Abstract
The theory of incentives and matching theory can complement each other. In particular, matching theory can be a tool for analyzing optimal incentive contracts within a general equilibrium framework. We propose several models that study the endogenous payoffs of principals and agents as a function of the characteristics of all the market participants, as well as the joint attributes of the principal–agent pairs that partner in equilibrium. Moreover, considering each principal–agent relationship as part of a market may strongly influence our assessment of how the characteristics of the principal and the agent affect the optimal incentive contract. Finally, we discuss the effect of the existence of moral hazard on the nature of the matching between principals and agents that we may observe at equilibrium, compared to the matching that would happen if incentive concerns were absent.
Introduction
The optimal design of incentives is a prevalent question in economic and social relationships. In labor contracts, a worker’s decision about his effort is often not contractible and his employer may need to provide him with incentives to work. A manager may have an inclination not to maximize the firm’s profits but to use some of the company resources to obtain private benefits. An insurance company may fear that insured people are less cautious with the insured property than when they face the whole cost of the damage. These incentive problems appear because one party in the relationship (the “agent”) takes a decision that affects another party (the “principal”), and they are usually referred to as “moral hazard” or “agency” problems.^{Footnote 1}
The theory of incentives studies the best contract (for addressing moral hazard) for a given relationship, where this relationship is considered in isolation from any other. That is, it looks at a given principal and a given agent (or possibly several principals and/or agents) that intend to establish a relationship, and characterizes the optimal contract that the principal will propose, among the ones that the agent is ready to accept and considering that the agent will exert the effort that is best for him given the contract.
Some principal–agent relationships are indeed isolated partnerships. This is the case, for example, for the regulatory relationship between a government and an established monopoly. Here we can see that the government cannot look for an alternative firm to provide the service and the firm cannot look for an alternative government. However, most relationships take place in “markets”: a principal is typically not forced to hire a particular agent but she can possibly partner with any agent present in the market, and similarly for the agents. For example, an investor is not compelled to invest in a certain startup and a startup is not forced to receive financing from a specific investor; any investor and any startup will look for who to partner and sign a contract with. This means that not only is the contract in each relationship endogenous but so is the identity of the partners that establish relationships.
Simultaneously to the development of the theory of incentives, Gale and Shapley (1962) and Shapley and Shubik (1972) pioneered the development of the twosided “matching theory.” Twosided matching theory studies markets where players belong, from the outset, to one of two disjoint sets. The questions this theory addresses refer to the partnerships that are formed at the equilibrium and, in some models, the terms of the transfers between partners. One relevant advantage of matching theory is that it can successfully accommodate situations with heterogeneous players in either or both sides of the market.
Some recent papers^{Footnote 2} emphasize that the theory of incentives and matching theory can complement each other. In particular, matching theory can be used as a tool to study incentive contracts in a general equilibrium scenario that allows the consideration of discrete as well as continuous sets of heterogeneous principals and heterogeneous agents.
The analysis of optimal incentive contracts within a general equilibrium framework allows new questions to be addressed such as the equilibrium payoffs of principals and also the endogenous agents’ utility as a function of the characteristics of all the market participants; and the joint characteristics of the principal–agent pairs that decide to partner. Moreover, considering each principal–agent relationship not in isolation but as part of a market may strongly influence, and even reverse, the results on the effects of the characteristics of the principal and the agent on the optimal incentive contract. Indeed, the comparative static exercises performed in a principal–agent model to understand the effect of, say, the improvement of an agent’s characteristic on the terms of the contract do not take into account that such an improvement may lead the agent to be matched, at equilibrium, to a different principal. Hence, the optimal contract is not only affected by the change in the agent’s characteristic but also by the difference between the characteristics of the new versus the old principal.
Moreover, from a matching point of view, the existence of moral hazard problems may have a significant effect on the type of matching between principals and agents that we may observe at equilibrium, compared to the matching that would happen if incentive problems were absent. Under moral hazard, the gains that the participants get when they match are different, and that affects the equilibrium outcome.
In this paper, we present several models that address incentive problems in general equilibrium environments. As will be clear during the discussion, some models are simple versions from previous papers by several authors. Other models are original. They provide meaningful economic intuitions in relevant economic situations. Moreover, they allow us to foresee how to use incentive and matching theory together to address other questions.
The assignment market
The assignment market (Shapley and Shubik 1972) is a representation of a twosided market where each player can participate at most in one transaction. There is a (finite) set of n heterogeneous principals, \( P=\{1,2,\,....,n\}\) and a (finite) set of m heterogeneous agents, \( A=\{1,2,\,....,m\}\). A principal might be an employer, a lender, or a landowner. An agent might a worker, a borrower, or a tenant. Principals are denoted by i, \(i^{\prime }\), etc., whereas agents are denoted by j, \( j^{\prime }\), etc. In the assignment game, each principal can hire at most one agent, and each agent can work for at most one principal.^{Footnote 3} In addition, in the assignment game the attributes or characteristics of the participants of both sides are public information (there are no frictions).
If principal i does not hire any agent in A, she obtains profit \(\Pi _{i}^{o}\), which we normalize to zero: \(\Pi _{i}^{o}=0\). If agent j does not work for any principal in P, then he obtains an outside utility of \( U_{j}^{o}=U^{o}\). Thus, \(\Pi _{i}^{o}\) and \(U_{j}^{o}\) represent the value for principal i and agent j of staying unmatched. On the other hand, if principal i and agent j match, then they produce a joint surplus of \( S_{ij}\). Given the heterogeneity of principals and agents, the surplus \( S_{ij}\) typically depends on the identity and the attributes of both the principal and the agent. An important assumption is that principal i and agent j can share \(S_{ij}\) in any way they wish, that is, utility among them is fully transferable. Thus, any profile of payoffs \(\left( \Pi _{i},U_{j}\right) \) such that \(\Pi _{i}+U_{j}\le S_{ij}\) is feasible for the pair \(\{i,j\}\).^{Footnote 4}
The formal definition of a matching in the market \(\{P,A,S,U^{o}\}\) is:
Definition 1
A feasible matching is a function \(\mu \) from \( P\cup A\) to \(P\cup A\) such that:

(a)
\(\mu (i)\in A\cup \left\{ i\right\} \) for any \(i\in P\),

(b)
\(\mu (j)\in P\cup \left\{ j\right\} \) for any \(j\in A\), and

(c)
for any \((i,j)\in P\times A\), \(\mu (i)=j\) if and only if \(\mu (j)=i\).
We say that principal i (resp. agent j) is matched if \(\mu (i)\in A\) (resp. \(\mu (j)\in P\)). If \(\mu (i)=i\) or \(\mu (j)=j\), then we say that principal i or agent j are unmatched.
A feasible matching \(\mu \) is optimal if it maximizes the gain of the whole set of players. If we denote \(S_{jj}\equiv U^{o}\) the utility obtained by an unmatched agent, we can then define:
Definition 2
A feasible matching \(\mu \) is optimal if \(\sum _{j\in A}S_{j\mu (j)}\ge \sum _{j\in A}S_{j\mu ^{\prime }(j)}\) for any feasible matching \(\mu ^{\prime }\).
In this market, an outcome consists of a feasible matching and a vector of feasible payoffs (that is, profits and utilities) for principals and agents. This vector describes how the joint surplus of any matched pair is shared among the partners.
Definition 3
A feasible outcome \(\left( \mu ;\Pi ,U\right) \) consists of a feasible matching \(\mu \), a vector of profit levels \(\Pi =\left( \Pi _{i}\right) _{i\in P}\), and a vector of utilities \(U=\left( U_{j}\right) _{j\in A}\) such that:

(a)
\(\Pi _{i}+U_{j}=S_{ij}\) if \(\mu (i)=j\),

(b)
\(\Pi _{i}=0\) if \(\mu (i)=i\), and \(U_{j}=U^{o}\) if \(\mu (j)=j\).
In the market \(\{P,A,S,U^{o}\}\), the matching between principals and agents as well as the sharing of the surplus of any partnership are endogenous. Any principal or any agent can look for an alternative partner and can sign a different contract. Therefore, we will focus on those outcomes that are stable. Moreover, in the assignment game, stability, pairwise stability, and competitive equilibrium are equivalent concepts. In particular, it is easy to define competitive equilibria in this market and show that an outcome is stable (or pairwise stable) if and only if it is a competitive equilibrium (Shapley and Shubik 1972). In this paper, we will refer to stable outcomes as competitive equilibrium outcomes, or simply as equilibrium outcomes.
An equilibrium outcome is individually rational. Moreover, if the outcome is a competitive equilibrium, it is not possible for a principal and an agent (who are possibly not matched under that outcome) to form a partnership and share the surplus in such a way that they are both better off under the new partnership than under the previous outcome. We could say that an equilibrium outcome is “divorceproof.”
Definition 4
A feasible outcome \(\left( \mu ;\Pi ,U\right) \) is a competitive equilibrium if

(a)
\(\Pi _{i}\ge 0\) for all \(i\in P\), \(U_{j}\ge U^{o}\) for all \(j\in A\), and

(b)
\(\Pi _{i}+U_{j}\ge S_{ij}\) for all \((i,j)\in P\times A\).
Shapley and Shubik (1972) prove that equilibrium outcomes always exist in the assignment game. They also show the following results on the set of equilibrium outcomes, which will be useful in the following sections.
Proposition 1

(a)
If \(\left( \mu ;\Pi ,U\right) \) is an equilibrium outcome then \(\mu \) is an optimal matching.

