Contract theory

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:

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

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.[2][3] 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.[4][5] The moral hazard model with risk-neutral but wealth-constrained agents has also been extended to settings with repeated interaction and multiple tasks.[6] 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.[7] Moreover, contract-theoretic models with hidden actions have been directly tested in laboratory experiments.[8]

A study on the solution to moral hazard concludes that adding moral sensitivity to the principal-agent model increases its descriptiveness, prescriptiveness, and pedagogical usefulness because it induces employees to work at the appropriate effort for which they receive a wage. The theory suggests that as employee work efforts increase, so proportional premium wage should increases also to encourage productivity.[9]

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.[10]

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.[11][12] More recently, adverse selection theory has been tested in laboratory experiments and in the field.[13][14]

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.[15][16][17]

In signalling models, one party chooses how and whether or not to present information about itself to another party to reduce the information asymmetry between them.[18] In signaling models, the signaling party agent and the receiving party principal have access to different information. The challenge for the receiving party is to decipher the credibility of the signaling party so as to assess their capabilities. The formulation of this theory began in 1973 by Michael Spence through his job-market signaling model. In his model, job applicants are tasked with signalling their skills and capabilities to employers to reduce the probabilities for the employer to choose a lesser qualified applicant over a qualified applicant. This is because potential employers lack the knowledge to discern the skills and capabilities of potential employees.[19]