Econ 277A: Economic Development I Semester II, 2011-12 Tridip Ray ISI, Delhi Final Exam (06 May 2012) There are 2 questions; you have to answer both of them. You have 3 hours to write this exam. 1. [30 points] Consider a community of a continuum of identical farmers of measure 1. Assume that all uncertainty is idiosyncratic, that is, shocks that a ect the state of the harvest on a particular plot. There are two possible outputs, H and L; with H > L; and the high output (H) is produced with probability 0 < p < 1: The farmers are risk-averse; their utility function u () is strictly concave with u 0 () > 0 and u 00 () < 0: Farmers live for an in nite number of periods, and discount the future by a discount factor 0 < < 1: Mutual insurance at the community level works as follows. Under any insurance scheme suppose that X is a farmer s net consumption when output is H and Y is his net consumption when output is L: That is, when a farmer enjoys a good harvest, he contributes H X > 0 to the community pool, and receives Y L > 0 from the community pool when he su ers a bad harvest. Note that mutual insurance schemes are rarely contracts that are written down in black and white and routinely backed or enforced by the law. Rather they are informal arrangements, set in the context of a social norm that encourages reciprocity and punishes deviations through social sanctions such as ostracization or public rebuke. For our mutual insurance scheme, if a farmer defaults voluntarily (that is, his harvest is good but he decides not contribute to the community pool), he is excluded from future access to insurance. Further he su ers a loss S; which is a measure of the utility damage imposed by social sanctions (such as ostracization or public rebuke). (a) [2 points] An insurance scheme must be feasible. Write down, with a brief and clear explanation, the feasibility constraint. (b) [8 points] An insurance scheme must be self-enforcing, where contributions rely on self-interest 1
of farmers, given the future consequences of a voluntary default. Write down, with a brief and clear explanation, the enforcement constraint. (c) [10 points] The insurance scheme is de ned by the two endogenous variables, X and Y ; the other variables, H; L; p; and S; are treated as parameters. Rewrite the enforcement constraint with the expressions involving only the endogenous variables on the left-hand side (LHS) of the inequality. Note that the feasibility constraint implicitly de nes Y as a function of X: Hence the LHS of the enforcement constraint, combined with the feasibility constraint, becomes a function of X only. Call this function F (X): (i) Complete insurance occurs when X = Y; that is, regardless of whether the harvest is good or bad a farmer s net consumption is the same. What are the values of X and Y under complete insurance? Show that F 0 (complete insurance) > 0: (ii) Assume that u 0 (H) + p 1 [u 0 (H) u 0 (L)] < 0: (Assumption 1.1) What are the values of X and Y under no insurance? Show that, under Assumption 1.1, F 0 (no insurance) < 0: (iii) Argue clearly that F () achieves a maximum in between the points of complete insurance and no insurance. (iv) With the above information, plot F (X) for values of X in between the points of complete insurance and no insurance. (d) [10 points] To answer the following parts, consider that only the social sanction parameter S varies while all the other parameters, H; L; p and ; remain unchanged. (i) Show that complete insurance is enforcable only for high values of social sanction. (ii) Incomplete insurance refers to a situation where X 6= Y. Show that, for moderate values of social sanction, only incomplete insurance is enforcable. Compare the values of X and Y under an enforcable incomplete insurance scheme. What kind of relationship do you expect to see then between individual consumption and individual income? (iii) Show that for low values of social sanction no insurance is possible at all. 2
2. [70 points] Consider a one-period model of a credit market under adverse selection. Technology and Preferences: All agents live in a village with a large population normalised to unity and are endowed with one unit of labour and a risky investment project. The project requires one unit of capital and one unit of labour. Agents do not have personal wealth and need to borrow to launch their projects. The outcomes of the project are success (S) and failure (F). There are two types of borrowers, risky and safe, characterised by the probability of success of their projects, p r and p s respectively, where 0 < p r < p s < 1: Risky and safe borrowers exist in proportions and 1 the population. The outcomes of the projects are independently distributed for the same type as well as across di erent types. The return of a project of a borrower of type i is R i > 0 if successful and 0 if it fails. Assume that risky and safe projects have the same mean return, that is, p r R r = p s R s R; but risky projects have a greater spread around the mean. Borrowers are risk-neutral and maximise expected returns. Borrowers of both types have an reservation payo u: The lending side is represented by risk-neutral banks whose opportunity cost of capital is 1 per unit. We assume that the village is small relative to the credit market, and so the supply of loans is perfectly elastic at the rate : Information and Contracting: The type of a borrower is unknown to the lenders. However, borrowers know each other s types. There is no moral hazard and agents supply labour to the project inelastically. The outcome of a project (whether it is a success or a failure ) is observable by the bank; so the credit contracts are contingent on the outcomes. in There is a limited liability constraint: in case their projects fail, borrowers are liable up to the amount of collateralisable wealth they posses, w. For simplicity we take w = 0: Assumptions: Assume rst that the projects are socially productive in terms of expected returns given the opportunity costs of labour and capital: R > + u: (Assumption 2.1) De ne p p r +(1 We assume further that ) p s ; the average probability of success for the entire population. R < p s + u: (Assumption 2.2) p 3
