Gale and Hellwig, REStud, 85Diamond, REStud, 84Stiglitz and Weiss, AER, 81Gale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Advanced Macroeconomics IECON 525a - Fall 2009Yale UniversityGuillermo L. Ordo˜nezWeek 2Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81QuestionWhy is debt the primary source of external finance?Gale and Hellwig show this is the case with ex-post hiddeninformation with costly state verification.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81EnvironmentE and L are risk neutral. E has all bargaining power.E has zero wealth. L has deep pockets. Risk free rate 0.E has monopoly access to a project that costs I .The project generates a random profit y ∈ [yL, yH], with cdf F andpdf f , smooth.The project has a positive NPV.ZyHyLyf (y)dy > IGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Information StructureHidden Information: Cash flow privately observed by E.Costly State Verification: L observes the realized y only by payingan auditing cost C .Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting ProblemE maximizes expected payoffs subject to L breaking even.Assumption: Deterministic audits a(y) = {0, 1} (not WLOG)WLOG, confine attention to direct mechanism design.Contract: {P(y), a (y )} enforced by a court without renegotiation.P : Y → R (payment for each report ˆy)a : Y → {0, 1} (audit decision at each report ˆy)Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting ProblemE maximizesZyHyL[y − P(y)]f (y )dysubject toLL: (limited liability)P(y) ≤ y , for all yPC: (participation constraint)ZyHyL[P(y) − a(y )C ]f (y)dy = IIC: When a(y ) = 0, P(y ) = R (constant since no info about y).P(y) = (1 − a(y))R + a(y)S(y ) = R − a(y )[R − S (y )]a(y )S (y ) ≤ RGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting ProblemHence E dec ides a(y) and S(y), (pinning down R uniquely from PC).We can rewrite the problem as an optimal control problem, decidingthe ˆy to stop auditing and the payments when auditing S(y ).Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting Problem as Optimal ControlDefineH(y) ≡ZyyL[P(ey) − a(ey)C ]f (ey)deyHence, PC can be written as H(yH) = IGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting Problem as Optimal Controlmaxa(y ),S(y )ZyHyL[y − R + a(y)(R − S(y ))]f (y )dysubject to PCH0(y) = f (y )[R − a(y)(R − S(y) + C)]H(yL) = 0 H(yH) = ILL: y − R + a(y)[R − S(y )] ≥ 0IC: R − a(y)S(y) ≥ 0Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Contracting Problem as Optimal ControlL = f (y)[y − R + a(y)(R − S(y ))] + πf (y )[R − a(y)(R − S(y) + C )]+λ[y − R + a(y)(R − S(y ))] + µ[R − a(y )S(y )]Take derivatives∂L∂a(y)= (f (y ) + λ)[R − S(y)] − πf (y )[R − S(y) + C ] − µS(y )∂L∂S(y)= a(y)[f (y )π − f (y ) − λ − µ]Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Solution a(y ) = 1If a(y ) = 1,∂L∂a(y )> 0,[R − S (y )][λ + f (y )(1 − π)] > πf (y)C + µS(y) ≥ 0Hence R − S(y) > 0 and µ = 0.We know, π0= −∂L∂H= 0, ∀y . This implies π(y) = π∗, ∀y.Furthermore, from∂L∂S(y)≥ 0, π∗≥ 1 (in fact, strict).Since λ∗> f (y )(π∗− 1) ≥ 0, then S(y) = yGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Standard debt is optimalFor all y s.t. a∗(y) = 1, P∗(y) = yFor all y s.t. a∗(y) = 0, P∗(y) = RHence, audit whenever y < R and get P(y) = y . For y ≥ R, noauditing and payment is P(y ) = R.Standard Debt Contract is Optimal.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Standard Debt Contract Audit NO AuditP(y) y P(y)=y y* R=y* Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81IssuesWhy restrict to a ∈ {0, 1}? It is more efficient to introduce publicrandomization. (Mookherjee and Png, QJE, 89).Not renegotiation proof.Krasa and Villamil (Ecta, 00) show the standard debt contract isoptimal when renegotiation is possible.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Delegated MonitoringWithout intermediaries there is a duplication of monitoring efforts.Monitors can also lie. Who monitor the monitor?Important to recognize delegation costs.Diversification is key despite risk neutrality for all the agents.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81ModelN risk neutral E’s with 0 initial wealth. A project costs $1 andproduces y ∈ [0, ∞) with expectation greater than 1.Many risk neutral L. Each has available $1m< 1. Risk free rate =0.Only E can freely observe the realization of y. L should pay C toobserve y.Monitoring represents a non-monetary cost to E. Non critical, asshown by Williamson (JME, 86)Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityGale and Hellwig, REStud, 85 Diamond, REStud, 84 Stiglitz and Weiss, AER, 81Optimal Contract is