Debt Contracts. Ram Singh. April 1, Department of Economics. Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

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1 Debt Contracts Ram Singh Department of Economics April 1, 215 Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

2 Debt Contracts I Innes (199, JET) Suppose Let There is a risk-neutral Bank B, and A risk-neutral Entrepreneur E The Entrepreneur has project/ideas but no money; Bank has money but no ideas They sign a contract; B lends I amount to E. Contract can be a debt contract or some other contract. q [, denotes the output/revenue/profit from the project q = q(e, θ), where Θ is the set of states of nature and captures randomness. e E R and θ Θ Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

3 Debt Contracts II Let q(e,θ) e. F (q e) is a conditional cumulative distribution of q f (q e) is the associated conditional density function Note: q(e,θ) e F e (q e) and q(e,θ) e > F e (q e) <. Assume: E(q ) = and the Monotone Likelihood Ratio Property (MLRP) holds. That is, d dq [f e(q e) d ], i.e., f (q e) ln f (q e) [ ]. dq e Payoff functions: Risk-neutral Bank s payoff function is V (x) = x, V >, V = Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

4 Debt Contracts III Let Risk-neutral Entrepreneur s payoff function is u(w, e) = u(w) ψ(e), u >, u =, i.e., u(w, e) = w ψ(e), where ψ(e) is the (money) cost of effort e, ψ > and ψ. r(q) be the contract, repayment schedule. Definition Limited Liability Contract is a repayment schedule r(q), such that: r(q) q Definition Debt Contract is a repayment schedule r(q), such that: r(q) q, i.e., two sided limited lability r (q), i.e., monotonicity Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

5 FB Repayment Schedule I Let the entrepreneur make the offer to B. Under the FB, the entrepreneur will solve: max r(q),e [q (r(q))]f (q e)dq ψ(e) s.t. IR, i.e., r(q)f (q e)dq I. Clearly, IR will bind. So, the entrepreneur solves: max e [q I]f (q e)dq ψ(e) Assuming that this programme is strictly concave, the FB effort, e, solves the following FOC is given by qf e (q e)dq ψ (e) = (.1) Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

6 SB Repayment Schedule I Under a two-way limited liability SB contract, the entrepreneur will solve s.t. e solves (.3) max r(q),e [q (r(q))]f (q e)dq ψ(e) (.2) [q (r(q))]f e (q e)dq = ψ (e) (.3) r(q)f (q e)dq = I (.4) r(q) q (.5) where r(q) q is the two-way limited liability constraint. Does the above programme have a solution? If there is a solution, is it unique? Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

7 SB Repayment Schedule II To ensure a solution, assume there exists an effort level e max, such that lim [q (r(q))]f (q e)dq ψ(e) < e e max [q (r(q))]f (q )dq ψ() So, the entrepreneur s effort level can be restricted to [, e max ]. Note: we have the following: For all e, [ ] [q (r(q))]f (q e)dq ψ(e) qf (q e)dq ψ(e) Further, for all e [, e max ], [ (r())]f (q e)dq ψ(e) [q (r(q))]f (q e)dq ψ(e). Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

8 SB Repayment Schedule III Let, k = Further, we have for all e [, e max ], [q k ]f (q e)dq ψ(e) qf (q e max )dq. [ (r())]f (q e)dq ψ(e) Note [ qf (q e)dq ψ(e) ] and [ [q k ]f (q e)dq ψ(e) ] are continuous functions of e so, they are bounded on the compact set [, e max ]. Therefore, for given r(q), there is exists at least one solution to the following: { } max [q (r(q))]f (q e)dq ψ(e). e Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

9 SB Repayment Schedule IV We will assume that (.2) has unique solution. The Lagrangian associated with IR and IC with (.2) is: [ ] L = [q (r(q))] f (q e)dq ψ(e) + µ [q (r(q))]f e (q e)dq ψ(e) [ ] + λ r(q)f (q e)dq I or, re-writing L = + [ r(q) q ] f (q e) 1 λ µ f e(q e) [ 1 + µ f e(q e) f (q e) f (q e)dq ] f (q e)dq ψ(e) µ ψ (e) λi (.6) Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

10 SB Repayment Schedule V It can be shown that IC will bind, i.e, µ >. So, the optimum repayment schedule is: [ ] q q 1 + µ fe(q e) r f (q e) (q) = [ < λ ] q 1 + µ fe(q e) f (q e) > λ (.7) [ ] q q fe(q e) r f (q e) (q) = > λ 1 [ µ ] q fe(q e) (.8) f (q e) < λ 1 µ We know that there exists a value of revenue, say q = Z such that for all q > Z, fe(q e) f (q e) > λ 1 µ. (Why?) So, the optimum contract is { r if q > Z (q) = q if q < Z (.9) Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

11 SB Repayment Schedule VI Under the optimum LL contract, e(r (q)) solves the following FOC: A comparison of (.1) and (.1) shows that e(r (q)) < e. Z qf e (q e)dq ψ (e) =. (.1) Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

12 Optimum Debt Contract I A debt contract is monotonic. Let, { r D D if q > D (q) = q if q D (.11) that is, r D (q) = min{q, D}. Under a debt contract, the entrepreneur s payoff is [q r D (q)]f (q e)dq ψ(e) = D [q q]f (q e)dq+ D So, he wants to choose minimum value of D such that D where e D solves the following FOC: qf (q e D )dq + [1 F(D e D )]D = I, D (q D)f e (q e)dq = ψ (e). [q D]f (q e)dq ψ(e). Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

13 Optimum Debt Contract II Note: [q r D (q)]f (q e)dq ψ(e) is continuous in e and D, Question in view of Maximum theorem, e D is a continuous function of D. Can a Debt contract can induce the FB effort? Comparison of (.1) and (.12) shows that r D (q) cannot induce the FB effort; r D (q) < r FB. (.1) and (.12) shows that r D (q) will be different from the SB effort. Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

14 Optimum Debt Contract III Question Is a Debt better than the optimum payment schedule discussed above? ( Assume the above, problem is strictly concave) Innes (199) showed that, DC is the most efficient among the class of monotonic contracts. Ram Singh (Delhi School of Economics) Debt Contracts April 1, / 14

Moral Hazard. Two Performance Outcomes Output is denoted by q {0, 1}. Costly effort by the agent makes high output more likely.

Moral Hazard. Two Performance Outcomes Output is denoted by q {0, 1}. Costly effort by the agent makes high output more likely. Moral Hazard Two Performance Outcomes Output is denoted by q {0, 1}. Costly effort by the agent makes high output more likely. Pr(q = 1 a) = p(a) with p > 0 and p < 0. Principal s utility is V (q w) and