(b)
Let \(\left( \mu ;\Pi ,U\right) \) be an equilibrium outcome and \(\mu ^{\prime }\) an optimal matching. Then \(\left( \mu ^{\prime };\Pi ,U\right) \) is also an equilibrium outcome.
Proposition 1 allows the study of the characteristics of the equilibrium matchings by analyzing the properties that make a matching optimal. It also states that principals or agents have the same set of equilibrium payoffs independently of the equilibrium matching.
Moreover, in the set of equilibrium payoffs, there is a polarization of interests between the two sides of the market, that is, if principals are better off in some equilibrium outcome than in another equilibrium outcome, then agents are better off in the second than in the first outcome. In fact, the set of equilibrium payoffs is endowed with a complete lattice structure under each partial order, where one is the dual of the other. In particular, there exist one and only one maximal element and one and only one minimal element in each lattice. Due to the polarization of interests between principals and agents, the best outcome for the principals is the worst outcome for the agents, and vice versa. Formally,
Proposition 2
In the set of equilibrium outcomes, there exist a unique principaloptimal payoff \(\left( \Pi ^{+},U^{}\right) \) and a unique agentoptimal payoff \(\left( \Pi ^{},U^{+}\right) \). Then, for any equilibrium outcome \(\left( \mu ;\Pi ,U\right) \),
In Sects. 4–6, we study equilibrium outcomes in several principal–agent markets. In any such market, when principal i and agent j establish a partnership (i.e., when principal i hires agent j) the surplus of the relationship \(S_{ij}\) will be the result of the contract signed by the partners. To better understand the analysis developed in the next sections, it is useful to make two remarks.
First, we use the assignment game as a tool to analyze the principal–agent markets. To be able to apply the results obtained for the assignment game, the surplus must be transferable inside a partnership. For this reason, we are going to assume particular functions for the preferences of principals and agents: the principals will be risk neutral and the agents will have constant absolute riskaverse (CARA) preferences. We will discuss principal–agent markets where the surplus is not fully transferable in Sect. 7.
Second, all the contracts in an equilibrium outcome are Pareto optimal. The optimality of the contracts is due to the possibility that the same pair can block the initial outcome with a different contract. Therefore, before we move to the analysis of principal–agent markets, we address the characteristics of the Paretooptimal contracts in any principal–agent relationship in the next section. We will recall the results under symmetric information among the participants as well as the optimal contracts in situations with moral hazard for the classic agency model.
Paretooptimal contracts in a principal–agent relationship
In this section, we consider an isolated partnership. Thus, we drop the subscript i and j for principals and agents. We can think of any such relationship as a principal hiring an agent to perform a task, which we refer to as effort, \(e\in E,\) in exchange for a wage, w. The final output of the relationship, x, depends on the effort e that the agent devotes to the task and a random variable for which both participants have the same prior distribution.
We assume that the principal is risk neutral, whereas the agent has CARA preferences (an exponential utility). Formally, an agent that receives salary w and exerts effort e obtains a utility of:
where \(r\ge 0\) is the coefficient of absolute risk aversion. Additionally, we assume that the cost of effort v(e) takes a quadratic form:
Concerning the output x, we assume that it is linear in the effort e and a random variable \(\varepsilon \):
where \(\alpha \ge 0\), \(\sigma >0\), and \(\epsilon \sim N(0,1)\). Thus, the expected output of the production process is \(\alpha +e\).
Finally, we assume that the contract is linear in the realized output. That is, we restrict attention to linear wage schemes of the form \(w=F+sx\), where F is a fixed payment and s is the share of the output that goes to the agent.^{Footnote 5}
It is convenient to express the utility of the agent as a function of the contract \(\left( F,s\right) \) and the effort e in terms of the agent’s certain equivalent income:^{Footnote 6}
The Paretooptimal contracts are the result of the principal maximizing her profits subject to the agent obtaining a certain utility level (equivalently, we can maximize the utility of the agent subject to the principal attaining a certain level of profit). The utility obtained by the agent in the market will be endogenous, but in this section we are going to denote it by \({\underline{U}}\). We note that in general \({\underline{U}}\) will be different from \(U^{o}\).
Paretooptimal contracts under symmetric information
If effort is contractible, the optimal contract \(\left( F,s,e\right) \) is the solution to
where we have taken into account that the principal maximizes expected profit. The constraint PC is the agent’s participation constraint, which ensures that he obtains at least \({\underline{U}}\). It is easy to see that PC is binding and it determines the fixed part of the sharing rule F. Using the PC, the program simplifies to:
The principal’s profit is decreasing in s, which gives the firstbest sharing rule under symmetric information \(s^{\mathrm{SI}}=0\). Moreover, the firstorder condition (FOC) with respect to e gives \(e^{\mathrm{SI}}=\frac{1}{v}\). Therefore, the optimal contract under symmetric information is:
For any such contract, the joint surplus is
which does not depend on \({\underline{U}}\). An increase in one unit in the level of utility of the agent \({\underline{U}}\) translates into a decrease of exactly one unit in the profit of the principal \(\Pi \). Therefore, the utility is fully transferable.
Paretooptimal contracts under moral hazard
If effort is not contractible, then the agent will choose the effort that maximizes his utility once the contract (F, s) is signed. His incentivecompatibility constraint (ICC) is the solution to
i.e., the ICC gives
The principal maximizes the same program as before, but taking into account the ICC. The binding PC determines the fixed fee, F. Using this expression for F, we conclude that the principal solves:
The FOC of this program leads to \(s^{\mathrm{MH}}=\frac{1}{1+rv\sigma ^{2}}\in \left( 0,1\right] .\)
Notice that \(s^{\mathrm{MH}}\) summarizes the standard conclusions of moral hazard problems: the power of the incentives is decreasing in the cost of the effort v and in the variance of the outcome \(\sigma ^{2}\) (as long as \( r>0).\) In addition, it is decreasing in the agent’s risk aversion (measured by r). Since a higher \(s^{\mathrm{MH}}\) translates into a higher expected output through the ICC, the previous expression reflects the tradeoff between efficiency (optimal risksharing would require \(s^{\mathrm{MH}}=s^{\mathrm{SI}}=0\)) and incentives.
Therefore, the optimal contract under moral hazard is
and leads to the effort
The joint surplus under moral hazard is, after some easy calculations,
Again, the surplus is independent of \({\underline{U}}\): the principal can give or take away utility directly through the fixed part of the contract F.
Contracts in a principal–agent market
We now go back to consider a principal–agent market. The set of heterogeneous, riskneutral principals is \(P=\{1,2,\,....,n\}\) and the set of heterogeneous agents, with a CARA utility function, is \( A=\{1,2,\,....,m\} \). Each participant knows the characteristics of all the principals and agents. We address questions such as the nature of the endogenous matching between principals and agents (who is hired by whom), the effect of the moral hazard on the nature of the matching, and the endogenous level of profit and utility that the participants obtain.
The participants can be heterogeneous in various characteristics. First, agents can differ in their degree of risk aversion and in their cost of exerting effort, so agent j’s utility function is
Second, both principals and agents can have a heterogeneous influence on the output. In particular, depending on the identity of the agent and/or the principal, the output can be more or less volatile. Thus, the output that is obtained in a partnership between principal i and agent j when the agent exerts effort e is:
where \(\alpha \ge 0\), \(\sigma _{ij}>0\), and \(\epsilon \sim N(0,1)\).
The total surplus obtained in a partnership depends on the principal’s and the agent’s characteristics. Suppose that we consider characteristics \(c_{i}\) and \(c_{j}\), and let us denote the total surplus by \(S\left( c_{i},c_{j}\right) \). Then, we say that the matching is positive assortative (PAM) if a principal with a higher value of \(c_{i}\) is matched with an agent with a higher value of \(c_{j}\): if \(c_{i}\ge c_{i^{\prime }}\) then \( c_{\mu \left( i\right) }\ge c_{\mu \left( i^{\prime }\right) }\). Similarly, we have a negative assortative matching (NAM) if \(c_{i}\ge c_{i^{\prime }}\) implies \(c_{\mu \left( i\right) }\le c_{\mu \left( i^{\prime }\right) }\). For instance, imagine that \(c_{i}\) and \(c_{j}\) are characteristics that improve the total surplus attained in a partnership: \(S\left( c_{i},c_{j}\right) \ge S\left( c_{i^{\prime }},c_{j}\right) \) if and only if \(c_{i}\ge c_{i^{\prime }}\,\) and \(S\left( c_{i},c_{j}\right) \ge S\left( c_{i},c_{j^{\prime }}\right) \) if and only if \(c_{j}\ge c_{j^{\prime }}\). Then, the matching is PAM if “good” principals are matched with “good” agents and “bad” principals are matched with “bad” agents. On the other hand, if the matching is NAM, “good” principals are matched with “bad” agents and “bad” principals are matched with “good” agents.
From the analysis of Sect. 2, we know that the equilibrium matching is PAM if and only if PAM is an optimal matching. Moreover, since Becker (1973), we also know that in markets with a transferable utility and where agents of each side of the market differ in a onedimensional characteristic, a sufficient condition for PAM to be an optimal matching is that there is type–type complementarity in the production of surplus. Similarly, a sufficient condition for NAM is type–type substitutability. If the surplus function is differentiable (as is the case in our model), then a sufficient condition for PAM (NAM) is that the crosspartial derivative of the surplus function with respect to the characteristic of the principal and the characteristic of the agent is positive (negative).
The first subsection will discuss the characteristics of the equilibrium outcomes under symmetric information in several scenarios concerning the heterogeneity of principals and agents. All the examples correspond to scenarios where principals and agents are heterogeneous with respect to characteristics that we can consider “vertical characteristics,” in the sense that we can rank, say, the agents (resp. the principals) from best to worst. For instance, the cost of exerting effort is a vertical characteristic: having a lower cost cannot be bad.^{Footnote 7} The next subsection will analyze the same scenarios when moral hazard is present in each of the partnerships.
A principal–agent market under symmetric information
We present three examples where under symmetric information any matching can be an equilibrium because the crosspartial derivative of the surplus with respect to the characteristics of the principal and agent is zero in the three scenarios, while other characteristics of the equilibrium may be different. This will facilitate the comparison with the results in the same environments when moral hazard is present.
Heterogeneous principals in the variance of their project and heterogeneous agents in their degree of risk aversion
Consider a situation where principals differ in the variance of their project, whereas agents differ in their degree of risk aversion. Each side of the market is similar in any other respect. Formally, \(v_{j}=v\) and \(\sigma _{ij}^{2}=\sigma _{i}^{2}\) for all \(i\in P\) and \(j\in A\). Then,
thus, any matching is an equilibrium matching.
In this scenario, any principal fully ensures the agent she hires, so principals do not care about the risk aversion of the agent they are matched with; hence, also the agents do not care about the variance of the principals’ project.^{Footnote 8} In particular, at equilibrium, all the matched principals obtain the same level of profits and all the matched agents obtain the same utility level. Indeed, in an outcome where \(U_{j}>U_{j^{\prime }}\), the principal \(\mu (j)\) and the agent \(j^{\prime }\) could deviate because \(\Pi _{\mu (j)}+U_{j^{\prime }}=S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},r_{j}\right) U_{j}+U_{j^{\prime }}=S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},r_{j^{\prime }}\right) U_{j}+U_{j^{\prime }}<S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},r_{j^{\prime }}\right) \), so \(\mu (j)\) and \(j^{\prime }\) together can produce more than the sum of the surplus they obtain at the outcome. And a similar reasoning holds if \(\Pi _{i}>\Pi _{i^{\prime }}\) for some principals i and \( i^{\prime }\).
Heterogeneous principals in the variance of their project and heterogeneous agents in their ability
Imagine that principals differ in the variance of their project and agents are heterogeneous in terms of their cost of effort: \(r_{j}=r\) for all \(j\in A\) but \(v_{j}\) can differ among agents. The parameter \(v_{j}\) can be thought of as the inverse of the ability of the agent. Then,
Also in this case, the crosspartial derivative of \(S^{\mathrm{SI}}\) with respect to \( \sigma _{i}^{2}\) and \(v_{j}\) is zero, and any matching is optimal.
As in Sect. 4.1.1, the variance of the principal’s project does not matter in the expression of \(S^{\mathrm{SI}}\left( \sigma _{i}^{2},v_{j}\right) \). Hence, all matched principals obtain the same profit level at equilibrium. However, this is not true for the agents. As is intuitive, a matched agent with higher ability (that is, a lower \(v_{j}\)) enables obtainment of higher surplus, hence, at equilibrium he obtains a higher utility level than an agent with lower ability. To check this property, consider an outcome where \(v_{j}>v_{j^{\prime }}\) but \(U_{j}\ge U_{j^{\prime }}\). Then, the principal \(\mu (j)\) and the agent \(j^{\prime }\) could deviate because \(\Pi _{\mu (j)}+U_{j^{\prime }}=S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) U_{j}+U_{j^{\prime }}\le S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) <S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j^{\prime }}\right) \).
Heterogeneity in the variance that principals and agents induce in the project
We now consider a situation where both the principal and the agent influence the volatility of the project. We can think of a market where principals may have more or less risky projects, and agents may be more or less precise in their job. Formally, \(r_{j}=r\) and \(v_{j}=v\) for all \(i\in P\) and \(j\in A\). Moreover, assume for simplicity \(\sigma _{ij}^{2}=\sigma _{i}^{2}+\sigma _{j}^{2}\) for all \(i\in P\) and \(j\in A\). Then again, the crosspartial derivative of the total surplus with respect to \(\sigma _{i}^{2}\) and \(v_{j}\) is zero, and any matching is efficient (Li et al. 2013).^{Footnote 9}
A principal–agent market under moral hazard
In this subsection, we analyze the consequences of the moral hazard problem by studying the same markets as above but when the effort is not verifiable.
Heterogeneous principals in the variance of their project and heterogeneous agents in their degree of risk aversion
When principals are heterogeneous in the risk of the project, \(\sigma _{ij}^{2}=\sigma _{i}^{2},\) and agents are heterogeneous in their degree of risk aversion, \(r_{j},\) the joint surplus when the partnerships are subject to moral hazard is:
Now, the volatility of the project and the agent’s degree of risk aversion have a negative impact on the joint surplus. A “good” principal is one with a lowvolatility project, and a “good” agent has a low degree of risk aversion. Moreover:
The main implication is (see, Wright 2004; Serfes 2005, 2008).
Proposition 3
Under moral hazard, if principals are heterogeneous in the risk of the project, \(\sigma _{i}^{2},\) and agents are heterogeneous in their degree of risk aversion, \(r_{j},\) then:

