Let us consider a risk-neutral agent. As we have already seen, (4.3) and (4.4) now take the following forms:
Let us also assume that the agent’s transfer must always be greater than some exogenous level −l, with l ≥ 0. The framework is quite similar to that of section 3.5, and we refer the reader to that section for some motivations and discussions of the origins of such limited liability constraints on transfers. Limited liability constraints in both states of nature are thus written as
These constraints obviously reduce the set of incentive feasible allocations and may prevent the principal from implementing the first-best level of effort even if the agent is risk-neutral. Indeed, when he wants to induce a high effort, the principal’s program is written as
A first observation is that the transfers (4.12) and (4.13), allowing the imple- mentation of the first-best, may not satisfy the newly added limited liability con- straints. Proposition 4.2 summarizes the solution to (P).
Proposition 4.2: With limited liability, the optimal contract inducing effort from the agent entails:
- For only (4.15) and (4.16) are Optimal transfers are given by (4.12) and (4.13). The agent has no expected limited liability rent; EUSB = 0.
- For , (4.15) and (4.18) are Optimal transfers are then given by:
Moreover, the agent’s expected limited liability rent EUSB is non-negative:
First, we note that only the limited liability constraint in the bad state of nature may be binding. Indeed, since inducing effort requires the creation of a positive wedge between t¯ and t, (4.18) necessarily implies (4.17). When the limited liability constraint (4.18) is binding, the principal is limited in his punishments to induce effort. The risk-neutral agent does not have enough assets to cover the punishment requested by the principal if q is realized in order to induce effort provision. The principal uses rewards when a good state of nature q¯ is realized. As a result, the agent receives a non-negative ex ante limited liability rent described by (4.21). Compared with the case without limited liability, this rent is actually the additional payment that the principal must incur because of the conjunction of moral hazard and limited liability.
As the agent is endowed with more assets, i.e., as l gets larger, the conflict between moral hazard and limited liability diminishes and then disappears when- ever l is large enough. In this case, the agent avoids bankruptcy even when he has to pay the optimal penalty to the principal in the bad state of nature.
Let us now assume that l = 0 so that only non-negative transfers are feasible. Therefore, we model a contractual environment in which the agent owns no asset at the time of starting the relationship with the principal. When the principal induces effort from the agent, the principal’s expected utility can be computed as
If the principal gives up the goal of inducing effort from the agent, he can choose t = t¯ = 0 and instead obtain the expected utility level (4.8). It is worth inducing effort if V1SB is greater than V0, i.e., when
The left-hand side of (4.23) is the gain of inducing effort, i.e., the gain of increasing the probability of a high production level. The right-hand side is instead the second-best cost CSB of inducing effort, which is the disutility of effort ψ plus the limited liability rent . This second-best cost of implementing effort obvi- ously exceeds the first-best cost. As it can easily been seen by comparing (4.23) and the right-hand side of (4.9) (taken for the case of risk neutrality, i.e., for h(ψ) = ψ), limited liability and moral hazard together make it more costly to induce effort.
Figure 4.3 describes the reduced subset of values of B, justifying a high effort from the agent when limited liability and moral hazard interact.
Moral hazard justifies an underprovision of effort when the benefit B = ΔπΔS of a strictly positive effort lies between ψ and . Our analysis is summa-rized in proposition 4.3.
Proposition 4.3: With moral hazard and limited liability, there is a trade- off between inducing effort and giving up an ex ante limited liability rent to the agent. The principal chooses to induce a high effort from the agent less often.
Figure 4.3: First-Best and Second-Best Efforts with Moral Hazard and Limited Liability
Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.