Sometimes the set of incentive-feasible contracts is constrained by some exogenous limits on the feasible transfers between the principal and the agent. These exoge- nous financial constraints could reveal the existence of previous financial contracts that the agent might have already signed. Those constraints will of course affect the usual rent-efficiency trade-off.
A first possible limit is that the net transfer of the agent, taking into account his own asset holding l, should not be lower than zero. This leads to the following limited liability constraints on transfers:
A possible motivation for this type of constraint is that the agent can use the transfer received from the principal to cover a debt of level −l. The production cost θq being already sunk, it does not enter into the left-hand sides of (3.75) and (3.76).
A second limit on transfers arises when the agent’s information rent itself must be greater than an exogenous value −l. This leads to the following limited liability constraints on rents:
The production cost hq is now incurred when the transfer t takes place. Again, the interpretation is that contracting with the principal may involve negative rents U or U¯ as long as those losses can be covered by the agent’s own liabilities l. To assess the impact of these limited liability constraints, let us go back to the framework of section 2.11. When contracting takes place ex ante, we have seen that the first-best outcome can still be obtained provided that the inefficient risk- neutral agent receives a negative payoff, . Obviously this negative payoff may conflict with the constraint (3.78).
With ex ante contracting, we have already seen that the relevant incentive and participation constraints are, respectively,
Adding the limited liability constraints, the principal’s program is written as
where limited liability constraints are either on transfers or on rents.
The next two propositions summarize the features of the optimal contract with a limited liability constraint on rents and transfers, respectively.18 We index with a superscript L, meaning limited liability, the second-best optimal contracts in these environments. We first focus on limited liability constraints on rents.
Proposition 3.4: Assume ex ante contracting and limited liability on rents. Then, the optimal contract entails
- For , only (3.79) and (3.80) are binding and the first-best outcome of section 11.1 remains optimal.
- For , (3.79), (3.80), and (3.78) are all The efficient agent produces efficiently qL = q∗, and the inefficient agent’s production is distorted downward from the first-best with and ,
- For , only (3.79) and (3.78) are The efficient agent produces efficiently qL = q∗, and the inefficient agent’s production is equal to the second-best output with the ex post participation con- straints, , defined in (2.28).
A limited liability constraint on ex post rents may reduce the efficiency of ex ante contracting. If the limited liability constraint on the inefficient type is stringent enough, the principal must reduce the inefficient agent’s output to keep the limited liability constraint satisfied. The agent is then subject to less risk on the allocation of ex post rents. When the limited liability constraint is even harder, the principal must give up his desire to hold the ex ante participation constraint binding. The limited liability constraint then implies an ex ante information rent. Indeed, when l is small enough, the agent’s expected utility becomes , which is then strictly positive.
Remark: Note the similarity of the solution obtained in Propo- sition 3.4 with that obtained when the agent is risk averse in section 2.11.2 (proposition 2.5). The limited liability constraint on rents plays a similar role as the agent’s risk aversion. Indeed, in both cases, the principal finds it costly to create a wedge between U and U¯, and reducing this cost calls for incentives that are lower powered than one would find with risk neutrality and unlimited transfers. More precisely, with a limited liability constraint on rents, everything happens as if the agent has an infinite risk aversion below a wealth of −l.
Let us now turn to the case of limited liability constraints on transfers. Restrict- ing the analysis to a few particular cases, we have the following characterization of the optimal contract.
Proposition 3.5: Assume ex ante contracting and limited liability on transfers. Then the optimal contract entails:
- For , only (3.80) is binding and the first-best outcome of section 11.1 remains optimal.
- For , (3.79), (3.80), and (3.76) are all binding. The efficient agent produces efficiently qL = q∗, and the inefficient agent’s production is distorted upwards from the first- best, with , and
- For , there is bunching such that both types produce the same output qL and (3.75), (3.76), (3.79), and (3.80) are all The constant output target qL is given by
The limited liability constraints on transfers give rise to allocative distortions that are rather different from those highlighted in proposition 3.4. As the limited liability constraint (3.76) is more stringent, it becomes quite difficult to create the wedge between U and U¯ that is necessary to ensure incentive compatibility. However, to relax the limited liability constraint (3.76), the principal now increases the inefficient type’s output. Indeed, using the information rent to rewrite (3.76), we obtain
Therefore, distorting the inefficient type’s output upward relaxes this lim- ited liability constraint. A limited liability constraint on transfers implies higher- powered incentives for the agent. It is almost the same as what we would obtain by assuming that the agent is a risk lover. The limited liability constraint on transfers somewhat convexifies the agent’s utility function.
Of course, the principal cannot indefinitively raise the inefficient agent’s out- put without conflicting with the implementability condition. Hence, some bunch- ing emerges. In this case, the agent receives a fixed payment that covers their cost in expectation. This transfer also satisfies the limited liability constraints (3.75) and (3.76), which both take the same form.
It is interesting to note that the limited liability constraints on transfers can only be binding when l is negative. This arises, for instance, when the transfer received by the agent from the principal must be used to reimburse a loan contract worth l.
Sappington (1983) derived the optimal contract under adverse selec- tion and limited liability constraints. Lewis and Sappington (2000) also provided a model involving moral hazard elements. Che and Gale (2000) and Lewis and Sappington (2001) analyzed models where the agent is pri- vately informed both of his ability and of his wealth. These latter papers showed that the wealth level and the agent’s ability are complements in deter- mining the probability that trade occurs with the agent.
Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.