(a)
the equilibrium matching is PAM if \(r_{j}\sigma _{i}^{2}\ge 1/v\) for all \(i\in P\) and \(j\in A\),

(b)
the equilibrium matching is NAM if \(r_{j}\sigma _{i}^{2}\le 1/v\) for all \(i\in P\) and \(j\in A\).
Proposition 3 shows that the moral hazard problem not only distorts the optimal contract inside a partnership but it can also change the nature of the equilibrium matching. Under symmetric information (Sect. 4.1.1), any matching can be part of an equilibrium outcome. However, only PAM can arise as an equilibrium matching if \( r_{j}\sigma _{i}^{2}\ge 1/v\) for all \(i\in P\) and \(j\in A\), and only NAM if \(r_{j}\sigma _{i}^{2}\le 1/v\) for all \(i\in P\) and \(j\in A\).
PAM emerges as an equilibrium matching if the degree of volatility and risk aversion in the market is high. On the other hand, in markets where the volatility and/or the degree of risk aversion are very low, the equilibrium matching is NAM. Of course, many markets do not satisfy either of the two sufficient conditions highlighted in Proposition 3. In those markets, we can have equilibrium matchings that are neither PAM nor NAM. Moreover, the equilibrium matching depends not only on the degree of the volatility of the projects and the risk aversion of the agents; it is also a function of the distribution of these characteristics on the population of the participants.
We now discuss how considering that principal–agent relationships are part of a market may modify some of the implications of the comparative statics exercises that are often conducted in principal–agent models.^{Footnote 10} One robust implication from this model is that there exists a negative relation between risk and incentives: the more volatile the project, and the more riskaverse the agent, the lower the power of the incentives in a moral hazard situation. In particular, in the CARA model that we analyze, the share \(s^{\mathrm{MH}}(\sigma ^{2},r)=\frac{1}{1+rv\sigma ^{2}}\) is decreasing in both \(\sigma ^{2}\) and r.
Let us now take into account that there is an endogenous matching between principals and agents (see Serfes 2005, 2008 for a more extensive discussion). Denote \(r_{j}(\sigma _{i}^{2})=r_{\mu (i)}\) the endogenous relationship in the matching between the volatility of the project and the agent’s level of risk aversion. The power of incentives as a function of the volatility of the project is:
If the matching is PAM, then \(r_{j}^{\prime }(\sigma _{i}^{2})>0\); hence, \( r_{j}(\sigma _{i}^{2})v\sigma _{i}^{2}\) increases with \(\sigma _{i}^{2}\), which implies that the incentives have less power as \(\sigma _{i}^{2}\) increases. This result is similar to the comparative statics result in a model where the principal–agent match is given. However, if the matching is NAM, then \(r_{j}^{\prime }(\sigma _{i}^{2})<0\). Therefore, \(r_{j}(\sigma _{i}^{2})v\sigma _{i}^{2}\) can be increasing, decreasing, or it may have any other shape, depending on the distribution of the attributes of the population of principals and agents.
Finally, it is also interesting to discuss the changes in the equilibrium profit and utility level as a function of the characteristics. Remember that under symmetric information (Sect. 4.1.1) all matched principals obtain the same profit and all the matched agents get the same utility level. However, this is no longer true under moral hazard. The higher the variance of her project, the lower the profit that a principal obtains. Similarly, the higher the agent’s risk aversion, the lower his equilibrium utility level.^{Footnote 11}
The fact that the “bargaining power” of principals and agents is endogenous in the market has other important implications for the empirical analysis. For instance, we have seen that in a PAM, an agent’s bonus is decreasing in his degree of risk aversion. Following the discussion in Serfes (2008), in an isolated principal–agent relationship where the principals have the bargaining power, bonuses and fixed salaries should be negatively correlated. Hence, a lower bonus should imply a higher fixed salary. However, in the equilibrium in a market, higher risk aversion also implies a lower level of utility and the negative correlation between fixed and variable payment may no longer hold.
Heterogeneous principals in the variance of their project and heterogeneous agents in their ability
The agents’ cost parameter v plays a role similar to the agents’ degree of risk aversion r in the optimal contract. However, the analysis when agents are heterogeneous in terms of ability (in our model, in terms of their cost parameter) is simpler. When \(r_{j}=r\) and \(\sigma _{ij}^{2}=\sigma _{i}^{2}\) for all \(i\in P\) and \(j\in A\), then
which crosspartial derivative is positive. Therefore (see Li and Ueda 2009):
Proposition 4
Under moral hazard, if principals are heterogeneous in the risk of the project, \(\sigma _{i}^{2},\) and agents are heterogeneous in their cost parameter, \(v_{j},\) then the equilibrium matching is PAM.
Proposition 4 states that we should expect more able agents (those with lower costs) matched with firms whose projects have lower variance. Li and Ueda (2009) use the proposition to provide an explanation for the fact that safer firms receive funding from more reputable venture capitalists (see also Sørensen 2006), a conclusion that cannot be derived in a model where moral hazard is not present.
In this model, we illustrate how to study the sensitivity of a principal’s (resp. an agent’s) payoff to her (resp. his) own characteristic. This exercise is easier in a model where the set of principals and the set of agents are continuous because, in contrast to the discrete assignment game, the scheme of equilibrium payoffs is unique.^{Footnote 12} Moreover, to discuss the sensitivity in terms of a “positive” characteristic: denote \(c_{i}\) and \(c_{j}\) the characteristic of principal i and agent \(j\,,\) respectively, and suppose that \(\sigma _{i}^{2}={\overline{\sigma }}^{2}c_{i}\) and \(v_{j}={\overline{v}} c_{j}\). Thus, the higher the parameter \(c_{i}\) or \(c_{j}\), the better the principal or the agent.
As we mentioned in Sect. 4.1.2, a principal’s profit is independent of her type under symmetric information, that is,
However, an agent with higher ability obtains a higher level of utility:^{Footnote 13}
Similarly, under moral hazard, we obtain:
Therefore,
and, in this model, while the principal’s characteristic is irrelevant under symmetric information, it has a strong influence on the principal’s profit under moral hazard. On the other hand, the (positive) effect of the characteristic in agent’s utility is stronger under symmetric than under moral hazard. This illustrates that the asymmetry of information is often detrimental not only to the principal’s profit but also to the agent’s equilibrium utility level.
Heterogeneity in the variance that principals and agents induce in the project
When the heterogeneity among principals and among agents derive from the influence that both have on the volatility of the project (and assuming \( \sigma _{ij}^{2}=\sigma _{i}^{2}+\sigma _{j}^{2}\)), then
Therefore, the crosspartial derivative is positive and Proposition 5 follows.
Proposition 5
Under moral hazard, if both principals and agents are heterogeneous in their influence on the volatility of the project, \(\sigma _{ij}^{2}=\sigma _{i}^{2}+\sigma _{j}^{2}\), then the equilibrium matching is PAM.
In this market, the principals with relatively safe projects end up hiring agents who are relatively precise in their job, whereas risky projects are carried out by agents who induce further volatility in the output. Also in this model, moral hazard considerations have a strong influence on the nature of the matching. The effect of the project’s volatility on the total surplus is only indirect, through the bonus \(s^{\mathrm{MH}}\left( \sigma _{i}^{2},\sigma _{j}^{2}\right) \) that the agent receives in the optimal contract. A higher \(\sigma _{i}^{2}\), that is, a riskier project, weakens the incentives that the agent receives. This happens because the cost of the bonus (versus paying a fixed fee) increases with the volatility of the output. More importantly for the nature of the matching, given that \(\frac{ \partial ^{2}s^{\mathrm{MH}}}{\partial \sigma _{i}^{2}\partial \sigma _{j}^{2}}>0\), the effect is less negative for agents with high \(\sigma _{j}\), that is, for less precise agents. Therefore, efficiency (or optimality) requires that risky projects are carried out by less precise agents.^{Footnote 14}
Beyond twosided onetoone partnerships
The environments discussed in the previous section involve twoparty partnerships and use the twosided onetoone assignment game as a tool. The analysis of more general environments where more than two parties can form partnerships can be complex and the existence of equilibrium or stable outcomes may be problematic.^{Footnote 15} However, some of the tools that we have used, and other tools provided in the literature, can be useful for particular environments. In this section, we present two examples.
A simple owner–principal–agent market
Suppose an environment where production requires the partnership between three parties: an owner (a landlord who owns the land, an owner of the permit to have a business, or a shareholder who provides the financial resources), a principal (who brings or run a project), and an agent (who works on the project). Thus, this market corresponds to a “threesided” (instead of twosided) onetoone game.
To make the model very simple, assume all the owners are identical and risk neutral. Moreover, the principals’ profit, agents’ utility, and production functions are as in Sect. 4. Finally, the number of owners is larger than the number of principals and than the number of agents.
An equilibrium is this simple threesided market consists of a set of threeparty (an owner, a principal, and an agent) partnerships and some isolated players, as well as an individually rational sharing of the surplus in each partnership, such that it is not possible for an owner, a principal, and an agent to form a partnership and share the surplus in such a way that they are all better off under the new partnership than under the previous outcome.
A model with these characteristics is easy to analyze because, at equilibrium, it is necessarily the case that the payoff of all the owners is zero. Indeed, consider an outcome where an owner, say k, obtains positive equilibrium profits. This owner is necessarily matched with a principal (say i) and an agent (j). But then, some unmatched owner (say \(k^{\prime }\)) could make offers to i and j, form a new partnership, obtain the same total surplus as the team \(\left\{ k,i,j\right\} ,\) and share it so that the three partners, \(k^{\prime },\) i, and j, are better off than before.
Given that the owners are identical and obtain zero profits, they are like “dummies” in this model. In fact:

