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

Transcription

1 Optimal Student Loans and Graduate Tax: under Moral Hazard and Adverse Selection Robert Gary-Bobo Sciences Po, 2014

2 Introduction (1) Motivation Preview of Results Literature Student loans are quantitatively important: in the US, outstanding student debt reached $ 1 trillion in In the UK, the rise in tuition fees caused a sharp increase in student debt. Economic downturn caused a rise in the number of students with repayment problems (for instance in the US). Debates on the optimal design of these loans: should student-loans be income contingent? Should there be interest-rate subsidies?

3 Introduction(2) Motivation Preview of Results Literature In Australia and the UK, student loans are income contingent. In France, student loans play a negligible role but a severe shortage of public funds should be expected and a substantial rise in tuition fees, going hand in hand with the development of student loans, may help universities. Student loans and tuition fees are not necessarily unfair or regressive. If loan repayments are based on the graduates s earnings, income-contingent loans are close to a graduate income tax, and may play a role in redistribution.

4 Introduction(3) Motivation Preview of Results Literature There is an important econometric literature on the impact of credit constraints on college (i.e., university) attendance; Carneiro and Heckman (2002), Keane and Wolpin (2001), Stinebrickner and Stinebrickner (2008). A few quantitative studies have been devoted to student-loan policies in the US: Ionescu (2009), Lochner and Monge-Naranjo (2010), Chatterjee and Ionescu (2011); S. Gregoir in France. To the best of our knowledge, there is very little work on the microeconomic theory of student loans. See however, Cigno and Luporini (2009), Del Rey Racionero (2010), Garcia-Penalosa and Wälde (2000), Hanushek et al. (2004),

5 Introduction 4 Motivation Preview of Results Literature We propose a normative study of student loans under asymmetric information. We study the combined effects of four ingredients : risk aversion, adverse selection, ex ante moral hazard and ex post moral hazard, on the optimal design of student loans. We explore the structure of the set of second-best optimal (interim incentive-efficient) allocations of credit. Students are risk-averse and future earnings are subject to risk. Individual "talent" and individual "effort" are not (perfectly) observed by the lender.

6 Preview of model (1) Motivation Preview of Results Literature We consider an economy with two-dimensional, unobservable types: students are characterized by an ex ante type and an ex post type. The ex ante type represents cognitive or academic skills. The ex post type represents job-markets skills and opportunities.

7 Preview of model (2) Motivation Preview of Results Literature Students choose the length (or quality) of studies in a menu of contracts". The ex ante type represents cognitive or academic skills. The ex post type is revealed later, after graduation.

8 Preview of model (3) Motivation Preview of Results Literature Ex ante effort is study effort not directly observable. This determines the ex ante moral hazard problem. Study effort influences the probability of graduation (success of failure). Ex post effort is effort exerted at work, also not observable by public authorities. ex post effort affects earnings: this is the source of the ex post moral hazard problem. Government observes education, success and individual earnings.

9 Preview of model (4) Motivation Preview of Results Literature Success (graduation), education and ex post type determine the student s potential wage. Two sources of wage-risk: random academic success and ex post types. We characterize second-best optima in this model. The optima can be implemented by a combination of structured loans and income taxes.

10 Preview of results (1) Motivation Preview of Results Literature The optimal menu of loan contracts exhibits incomplete insurance, because of ex ante moral hazard. The talented types bear more risk than necessary for study effort incentives, due to self-selection constraints. Second-best optimal loan repayments are always income-contingent, even in the presence of an income tax.

11 Preview of results (2) Motivation Preview of Results Literature We find a Non-decomposition Theorem: the second-best cannot be implemented by the sum of an income tax, depending only on earnings, and loan repayments, depending only on education. Either the income tax depends on education (graduate tax) or the loan repayments depend on income. Since the budget is balanced, second-best optima entail cross-subsidies between types.

12 Preview of results (3) Motivation Preview of Results Literature The optimal menu of contracts exhibits an Equal Treatment Property (i.e., interim equalization). The post-study but pre-work expected utilities are equal. Equal treatment is the result of the interplay of self-selection constraints (adverse selection problem) and effort incentives (ex ante moral hazard problem). The student types are ex ante unequal because they differ in their probabilities of success. The student types are ex post unequal because the income tax trades off incentives and redistribution.

