# Contract theory

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.

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.

## 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.

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

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

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

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

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

$\hat{e} = \arg \max_e E \left[ u(w(y(e))) - c(e) \right] \geq \bar{u}$,

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

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

$u(\cdot)$ 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.

In adverse selection models, the principal is not informed about a certain characteristic of the agent. For example, health insurance is more likely to be purchased by people who are more likely to get sick.

## 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.

Because it would be impossibly complex and costly for the parties to an agreement to make their contract complete,[1] 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.