Local Private-Information Contracts

Consider a collection of agents with stochastically fluctuating heterogeneous endowments. It seems natural that credit is the appropriate market mechanism for insuring such fluctuations: individuals save when endowment is high, and deplete their savings or borrow when their endowment is low. The first welfare theorem immediately would establish the efficiency of a credit equilibrium. However, it is straightforward that this is not the case: in a credit equilibrium, consumption fluctuates, while efficiency dictates that consumption be constant for each individual, conditional on his initial endowment, and this is physically feasible. We can contemplate the insurance problem from the perspective of a contract that is explicitly set up to atttempt to effect this redistribution. If endowments are unobservable to the contract, and the contract must base its redistribution on reports of individuals, then an incentive constraint must be satisfied by the contract. Green (1985) demonstrated that an incentive compatible contract that partially insures endowment is feasible in restricted settings in which endowment is i.i.d.. The contract works by increasing promised future utility in return for truthful revelation of high endowment. An extensive literature has expanded on this idea. The striking feature of Green's contract is its resemblance to credit. The promise of future utility behaves essentially like a stock of debt. Signalling high endowment to the contract today entails lowering consumption below endowment, but the contract compensates for reduction in consumption essentially by paying interest on it. The apparent inefficiency of credit equilibria is thus explained: first-best efficiency is not attainable because an incentive constraint must be satisfied. The impetus is then to characterize asset equilibria more broadly in these terms. In serially correlated settings, and in settings in which endowment realizations are unbounded, this reasoning is less successful. In those environments, incentive compatibility breaks down if a contract is structured like credit. That is because the incentive is to overstate the negative realizations, and with no lower bound on realizations (as in a standard Brownian process) there is no check on reports. Adding borrowing constraints is a standard but somewhat ad-hoc solution. The purpose of this paper is to demonstrate that the credit-like feature of the contract extends in an appropriate sense to such wider environments when an additional feature is added: delayed and sporadic observability. In this somewhat abstract treatment, observability occurs at boundaries that endowments hit at random stopping times. This generates boundary conditions that reinstate incentive compatibility. The boundary conditions induce nonlinearity that causes the contract to deviate from a credit-like arrangement near the boundaries. However, away from the boundaries the credit-like features of the model remain; in particular the contract is locally linear and for incentive reasons does not locally achieve full consumption smoothing. The first part of the paper quantitatively characterizes the degree to which the local credit-like character of the contract approximates the optimal contract. Continuous time methods are ideal for this analysis because the boundary effects can be isolated from the primary properties of the contract. This construction differs from a related literature in which there is hidden information about endowments, as here, but agents are presumed able to save and borrow externally and privately at a fixed interest rate. Such external saving and borrowing would be equally constrained by the privacy of information about endowment: here, the credit equilibrium is the same thing as the private-information contract. The second part of the paper then uses the locally linear approximation of the exact contract to ask whether it can be supported in the sense of satisfying a nondefection constraint. The contract can in fact be supported in this fashion. The locally linear contract is therefore decentralizable. This demonstrates that economies with many agents can support locally credit-like equilibria, but a full characterization of credit equilibria requires explicitly incorporating noncooperative underpinnings.