13 Preview of results (4) Motivation Preview of Results Literature We also provide a complete characterization of the set of second-best optima in an economy with risk, risk-aversion, moral hazard and adverse selection. Equal treatment result is not the familiar separating menu à la Rothschild-Stiglitz. Separating optima appear when the weight of the talented types is raised above their true frequency is the welfare function. When the weight of student types is in the neighborhood of type frequencies, the second-best optima exhibit the equal treatment property.

14 Motivation Preview of Results Literature Literature: Moral Hazard and Adverse Selection combined Microeconomic models of banking and insurance are formally close (e.g., Rothschild and Stiglitz (1976), Bester (1985)). Classic theories treat adverse selection and moral hazard separately (e.g., Freixas and Rochet (1998)). Principal-Agent theory under risk-neutrality: Picard (1987), Guesnerie, Caillaud and Rey (1992). Optimal regulation theory: some special cases: McAfee and McMillan (1986), Baron and Besanko (1987), Laffont and Rochet (1998). Extensions of Rothschild-Stiglitz: Chassagnon and Chiappori (1997). Some other technical contributions: Faynzilberg and Kumar (2000), Sung Jaeyoung (2005).

15 Motivation Preview of Results Literature Literature (2): Optimal Taxation with Human Capital Optimal Taxation Theory with Uncertain Earnings: see the survey of Boadway and Sato (2012). Optimal Taxation Theory with Education and Human Capital: Gianni de Fraja (2002), Fleurbaey et al. (2001), Bovenberg and Jacobs (2005), Anderberg (2009), Grochulski and Piskorski(2010), Findeisen and Sachs (2012).

16 Basic Assumptions Assumptions First-best Optimality A population of students with same VNM utility u(.), strictly concave and differentiable. Two-dimensional student type (i, k), i, k = 1, 2. Type i is ex ante type. Type k is ex post type. Type i is known when education is chosen. Type k is revealed after graduation. Probability (frequency) of ex ante type i is denoted λ i. There are 3 building blocks: education, job market and the loan contract.

17 Basic Assumptions Assumptions First-best Optimality A population of students with same VNM utility u(.), strictly concave and differentiable. Two-dimensional student type (i, k), i, k = 1, 2. Type i is ex ante type. Type k is ex post type. Type i is known when education is chosen. Type k is revealed after graduation. Probability (frequency) of ex ante type i is denoted λ i. There are 3 building blocks: education, job market and the loan contract.

18 Basic Assumptions Assumptions First-best Optimality A population of students with same VNM utility u(.), strictly concave and differentiable. Two-dimensional student type (i, k), i, k = 1, 2. Type i is ex ante type. Type k is ex post type. Type i is known when education is chosen. Type k is revealed after graduation. Probability (frequency) of ex ante type i is denoted λ i. There are 3 building blocks: education, job market and the loan contract.

19 Basic Assumptions Assumptions First-best Optimality A population of students with same VNM utility u(.), strictly concave and differentiable. Two-dimensional student type (i, k), i, k = 1, 2. Type i is ex ante type. Type k is ex post type. Type i is known when education is chosen. Type k is revealed after graduation. Probability (frequency) of ex ante type i is denoted λ i. There are 3 building blocks: education, job market and the loan contract.

20 Basic Assumptions Assumptions First-best Optimality A population of students with same VNM utility u(.), strictly concave and differentiable. Two-dimensional student type (i, k), i, k = 1, 2. Type i is ex ante type. Type k is ex post type. Type i is known when education is chosen. Type k is revealed after graduation. Probability (frequency) of ex ante type i is denoted λ i. There are 3 building blocks: education, job market and the loan contract.

21 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

22 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

23 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

24 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

25 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

26 Assumptions First-best Optimality Basic Assumptions (2): Education Student learns ex ante type i. Student chooses education q in set {1, 2}. Interpretation q = 2 means long (or high quality) studies. Cost of education is γ q. We assume γ 2 γ 1. Let q i be the education chosen by type i. Each student is successful or fails; success may be interpreted as graduation. Type i student chooses study effort e i ex ante in set {0, 1}. Cost of effort is c i e i.

27 Assumptions First-best Optimality Basic Assumptions (2): Education, ctd. Probability of success, p i (e i ) = Pr(success i and e i ) Notation P i = p i (1), p i = p i (0). Type 2 is more likely to succeed given the effort level. Assumption 1. 0 < p i < P i < 1 and P 2 > P 1, p 2 > p 1.

