 u 1 . With M H 1 , we then obtain the following string of inequalities:

U 2 u 2 U 1 u 1 K 1 K 2 :
It follows that M H 2 is satis…ed.

Q.E.D.

Proof of Lemma 3 :

As in the proof of Lemma 2, adding up the IC i constraints and P 2 > P 1 immediately yields U 2 U 1 u 2 u 1 : this is property D. Given IC, since P 2 > P 1 , if U 1 > U 2 we …nd a contradiction. IC above shows that U 2 U 1 implies u 1 u 2 0. This yields D 0 .

The proof of Lemma 3a is trivial, since P 2 > P 1 . Lemma 3b follows from the fact that u 1 = u 2 and IC imply U 2 U 1 0 U 2 U 1 and therefore U 2 = U 1 . But under IC, we also have that U 2 = U 1 implies u 1 = u 2 . Q.E.D.

  Assumption 5 is not needed for the proof of Lemma 3. The IC i constraints are an added di¢ culty, but we can in fact ignore them, as shown by Lemma 4.

	Lemma 4. Under Assumption 5,
	(a) if IC i , i = 1; 2 and M H 1 hold, then IC 1 is satis…ed.
	(b) if IC 2 is satis…ed, and if, in addition, IC 1 and M H 1 are binding, then, IC 2 is
	satis…ed.

IC 1 and IC 2 are simultaneously binding, then U 2 = U 1 and u 1 = u 2 : we get equal treatment (but not necessarily full insurance).

(b) Under IC 1 and IC 2 , we have u 1 = u 2 if and only if U 2 = U 1 .

Lemma 3 says that if equal treatment does not hold, then, either IC 1 or IC 2 is binding, or none of them, but not both.

  constraint, and we can show that this happens only if 2 > 2 . IC 1 is the binding constraint; IC 2 is slack;(b) constraints M H 1 , RC, ICX 1 and ICX 2 are binding;(c) we have U 2 > U 1 > u 1 > u 2 and, necessarily, 2 > 2 .(d) The second-best solution is fully determined by the following 4 equations: IC 1 , M H 1 and RC, expressed as equalities, and the condition,

	Proposition 9. (Separating Second-best Optima) Under Assumptions 1-5, for all < 1=2,
	if a second-best optimum has only one binding IC constraint, then,
	(a) 2
	1 2

This property could also be de…ned as interim equalization.

See Rothschild and Stiglitz (1976),Wilson (1977),[START_REF] Miyazaki | The Rat Race and Internal Labor Markets[END_REF],[START_REF] Spence | Product Di¤erentiation and Performance in Insurance Markets[END_REF].

It is well-known that microeconomic models of insurance and models of banking are formally close.

The government observes success (i.e., graduation). This assumption leads to some simpli…cations, because it allows us to reduce the number of ex post incentive constraints taken into consideration, again without losing anything essential in the analysis.

The fact that zero ex post e¤ort is the "norm"for middle-range jobs is just a convenient normalization.

This is a restriction because we could have considered partial funding, but this would have required consideration of a third dimension of types: initial wealth or parental wealth. Under the simplifying assumption that all students have the same initial wealth, we would have then studied the possible impact of credit rationing on incentives.

The government may collect some taxes on tuition payments i , so that the appropriate de…nition of should be net of any such taxes.

Note that Ew 2 > Ew 1 since under Assumption 2, (1 )W 2 + w 2 > (1 )W 1 + w 1 > (1 )w 1 + W 1 .

This case is studied in a working paper; see,Gary-Bobo and Trannoy (2012).

On this notion, see also[START_REF] Jean | The Theory of Incentives: The Principal-Agent Model[END_REF]

The cost of ex ante e¤ort could depend on q i . For instance, we could easily handle the following more general formulation: C i (q; e i ) = d(q) + c i e i , but this would not change the conclusions.

1=K i measures the e¢ ciency of e¤ort.

This is shown in Section 5 below.

This is because we must have P i (U i u i ) + u i P i (U j u j ) + u j for all i 6 = j.

But q itself may be a function of the interest rate on government bonds, and the latter rate would thus of course play a role in the resource constraint.

This point is also shown by the proof of Proposition 1, in the Appendix: full equality is not a consequence of …rst-best optimality, whereas full insurance is implied by …rst-best e¢ ciency. The two things are di¤erent here. In the next subsection, Proposition 2'shows that equal treatment is a second-best device: it is imposed to take care of ex ante self-selection constraints, even when it is not a requirement of …rst-best optimality.

