by Yu Fu WONG

Last year marked a milestone in the development of contract theory. The Nobel Memorial Prize in Economic Sciences 2016 was awarded jointly to Oliver Hart and Bengt Holmström, “for their contributions to contract theory”.

In the same year, the John Bates Clark Medal, which acknowledges to economists under the age of forty, was bestowed upon Yuliy Sannikov for his new methodologies in game theory and contract theory. It was no coincidence—the two most prestigious awards in economics went to them, in recognition of the growing prevalence of the theory of incentives in the last thirty years. As the faculty at TSE have also made remarkable progress in the field, it is of interest for fellow students to understand the basics of contract theory and the accomplishments of the three laureates.

Contracts are ubiquitous in our daily life. Property owners rent us their apartments, conditional on paying the rent punctually and keeping the furniture intact. In employment agreements, enterprises will compensate us for our labour upon satisfactory performance. For the blessed ones, the bride and groom promise to abide by certain vague code of honour, contingent on each other’s behaviour. For the less fortunate ones, nothing is more vivid than clicking on “I agree” in order to use the latest apps and gadgets without reading the forty-page small print, which now comes into legal effect. Loosely speaking, a contract is an arrangement which specifies some actions or allocations conditional on some observations in the future, usually between two parties.

We can gain many insights studying one common type of informal contract in our early years. It usually reads, “if you behave yourself, you will get ice-cream.” When the young boy behaves, the mother offers ice-cream, with the converse implicitly understood. It is ambiguous how well the boy has to behave to be rewarded and, in that case, how much ice-cream the mother will give in return. When the child does not meet the standard, he may bargain for a smaller portion; even when he does, there might not be any left in the fridge. Some smart kids may realize factors other than their behaviour in this reward scheme, the expiration date of the ice-cream for instance. This example demonstrates the simplicity, incompleteness, negotiability, possibility of default, and lack of commitment of contracts in the society.

Incomplete Contract Theory

Oliver Hart and John Moore are generally credited for laying the foundation of incomplete contract theory. Incomplete contracts, or simple contracts, focus on the difficulties and imperfections of real-life contracts. Hart has also studied extensively their implications in capital structure, management finance, and public economics. For an illustration, we shall take the example of the ice-cream reward scheme.

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This contract between mother and son is incredibly simple. Both the conditions and outcomes are quite vague as well. Hart argues that such contracts are popular because it is costly to specify the definite allocation for all potential observations. While it may be possible to devise a complex scheme to provide incentives more efficiently, the parties seek simple, incomplete, sub-optimal (constrained-optimal) ones in face of the staggering transaction cost. In our case, it is almost impossible to explain to a child the amount of ice-cream he will get as a function of his behaviour, which is defined formally and precisely in all possible situations.

With such simple contracts, the parties may not be able to commit to the plan. On the one hand, if the child does not behave, the soft-hearted mother may offer a smaller portion to her weeping child, despite their original agreement. On the other hand, even if the child behaves, the mother may not be able to uphold her end when the fridge is empty. Hart shows that the temptation of renegotiation and the transfer of idiosyncratic risk will undermine social surplus in general. Even when there is enough ice-cream, the mother will tend to give out more than she was planning to at the beginning, because the ice-cream purchase is sunk. This issue of excessive incentive is known as the hold-in problem. A parallel dilemma of insufficient incentive is called the hold-up problem. In the context of corporate finance, firms are tempted too often to restructure its debt with small investors who face a hold-in problem. In education economics, workers tend to underinvest in human capital because part of this surplus is captured by their employers, resulting in a hold-up problem.

Complete Contract Theory

Despite the inefficiencies in simple contracts, Eric Maskin and Jean Tirole proved that the hold-up problem can be solved by writing more complex ones. A natural extension of the study of incomplete contracts is to examine the optimal, potentially complicated, contract for a given information structure. This field of research is known as complete contract theory. Bengt Holmström is perhaps best known for his work on the optimal use of information, which balances risk exposure and incentive provision. He has also looked into various forms of contractual possibilities, such as promotions, multi-tasking, and teamwork.

Before 1987, most of the literature in contract theory was limited to static cases, which presented a few challenges. For one, James Mirrlees shows that the optimal contract does not exist when the agent’s utility is unbounded from below—at the limit, which means that the principal can motivate the agent efficiently by punishing him only in the worst scenarios. This contradicts the intuition that the agent is motivated by ordinary circumstances. For another, the analytical complexity grows exponentially when the number of periods in the economic model increases. Under this context, the seminal paper of Bengt Holmström and Paul Milgrom on dynamic contracting sparked numerous discussions in contract theory that continue to this day.


In their paper, the agent can exert costly effort to imperfectly improve the outcome in each period. The contract is terminated after a large number of periods and the principal compensates the agent based on the history of observations. Under appropriate assumptions, the agent’s expected value follows a Brownian motion, and the optimal repayment scheme counts the occurrence of each observation, i.e. the optimal contract is effectively a piece-rate contract. The linear compensation function motivates the agent to exert effort in all circumstances, not only in the worst ones. The mathematical elegance of Brownian motion offers analytical convenience and tractability—the authors were able to isolate the compensation into four components: reservation value, cost of effort, incentive provision, and risk-premium. Since the publication, thousands of papers followed their framework to examine various contractual possibilities.

Dynamic Contract Theory

One deadly criticism of the model of Holmström and Milgrom is that it only considers compensations at the very end. In real life, rents are settled, wages are paid, and relationships are maintained throughout the contracts. This model is defensible at best for impatient agents; however, it cannot capture truly the dynamic tradeoff between efficiency and incentives. This flaw is similar to that of the folk theorem in game theory, where the same conflict de-materializes when players become patient. After twenty years of research, dynamic contracts and repeated games in discrete time seemed to have reached their pinnacles with limited progress. And this is where Yuliy Sannikov came into play.

Sannikov rose into the spotlight in 2004 with his PhD dissertation on repeated games with imperfect observations in continuous time. Instead of discrete time, he attacked the problem with the arsenal of continuous-time methodologies. With the martingale representation theorem, he was able to solve for the set of all public perfect equilibria for any discount rate. In application to contract theory, he has provided a tractable model to study the balance between efficiency, risk-sharing, and moral hazard in both the short run and the long run. With his novel tools for dynamic contracts, he has been working on capital structure, compensation schemes, structural learning, and many other areas.

Capture d_écran 2017-06-06 à 14.31.17Sannikov’s methodological breakthrough has revitalized contract theory and game theory. Many factors that were once ignored in the name of tractability can now be analyzed in simple models. Here at TSE, Bruno Biais, Thomas Mariotti, Guillaume Plantin, and Jean-Charles Rochet have accomplished a connection between the continuous-time and discrete-time approaches, and gained new insights into the link between asset pricing and corporate finance from the optimal repayment scheme for managers. Macroeconomists can look into dynamic taxation incidence, or equivalently the social contract, which balances between wealth redistribution and moral hazard. Public economists have new tools to inspect the role of the government’s reputation in implementing new policies under a relational contract. The list grows longer by the day.

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