Dynamics under Full Commitment in Agency Theory

Contracts are often repeated over time. Examples of such long-term relationships abound and span all areas of contract theory. Let us describe a few. The insurance contract of an agent entails bonuses and maluses that link his current coverage and risk premium to his past history of accidents. Labor contracts often continue to reward the past performances of an agent in the future, either in monetary terms or by means of promotions. Lastly, in many regulated sectors, regulatory contracts often stipulate the current price caps that apply to a given firm as a function of the past realizations of its costs.

In view of the analysis of the previous chapters, the general framework to understand those repeated contractual relationships must be one where the principal controls several activities performed by the agent at different points in time. In an adverse selection setting, the reader will probably have rec- ognized the multi-output framework of section 2.10.3. Under moral hazard, the setting is akin to a multitask model along the lines of section 5.2. With respect to those general frameworks, the repeated contracting setting nevertheless has its own peculiarities that are worth studying. A few of these peculiari- ties are that the principal and the agent’s utility functions are separable over time, that the information structures may change as time passes, and that the arrow of time introduces a natural asymmetry between today’s and tomorrow’s relationships.

Indeed, the analysis of long-term contractual relationships raises a number of new questions. How do the repeated contractual relationships compare with the one-shot relationships studied in previous chapters? How does the repetition of the relationship change the terms of the static trade-offs? How should we model changes in information structures? What are the benefits, if any, of long lasting relationships? Does the past history of the agent’s performance play any role in current compensation, and if so, why? Are the implicit incentives provided by the desire of the agent to build a reputation vis-à-vis the labor market good disciplining devices?

To answer these questions, we remain in the general framework of this vol- ume1 and assume that the principal has the complete ability to commit to the contract he proposes to the agent. The principal designs the mechanism that is going to be played by the agent over time and sticks to its rules, no matter what happens during their relationship.

Even though the information structures associated with moral hazard and adverse selection dynamic models may look somewhat different, they can neverthe- less be classified within three broad categories that give rise to similar conclusions in both paradigms.

  • Permanent Shocks: In an adverse selection setting, the agent’s private information on the value of the trades that can be performed at different points of time may be constant over time. For instance, a regulated firm has a constant technology over the whole length of the regulated contract. A worker has a constant ability over the length of the labor contract. An insuree has a driving ability, e., a probability of having an accident, that does not change over his entire life. In such a setting, the optimal long-term contract is obtained in a straightforward manner as the replica of the one-shot optimal contract described in chapter 2. Indeed, under the assumptions of separability of the principal’s and the agent’s utility functions between today’s trades and tomorrow’s trades, the intertemporal benefit of a given profile of trades and its intertemporal cost, including the informational cost, is obtained as the discounted sum of the per-period benefits and costs of the volumes of trade chosen in the different periods. Within each period, the optimal trade-off between rent extraction and allocative efficiency is similar to that in a one-shot static relationship. Thus, the optimal long-term contract is obtained as the replica of the optimal static contract.

Importantly, one way of implementing this long-term optimal contract is given by the dynamic version of the revelation principle under full commitment. With a direct mechanism, the principal requests that the agent reveals his type once and for all, before any trade takes place. The principal then commits to replicating the one-shot optimal contract in each period of their relationship.3

Because of these analogies between the optimal dynamic contract and a static optimal contract, these models have sometimes been grouped under the general terminology of false dynamics.

  • Correlated Shocks: Still in an adverse selection setting, let us now turn to the more general case where the agent has private information on the values of trade with the principal in all periods and those values are correlated over time. For instance, the cost of producing a good for a seller may be the sum of two components, a permanent component linked to the production technology and a transient component linked to short-term shocks on the price of inputs. Similarly, because of learning by doing, a worker’s ability may change over time but still with some correlation across Those contractual settings are interesting because the mere realizations of the first-period volumes of trade convey some information about the future values of trade. The revelation principle still applies to these contractual relationships if one requests the agent to report any information he learns during the course of actions. In a direct revelation mechanism, the agent decides to report his type truthfully to the principal in any given period, knowing that the principal uses this information possibly to update his beliefs on the agent’s future types and may therefore specify different continuations of the long-term contract, depending on this latter report. This effect allows us to derive dynamic incentive constraints in a simple model with two periods and a risk-neutral agent. We then show how the principal should design the intertemporal contract by using earlier revelations of information in order to improve the terms of the rent extraction-efficiency trade-off in this dynamic setting.

It is interesting to observe the link between these latter models and the model of informative signals that improve contracting, which we already studied in section 2.14. In section 2.14, we analyzed how the principal may benefit from exogenous signals that are correlated with the agent’s information to improve the terms of the rent extraction-efficiency trade-off in a static model. In dynamic rela- tionships with correlated shocks, the past history of performances offers an endoge- nous signal that is correlated with the agent’s current type. History-dependent con- tracts are useful to take into account the informativeness of earlier performances. As a corollary, the optimal long-term contract is no longer obtained as the replica of the one-shot optimal contract.

Again, it is worth pointing out the moral hazard counterpart to this model. Let us assume that the agent is performing a different effort at each date of his relationship with the principal but that the random stochastic productions at the different dates are correlated. In such a framework, it is a simple corollary of the Sufficient Statistic Theorem of section 4.6.1 that past performances should be used to compute current compensations.

The correlation of shocks also plays an important role in the emergence of implicit incentives in moral hazard environments. We analyze a model of career concerns, where the agent’s current performance and ability affect the future rewards he may receive from the market. The agent’s desire to build a reputation for being efficient provides incentives to exert effort but, of course, this disciplin- ing device is only an imperfect substitute for explicit incentives contingent on performance.

  • Independent Shocks: Let us now envision a case where there is no cor- relation across periods among the values of trade. For instance, an agent may be looking for insurance against independently distributed income shocks, or a seller may be subject to one-period independent shocks on his In such a model, the past history of the agent’s performances loses any informative role on the cur- rent values of trade. It does not mean that history plays no longer any role. Indeed, history may allow the principal to smooth the cost of incentive compatibility over time.

To stress this new effect, we develop a two-period simple moral hazard model in the context of an efficiency-insurance trade-off. The model is basically a twice-replica of that in chapter 4.4 We derive the dynamic incentive compatibility constraint and optimize the principal’s intertemporal objective function. We show that the optimal contract exhibits a martingale property, linking current compen- sations with future rewards and punishments. The source of this property comes from the desire of the principal to smooth the cost of incentive compatibility over time. We discuss how this smoothing can be somewhat perturbed if the agent can save part of his wealth or can end the relationship in any given period. Then, we develop an infinitely repeated version of the model to explore the intertemporal distribution of utilities achieved in the long run or the behavior of the contract as the discount factor goes to one. We show that agency problems disappear in the limit by means of a complete diversification of the risk borne by the agent in any given period.

Section 8.1 presents the dynamics of repeated adverse selection relation- ships for a two-period example. In this section we make various assumptions on the information structure and derive some of the conclusions stressed above. In section 8.1.4, we briefly discuss the full commitment assumption. section 8.2 deals with the case of repeated moral hazard, both in a two-period model with vari- ous contractual limits but also in an infinitely repeated setting. Here we provide the basic dynamic programming methods necessary to analyze such a setting. In section 8.3, we analyze implicit incentives in a moral hazard framework.

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.

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