# Contract theory

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In economics, contract theory studies how economic actors can and do construct contractual arrangements, generally in the presence of asymmetric information. Because of its connections with both agency and incentives, contract theory is often categorized within a field known as Law and economics. One prominent application of it is the design of optimal schemes of managerial compensation. In the field of economics, the first formal treatment of this topic was given by Kenneth Arrow in the 1960s. In 2016, Oliver Hart and Bengt R. Holmström both received the Nobel Memorial Prize in Economic Sciences for their work on contract theory, covering many topics from CEO pay to privatizations.

Economics is the social science that studies the production, distribution, and consumption of goods and services.

An incentive is a contingent motivator. Traditional incentives are extrinsic motivators which reward actions to yield a desired outcome. The effectiveness of traditional incentives has changed as the needs of Western society have evolved. While the traditional incentive model is effective when there is a defined procedure and goal for a task, Western society started to require a higher volume of critical thinkers, so the traditional model became less effective. Institutions are now following a trend in implementing strategies that rely on intrinsic motivations rather than the extrinsic motivations that the traditional incentives foster.

Law and economics or economic analysis of law is the application of economic theory to the analysis of law that began mostly with scholars from the Chicago school of economics. Economic concepts are used to explain the effects of laws, to assess which legal rules are economically efficient, and to predict which legal rules will be promulgated.

## Contents

A standard practice in the microeconomics of contract theory is to represent the behaviour of a decision maker under certain numerical utility structures, and then apply an optimization algorithm to identify optimal decisions. Such a procedure has been used in the contract theory framework to several typical situations, labeled moral hazard , adverse selection and signalling . The spirit of these models lies in finding theoretical ways to motivate agents to take appropriate actions, even under an insurance contract. The main results achieved through this family of models involve: mathematical properties of the utility structure of the principal and the agent, relaxation of assumptions, and variations of the time structure of the contract relationship, among others. It is customary to model people as maximizers of some von Neumann–Morgenstern utility functions, as stated by expected utility theory.

In economics, moral hazard occurs when someone increases their exposure to risk when insured, especially when a person takes more risks because someone else bears the cost of those risks. A moral hazard may occur where the actions of one party may change to the detriment of another after a financial transaction has taken place.

Adverse selection is a term commonly used in economics, insurance, and risk management that describes a situation where market participation is affected by asymmetric information. When buyers and sellers have different information, it is known as a state of asymmetric information. Traders with better private information about the quality of a product will selectively participate in trades which benefit them the most, at the expense of the other trader. A textbook example is Akerlof's market for lemons.

In contract theory, signalling is the idea that one party credibly conveys some information about itself to another party. Although signalling theory was initially developed by Michael Spence based on observed knowledge gaps between organisations and prospective employees, its intuitive nature led it to be adapted to many other domains, such as Human Resource Management, business, and financial markets.

## Main models of agency problems

### Moral hazard

In moral hazard models, the information asymmetry is the principal's inability to observe and/or verify the agent's action. Performance-based contracts that depend on observable and verifiable output can often be employed to create incentives for the agent to act in the principal's interest. When agents are risk-averse, however, such contracts are generally only second-best because incentivization precludes full insurance.

In contract theory and economics, information asymmetry deals with the study of decisions in transactions where one party has more or better information than the other. This asymmetry creates an imbalance of power in transactions, which can sometimes cause the transactions to go away, a kind of market failure in the worst case. Examples of this problem are adverse selection, moral hazard, and monopolies of knowledge.

The typical moral hazard model is formulated as follows. The principal solves:

${\displaystyle \max _{w(\cdot )}E\left[y({\hat {e}})-w(y({\hat {e}}))\right]}$

subject to the agent's "individual rationality (IR)" constraint,

${\displaystyle E\left[u(w(y(e)))-c(e)\right]\geq {\bar {u}}}$

and the agent's "incentive compatibility (IC)" constraint,

${\displaystyle E\left[u(w(y({\hat {e}})))-c({\hat {e}})\right]\geq E\left[u(w(y(e)))-c(e)\right]\ \forall e}$,

where ${\displaystyle w(\cdot )}$ is the wage as a function of output ${\displaystyle y}$, which in turn is a function of effort:${\displaystyle e}$.

${\displaystyle c(e)}$ represents the cost of effort, and reservation utility is given by ${\displaystyle {\bar {u}}}$.

${\displaystyle u(\cdot )}$ is the "utility function", which is concave for the risk-averse agent, is convex for the risk-prone agent, and is linear for the risk-neutral agent.

If the agent is risk-neutral and there are no bounds on transfer payments, the fact that the agent's effort is unobservable (i.e., it is a "hidden action") does not pose a problem. In this case, the same outcome can be achieved that would be attained with verifiable effort: The agent chooses the so-called "first-best" effort level that maximizes the expected total surplus of the two parties. Specifically, the principal can give the realized output to the agent, but let the agent make a fixed up-front payment. The agent is then a "residual claimant" and will maximize the expected total surplus minus the fixed payment. Hence, the first-best effort level maximizes the agent's payoff, and the fixed payment can be chosen such that in equilibrium the agent's expected payoff equals his or her reservation utility (which is what the agent would get if no contract was written). Yet, if the agent is risk-averse, there is a trade-off between incentives and insurance. Moreover, if the agent is risk-neutral but wealth-constrained, the agent cannot make the fixed up-front payment to the principal, so the principal must leave a "limited liability rent" to the agent (i.e., the agent earns more than his or her reservation utility).

