On the first-order approach in principal-agent models with hidden borrowing and lending

We provide su¢ cient conditions for the validity of the …rst-order approach for two-period dynamic moral hazard problems where the agent can save and borrow secretly. The …rst-order approach is valid if the following conditions hold: i) the agent has nonincreasing absolute risk aversion utility (NIARA), ii) the output technology has monotone likelihood ratios (MLR), and iii) the distribution function of output is log-convex in e¤ort (LCDF). Moreover, under these three conditions, the optimal contract is monotone in output. We also investigate a few possibilities of relaxing these requirements. Keywords: Moral Hazard, Hidden Savings, First-Order Approach, Log-Convexity. JEL: C61, D81, D82, D86, E21. This paper merges and replaces two previous papers: “Optimal Income Taxation and Hidden Borrowing and Lending: The First-Order Approach in Two Periods,”by Á. Ábrahám and N. Pavoni, and “The First Order Approach to Moral Hazard Problems with Hidden Saving,”by S. Koehne. We are grateful to Ian Jewitt for many useful suggestions. We also thank Christian Hellwig, two anonymous referees, Bruno Biais, Kaiji Chen, Jacques Cremer, Bruno Jullien, Narayana Kocherlakota, Chris Phelan, Bernard Salanié, Ernst-Ludwig von Thadden, and participants at numerous seminars for their comments. Sebastian Koehne gratefully acknowledges …nancial support from the German Science Foundation (DFG). Nicola Pavoni thanks the Marie Curie Fellowship MCFI-2000-00689 and the Spanish Ministry of Science and Technology Grant BEC2001-1653. yEuropean University Institute. E-mail address: [email protected] zInstitute for International Economic Studies, Stockholm University. E-mail address: [email protected] xBocconi University, IGIER, IFS, and CEPR. E-mail address: [email protected]