28 Assumptions First-best Optimality Basic Assumptions (2): Education, ctd. Probability of success, p i (e i ) = Pr(success i and e i ) Notation P i = p i (1), p i = p i (0). Type 2 is more likely to succeed given the effort level. Assumption 1. 0 < p i < P i < 1 and P 2 > P 1, p 2 > p 1.

29 Assumptions First-best Optimality Basic Assumptions (2): Education, ctd. Probability of success, p i (e i ) = Pr(success i and e i ) Notation P i = p i (1), p i = p i (0). Type 2 is more likely to succeed given the effort level. Assumption 1. 0 < p i < P i < 1 and P 2 > P 1, p 2 > p 1.

30 Assumptions First-best Optimality Basic Assumptions (3): Job Market Skills In case of failure, the students gets a basic job with basic wage w. In case of success, a third chance move determines ex post type k. Ex post type k determines the possibility of occupying a top job or a middle-range job. Distribution of ex post type depends on q and success, Pr(k = q q, success) = 1 π. We assume that π < 1/2. Interpretation: With a small probability π, an individual choosing q will draw a type k q.

31 Assumptions First-best Optimality Basic Assumptions (3): Job Market Skills In case of failure, the students gets a basic job with basic wage w. In case of success, a third chance move determines ex post type k. Ex post type k determines the possibility of occupying a top job or a middle-range job. Distribution of ex post type depends on q and success, Pr(k = q q, success) = 1 π. We assume that π < 1/2. Interpretation: With a small probability π, an individual choosing q will draw a type k q.

32 Assumptions First-best Optimality Basic Assumptions (3): Job Market Skills In case of failure, the students gets a basic job with basic wage w. In case of success, a third chance move determines ex post type k. Ex post type k determines the possibility of occupying a top job or a middle-range job. Distribution of ex post type depends on q and success, Pr(k = q q, success) = 1 π. We assume that π < 1/2. Interpretation: With a small probability π, an individual choosing q will draw a type k q.

33 Assumptions First-best Optimality Basic Assumptions (3): Job Market Skills In case of failure, the students gets a basic job with basic wage w. In case of success, a third chance move determines ex post type k. Ex post type k determines the possibility of occupying a top job or a middle-range job. Distribution of ex post type depends on q and success, Pr(k = q q, success) = 1 π. We assume that π < 1/2. Interpretation: With a small probability π, an individual choosing q will draw a type k q.

34 Assumptions First-best Optimality Basic Assumptions (3): Ex Post Effort Ex post effort of type (i, k) is ε ik, chosen in the set {0, 1}. The wage of an individual depends on education q and ex post effort ε ik only w = ω(q, ε). We denote, ω(q, 1) = W q and ω(q, 0) = w q. A successful type (i, k) can earn a top salary in a top job only if ex post effort is high... Assumption 2. w 2 > w 1 w, W 2 w 2 W 1 w 1 > 0.

35 Assumptions First-best Optimality Basic Assumptions (3): Ex Post Effort Ex post effort of type (i, k) is ε ik, chosen in the set {0, 1}. The wage of an individual depends on education q and ex post effort ε ik only w = ω(q, ε). We denote, ω(q, 1) = W q and ω(q, 0) = w q. A successful type (i, k) can earn a top salary in a top job only if ex post effort is high... Assumption 2. w 2 > w 1 w, W 2 w 2 W 1 w 1 > 0.

36 Assumptions First-best Optimality Basic Assumptions (3): Ex Post Effort Ex post effort of type (i, k) is ε ik, chosen in the set {0, 1}. The wage of an individual depends on education q and ex post effort ε ik only w = ω(q, ε). We denote, ω(q, 1) = W q and ω(q, 0) = w q. A successful type (i, k) can earn a top salary in a top job only if ex post effort is high... Assumption 2. w 2 > w 1 w, W 2 w 2 W 1 w 1 > 0.

37 Assumptions First-best Optimality Basic Assumptions (3): Disutility of Ex Post Effort The disutility of effort at work is β ik ε ik. (Total cost of effort is c i e i + β ik ε ik.) Type k affects disutilities as follows, β 22 = β 12 = b > 0, β 11 = β 21 = B > 0. We assume that B >> b and B is so large that a type (i, 1) will never find it worthwhile to exert high ex post effort. These assumptions generate an elementary form of the optimal income taxation problem à la Mirrlees.