(i)
Take any equilibrium \(\left( \mu ;\Pi ,U\right) \), with \(\Pi =\left( \Pi _{i}\right) _{i\in P}\) and \(U=\left( U_{j}\right) _{j\in A}\), in the twosided principal–agent matching market. Consider the following outcome in the threesided owner–principal–agent market: (a) a partnership \(\left\{ k,i,j\right\} \) is formed if and only if \(j=\mu (i)\), where k is any owner; (b) if the partnership \(\left\{ k,i,j\right\} \) is formed, then the owner obtains zero profits, principal i gets \(\Pi _{i}\) and agent j gets \(U_{j}\). This is an equilibrium outcome.

(ii)
And similarly, given an equilibrium in the threesided market (which involves zero payoff for the owners), the restriction of the partnership and the payoffs to the sets of principals and agents constitutes an equilibrium in the twosided market.
We note that the previous result holds because the market is particularly simple, not only due to the existence of many identical owners but also because only one of the two “important” partners (the agent) is subject to moral hazard. If both the principal and the agent are subject to moral hazard then the owners can play the role of “residual claimant” in the relationships because they can break the budgetbalance constraint, even if there are still many identical owners and they obtain zero benefits at equilibrium. Thus, the existence of owners would improve the efficiency of the production by the principal and the agent and the previous equivalence would no longer hold.^{Footnote 16} But the approach that we have proposed can still be useful for analyzing such markets.
A market where each principal hires two agents from a single pool
We consider again an environment where there are only two sets of participants: a set of identical riskneutral principals P and a set of riskaverse agents A with CARA utility function. We now assume that the coefficient of risk aversion r is the same for all the agents and that their disutility of effort is \(\frac{1}{2}ve^{2}\).
However, we study a production function that requires that each principal has to fill up two positions, hence she needs to hire two agents. Thus, this is a twosided manytoone matching problem.^{Footnote 17}
Each agent makes an effort in the production and the identity of the two agents hired will determine the volatility of the project. That is, the variance is not a characteristic of the principal but of the team of agents. In particular, when a principal hires agents j and k, the output is
where \(e_{j}\) and \(e_{k}\) are the efforts exerted, respectively, by agents j and k, \(\alpha \ge 0\), \(\sigma _{jk}>0\), and \(\epsilon \sim N(0,1)\).
Given that the principals are risk neutral and the agents have a CARA utility function, the utility is still transferable among the participants in any partnership. Therefore, the contracts between the principal and agents j and k maximize the total surplus.
Under symmetric information, and similarly to the case when the principal hires only one agent, the variable part of the optimal contract is zero, \(s_{j}^{\mathrm{SI}}\left( \sigma _{jk}^{2}\right) =s_{k}^{\mathrm{SI}}\left( \sigma _{jk}^{2}\right) =0\) and the effort requested is \(e_{j}^{\mathrm{SI}}=e_{k}^{\mathrm{SI}}= \frac{1}{v}.\) Total surplus for a partnership \(\{i,j,k\}\) is
If the agents are subject to a (team) moral hazard problem, and they do not cooperate, then the optimal contracts solve (when agents’ utility in the market is \(U_{j}\) and \(U_{k}\)):
where the expression for \(e_{j}\) and \(e_{k}\) corresponds to the ICC of the agents (the Nash equilibrium in efforts). The optimal contracts involve \( s_{j}^{\mathrm{MH}}\left( \sigma _{jk}^{2}\right) =s_{k}^{\mathrm{MH}}\left( \sigma _{jk}^{2}\right) =\frac{1}{1+rv\sigma _{jk} {{}^2} }\).^{Footnote 18} Therefore, total surplus under moral hazard for a partnership \( \{i,j,k\}\) is
To discuss the characteristics of the equilibrium outcomes both under symmetric information and under moral hazard, first note that at equilibrium all principals necessarily obtain the same profits \(\Pi ^{eq}\) because they are identical. For instance, if there are fewer agents than twice the number of principals, then some principal will certainly remain unmatched at equilibrium and all the principals (matched or unmatched) will obtain \(\Pi ^{eq}=0\). In any case, at equilibrium, any surplus beyond \(\Pi ^{eq}\) generated in any partnership goes to the agents. Thus, even though it is the principals who are competing to create the partnerships, the equilibrium characteristics of the matching correspond to the characteristics of the equilibrium in the onesided onetoone matching problem among the agents.
In the onesided onetoone matching problem, there is a unique set of players (in our case, the set of agents A) and any two agents can form a partnership if they so decide. An outcome corresponds to a matching between agents (which can also be identified by a partition of the set of agents in either pairs of agents or singletons) and a sharing of the surplus obtained by any pair. In the onesided onetoone matching model with a finite number of agents, equilibria may not exist.^{Footnote 19} But equilibria always exist if there is a continuum of agents.^{Footnote 20} Thus, for this model and for simplicity, we are going to assume that there is a continuum of agents A and a continuum of principals P. The definitions and properties of the assignment game are easily extended to the continuous framework.
If there is symmetric information in the market, even though the agents may be different in their effect on the variance of the project, the variance is in fact irrelevant because \(S^{\mathrm{SI}}\left( \sigma _{jk}^{2}\right) \) is constant. Therefore, in terms of the surplus, agents are identical. This implies that any matching (both in the onesided and in the twosided matching models) is optimal, hence, any matching is an equilibrium matching. Moreover, all the agents obtain the same level of utility. For instance, each agent obtains a utility of \(\frac{1}{2}\left( \alpha +\frac{1}{v} \right) \) if there are more principals than half the number of agents.
To study the market equilibrium when both agents are subject to moral hazard, let us assume that each \(j\in A\) is characterized by a parameter \( \sigma _{j}^{2}>0\) so that the variance of a team formed by j and k is \( \sigma _{jk}^{2}=\sigma _{j}^{2}+\sigma _{k}^{2}+\gamma \sigma _{j}^{2}\sigma _{k}^{2}\) with \(\gamma \in {\mathbb {R}} \) and \(\left \gamma \right \) not too large. Then, the surplus \( S^{\mathrm{MH}}\left( \sigma _{j}^{2},\sigma _{k}^{2}\right) \) depends on the types \( \sigma _{j}^{2}\) and \(\sigma _{k}^{2}\) working for the principal.
As it happens in the twosided models, in the onesided models there is PAM when agents’ characteristics are complementary: regardless of the distribution of types, “good” agents partner with “good” agents, and “bad” agents partner with “bad” agents. In fact, if the surplus function is strictly supermodular, then there is segregation among agents: every agent matches with someone identical to themselves.^{Footnote 21} On the other hand, if the surplus function is strictly submodular, then at equilibrium there is NAM among agents.^{Footnote 22}
Therefore, taking into account the expression for \(S^{\mathrm{MH}}\left( \sigma _{j}^{2},\sigma _{k}^{2}\right) \), there is segregation if
In this case, each principal hires at equilibrium two identical agents. It is more efficient that highvariance agents go together and lowvariance agents go together, because this matching minimizes the distortion in incentives for the team. This happens when \(\gamma \) is negative, or it is positive but small enough.
Similarly, the surplus is strictly submodular and the equilibrium is NAM if
Therefore, if \(\gamma >0\) and large, then we should see at equilibrium that a principal who hires an agent with very low variance also hires an agent with very high variance. In this case, hiring two highvariance agents is very costly, because providing incentives is very expensive. Thus, it is better to mix high and lowvariance agents. NAM is also more likely if v is small, because a lower v means a higher efforts in the optimal contracts, which makes dealing with very high variances more expensive and hence less efficient.
Again, the moral hazard problem has important consequences for the type of matching that takes place in the market. And it also affects in a new way the relationship between an agent’s level of variance and the power of the incentives he receives in the market. We now discuss this fact in brief.
In an isolated multiagent moral hazard situation, when the variance of one of the agents increases, the total variance of the team increases and the incentives for both partners decrease. That is, we should observe that the power of the incentives decreases with the variance. However, in the market, taking into account the assignment, one has to be more careful.
To see how incentives change with an agent’s volatility, consider a market where (3) holds so that the matching is NAM. Moreover, \( \sigma _{j}^{2}\) is distributed according to a uniform distribution in \( \left[ {\underline{\sigma }}^{2},{\overline{\sigma }}^{2}\right] \). This means that the equilibrium partner \(\mu \left( j\right) \) of j has an associated variance of \(\sigma _{\mu (j)}^{2}={\overline{\sigma }}^{2}+{\underline{\sigma }} ^{2}\sigma _{j}^{2}\). The power of incentives given to agent j is
Therefore, \(\frac{\partial s_{j}^{\mathrm{MH}}}{\partial \sigma _{j}^{2}}\) is proportional to \(\gamma \left( \sigma _{j}^{2}\sigma _{\mu (j)}^{2}\right) \) which, given that \(\gamma >0\) if NAM, is negative if and only if \(\sigma _{j}^{2}<\sigma _{\mu (j)}^{2}\).
