First Order Approach for Principal-Agent Models with Hidden Borrowing and Lending: The Two Period Case

In this paper we provide sufficient conditions for the validity of the first-order condition approach (FOCA) for two period dynamic moral hazard problems where the agent can save and borrow secretly, and we characterize the optimal contract. Recently, dynamic principal-agent models became very popular instruments to study several diverse issues such as design of optimal social insurance schemes (e.g. unemployment insurance, disability insurance), bank-firm financing relationships or efficient compensation contracts. Most of these models assume that the agent's consumption-savings decision is observable (and contractible) by the principal. However, it is also well known that this assumption is potentially very dangerous, because if the agent had a hidden opportunity to save then he would deviate from the optimal contract by doing so. Therefore, the opportunity of hidden savings will lead to a different optimal contract (Rogerson, 1985a). Moreover, in most of the above-mentioned examples, the possibility of non-contractible borrowing and lending seems to be empirically relevant. Since the seminal works of Mirrlees (1972) and Holmstrom (1979) it became clear that the study of the moral hazard models is much easier if one can rely on the FOCA. Obviously, this is even more true in a dynamic environment where the agent has secret access to the credit market. Rogerson (1985b) and Jewitt (1988) provide conditions for the validity of the FOCA in the static principal-agent model. The strategy consists of showing that in the optimal contract the agent problem is concave hence the first order conditions are actually sufficient for the optimality of the agent's effort decisions. It is not known under what conditions the FOCA can be applied to multi-period principal-agent problems with hidden asset accumulation. In fact, Kocherlakota (2003) finds that under some conditions (linearity of the effort cost and effort effectiveness), the agent problem is not concave in the optimum. In this paper, we provide sufficient conditions under which the agent's problem is concave and therefore the incentive compatibility constraints for effort and asset decisions can be replaced by their necessary and sufficient first-order conditions. We show that within the class of non-increasing absolute risk aversion (NIARA) utility functions in consumption, a simple, strongly concave, version of the spanning condition of Grossman and Hart (1983) guarantees that the agent's problem is globally concave. Our conditions imply that most of the utility functions and many effort specifications used in applications allow for a first-order-condition representation of the problem. The use of first-order conditions enables us to characterize the optimal contract in great detail. First, we show that under the assumptions needed for the FOCA the optimal consumption is monotonic in output. Then, we characterize the optimal contract further and analyze the progressivity of the optimal payment scheme. It turns out that CARA utilities with concave likelihood ratios lead to progressive schemes, that is the higher the agent's output the bigger part is taken by the principal. On the other hand, convex likelihood ratios with CRRA utilities with risk aversion larger or equal than one induce regressive schemes. In the case of log utility, this result is in sharp contrast with the case where asset holdings are observable, because there the optimal contract induces proportional schemes. In general, the possibility of hidden savings will induce more regressive optimal payment schemes compared to the observable asset accumulation case. Unfortunately, our analytical results for the two-period model cannot be easily extended for a framework with more than two periods. In another work (Abraham and Pavoni, 2004) we show that in a general multiperiod setting the FOCA is crucial to get a tractable recursive reformulation of the original problem. There, we then verify the validity of the FOCA approach ex post, numerically. Finally, note that Williams (2003) also gives sufficient conditions for the validity of the FOCA but for a different set of continuous time principal-agents models. It is not clear how his conditions map in a more standard discrete time moral hazard framework.