38 Assumptions First-best Optimality Basic Assumptions (3): Disutility of Ex Post Effort The disutility of effort at work is β ik ε ik. (Total cost of effort is c i e i + β ik ε ik.) Type k affects disutilities as follows, β 22 = β 12 = b > 0, β 11 = β 21 = B > 0. We assume that B >> b and B is so large that a type (i, 1) will never find it worthwhile to exert high ex post effort. These assumptions generate an elementary form of the optimal income taxation problem à la Mirrlees.

39 Assumptions First-best Optimality Basic Assumptions (3): Disutility of Ex Post Effort The disutility of effort at work is β ik ε ik. (Total cost of effort is c i e i + β ik ε ik.) Type k affects disutilities as follows, β 22 = β 12 = b > 0, β 11 = β 21 = B > 0. We assume that B >> b and B is so large that a type (i, 1) will never find it worthwhile to exert high ex post effort. These assumptions generate an elementary form of the optimal income taxation problem à la Mirrlees.

40 Assumptions First-best Optimality Basic Assumptions (3): Disutility of Ex Post Effort The disutility of effort at work is β ik ε ik. (Total cost of effort is c i e i + β ik ε ik.) Type k affects disutilities as follows, β 22 = β 12 = b > 0, β 11 = β 21 = B > 0. We assume that B >> b and B is so large that a type (i, 1) will never find it worthwhile to exert high ex post effort. These assumptions generate an elementary form of the optimal income taxation problem à la Mirrlees.

41 Assumptions First-best Optimality Basic Assumptions (4): Loan contracts We consider a public lending system. A loan covers the cost γ q. The Menu of contracts is (q, R q1, R q2, r q ) q=1,2. A student choosing education q must repay, (i) r q in case of failure; (ii) R q1 is earnings are w q ; (iii) R q2 is earnings are W q ;

42 Assumptions First-best Optimality Basic Assumptions (4): Loan contracts We consider a public lending system. A loan covers the cost γ q. The Menu of contracts is (q, R q1, R q2, r q ) q=1,2. A student choosing education q must repay, (i) r q in case of failure; (ii) R q1 is earnings are w q ; (iii) R q2 is earnings are W q ;

43 Assumptions First-best Optimality Basic Assumptions (4): Loan contracts We consider a public lending system. A loan covers the cost γ q. The Menu of contracts is (q, R q1, R q2, r q ) q=1,2. A student choosing education q must repay, (i) r q in case of failure; (ii) R q1 is earnings are w q ; (iii) R q2 is earnings are W q ;

44 Assumptions First-best Optimality Basic Assumptions (4): Resource Constraint Assuming that q = i for all i, the resource constraint can be written, λ i {p i (e i )[(1 π)x i + πy i ] + (1 p i (e i ))r i γ i } 0, i where X i = ε ii R i2 + (1 ε ii )R i1, Y i = ε ij R i2 + (1 ε ij )R i1.

45 First-Best Optimality (1) Assumptions First-best Optimality The interim expected utility of a successful student with education q i = i can be written, U i = π(v i β ij ε ij ) + (1 π)(v i β ii ε ii ), where ex post utilities are, v i = u[ω(q i, ε ij ) Y i ] and V i = u[ω(q i, ε ii ) Y i ]. The utility of an unsuccessful type is u i = u(w r i ). The ex ante utility of a type i student is simply, Eu i = p i (e i )U i + (1 p i (e i ))u i c i e i.

46 First-Best Optimality (1) Assumptions First-best Optimality The interim expected utility of a successful student with education q i = i can be written, U i = π(v i β ij ε ij ) + (1 π)(v i β ii ε ii ), where ex post utilities are, v i = u[ω(q i, ε ij ) Y i ] and V i = u[ω(q i, ε ii ) Y i ]. The utility of an unsuccessful type is u i = u(w r i ). The ex ante utility of a type i student is simply, Eu i = p i (e i )U i + (1 p i (e i ))u i c i e i.

47 First-Best Optimality (1) Assumptions First-best Optimality The interim expected utility of a successful student with education q i = i can be written, U i = π(v i β ij ε ij ) + (1 π)(v i β ii ε ii ), where ex post utilities are, v i = u[ω(q i, ε ij ) Y i ] and V i = u[ω(q i, ε ii ) Y i ]. The utility of an unsuccessful type is u i = u(w r i ). The ex ante utility of a type i student is simply, Eu i = p i (e i )U i + (1 p i (e i ))u i c i e i.