Assume nally that the credit market is competitive so that banks are subjected to a zero-pro t constraint on each loan. We are going to focus on two types of credit contracts: individual liability contracts and joint liability contracts. Individual Liability Lending: This is a standard debt contract between a borrower and the bank with a xed repayment r > 0 in case of success and zero repayment in case of failure. (a) [5 points: 1 + 1 + 2 + 1] First, to set the benchmark, consider that the bank has full information about a borrower s type. (i) Write down the expressions for expected pro ts of the bank from each type of borrower. (ii) Write down the expressions for expected payo s to each type of borrower. (iii) What repayment, r i ; i = r; s; will the bank o er to type i borrower? Will the type i borrower accept this o er? Give clear explanations for your answers. (iv) What will be the average repayment rate (rate at which the bank gets repaid)? (b) [10 points: 2 + 4 + 2 + 2] Now consider that the bank cannot identify a borrower s type. (i) Explain clearly that the separating repayments that you have identi ed in part (a) will not work. [We therefore turn to pooling individual liability contracts.] (ii) Explain clearly that a pooling contract does not exist that attracts both types of borrowers. (iii) Find out the repayment, r; under the unique pooling contract. borrower will accept this contract? What type of (iv) Explain clearly that both the repayment rate (rate at which the bank gets repaid) and welfare are strictly less than that under full-information. 4
Joint Liability Lending: This involves asking the borrowers to form groups of size two and stipulating an individual liability component (that is, repayment) r > 0 and a joint liability component c > 0: As in standard debt contracts, if the project fails then, owing to the limited liability constraint, the borrower pays nothing to the bank. But if the project is successful then, apart from repaying her own debt r; the borrower has to pay an additional joint liability payment c if her partner s project fails. (c) [12 points: 4 + 4 + 4] Group Formation: Positive Assortative Matching Positive assortative matching refers to the property where borrowers choose partners of the same type at the group formation stage. (i) Write down, with a brief explanation, the expression of the expected payo of a borrower of type i when her partner is type j from a joint liability contract (r; c). (ii) What is the net expected gain of a risky borrower from having a safe partner? What is the net expected loss of a safe borrower from having a risky partner? (iii) Explain clearly that group formation will display positive assortative matching. (d) [7 points: 5 + 2] (i) As a result of positive assortative matching, argue that an indi erence curve of a borrower of type i in the (r; c) plane is represented by the straight line rp i + c (1 p i ) p i = k; where k is some constant. Argue that this straight line also represents an iso-pro t curve of the bank from lending to a borrower of type i: (ii) As k increases what happen to the borrower s expected payo and to the bank s expected pro t? (e) [24 points: 2 + 1 + 1 + 3 + 15 + 2] Separating Joint Liability Contracts Suppose the bank o ers the pair of joint liability contracts (r r ; c r ) and (r s ; c s ) designed for groups consisting of risky and safe borrowers respectively. (i) Write down the zero-pro t condition of the bank for each type of loan contract separately. (ii) Write down the participation constraint for each type of borrower. (iii) The limited liability constraint requires that a borrower cannot make any transfer to the bank when her project fails, and that the sum of individual and joint 5
liability payments, r + c; cannot exceed the realised revenue from the project when it succeeds. Write down the limited liability constraint for each type of borrower. (iv) The incentive compatibility constraint for each type of borrower requires that it is in the self-interest of a borrower to choose a contract that is designed for her type. Write down the incentive compatibility constraint for each type of borrower separately. (v) A separating joint liability contract must satisfy all the 4 conditions above ((i) to (iv)). In the (r; c) plane draw the two iso-pro t curves of the bank representing zero pro t from lending to the two types of borrowers. [Clearly label each zero iso-pro t curve that arises from lending to the risky and safe borrowers.] Let (^r; ^c) denote the contract that satis es the zero-pro t conditions for both risky and safe borrowers. Assume that (^r; ^c) satis es the limited liability constraint. In the same gure show, with clear labeling, the limited liability constraint for each type of borrower. In the gure show the respective sets of separating joint liability contracts for safe and risky borrowers. Explain clearly how the contracts satisfy all the 4 conditions ((i) to (iv)). (vi) Explain clearly that the average repayment rate (rate at which the bank gets repaid) and welfare under these contracts are equal to their full-information levels. (f) [12 points: 5 + 5 + 2] Pooling Joint Liability Contracts Suppose the bank o ers the same joint liability contract (r; c) to all borrowers. (i) Explain clearly how the 4 conditions, (e)(i) (e)(iv), get modi ed for the pooling joint liability contracts. (ii) Assume that (^r; ^c), as de ned in (e)(v), satis es the limited liability constraint. Explain clearly how (^r; ^c) satis es all the required conditions for a pooling joint liability contract. (iii) Explain clearly that the average repayment rate (rate at which the bank gets repaid) and welfare under (^r; ^c) are equal to the full-information levels. 6