This has been discussed by[START_REF] Spence | Job Market Signaling[END_REF], although in the present paper, we model screening by means of self-selection in a menu of contracts as in[START_REF] Rothschild | Equilibrium in Competitive Insurance Markets[END_REF].

, such that ( ) > 0. In addition, ( ) > 0 for all values of 2 2 since 2 ! 0 implies P ! P 1 < P . We conclude that there exists an interval ( 2 ; 2 ) of values of 2 , including

under Asymmetric Information". This research has been supported by a research grant of the ANR "programme blanc".

the College Drop-out Decision: A Direct Approach Using a New Panel Study", American Economic Review, 98, 2163-2184. Sung Jaeyoung (2005), "Optimal Contracts under Adverse Selection and Moral Hazard: A Continuous-time Approach", Review of Financial Studies , 18(3), 1021-1073. Wilson, Charles (1977), "A Model of Insurance Markets with Incomplete Information", Journal of Economic Theory, 16, 167-207.

Appendix: Proofs

Maximizing a weighted sum of expected utilities.

Our goal is to characterize all …rst-best and second-best Pareto optima in our model. To this end we consider weights for ex ante types that may di¤er from their frequencies in the population. Let 1 and 2 be the weights of type i = 1 and type i = 2 in the welfare function.

We assume 1 + 2 = 1 without loss of generality, and i > 0 for all i.

Under the assumptions made above, the …rst-best optimality problem can be rewritten as follows: maximize i i [P i U i + (1 P i )u i ] with respect to (V i ; v i ; u i ) i=1;2 , subject to the resource constraint RC, exactly as de…ned in the text above. Second-best optima are obtained while maximizing the same objective function, subject to the resource constraint, plus all incentives and moral hazard constraints.

First-best e¢ ciency

Proof of Proposition 1.

The …rst-best optimality problem is a convex programming problem, since z(:) is a convex function and the objective is a linear function of utility levels V i , v i , and u i . To write the …rst-order necessary conditions for optimality, let denote the Lagrange multiplier of the resource constraint. We …nd, for i = 1; 2,

for all i. This immediately yields,

It follows that …rst-best optimality implies full insurance, that is, for all i,

Proof of Lemma 4 :

(a) If IC 1 holds, then,

and since under IC, U 2 U 1 0, and we assumed P 1 > p 1 , we also have

and rearranging terms we get the equivalent inequality,

but this is exactly IC 1 .

(b) Given that M H 1 is binding, IC 2 can be expressed as follows,

Combining IC 1 and M H 1 , holding as equalities, we easily obtain,

Substituting the value of u 1 derived from (IC 1 + M H 1 ) in (IC 2 + M H 1 ) yields, after some rearrangement of terms,

Dividing both sides by (P 2 p 2 ) > 0 and rearranging terms, we obtain,

From condition D and M H 1 we know that U 2 u 2 U 1 u 1 = K 1 . In addition, (P 2 P 1 )=(P 2 p 2 ) = 1 + (p 2 P 1 )=(P 2 p 2 ) > 0. Hence, the following string of inequalities:

since, by Assumption 5, K 1 K 2 . This shows that condition Z is satis…ed, hence, IC 2 is satis…ed when IC 2 holds and when IC 1 and M H 1 are binding.

Q.E.D.

Optimal graduate tax

Before the proof of Propositions 2 an 2', we state a useful preliminary result. Lemma 5 gives indications on the constraints that must be binding at a second-best optimum. There is an asymmetry between the two IC i constraints. Lemma 5 shows that, if a single IC constraint is binding at the optimum, this constraint must be IC 1 .

Lemma 5. At a second-best optimum, (a) RC is binding;

(b) if IC 1 and IC 2 are binding, then, M H 1 must be binding;

(c) if IC 2 is binding, then, IC 1 must be binding too.

Proof of Lemma 5 :

Using Lemmata 1-4, the second-best optimality problem can be further simpli…ed. The benevolent public banker should maximize i i

To study this problem, we will also temporarily ignore (i.e., relax) constraint IC 2 and check at the end that it is indeed satis…ed. Let , , 1 and 2 be the nonnegative Lagrange multipliers of, respectively, constraints RC, M H 1 , IC 1 and IC 2 . The …rst-order conditions (i.e., Kuhn-Tucker conditions) for the second-best optimality problem are the following. For i; j = 1; 2,

with, by de…nition,

and with the complementary slackness conditions, i.e.,

where j 6 = i. These conditions are necessary and su¢ cient for an optimum, because as noted above, the problem is convex. It follows from this that, if we …nd a solution in which all multipliers are nonnegative, we have found the solution.