48 Assumptions First-best Optimality First-Best Optimality (2): Optimal Effort To simplify the analysis, we focus on the case in which first-best optimal effort is high for both types ex ante, i.e., (e 1, e 2 ) = (1, 1) and (ε i1, ε i2 ) = (0, 1), that is, high ex post types should exert high ex post effort. Under these assumptions, 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 ),

49 Assumptions First-best Optimality First-Best Optimality (2): Optimal Effort To simplify the analysis, we focus on the case in which first-best optimal effort is high for both types ex ante, i.e., (e 1, e 2 ) = (1, 1) and (ε i1, ε i2 ) = (0, 1), that is, high ex post types should exert high ex post effort. Under these assumptions, 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 ),

50 Assumptions First-best Optimality First-Best Optimality (3): Inverse Utility Now, define the inverse utility z(x) = u 1 (x). By definition, z(u) is the minimal amount of resources needed to provide utility u. We have, r i = w z(u i ), R 22 = W 2 z(v 2 ), R 21 = w 2 z(v 2 ), etc... The expected amount of resources needed to provide Eu i is E(z i ) = P i [(1 π)z(v i ) + πz(v i )] + (1 P i )z(u i ).

51 Assumptions First-best Optimality First-Best Optimality (3): Inverse Utility Now, define the inverse utility z(x) = u 1 (x). By definition, z(u) is the minimal amount of resources needed to provide utility u. We have, r i = w z(u i ), R 22 = W 2 z(v 2 ), R 21 = w 2 z(v 2 ), etc... The expected amount of resources needed to provide Eu i is E(z i ) = P i [(1 π)z(v i ) + πz(v i )] + (1 P i )z(u i ).

52 Assumptions First-best Optimality First-Best Optimality (3): Inverse Utility Now, define the inverse utility z(x) = u 1 (x). By definition, z(u) is the minimal amount of resources needed to provide utility u. We have, r i = w z(u i ), R 22 = W 2 z(v 2 ), R 21 = w 2 z(v 2 ), etc... The expected amount of resources needed to provide Eu i is E(z i ) = P i [(1 π)z(v i ) + πz(v i )] + (1 P i )z(u i ).

53 Assumptions First-best Optimality First-Best Optimality (3): Inverse Utility Now, define the inverse utility z(x) = u 1 (x). By definition, z(u) is the minimal amount of resources needed to provide utility u. We have, r i = w z(u i ), R 22 = W 2 z(v 2 ), R 21 = w 2 z(v 2 ), etc... The expected amount of resources needed to provide Eu i is E(z i ) = P i [(1 π)z(v i ) + πz(v i )] + (1 P i )z(u i ).

54 Assumptions First-best Optimality First-Best Optimality (4): Surplus Define the expected wages, Ew 1 = (1 π)w 1 + πw 1, and Ew 2 = (1 π)w 2 + πw 2. The social surplus of education q = i assigned to type i is defined as, S i = P i Ew i + (1 P i )w γ i We assume that assigning education q = i to type i generates a greater surplus than assigning education q i to type i.

55 Assumptions First-best Optimality First-Best Optimality (4): Surplus Define the expected wages, Ew 1 = (1 π)w 1 + πw 1, and Ew 2 = (1 π)w 2 + πw 2. The social surplus of education q = i assigned to type i is defined as, S i = P i Ew i + (1 P i )w γ i We assume that assigning education q = i to type i generates a greater surplus than assigning education q i to type i.

56 Assumptions First-best Optimality First-Best Optimality (4): Surplus Define the expected wages, Ew 1 = (1 π)w 1 + πw 1, and Ew 2 = (1 π)w 2 + πw 2. The social surplus of education q = i assigned to type i is defined as, S i = P i Ew i + (1 P i )w γ i We assume that assigning education q = i to type i generates a greater surplus than assigning education q i to type i.

57 Assumptions First-best Optimality First-Best Optimality (5): Optimization Problem The terms R ij, r i can be eliminated from the resource constraint, using inverse utility z(.). The first-best utilitarian optimum problem: Maximize λ i [PU i + (1 P i )u i c i ], subject to the resource constraint RC: λ i {S i E(z i )} 0. i i

58 Assumptions First-best Optimality First-Best Optimality (5): Optimization Problem The terms R ij, r i can be eliminated from the resource constraint, using inverse utility z(.). The first-best utilitarian optimum problem: Maximize λ i [PU i + (1 P i )u i c i ], subject to the resource constraint RC: λ i {S i E(z i )} 0. i i

59 Assumptions First-best Optimality First-Best Optimality (6): Characterization We can state the following result, Proposition 1. First-best Pareto Optimality implies full insurance, that is, for all i, V i = v i = u i, and in addition, the standard utilitarian first-best optimum exhibits full equality, i.e., V 1 = V 2, v 1 = v 2 and u 1 = u 2.

