The First-Order Approach to Moral Hazard Problems with Hidden Saving

Moral hazard models with hidden saving decisions are useful to study such diverse problems as unemployment insurance, income taxation, executive compensation, or human capital policies. How can we solve such models? In general, this is very difficult. Under the conditions derived in this paper, however, we can replace the incentive constraint with the associated first-order condition. This allows the application of simple Lagrangian methods and yields a precise characterization of optimal contracts. To obtain tractable conditions for the validity of this approach, the paper draws on the concept of log-convexity. Since logconvexity, unlike convexity, is preserved under multiplication, the paper is able to separate the assumptions on the output distribution from the assumptions on the agent’s preferences in a sense, even though the interaction between these two is important for the agent’s incentives. The first-order approach is valid if the following conditions hold: a) the agent has nonincreasing absolute risk aversion (NIARA) utility, b) the output technology has monotone likelihood ratios (MLR), and c) the distribution function of output is log-convex in effort (LCDF). Finally, the paper shows how the curvature of optimal wage schemes can be used to relax the above conditions.