Proof of Lemma 5(a) Adding equations FOC1 to FOC4, we easily …nd,

so that > 0, since z 0 (:) > 0. Hence, by CS 2 , it follows that RC is binding.

Suppose that M H 1 is slack, i.e., U 1 > u 1 + K 1 , at the second-best optimum, then by CS 1 , we have = 0. With = 0, F OC1 and F OC3 form a linear system in ( 1 ; 2 ); that is,

This system has a nonzero determinant, equal to P 2 P 1 > 0, and a unique solution

Since b > 0 and since M H 1 implies U > u + K 1 , it follows that Ez 0 1 > z 0 (U ) and thus, z 0 (u) Ez 0 1 < 0. We conclude that 2 < 0. This is a violation of Kuhn-Tucker conditions, since all multipliers must be non-negative. We have found a contradiction. This proves 5b.

Proof of Lemma 5(c):

If IC 2 is binding, and IC 1 is slack, then, by CS 31 , we have 1 = 0.

Using F OC2 and F OC4, we easily obtain, 2 z 0 (u 2 ) = 2 + 2 = 2 Ez 0 2 and therefore, z 0 (u 2 ) = Ez 0 2 . Using the de…nition of Ez 0 i given above, Ez 0

we …nd Ez 0 2 > z 0 (U 2 ). But IC and M H 1 together imply

It follows that we have a contradiction since U 2 > u 2 and hence

Proof Propositions 2 and 2':

The FOCs can be rewritten,

Our optimum candidate exhibits equal treatment,

and both IC constraints are binding. By Lemma 5 above, M H 1 must then be binding too.

This imposes U ( ) = u( ) + K 1 . Adding the four F OCb equations easily yields, = ( ), satisfying the relation,

where, by de…nition, P = P 1 1 + P 2 2 , and where,

for i = 1; 2. We know that RC is binding and it follows that (U ( ); u( )) is fully determined by the intersection of RC and M H 1 . It is easy to check that the solution (U ( ); u( )) is a continuous function of for 0, and this solution does not depend on 2 .

We must check that the associated multipliers i ( ) are nonnegative. F OC2b and F OC4b provide us with a linear system of equations for ( 1 ; 2 ). The determinant of this system is P 2 P 1 > 0, so that there is a unique solution ( 1 ( ); 2 ( )). We easily derive,

Remark …rst that Ez 0 2 > Ez 0 1 > z 0 (U ( )) > z 0 (u( )), since U ( ) > u( ) and < 1=2. Using these inequalities, expression F OC above, and the fact that z 0 > 0, we derive 1 > ( )z 0 (u( )); and 1 < ( )Ez 0 2 :

(28)

From these remarks, we immediately obtain 1 ( ) > 0. Note that this property is true for all values of 2 in (0; 1) since 1 ( ) does not depend on 2 . For the same reasons, ( ), given by equation F OC, does not depend on 2 , and thus, we always have Ez 0 2 > z 0 (u( )) and ( ) > 0. Given the inequalities derived above, for any 2 2 , we also have ( )Ez 0

2

( 2 = 2 ) > 0, and therefore, using P 2 > P 1 , we …nd

In particular, if 2 = 2 , then 2 ( ) > 0. By continuity, there exists an interval of values of 2 , including 2 , such that 2 ( ) is positive. By Lemma 4, we know that both IC i constraints are satis…ed. By Lemma 2, we know that M H 2 is also satis…ed.

Finally, we must check that the multiplier = ( ) > 0. Adding F OC3b and F OC4b,

we derive an expression for ( ), that is,

where by de…nition, P = 1 P 1 + 2 P 2 . This can be rewritten

Since we have 1 > ( )z 0 (u( )), this means that there is an interval of values of 2 , around 2 , such that all Lagrange multiplers are positive. Therefore, we have found the optimal solution for these values of 2 and .

Q.E.D.

Generic non-decomposability of the graduate tax

Proof of Proposition 8.