60 Incentive Constraints (1) Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Types (i, k) and efforts (e i, ε ik ) are not observed by public authorities. For the sake of simplicity, we assume that the second-best optimal effort levels are the same as the first-best effort levels. This will be true if P i p i is large enough, c i is small enough for each i and if b is not too large.

61 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Ex Post Incentives Suppose that q i = i for all i (students reveal ex ante types). In the job market stage, a student with q = 1 and k = 2 will not choose a job with a middle-range salary ex post iff, v 1 b V 1. (ICX 1 ) A student with q = 2 and k = 2 will not mimic type k = 1 ex post iff, V 2 b v 2. (ICX 2 ) The other constraints, i.e., V 1 v 1 B and v 2 V 2 B, are satisfied since B is large.

62 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Ex Post Incentives Suppose that q i = i for all i (students reveal ex ante types). In the job market stage, a student with q = 1 and k = 2 will not choose a job with a middle-range salary ex post iff, v 1 b V 1. (ICX 1 ) A student with q = 2 and k = 2 will not mimic type k = 1 ex post iff, V 2 b v 2. (ICX 2 ) The other constraints, i.e., V 1 v 1 B and v 2 V 2 B, are satisfied since B is large.

63 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Ex Post Incentives Suppose that q i = i for all i (students reveal ex ante types). In the job market stage, a student with q = 1 and k = 2 will not choose a job with a middle-range salary ex post iff, v 1 b V 1. (ICX 1 ) A student with q = 2 and k = 2 will not mimic type k = 1 ex post iff, V 2 b v 2. (ICX 2 ) The other constraints, i.e., V 1 v 1 B and v 2 V 2 B, are satisfied since B is large.

64 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints: Revelation and Obedience We apply the Extended Revelation Principle (see Laffont and Martimort (2002)). Revelation and obedience constraints can be written as follows. P i U i + (1 P i )u i P i U j + (1 P i )u j (IC i ) P i U i + (1 P i )u i c i p i U i + (1 p i )u i (MH i ) P i U i + (1 P i )u i c i p i U j + (1 p i )u j (IC i )

65 The Second-best Optimality Problem Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax By definition, the second best optimality problem is the following, Maximize λ i [P i U i + (1 P i )u i c i ], i subject to RC, ICX i, IC i, MH i and IC i, i = 1, 2, and U 1 = (1 π)v 1 + π(v 1 b), (EU 1 ) and U 2 = (1 π)(v 2 b) + πv 2. (EU 2 )

66 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (1): Ex Post Moral Hazard Our first Lemma shows that ICX i constraints must be binding, Lemma 1. The ex post incentive constraints must be binding at any second-best optimum, that is, v 1 = V 1 + b, and v 2 = V 2 b Given this result, we derive, U 1 = V 1 and U 2 = V 2 b.

67 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Ex Ante Moral Hazard It is easy to see that (MH i ) can be rewritten as We assume, U i u i K i where K i = c i P i p i Assumption 3: K 1 K 2.

68 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Ex Ante Moral Hazard, ctd. We can prove the following useful Lemma, Lemma 2. Under Assumption 3, if IC 1, IC 2 and MH 1 hold, then, MH 2 is satisfied.

69 Self-selection Constraints (1) Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Using (IC 1 ) and (IC 2 ), we get the property: P 2 1 P 2 (U 2 U 1 ) u 1 u 2 P 1 1 P 1 (U 2 U 1 ) (IC) We get the following results: Lemma 3. (IC) constraints imply, U 2 u 2 U 1 u 1. (D) U 2 U 1 and u 1 u 2 (D )

70 Self-selection Constraints (2) Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We also have, Lemma 3, ctd. a) If (IC 1 ) and (IC 2 ) are simultaneously binding, then U 1 = U 2 and u 1 = u 2 (equal treatment but not necessarily full insurance). b) Under (IC 1 ) and (IC 2 ), then U 1 = U 2 if and only if u 1 = u 2.

