Efficient Allocations with Moral Hazard and Hidden Borrowing and Lending

Abstract

In this paper we develop a recursive approach to study efficient allocations in a dynamic moral hazard setting, where agents can borrow and lend and their decisions about effort, consumption and savings are private information. The recursive formulation of the problem is based on a generalized first order approach, whose validity is verified using a parsimonious numerical procedure based on the recursive formulation itself. In contrast with previous findings, we show that the second best allocation is welfare improving with respect to the case where the agents can self insure themselves only through borrowing and lending. Thanks to the recursive formulation, we are able to quantify the efficiency gains in a number of numerical examples. We find that welfare gains are substantial and do not vary monotonically with the credit market return. We also identify the main observational characteristics of the constrained efficient allocation.

A solvable continuous time dynamic principal–agent model

“…Thus I solve for a candidate optimal contract using the necessary optimality conditions, then show ex-post that the contract is truly incentive compatible and therefore implementable. Werning (2001), Abraham and Pavoni (2008), and Farhi and Werning (2013) follow the same approach, and here I am able to verify incentive compatibility analytically. In general with hidden state variables, the contract must condition on an additional endogenous state variable, capturing the shadow value (in terms of the agent's marginal utility) of the hidden state.…”

“…Sannikov (2008) considers dynamic payments but has no state variables other than promised utility. the approach of Werning (2001), Abraham and Pavoni (2008), Farhi and Werning (2013), Kapicka (2013), and Pavan, Segal, and Toikka (2014) in discrete time dynamic moral hazard problems. 5 Unlike the hidden action case and Williams (2011), with hidden savings I cannot provide useful conditions which guarantee the validity of the first-order approach ex-ante.…”

Unemployment Insurance

“…6 There is no set of known conditions in the infinite horizon problem UIP that are sufficient to guarantee that the first-order approach is valid with hidden savings. Abraham and Pavoni (2003) point out, though, that it is possible to verify whether a particular solution to the first-order approach problem is actually a solution to the true problem. They use a two-step numerical procedure in their analysis of optimal unemployment insurance with hidden borrowing and lending.…”

“…(He interprets this falling differential as implying that unemployment benefits should be increasing in the duration of unemployment.) His paper does not have the kind of explicit verification step contained in Abraham and Pavoni (2003). Hence, his paper contains no information about whether his characterization of the solution to the first-order approach problem carries over to the true problem UIP or not.…”

“…It is not known, though, how to translate their recursive formulation into a practical computational procedure when savings can take on a continuum of values. Werning (2002) and Abraham and Pavoni (2003) attack the problem by using a computationally feasible first-order approach that replaces the agent's incentive constraints with the corresponding first order conditions. However, I show that even in simple examples, the first-order approach may not be valid because the agent's decision problem is intrinsically non-concave in effort and savings.…”