One way to think about the different types of contracts modelled in contract theory is to divide contract theory into three groups: complete contacts, comprehensive contracts and incomplete contracts. Complete contracts are those which are written in a zero transaction cost, Arrow-Debreu type, world. Such contracts can be made continent on all variables in all states of the world. They result in the first-best being achieved in all states of world. Comprehensive contracts are those written under conditions of asymmetric information, that is, in a world with moral hazard and/or adverse selection. Such contracts are "constrained optimal" in that they are optimal given the existence of the information asymmetry. Comprehensive contracts maximise the objective function of the agent subject to the informational constraint. Incomplete contracts do not maximise the objective function of the agent, they result in "money being left on the table", even taking into account any informational asymmetries. A standard incomplete contracts model will be a symmetric information model and thus neither moral hazard or adverse selection are driving the model's results. The issue for incomplete contracts is generally argued to be one of "non-verifiability" rather than asymmetric information. That is, the informational problem with incomplete contracts is between that contracting parties and the courts rather than between the contracting parties themselves, as in asymmetric information models.
In the law and economics literature it is argued that there are two forms of incompleteness: obligationally incomplete [OI] and informationally incomplete (or insufficiently state contingent) [II] contracts. A contract is obligationally incomplete if it does not fully describe the obligrations of each party in every state of the world. That is, the contract has a "gap" and thus will be silent on what should happen in any state of the world which falls within the "gap". The problem here is why should a contract be ever be obligationally incomplete since it should be possible to complete a contract with an obligration that applies to a broadly enough defined set of contingencies at a reasonable cost. A contract is informationally incomplete if it fails to describe an efficient set of obligations in each possible state of the world. As Oliver Hart puts it
[...] the contract might not specify what is to happen if the supplier's factory burns down, because this is not anticipated [OI]; or the contract might say that the supplier must always supply one widget, rather than a number of widgets that varies with the state of the world, because it is too costly to distinguish between different states of the world [II].A problem here is that if a contract with an inefficient set of obligations specified is incomplete then why are asymmetric information contracts not incomplete?
Eric Maskin writes that
I will consider a contract to be “incomplete” if it is not as fully contingent on the state of the world” (the resolution of uncertainty about the future) as the parties to the contract might like it to be.This seems to mean that asymmetric information contracts are incomplete. But in a footnote Maskin says
This definition is so broad that it covers many contracts in the literature that are not normally considered "incomplete", e.g., insurance contracts with adverse selection.So moral hazard and adverse selection contracts are not, it seems, incomplete contracts. The issue is that asymmetric contracts are verifiable, the variable on which the incentive contract is written is assumed to be observable and verifiable so that the courts can fully enforce the contract. As noted above the standard assumption for incomplete contracts is that they are incomplete because some relevant variable is not verifiable, to the courts. Another way to look at this is that if performance of the terms of a contract would result in the gains from trade not being fully exploited, given the information that the contracting parties and the courts have available to them at the time performance takes place, then the contract is incomplete. Under the assumptions of complete or comprehensive contracting any gains from trade available are always exploited to the fullest extent possible.
But the assumption of non-verifiability has its own problems as Maskin and Tirole have pointed out. Maskin and Tirole argue that information which is observable to the contracting parties (symmetric information) can be made verifiable (to a third party) by the use of ingenious revelation mechanisms. The contracting parties write into their contract a game which when played gives the appropriate incentives for them to truthfully reveal their private information in equilibrium. This undermines the non-verifiability approach to incomplete contracts.
To deal with the Maskin and Tirole critique, Hart and Moore developed the 'reference point' approach to incomplete contracts. Very briefly the Hart and Moore reference point theory argues that when the parties meet at date 0 there is uncertainty about the state of the world. This uncertainty is resolved shortly before date 1. There is symmetric information throughout, but the state is not verifiable. A date 0 contract serves as a reference point for the contracting parties' feelings about entitlements at date 1. Specifically, neither party feels entitled to an outcome outside those permitted by the contract but within the contract there can be disagreement about the appropriate outcome. To simplify matters, it is supposed that each party feels entitled to their best possible outcome permitted by the contract. Of course, this means that usually at least one party will be disappointed or "aggrieved" by any particular outcome. Hart and Moore assume that no outcome from a transaction is perfectly contractible even at date 1. In particular, they assume that each party has the discretion to provide "perfunctory" performance rather than "consummate" performance. Performing at the lower perfunctory level rather than the higher consummate level is referred to as shading and it is assumed that shading cannot be penalised by a court. A court can, however, enforce the perfunctory level of performance. When a party is aggrieved he shades, by an amount theta times his level of aggrievement where 0< theta <=1. Consummate performance does not cost significantly more than perfunctory performance to whomever is providing the good or service, and a party will provide consummate performance if he feels "well treated" but not otherwise. Shading hurts the other party and causes a deadweight loss. The important point here is that the reference point approach does not suffer from the Maskin and Tirole critique but get around it by introducing a number of ad hoc behavioural assumptions, e.g. aggrievement and shading.
After all of this we find that M'hand Fares argues that the difference between complete and incomplete contracts is not verifiability at all but the ability to commit to not renegotiating the initial contract.
In the controversy on the theoretical foundations of the property rights approach, Hart and Moore [...] and Maskin and Tirole [...] point out that a key distinction between complete and incomplete contracting is the ability of risk neutral parties to commit not to renegotiate the initial contract. The renegotiation design issue in contract solutions to the hold-up problem restates this view in a more general fashion as a contrast between (i) a world where contract can determine the entire relationship between the parties and (ii) a world where contract can only influence an existing underlying game between them, that is, the renegotiation game. This implies that the capacity of a contract to influence this game defines its 'incompleteness' or 'degree of incompleteness' [...]: the more a contract is able to design the renegotiation game, the less it is incomplete.So in the end we are left with Jean Tirole's point that,
[f]or all its importance, there is unfortunately no clear definition of "incomplete contracting" in the literature. While one recognizes one when one sees it, incomplete contracts are not members of a well-circumscribed family; at this stage an incomplete contract is rather defined as an ad hoc restriction on the set of feasible contracts in a given model. The concept of "ad hoc restriction" is of course subjective: to give it some content, we will [...] take the standard approach to contract theory as the benchmark. The methodology developed in the last thirty years to treat moral hazard, adverse selection, and implementation problems provides a well-defined delineation of the set of feasible outcomes by incentive constraints. Incomplete contracting then relates to a focus on a subset of feasible outcomes through the imposition of restrictions on the set of allowable contracts.Thus I'm left asking, When is a contract incomplete?