If the ICX i are the only constraints, Proposition 6 tells us that we must have Ez 0 2 = Ez 0 1 at the optimum, or equivalently,

We can easily prove by contradiction that the latter condition implies

To prove condition H8, suppose …rst that V 1 V 2 . Then, from equation G8, we have,

but this is impossible since our assumption also implies V 2 b < V 1 + b, and it must be that the left hand side is non negative, while the right-hand side is strictly negative. We conclude

Then, given condition G8, we derive,

If the repayment schedule can be decomposed, equations ( 16) above must hold, and we must have = 0. Using (7), the de…nitions of R ij and the fact that ICX i constraints are binding, we …nd,

By Assumption 2, (W 2 W 1 ) > (w 2 w 1 ). Since z(:) is increasing and strictly convex, given condition H8 above, it must be that

Therefore, > 0: we conclude that the repayment schedule is not decomposable. Q.E.D.

Separating optima

Proof of Proposition 9:

If a second-best optimum has only one binding IC constraint, then, by Lemma 5, IC 1 must be binding and IC 2 is slack. Hence, 2 = 0. By Lemmas 2 and 4, we can neglect M H 2 and IC i constraints. By Lemma 1, ICX i constraints are binding. Thus, using (13), de…ne

Adding equations F OC1 to F OC4, we easily obtain, 1= = i i (P i Ez 0 i + (1 P i )z 0 (u i )) > 0. Hence, RC is binding. Suppose now that M H 1 is slack. Then, = 0. From F OC1b and F OC3b, we easily derive,

Hence, z 0 (u 1 ) = Ez 0 1 . But this leads to a contradiction, since M H 1 imposes u 1 + K 1 < U 1 and therefore, Ez 0 1 > z 0 (U 1 ) > z 0 (u 1 + K 1 ) > z 0 (u 1 ):

We conclude that M H 1 must be binding and U 1 = u 1 + K 1 . From F OC2b, and the requirement that 1 0, we derive

From F OC1b, we derive,

Combining A9 and B9, we obtain,

We must compare Ez 0 2 and Ez 0 1 . It is easy to see that Ez 0 2 > Ez 0 1 if and only if,

To prove that D9 is true, assume that Ez 0 1 Ez 0 2 . Then, using U 2 > U 1 , we must have,

(

but this leads to a contradiction since, obviously, U 1 + b > U 1 and < 1=2. This proves D9.

Now, it follows from C9 and D9 that we must have 2 (1 2 ) > (1 2 ) 2 , or 2 > 2 . Combining A9 and B9, and assuming that > 0, we obtain

The allocation U 1 , U 2 , u 1 , u 2 is determined by a system of four equations, the …rst three are obviously IC 1 , M H 1 and RC, expressed as equalities. To …nd the fourth equation, we eliminate Lagrange multipliers from F OC1b, F OC3b and F OC4b. More precisely, adding F OC1b and F OC3b yields 1 = 1 [P 1 Ez 0 1 + (1 P 1 )z 0 (u 1 )] 1 + 2 : (F 9)

On the other hand, substituting A9 yields,

or equivalently, P = 2 P 2 Ez 0 2 + 1 P 1 [P 1 Ez 0 1 + (1 P 1 )z 0 (u 1 )]: (G9)

Note that if G9 holds, since Ez 0 1 > z 0 (u 1 ), then, necessarily, E9 holds: this con…rms that > 0. Substituting the expression for , derived above, in G9, and rearranging terms, yields the fourth equation that we need to solve the problem, 2 1 2 [P 2 (1 P )Ez 0 2 P (1 P 2 )z 0 (u 2 )] (P 2 P 1 ) = P 1 Ez 0 1 + (1 P 1 )z 0 (u 1 ) (K)

The second-best optimum (U 1 ; u 1 ; U 2 ; u 2 ) is fully determined by RC, M H 1 ; IC 1 expressed as equalities and condition K. The condition 1 0 yields a lower bound on the values of 2 that can be derived from A9, that is, equivalently, from C9, we have,

Ez 0 2 : (H9)

Summing the F OCs, we found 1= = i i (P i Ez 0 i + (1 P i )z 0 (u i )). Since M H i implies Ez 0 i > z 0 (u i ), and since by D9, Ez 0 2 > Ez 0 1 , we must have 1= < Ez 0 2 or Ez 0 2 > 1. We conclude from H9 that 2 > 2 is required for this type of solution to be optimal. Q.E.D.