Figure 1 shows the power of the incentives as a function of \(\sigma _{j}^{2}\), when \(\left[ {\underline{\sigma }}^{2},{\overline{\sigma }}^{2}\right] =\left[ 1,3\right] \). The teams formed by agents with variance more to the center of the interval of individual variance (the team \(\left( k,\mu (k)\right) \) as compared to the team \(\left( j,\mu (j)\right) \)) are those teams with higher total variance, hence they receive fewer incentives. Below the mean, when the variance of an agent increases, each individual will receive lower incentives. This is the same comparative static as in a single principal–agent model. But above the mean, an increase in the variance of the agent will lead to an increase in his incentives (because his partner will have lower individual variance and the team total variance will decrease).
A market with repeated moral hazard
In the previous sections, we have studied several markets where principals and agents interact. One important feature of those markets is that they are static. Interactions between principals and agents only happen once. This is a natural hypothesis given that the assignment game, which constitutes our tool to model markets, is also a static model. However, we can also use the ideas and methodology derived from the assignment game to model some dynamic markets.
In this section, we propose a dynamic model where a set of principals and a set of agents meet every period. The model is in the same spirit as MachoStadler et al. (2014), but the particulars of the model and the objective are different. In that paper, the main objective is to show that the existence of a market strongly influences the principals’ choice of shortterm (ST) or longterm (LT) contracts when agents have industryspecific abilities and are subject to moral hazard.^{Footnote 23}
We consider that agents have industry and principalspecific characteristics and we analyze the influence of these characteristics on the equilibrium configuration of LT and ST contracts in the industry. Moreover, contrary to MachoStadler et al. (2014), we assume (as in the previous sections in this paper) that agents are riskaverse with a CARA utility function, which implies that LT and ST contracts are equally optimal in an isolated principal–agent relationship.^{Footnote 24} For simplicity, we consider that the agent consumes all his income at the end of the second period, so that we do not study his intertemporal consumption decision. Also for simplicity, we model the sets of principals and agents as continuous, instead of discrete, sets.
We model the economy as an overlapping generation model where at each period t, with \(t=1,2,...\), principals (firms) contract with agents (workers) to develop projects. Principals are infinitely lived, riskneutral players, and the set of principals is constant for all periods. They are heterogeneous in the potential return R of the technology they own. For a given principal i, the attribute \(R_{i}\) is the same across periods and it is distributed in the interval \(\left[ {\underline{R}},{\overline{R}}\right] \), with \({\underline{R}}>0\), according to the distribution function G(R). On the other hand, agents live for two periods, and their preferences are represented through a CARA utility function with the same coefficient of risk aversion r. Both principals and agents discount the future according to the discount factor \(\delta \in (0,1)\).
At any period t, a generation of agents is born. Thus, in period t the market is composed of the set of principals, the set of agents that enter the market during this period and the set of older agents that entered the market in period \(t1\). In period 1, there is a set of agents who are already old.
To run its project, a principal must hire a nontrained (junior) agent and a trained (senior) agent. To become trained, that is, senior, an agent must have worked in this market in the first period of his life. We assume that the measure of the set of agents born in any period is larger than the measure of the set of principals, so there are more junior agents than nontrained positions to fill in the market.
The output x for principal i from the project follows the production function:
where \(\sigma >0\) and \(\epsilon \sim N(0,1)\). Parameter p refers to the characteristic of the senior agent, and e to his effort. The senior’s cost of effort is \(v(e)=\frac{1}{2}ve^{2}\). We assume that the noncontractible effort of the senior agent is crucial for determining the output, whereas the junior agent performs a routine job whose cost is normalized to zero.
The senior agent’s productivity, p, summarizes his ability/productivity. We assume that this productivity takes the form \(p=p_{I}+\theta \), where \( p_{I}\) is the senior industryspecific ability (the same for all principals) and \(\theta \) is the senior principalspecific ability.^{Footnote 25} Concerning the principalspecific ability, we assume that \(\theta =0\) when the senior agent works for a principal different that when junior, and \(\theta =\Theta >0\) when he works for the same principal than when junior. As for the industryspecific ability, all juniors are identical ex ante but during their work as juniors, their industryspecific talent becomes public. We assume that there is a proportion q of highability agents that have \(p_{I}=p_{H}\) and a proportion \((1q)\) of lowproductivity agents with \(p_{I}=p_{L}\). Industryspecific ability is important, so that \( p_{H}p_{L}>\Theta .\) Then, they are two types of agents but four possible levels of productivity: \(p\in \left\{ p_{H},p_{H}+\Theta ,p_{L},p_{L}+\Theta \right\} \).
A senior agent enters a relationship only if his expected utility is at least equal to \(U {{}^o} \,\). Similarly, a junior agent accepts a contract only if his expected intertemporal utility is at least \(U {{}^o} +\delta U {{}^o} \). For simplicity, we assume that \({\underline{R}}\) is high enough and all principals in \(\left[ {\underline{R}},{\overline{R}}\right] \) are active in the market; hence, we disregard the principals’ participation constraint.
Concerning the salaries, a junior agent working for a principal receives a fixed wage B. As above, the principal offers a linear contract \(w=F+sx\) to the senior agent, with \(s\in \left[ 0,1\right] .\) Thus, if he is hired by principal i, a senior agent with ability p selects the effort
In any period t, the expected profit of a principal i that runs her project with a junior agent, to whom she pays the salary B, and a senior agent of ability p, who is paid according to the payment scheme (F, s), is \(\left( 1s\right) R_{i}peBF.\)
Principals and agents can sign either ST or LT contracts. An ST contract between a principal and a junior agent consists of a salary B. An ST contract between a principal and a senior agent is an incentive scheme (F, s) that may depend on the potential of the principal’s project R and the agent’s productivity p. An LT contract between a principal and a junior agent in period t specifies the salary that the agent will receive during this period and the incentive scheme that will govern the relationship in period \(t+1\), which will be a function of the revealed ability of the agent. That is, an LT contract is a vector \(\left( B,F_{H},s_{H},F_{L},s_{L}\right) \) that implies a commitment by the principal to retain the agent as a senior and a commitment by the agent to work for the same principal in period \(t+1\).
We focus on stationary equilibria, that is, on equilibria where firms offer the same contracts every period. This allows us to do the analysis, taking into account the expected profits that principals make in one (in any) period. The only small arrangement we have to make is that we need to associate to the junior agent a cost of \(\frac{1}{\delta }B\) rather than B , because any possible deviation of the type of contract by a principal will have consequences in the next period.^{Footnote 26} We denote the oneperiod profit \(E{\widetilde{\pi }}\). A principal has an incentive to switch from contract C to contract \(C^{\prime }\) if and only if \(E{\widetilde{\pi }}(C)<E{\widetilde{\pi }}(C^{\prime })\). As was the case in the static models that we presented in the previous sections, all the equilibrium contracts must be Paretooptimal. Thus, before describing more characteristics of the equilibrium, we state the Paretooptimal LT and ST contracts.
Paretooptimal longterm contracts
Given that there are more junior agents than principals, and junior agents are ex ante identical, any principal can secure the services of a junior agent if he receives a total (twoperiod) discounted payment of \(\left( 1+\delta \right) U {{}^o} \). Therefore, principal i looks for the contract \( (B,F_{H},s_{H},F_{L},s_{L})\) that satisfies the agent’s PC (with equality at the optimum):
and she solves the following problem:
The previous program takes into account that an agent hired under an LT contract always acquires the principalspecific ability; hence, his productivity is either \(p_{H}+\Theta \) or \(p_{L}+\Theta \).
Proposition 6 states the characteristics of the candidate LT contract for principal i.^{Footnote 27}
Proposition 6
If in equilibrium principal i offers an LT contract, then:

(a)
\(s_{H}=\frac{R_{i}^{2}\left( p_{H}+\Theta \right) ^{2}}{R_{i}^{2}\left( p_{H}+\Theta \right) ^{2}+rv\sigma ^{2}},\) \(s_{L}=\frac{R_{i}^{2}\left( p_{L}+\Theta \right) ^{2}}{R_{i}^{2}\left( p_{L}+\Theta \right) ^{2}+rv\sigma ^{2}}\) , and the vector of fixed payments \((B,F_{H},F_{L})\) satisfies (PCLT).

(b)
Efforts are \(e_{H}=\frac{1}{v}\frac{R_{i}^{3}\left( p_{H}+\Theta \right) ^{3}}{R_{i}^{2}\left( p_{H}+\Theta \right) ^{2}+rv\sigma ^{2}}\) and \(e_{L}= \frac{1}{v}\frac{R_{i}^{3}\left( p_{L}+\Theta \right) ^{3}}{R_{i}^{2}\left( p_{L}+\Theta \right) ^{2}+rv\sigma ^{2}}\).