The moral hazard model with risk aversion was pioneered by Steven Shavell, Sanford J. Grossman, Oliver D. Hart, and others in the 1970s and 1980s. [1] [2] It has been extended to the case of repeated moral hazard by William P. Rogerson and to the case of multiple tasks by Bengt Holmström and Paul Milgrom. [3] [4] The moral hazard model with risk-neutral but wealth-constrained agents has also been extended to settings with repeated interaction and multiple tasks. [5] While it is difficult to test models with hidden action empirically (since there is no field data on unobservable variables), the premise of contract theory that incentives matter has been successfully tested in the field. [6] Moreover, contract-theoretic models with hidden actions have been directly tested in laboratory experiments. [7]

Sanford "Sandy" Jay Grossman is an American economist and hedge fund manager specializing in quantitative finance. Grossman’s research has spanned the analysis of information in securities markets, corporate structure, property rights, and optimal dynamic risk management. He has published widely in leading economic and business journals, including American Economic Review, Journal of Econometrics, Econometrica, and Journal of Finance. His research in macroeconomics, finance, and risk management has earned numerous awards. Grossman is currently Chairman and CEO of QFS Asset Management, an affiliate of which he founded in 1988. QFS Asset Management shut down its sole remaining hedge fund in January 2014.

Oliver Simon D'Arcy Hart is a British-born American economist, currently the Andrew E. Furer Professor of Economics at Harvard University. Together with Bengt R. Holmström, he received the Nobel Memorial Prize in Economic Sciences in 2016.

Paul Robert Milgrom is an American economist. He is the Shirley and Leonard Ely Professor of Humanities and Sciences at Stanford University, a position he has held since 1987. Milgrom is an expert in game theory, specifically auction theory and pricing strategies. He is the co-creator of the no-trade theorem with Nancy Stokey. He is the co-founder of several companies, the most recent of which, Auctionomics, provides software and services that create efficient markets for complex commercial auctions and exchanges.

### Adverse selection

In adverse selection models, the principal is not informed about a certain characteristic of the agent at the time the contract is written. The characteristic is called the agent's "type". For example, health insurance is more likely to be purchased by people who are more likely to get sick. In this case, the agent's type is his or her health status, which is privately known by the agent. Another prominent example is public procurement contracting: The government agency (the principal) does not know the private firm's cost. In this case, the private firm is the agent and the agent's type is the cost level. [8]

Health insurance is an insurance that covers the whole or a part of the risk of a person incurring medical expenses, spreading the risk over a large number of persons. By estimating the overall risk of health care and health system expenses over the risk pool, an insurer can develop a routine finance structure, such as a monthly premium or payroll tax, to provide the money to pay for the health care benefits specified in the insurance agreement. The benefit is administered by a central organization such as a government agency, private business, or not-for-profit entity.

In adverse selection models, there is typically too little trade (i.e., there is a so-called "downward distortion" of the trade level compared to a "first-best" benchmark situation with complete information), except when the agent is of the best possible type (which is known as the "no distortion at the top" property). The principal offers a menu of contracts to the agent; the menu is called "incentive-compatible" if the agent picks the contract that was designed for his or her type. In order to make the agent reveal the true type, the principal has to leave an information rent to the agent (i.e., the agent earns more than his or her reservation utility, which is what the agent would get if no contract was written). Adverse selection theory has been pioneered by Roger Myerson, Eric Maskin, and others in the 1980s. [9] [10] More recently, adverse selection theory has been tested in laboratory experiments and in the field. [11] [12]

Roger Bruce Myerson is an American economist and professor at the University of Chicago. He holds the title of The Glen A. Lloyd Distinguished Service Professor in Economics and the College and Harris Graduate School of Public Policy Studies. In 2007, he was the winner of the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel with Leonid Hurwicz and Eric Maskin for "having laid the foundations of mechanism design theory." He was elected a Member of the American Philosophical Society in 2019.

Eric Stark Maskin is an American economist and 2007 Nobel laureate recognized with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism design theory". He is the Adams University Professor at Harvard University. Until 2011, he was the Albert O. Hirschman Professor of Social Science at the Institute for Advanced Study, and a visiting lecturer with the rank of professor at Princeton University.

Adverse selection theory has been expanded in several directions, e.g. by endogenizing the information structure (so the agent can decide whether or not to gather private information) and by taking into consideration social preferences and bounded rationality. [13] [14] [15]

## Incomplete contracts

Contract theory also utilizes the notion of a complete contract, which is thought of as a contract that specifies the legal consequences of every possible state of the world. More recent developments known as the theory of incomplete contracts, pioneered by Oliver Hart and his coauthors, study the incentive effects of parties' inability to write complete contingent contracts, e.g. concerning relationship-specific investments. A leading application of the incomplete contracting paradigm is the Grossman-Hart-Moore property rights approach to the theory of the firm (see Hart, 1995).

Because it would be impossibly complex and costly for the parties to an agreement to make their contract complete, [16] the law provides default rules which fill in the gaps in the actual agreement of the parties.

During the last 20 years, much effort has gone into the analysis of dynamic contracts. Important early contributors to this literature include, among others, Edward J. Green, Stephen Spear, and Sanjay Srivastava.