71 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Incentive Constraints (2): Effort Incentives, ctd. We can also ignore the (IC i ) constraints to a certain extent: Lemma 4. Under Assumption 3, a) If (IC 1 ), (IC 2 ) and (MH 1 ) hold, then (IC 1 ) is satisfied. b) If (IC 2 ) is satisfied, and in addition, (IC 1 ) and (MH 1 ) are binding, then (IC 2 ) is satisfied.

72 The Graduate Tax Problem Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax A study of first-order conditions for optimality yields the following key result. Proposition 2. Under Assumptions 1-3, for all π < 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, MH 1, IC 1, IC 2, ICX 1 and ICX 2 are all binding. If, in addition, K 1 > K 2, then, MH 2, IC 1 and IC 2 hold as strict inequalities.

73 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Equal Treatment Property: Interpretation Equal treatment (or interim equalization) stems from the interplay of adverse selection and ex ante moral hazard. Under ex ante moral hazard alone, we would have U i u i = K i for all i. But this violates IC under Assumption 3, since K 1 > K 2 would then imply U 1 u 1 > U 2 u 2, a contradiction. To satisfy IC and MH i constraints, the best is to set equal differences, that is, U 1 u 1 = U 2 u 2 = K 1. Equal differences and IC imply u 1 = u 2. This proves the equal treatment property.

74 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Equal Treatment Property: Interpretation Equal treatment (or interim equalization) stems from the interplay of adverse selection and ex ante moral hazard. Under ex ante moral hazard alone, we would have U i u i = K i for all i. But this violates IC under Assumption 3, since K 1 > K 2 would then imply U 1 u 1 > U 2 u 2, a contradiction. To satisfy IC and MH i constraints, the best is to set equal differences, that is, U 1 u 1 = U 2 u 2 = K 1. Equal differences and IC imply u 1 = u 2. This proves the equal treatment property.

75 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Equal Treatment Property: Interpretation Equal treatment (or interim equalization) stems from the interplay of adverse selection and ex ante moral hazard. Under ex ante moral hazard alone, we would have U i u i = K i for all i. But this violates IC under Assumption 3, since K 1 > K 2 would then imply U 1 u 1 > U 2 u 2, a contradiction. To satisfy IC and MH i constraints, the best is to set equal differences, that is, U 1 u 1 = U 2 u 2 = K 1. Equal differences and IC imply u 1 = u 2. This proves the equal treatment property.

76 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Equal Treatment Property: Interpretation Equal treatment (or interim equalization) stems from the interplay of adverse selection and ex ante moral hazard. Under ex ante moral hazard alone, we would have U i u i = K i for all i. But this violates IC under Assumption 3, since K 1 > K 2 would then imply U 1 u 1 > U 2 u 2, a contradiction. To satisfy IC and MH i constraints, the best is to set equal differences, that is, U 1 u 1 = U 2 u 2 = K 1. Equal differences and IC imply u 1 = u 2. This proves the equal treatment property.

77 Ex Ante Inequality Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Student types are ex ante unequal, since P 2 > P 1. With Eu i = P i U i + (1 P i )U i c i, we get with c 1 c 2. Eu 2 Eu 1 = (P 2 P 1 )K 1 + (c 1 c 2 ) > 0,

78 Ex Post Inequality Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax From Proposition 2, we easily derive the following, Proposition 3. In the standard utilitarian case, the second-best 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.

79 Optimal Repayment Schedule Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Corollary 3. The optimal repayment schedule 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);

80 Generic Non-decomposition Property Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Can we decompose optimal repayments as the sum of an income tax T i = T (w i ), depending only on earnings, and a loan repayment L i = L(q i ), depending only on education? If the optimal schedule is decomposable, we should have, This implies, R 11 = T 1 + L 1, R 22 = T 2 + L 2, R 12 = T 2 + L 1, R 21 = T 1 + L 2. R 22 R 12 = L 2 L 1 = R 21 R 11, R 22 R 21 = T 2 T 1 = R 12 R 11,

81 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Generic Non-decomposition Property (2) Define, = R 22 R 12 (R 21 R 11 ). If the repayment schedule is decomposable, then, = 0. Using the definitions, we find, = z(v 1 ) Z (V 1 ) + z(v 2 )) z(v 2 ) + W 2 W 1 + w 1 w 2. that is, the property > 0 is generic. = W 2 W 1 + w 1 w 2 0.

82 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Justification of Income-contingent Loan Repayments (3) Proposition 4. (Non-decomposition Theorem) It is generically impossible to decompose the second-best optimal transfers R ij 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 (i.e., graduate tax).