(c)
The principal’s oneperiod profit under the optimal LT contract is:
$$\begin{aligned} E{\widetilde{\pi }}^{LT}\left( R_{i}\right)= & {} \frac{1}{2v}R_{i}^{4}\left( \frac{ q\left( p_{H}+\Theta \right) ^{4}}{R_{i}^{2}\left( p_{H}+\Theta \right) ^{2}+rv\sigma ^{2}}+\frac{\left( 1q\right) \left( p_{L}+\Theta \right) ^{4} }{R_{i}^{2}\left( p_{L}+\Theta \right) ^{2}+rv\sigma ^{2}}\right) \\&\frac{1}{ \delta }\left( 1+\delta \right) U {{}^o} . \end{aligned}$$
It is worth noticing that the profit function \(E{\widetilde{\pi }}^{LT}\left( R_{i}\right) \) is continuously differentiable and increasing in \(R_{i}\).
Paretooptimal shortterm contracts
All principals signing ST contracts hire similar junior agents, as they are indistinguishable ex ante. With respect to senior agents, principals can decide to hire highability or lowability agents (and a senior with principalspecific skills or not). If a senior agent is hired by the principal for whom he worked last period, he has a higher productivity than if hired by another principal. However, all seniors have the same value for the other principals in the market. As a consequence, all highability seniors have the same “equilibrium value” \(U_{H}\) in the market and all lowability seniors can obtain the same \(U_{L}\) .^{Footnote 28}
The equilibrium salary B that the junior agent will receive satisfies:
where the equality is due to the abundance of junior agents.
We now compute the Paretooptimal contract offered by principal \(R_{i}\) to a senior agent with industryspecific ability I, for \(I=H,L,\) who must receive \(U_{I}\). Denote \(p_{I}^{\sharp }\) the agent’s productivity: \( p_{I}^{\sharp }=p_{I}+\Theta \) if the agent worked last period as a junior for the same principal and \(p_{I}^{\sharp }=p_{I}\) otherwise. Then, the principal solves the following program:
Proposition 7
a) If principal \(R_{i}\) offers ST contracts and hires senior agents then:

(a)
If the productivity of the agent is \(p_{I}^{\sharp }\), the contract is \( s_{I}^{\sharp }=\frac{R_{i}^{2}p_{I}^{\sharp 2}}{R_{i}^{2}p_{I}^{\sharp 2}+rv\sigma ^{2}},\) \(F_{I}^{\sharp }=U_{I}\frac{1}{2v}\left( \frac{ R_{i}^{2}p_{I}^{\sharp 2}}{R_{i}^{2}p_{I}^{\sharp 2}+rv\sigma ^{2}}\right) ^{2}\left( R_{i}^{2}p_{I}^{\sharp 2}rv\sigma ^{2}\right) \).

(b)
If the productivity of the agent is \(p_{I}^{\sharp }\), the effort is \( e_{I}^{\sharp }=\frac{1}{v}\frac{R_{i}^{3}p_{I}^{\sharp 3}}{ R_{i}^{2}p_{I}^{\sharp 2}+rv\sigma ^{2}}\).

(c)
The expected principal’s oneperiod profit when she hires a highability or a lowability senior agent, also taking into account the cost of the junior agent, is
$$\begin{aligned} E{\widetilde{\pi }}_{H}^{ST}\left( R_{i},B,U_{H}\right)= & {} \frac{1}{2v} R_{i}^{4}\left( \frac{q\left( p_{H}+\Theta \right) ^{4}}{R_{i}^{2}\left( p_{H}+\Theta \right) ^{2}+rv\sigma ^{2}}+\frac{\left( 1q\right) p_{H}^{4}}{ R_{i}^{2}p_{H}+rv\sigma ^{2}}\right) \\&U_{H}\frac{1}{\delta }B \\ E{\widetilde{\pi }}_{L}^{ST}\left( R_{i},B,U_{L}\right)= & {} \frac{1}{2v} R_{i}^{4}\left( \frac{qp_{L}^{4}}{R_{i}^{2}p_{L}^{2}+rv\sigma ^{2}}+\frac{ \left( 1q\right) \left( p_{L}+\Theta \right) ^{4}}{R_{i}^{2}\left( p_{L}+\Theta \right) ^{2}+rv\sigma ^{2}}\right) \\&U_{L}\frac{1}{\delta }B. \end{aligned}$$
As it happens for the optimal LT contracts, the profit functions \(E {\widetilde{\pi }}_{H}^{ST}\left( R_{i},B,U_{H}\right) \) and \(E{\widetilde{\pi }} _{L}^{ST}\left( R_{i},B,U_{L}\right) \) are continuously differentiable and increasing in \(R_{i}\).
Equilibria
We now look for the equilibrium outcomes. We focus on equilibria where the lowability senior agents obtain \(U^{o}\), that is, \(U_{L}=U {{}^o} \). For the same reasons discussed in the previous subsection, this simplification does not have consequences for the total agents’ expected utility and for the form of the equilibrium contract.
In equilibrium, the set of principals is partitioned into a maximum of three subsets: the set of principals that offer LT contracts, the set of principals that offer ST contracts to juniors and to highability seniors, and the set of principals that offer ST contracts to juniors and to lowability seniors. We explore equilibria where ST contracts may appear.^{Footnote 29}
In equilibrium, highability agents should be more expensive than lowability agents, that is, \(U_{H}>U_{L}=U^{o}\) because every principal makes a higher profit with a high than with a lowability senior agent, and the number of highability senior agents is lower than the number of principals. Also, the willingness to pay for a high instead of a lowability senior increases with the attribute R of the principal. Therefore, if there are ST contracts, principals with a high R will hire highability seniors, whereas principals with a low R will hire lowability senior agents. Finally, it is intuitive that if there is an equilibrium where LT and ST contracts coexist, the principals using LT contracts should have an attribute R that is not too high (so that it is not worthwhile for them to pay as much as \(U_{H}\) every period) and not too low (so that they do not hire lowability agents every period).
Thus, we study the existence of equilibria where there are two thresholds \( R^{L}\) and \(R^{H}\) with \({\underline{R}}<R^{L}<R^{H}<{\overline{R}},\) such that principal i signs ST contracts with lowability seniors if \(R_{i}\in \left[ {\underline{R}},R^{L}\right] ,\) LT contracts if \(R_{i}\in \left( R^{L},R^{H}\right) \), and ST contracts with highability seniors if \( R_{i}\in \left[ R^{H},{\overline{R}}\right] \). At equilibrium, lowskilled agents obtain \(U_{L}=U {{}^o} \) and junior agents in ST contracts get \(\left( 1+\delta \right) U {{}^o} \) in total, that is, equation (4) holds, which implies \(B=U {{}^o} q\delta \left( U_{H}U^{o}\right) \). Moreover, the equilibria (that is, the parameters \(U_{H}\), \(R^{H}\), and \(R^{L}\)) must satisfy the following three properties:

(1)
There are as many principals with \(R_{i}\) in \(\left[ {\underline{R}},R^{L} \right] \) as in \(\left[ R^{H},{\overline{R}}\right] \), that is,
$$\begin{aligned} G(R^{L})=1G(R^{H})\text {.} \end{aligned}$$(5) 
(2)
If \(R_{i}=R^{L}\), then principal i is indifferent between using ST contracts hiring lowability seniors and using LT contracts, that is,
$$\begin{aligned} E{\widetilde{\pi }}_{L}^{ST}\left( R_{i}=R^{L},B=U {{}^o} q\delta \left( U_{H}U^{o}\right) ,U_{L}=U^{o}\right) =E{\widetilde{\pi }} ^{LT}\left( R_{i}=R^{L}\right) . \end{aligned}$$(6) 
(3)
If \(R_{i}=R^{H}\), then principal i is indifferent between using LT contracts and using ST contracts hiring highability seniors:
$$\begin{aligned} E{\widetilde{\pi }}_{H}^{ST}\left( R_{i}=R^{H},B=U {{}^o} q\delta \left( U_{H}U^{o}\right) ,U_{H}\right) =E{\widetilde{\pi }} ^{LT}\left( R_{i}=R^{H}\right) . \end{aligned}$$(7)
Proposition 8 shows that an equilibrium with the previous characteristics exists if and only if \(\Theta \) is low enough. In particular, \(\Theta \) needs to be lower than the unique threshold \(\Theta ^{o}\in \left( 0,p_{H}p_{L}\right) \) implicitly defined by Eq. (8):
Proposition 8

(a)
There is never an equilibrium where all firms use ST contracts.

(b)
There is an equilibrium where LT and ST contracts coexist if and only if \( \Theta \le \Theta ^{o}\).