83 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Complete Characterization of the Set of Second-best Optima (1) To characterize all optima, we consider a weighted sum of expected utilities, with weights α i > 0, i = 1, 2. We maximize α i [P i U i + (1 P i )u i c i ] i subject to RC, ICX i, IC i, MH i and IC i and EU i, i = 1, 2.

84 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Complete Characterization of the Set of Second-best Optima (2) Proposition 2. For all π < 1/2, there exists an open interval (λ 2, λ 2 ), including the frequency λ 2, such that, if α 2 (λ 2, λ 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) RC, MH 1, IC 1, IC 2, ICX 1 and ICX 2 are all binding. If, in addition, K 1 > K 2, then, MH 2, IC 1 and IC 2 hold as strict inequalities.

85 Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax Complete Characterization of the Set of Second-best Optima (3): Separating Optima Proposition 5. If the second-best optimum has only one binding IC constraint, then, IC 1 is binding, IC 2 is slack, MH 1 and RC are binding; we have U2 > U 1 > u 1 > u 2 and necessarily, α 2 > λ 2. The second-best solution is then determined by the following 4 equations: IC 1, MH 1, RC, expressed as equalities, and λ 2 [P 2 (1 P α )Ez 2 P α(1 P 2 )z (u 2 )] λ 1 α 2 P 2 P 1 = P 1 Ez 1 + (1 P 1)z (u 1 ) where P α = α 1 P 1 + α 2 P 2, and Ez i = (1 π)z (V i ) + πz (v i ).

86 Conclusion Introduction Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We characterized the second-best optimal student loans under risk aversion, adverse selection, ex ante and ex post moral hazard. The second-best optimal solution entails incomplete insurance, because of ex ante moral hazard. The second-best optimal solution is characterized by equal treatment (i.e., interim equalization) in the neighborhood of the standard utilitarian welfare function. Students are ex ante and ex post unequal. The optimal repayment schedule cannot be decomposed as the sum of a loan repayment depending only on education and an income tax, depending only on earnings: the optimal solution can be implemented by income-contingent repayments or by a graduate tax.

87 Conclusion Introduction Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We characterized the second-best optimal student loans under risk aversion, adverse selection, ex ante and ex post moral hazard. The second-best optimal solution entails incomplete insurance, because of ex ante moral hazard. The second-best optimal solution is characterized by equal treatment (i.e., interim equalization) in the neighborhood of the standard utilitarian welfare function. Students are ex ante and ex post unequal. The optimal repayment schedule cannot be decomposed as the sum of a loan repayment depending only on education and an income tax, depending only on earnings: the optimal solution can be implemented by income-contingent repayments or by a graduate tax.

88 Conclusion Introduction Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We characterized the second-best optimal student loans under risk aversion, adverse selection, ex ante and ex post moral hazard. The second-best optimal solution entails incomplete insurance, because of ex ante moral hazard. The second-best optimal solution is characterized by equal treatment (i.e., interim equalization) in the neighborhood of the standard utilitarian welfare function. Students are ex ante and ex post unequal. The optimal repayment schedule cannot be decomposed as the sum of a loan repayment depending only on education and an income tax, depending only on earnings: the optimal solution can be implemented by income-contingent repayments or by a graduate tax.

89 Conclusion Introduction Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We characterized the second-best optimal student loans under risk aversion, adverse selection, ex ante and ex post moral hazard. The second-best optimal solution entails incomplete insurance, because of ex ante moral hazard. The second-best optimal solution is characterized by equal treatment (i.e., interim equalization) in the neighborhood of the standard utilitarian welfare function. Students are ex ante and ex post unequal. The optimal repayment schedule cannot be decomposed as the sum of a loan repayment depending only on education and an income tax, depending only on earnings: the optimal solution can be implemented by income-contingent repayments or by a graduate tax.

90 Conclusion Introduction Incentive Constraints Preliminary Analysis of Incentive Constraints Main Results: Income-Contingent Loans and Graduate Tax We characterized the second-best optimal student loans under risk aversion, adverse selection, ex ante and ex post moral hazard. The second-best optimal solution entails incomplete insurance, because of ex ante moral hazard. The second-best optimal solution is characterized by equal treatment (i.e., interim equalization) in the neighborhood of the standard utilitarian welfare function. Students are ex ante and ex post unequal. The optimal repayment schedule cannot be decomposed as the sum of a loan repayment depending only on education and an income tax, depending only on earnings: the optimal solution can be implemented by income-contingent repayments or by a graduate tax.