(c)
If \(\Theta \ge \Theta ^{o}\), there are only LT contracts at equilibrium.
To illustrate Proposition 8, consider the situation where R is uniformly distributed over \(\left[ {\underline{R}},{\overline{R}}\right] =\left[ 2,6\right] ,\) \(vr\sigma ^{2}=1,\) \(p_{H}=\frac{3}{6},\) \(p_{L}=\frac{1}{6},\) and \(\Theta \in \left[ 0,\frac{2}{6}\right] \). In this situation, \(\Theta ^{o}=0.282.\) Figure 2 summarizes the equilibrium choice between ST and LT contracts in the space \((R,\Theta ).\)
As stated in Proposition 8, only LT contracts are signed at equilibrium if the principalspecific ability that a junior agent learns when working for a principal is very large, \(\Theta \ge \Theta ^{o}=0.282.\) In this case, even for the principal with the most profitable project \( {\overline{R}}=6\) it is not worthwhile using ST contracts to always catch a highproductivity (in terms of industryspecific ability) agent. She would prefer to sign an LT contract with junior agents and benefit from their acquired principalspecific ability.
On the other hand, if \(\Theta <\Theta ^{o}\), then there are three groups of principals. Principals with a high R choose ST contracts to make sure that they always hire highproductivity agents. Some of these highproductivity agents also have a principalspecific ability because they were hired by the same principal when junior, whereas others were working for other principals. They all receive a high salary at equilibrium. At the other extreme, principals with a low R choose ST contracts because junior agents are ready to accept low salaries if hired under these types of contracts. They hope to have high productivity and access a high salary when senior. For the principals with intermediate values of R, the principalspecific ability is important enough so that they prefer to always keep the same agents for both periods. The set of principals that prefer LT contracts at equilibrium increases with the importance of the principalspecific ability, so it is larger as \(\Theta \) increases.
In our model, the junior’s ability is unknown to everyone. Hermalin (2002) studies a competitive labor market where workers initially have private information about their ability while this ability becomes public when they become seniors. Highability workers value the option to entertain outside wage offers once their ability becomes known to the market. Then, offering ST contracts allows the screening of highability types from lowability ones (who prefer LT contracts).
That the ability of a senior agent is public information is another important hypothesis in the model. If the current principal has an informational advantage over the senior’s ability then the other principals will attempt to infer the worker’s quality by observing the principal’s job assignment or promotion decisions (see, e.g., Waldman 1984).
Finally, we assume that the agent can commit to not leaving the firm. If the contract cannot include buyout clauses (to be paid by the principal who wishes to hire the senior worker), penalties in the case of breaking the contract (that the worker would have to pay), or noncompete clauses (forbidding working for another firm in the market) then an agent may not be able to commit to staying in the firm that trained him as a junior (see, e.g., Mukherjee and Vasconcelos 2018).
Conclusion and extensions
In this paper, we analyze the optimal incentive scheme in principal–agent relationships in several market situations. We use the assignment game as a tool to embed the relationships in a general equilibrium framework. We first highlighted the importance of considering principal–agent relationships not as isolated partnerships but as part of a market. When not only the contract but also the identity of the partners are endogenous, some of the conclusions that one obtains in classic principal–agent theory may be reversed. Second, we have shown that the existence of moral hazard may alter the characteristics of the equilibrium matching in markets, compared to the situation where information about the effort of the agents is verifiable.
In all the models that we have seen in this paper, the surplus obtained in any partnership can be fully distributed among principals and agents. For instance, the cost for a principal of increasing by one unit the level of utility of an agent is also one. This is an important characteristic that makes the models share the main properties of the assignment game (with a discrete or a continuous number of agents). In particular, a matching is an equilibrium matching if and only if it is optimal. However, there are many relevant environments where this characteristic does not hold, especially when moral hazard problems are present, and then the surplus is not (at least, not fully) transferable.
Several papers study environments with a nontransferable utility to analyze partnerships and contracts in markets. Legros and Newman (2007) provide necessary and sufficient conditions for PAM and NAM in these markets. The monotonicity of the equilibrium matching requires not only the complementarity/substitutability of the surplus in types but also the complementarity/substitutability between an agent’s type and his partner’s payoff. Besley and Ghatak (2005) consider a market with two types of principals (profitoriented and missionoriented) and two types of agents (those who only care about the monetary reward and those that receive an intrinsic motivation if they work for a missionoriented firm) and study the market assignment and contracts. Dam and PérezCastrillo (2006) model the interaction between landowners and heterogeneous (poor) tenants who are subject to limited liability constraints and study the consequences of competition on the power of incentives, the efficiency of the relationships, and the effect of redistributive policies.^{Footnote 30} Legros and Newman (2007) also propose two applications: they study the market between a set of principals endowed with projects with heterogeneous risk characteristics and a set of agents who differ in initial wealth and have a declining absolute risk aversion, as well as a “marriage market” where agents are heterogeneous in their absolute risk tolerance (see also Chiappori and Reny 2016 and Gierlinger and Laczó 2018). AlonsoPaulí and PérezCastrillo (2012) analyze the owners’ choice between an incentive contract or a contract with a rigid effort of managers, who are heterogeneous in their ability, in an environment where there is uncertainty concerning market conditions. Ghatak and Karaivanov (2014) study the equilibrium sharecropping contracts in a model with endogenous matching and doublesided moral hazard.
Finally, we discuss some of the empirical literature related to situations with incentives and moral hazard using data from markets where the match principal–agent is not exogenous. Note that, unlike singleagent choices, matching outcomes depend on the preferences of other agents in the market.^{Footnote 31} Recently, several papers have proposed estimation strategies adapted for matching situations, and some empirical papers estimate twosided matching situations with (and without) transfers.^{Footnote 32}
Some papers use reducedform models, as the “probitcounterfactual” approach (see, Gompers et al. 2016). This approach uses data on the actual pairs to construct a plausible set of counterfactual pairs (control group) of available alternatives to the actual partner. Then, it estimates the likelihood of an agent being matched with an actual rather than an alternative partner.^{Footnote 33} Following this approach, Agrawal et al. (2008) use the spatial and social proximity of inventors to explain the access to knowledge. Hegde and Tumlinson (2014) find that ethnic proximity between U.S. venture capitalists and startup executives is positively related to the startup’s successful exit and exit revenue. Also, Gompers et al. (2016) find that, in addition to abilitybased characteristics, venture capitalists choose to collaborate with other venture capitalists for affinitybased characteristics and show that homophyly is detrimental for the investments.
Another empirical strategy, closely related to matching theory, explicitly introduces a stochastic structure at the level of the individual matches to cope with unobserved heterogeneity among the market participants. Given that monetary transfers between matched partners are often not observed, the goal is to provide estimations of the aggregate surplus that is divided between matched partners. In the TU games this approach was introduced by Choo and Siow (2006), who show that the matching surplus can incorporate latent characteristics (heterogeneity that is unobserved by the analyst). Building on that seminal paper, Chiappori et al. (2017) estimate the changes in the returns to education in the US marriage market, whereas Galichon and Salanié (2015) study the crossdifferential effect of variation in the attributes of the two sides of the market (such as whether education matters more for conscientious men/women than for extroverted ones).
Also related to matching theory, Fox (2018) proposes a maximum score estimator for matching situations where transfers are endogenous but not in the data. This maximizes the number of inequalities implied by pairwise stability that hold true. Fox’s (2018) maximum score method has been used in several papers. Bajari and Fox (2013) estimate the bidders’ valuation with data on the US auction of licenses of radio spectrum for mobile phone service and find that the final allocation of licenses was inefficient. Levine (2008) explores the matching of biotechnology innovations to marketing firms. In the case of bank mergers, Akkus et al. (2015) adapt Fox‘s (2018) maximum score estimator for the case of the availability of data on equilibrium transfers. They find that merger value arises from cost efficiencies in overlapping markets, the relaxing of regulation, and network effects exhibited by the acquirer–target matching. BanalEstañol et al. (2018) use the method to study the characteristics of the equilibrium matching between UK academics and firms for grant applications.
Given its interest both from the theoretical and the empirical perspective, we expect that considering moral hazard problems in markets will allow researchers to increase the understanding of important economic questions.
Notes
The concept of moral hazard was introduced by Arrow (1963) to depict a market failure that insurance companies had identified long before. Ross (1973) formalized it as a program where the principal maximizes her expected utility, taking into account not only that the agent has to accept the offer but also that he can choose the effort or decision that is best for him, given the contract. Pioneer papers include Mirrlees (1974, 1975), Harris and Raviv (1979) and Holmström (1979). Textbooks on the theory of incentives include MachoStadler and PérezCastrillo (1997) and Bolton and Dewatripont (2005).
There is an important literature that considers twosided matching models where utility is impossible to transfer, in the sense that each partner in a pair obtains a certain level of profit or utility that cannot be transferred to the other partner. This model was proposed in the seminal paper of Gale and Shapley (1962). See also Roth and Sotomayor (1990) for an excellent introduction to matching models.
Holmström and Milgrom (1987) show that the optimal contract is linear if the agent chooses efforts continuously to control the drift vector of a Brownian motion process and he observes his accumulated performance before acting. This setup has been extensively used to study many interesting questions (see, e.g., MachoStadler and PérezCastrillo 2018).
See, e.g., Bolton and Dewatripont (2005) for the details of the calculation.
In this paper, we keep a common framework where principals are risk neutral. Several authors examine the sorting patterns in a twosided matching market when principals are riskaverse and the main objective of the partnership is to share risks. In these environments, NAM tend to arise. For a general analysis, see Legros and Newman (2007) and Chiappori and Reny (2016).
Li et al. (2013) prove that this is also true if principals are riskaverse with the same degree of risk aversion.
This can be extremely important for empirical applications since most empirical analyses are conducted in a market. It can be an explanation for why the empirical evidence supporting the negative relationship between incentives and risk is far from overwhelming (see, e.g., Prendergast 2002).
In an environment with a continuous of principals and agents,
$$\begin{aligned} \frac{\partial \Pi _{i}}{\partial \sigma _{i}^{2}}=\frac{1}{2}\frac{r_{j}}{ \left( 1+r_{j}v\sigma _{i}^{2}\right) ^{2}}{\qquad \mathrm{and}\qquad }\frac{ \partial U_{j}}{\partial r_{j}}=\frac{1}{2}\frac{\sigma _{i}^{2}}{\left( 1+r_{j}v\sigma _{i}^{2}\right) ^{2}}\text {.} \end{aligned}$$We could also use the discrete assignment game and focus in one of the two extremes of the complete lattice of the set of equilibrium payoffs. Demange (1982), Leonard (1983), or Lemma 8.15 in Roth and Sotomayor (1990) show how to compute the precise levels of principals’ profits and agents’ utilities in the principaloptimal payoff and in the agentoptimal payoff.
We can compute the change in the utility level of the agent as a function of v as follows. Consider agent j and agent \(j^{\prime }\) such that \( v_{j^{\prime }}=v_{j}+\delta \). Denote \(i^{\prime }=\mu (j^{\prime })\). In an equilibrium, principal \(i^{\prime }\) and agent j do not have an incentive to deviate because \(S^{\mathrm{SI}}\left( \sigma _{i^{\prime }}^{2},v_{j}\right) \le \Pi _{i^{\prime }}^{\mathrm{SI}}+U_{j}^{\mathrm{SI}}\). Given that \( \Pi _{i^{\prime }}^{\mathrm{SI}}=S^{SI}\left( \sigma _{i^{\prime }}^{2},v_{j^{\prime }}\right) U_{j^{\prime }}^{\mathrm{SI}}\), then \(S^{\mathrm{SI}}\left( \sigma _{i^{\prime }}^{2},v_{j}\right) \le S^{\mathrm{SI}}\left( \sigma _{i^{\prime }}^{2},v_{j^{\prime }}\right) U_{j^{\prime }}^{\mathrm{SI}}+U_{j}^{\mathrm{SI}}\), that is, \(U_{j^{\prime }}^{\mathrm{SI}}U_{j}^{\mathrm{SI}}\le S^{\mathrm{SI}}\left( \sigma _{i^{\prime }}^{2},v_{j^{\prime }}\right) S^{\mathrm{SI}}\left( \sigma _{i^{\prime }}^{2},v_{j}\right) \). Dividing both sides of the equation by \(\delta \) and taking the limit when \(\delta \) goes to zero, we obtain \(\frac{\partial ^{\mathrm{SI}}U_{j}}{\partial v_{j}}\le \frac{\partial S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) }{\partial v_{j}}\). But we can take the other sense of the inequality as well, hence,
$$\begin{aligned} \frac{\partial U_{j}^{\mathrm{SI}}}{\partial v_{j}}=\frac{\partial S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) }{\partial v_{j}}=\frac{1}{2v_{j}^{2}} \text {.} \end{aligned}$$We can use a similar procedure for the following expressions.
In quite a different model, Li et al. (2013) also find that moral hazard pushes toward PAM in terms of risk. They study the equilibrium matching between principals and agents who are all riskaverse and heterogeneous in their degree of risk aversion. Moreover, the agents exert unverifiable efforts to increase the mean of the output and to reduce its volatility.
For example, Alkan (1988) shows that in the threesided onetoone matching market stable outcomes may not exist. Kelso and Crawford (1982) show that a sufficient condition for the existence of equilibrium in a twosided manytoone matching model is that agents are gross substitutes from each principal’s standpoint. If agents are complementary, then equilibria may fail to exist.
Holmström (1982) highlights the importance of the role of a residual claimant when moral hazard affects more than one participant in a relationship.
In our model, each principal hires several (two) agents who are subject to moral hazard. One can also analyze situations where it is the agent (subject to moral hazard) who contracts with several principals. One example of such a situation is found in von LilienfeldToal and Mookherjee (2016), who analyze a credit market. This paper also illustrates that an exogenous shock may have a general equilibrium effect in a market contracts which is absent in an isolated principal–agent relationship.
Note that the variable part of the contract s is the same for both partners, regardless of who contributes more (or less) to the variance of the project.
See, e.g., Talman and Yang (2011) for some sufficient conditions for the existence of equilibria.
See, for instance, Kremer (1993).
See Legros and Newman (2002) for a careful analysis of sufficient conditions for monotone matching.
In an isolated principal–agent relationship in their framework, if both participants are able to commit to the duration of the contract, an LT contract is always optimal (see, e.g., Chiappori et al. 1994). However, when there is a market, the sorting of workers with heterogeneous ability to firms which are heterogeneous in their profitability is also important and this can only be achieved with ST contracts. The paper shows that ST contracts are often offered at equilibrium, and they sometimes coexist with LT contracts.
See Chiappori et al. (1994). The intuition behind this result is that when the agent has a CARA utility function, the incentives do not depend on the income of the agent, obtained in particular in the previous periods.
Some authors refer to industryspecific as portable skills (Grosyberg et al. 2008). In addition to the agent’s ability, one can also think of portable resources such as carrying contacts, clients, or providers when moving to a new firm.
See MachoStadler et al. (2014) for a more careful explanation about this property.
The proofs of the results in this section are in the online Appendix, and in MachoStadler and PérezCastrillo (2020).
The market for seniors is an assignment game with a continuum of equilibria in terms of payoffs. If, for instance, we denote \(U_{H}\) the utility that a highability senior obtains in equilibrium when he works for a principal for whom he has no principalspecific ability, then he could obtain any \(U\in \left[ U_{H},U_{H}+\Theta \right] \) in equilibrium if he works as a senior for the same principal as a junior. Thus, we focus at the equilibrium that gives the principaloptimal payoff (see Proposition 2). If we would consider equilibria where the highability seniors obtain more than \( U_{H}\), then this increase in utility when senior would lead to a decrease in the fixed payment to all junior agents and, in expectation, a junior agent would obtain the same utility in both equilibria. The same comments hold for equilibria where the lowability agents would obtain more than at the principaloptimal payoff.
There is a trivial equilibrium where all principals sign LT contracts with their agents: if all the principals in the economy sign LT contracts, then no single principal has an incentive to deviate and offer a sequence of ST contracts because she can only hire the same agent that worked for her as a junior.
Barros and MachoStadler (1988) also analyze the effect of the principals’ competition for a good agent on the power of incentives and the efficiency of the relationship.
This is the case in other interesting economic situations such as in Nash equilibrium outcomes, or any other cooperative or noncooperative outcome that depends on the preferences of all the agents.
As argued by Akkus et al. (2015), even if other methods of estimation are econometrically superior, estimation methods based on random utility models (such as the probitcounterfactual approach) are widely understood and applied in different contexts.
References
Agrawal A, Kapur D, McHale J (2008) How do spatial and social proximity influence knowledge flows? Evidence from patent data. J Urban Econ 64(2):258–269
Akkus O, Cookson JA, Hortacsu A (2015) The determinants of bank mergers: a revealed preference analysis. Manag Sci 62(8):2241–2258
Alkan A (1988) Nonexistence of stable threesome matchings. Math Soc Sci 16:207–209
AlonsoPaulí E, PérezCastrillo D (2012) Codes of best practice in competitive markets for managers. Econ Theory 49:113–141
Arrow K (1963) Uncertainty and the welfare economics of medical care. Am Econ Rev 53:541–67
Bajari P, Fox JT (2013) Measuring the efficiency of an FCC spectrum auction. Am Econ J Microecon 5(1):100–146
BanalEstañol A, MachoStadler I, PérezCastrillo D (2018) Endogeneous matching in universityindustry collaboration: theory and empirical evidence from the United Kingdom. Manag Sci 64(4):1591–1608
Barros F, MachoStadler I (1988) Competition for managers and market efficiency. J Econ Manag Strat 7:89–103
Becker GS (1973) A theory of matching: part I. J Polit Econ 81:813–846
Besley T, Ghatak M (2005) Competition and incentives with motivated agents. Am Econ Rev 95(3):616–636
Bolton B, Dewatripont M (2005) Contract theory. MIT Press, Cambridge
Chiappori PA, Reny P (2016) Matching to share risk. Theor Econ 11(1):227–251
Chiappori PA, Salanié B (2016) The econometrics of matching models. J Econ Lit 54(3):832–861
Chiappori PA, MachoStadler I, Rey R, Salanié B (1994) Repeated moral hazard: the role of memory, commitment, and the access to credit markets. Eur Econ Rev 38:1527–1553
Chiappori PA, Salanié B, Weiss Y (2017) Partner choice, investment in children, and the marital college premium. Am Econ Rev 107(8):2109–2167
Choo E, Siow A (2006) Who marries whom and why. J Polit Econ 114(1):175–201
Dam K, PérezCastrillo D (2006) The principalagent matching market. Front Econ Theory Berkeley Electron 2(1):Article 1
Demange G (1982) Strategyproofness in the assignment market game. École Polytechnique, Laboratoire d’Économetrie, Paris
Fox JT (2018) Estimating matching games with transfers. Quant Econ 9(1):1–38
Gale D, Shapley L (1962) College admissions and stability of marriage. Am Math Mon 69:9–15
Galichon A, Salanié B (2015) Cupid’s invisible hand: social surplus and identification in matching models. SSRN 1804623
Ghatak M, Karaivanov A (2014) Contractual structure in agriculture with endogenous matching. J Dev Econ 110:239–249
Gierlinger J, Laczó S (2018) Matching to share risk without commitment. Econ J 128(613):2003–2031
Gompers PA, Mukharlyamov V, Xuan Y (2016) The cost of friendship. J Financ Econ 119(3):622–644
Graham BS (2011) Econometric methods for the analysis of assignment problems in the presence of complementarity and social spillovers. In: Benhabib J, Bisin A, Jackson, M (eds) Handbook of social economics, vol 1. North Holland, pp 965–1052
Gretsky N, Ostroy J, Zame W (1992) The nonatomic assignment model. Econ Theory 2:103–127
Groysberg B, Lee LE, Nanda A (2008) Can they take it with them? The portability of star knowledge workers’ performance. Manag Sci 54(7):1213–1230
Harris M, Raviv A (1979) Optimal incentive contracts with imperfect information. J Econ Theory 20:231–259
Hegde D, Tumlinson J (2014) Does social proximity enhance business partnership? Theory and evidence from ethnicity role in U.S. venture capital. Manag Sci 60(9):2355–2380
Hermalin BE (2002) Adverse selection, shortterm contracting, and the underprovision of onthejob training. Contrib Econ Anal Pol 1(1):1–21
Holmström B (1979) Moral hazard and observability. ACA Trans 10:74–91
Holmström B (1982) Moral hazard in teams. ACA Trans 13:324–340
Holmström B, Milgrom P (1987) Aggregation and linearity in the provision of intertemporal incentives. Econometrica 55:303–328
Kaneko M, Wooders M (1986) The core of a game with a continuum of players and finite coalitions: the model and some results. Math Soc Sci 12:105–137
Kelso AS, Crawford V (1982) Job matching, coalition formation and gross substitutes. Econometrica 50:1483–1504
Kremer M (1993) The Oring theory of economic development. Q J Econ 108:551–575
Legros P, Newman A (2002) Monotone matching in perfect and imperfect worlds. Rev Econ Stud 69:925–942
Legros P, Newman A (2007) Beauty is a beast, frog is a prince: assortative matching with nontransferabilities. Econometrica 75:1073–102
Leonard HB (1983) Elicitation of honest preferences for the assignment of individuals to positions. J Polit Econ 91:461–479
Levine AA (2008) Licensing and scale economies in the biotechnology pharmaceutical industry. Harvard University Press, Cambridge
Li F, Ueda M (2009) Why do reputable agents work for safer firms? Financ Res Lett 6:2–12
Li S, Sun H, Chen P (2013) Assortative matching of riskaverse agents with endogenous risk. J Econ 109(1):27–40
MachoStadler I, PérezCastrillo D (1997) An introduction to the economics of information: incentives and contracts. Oxford University Press, Oxford
MachoStadler I, PérezCastrillo D (2014) Copyright licensing under asymmetric information. Rev Econ Res Copyright Iss 11(2):1–26
MachoStadler I, PérezCastrillo D (2018) Moral hazard: base models and two extensions. In: Corchón LC, Marini MA (eds) Handbook of game theory and industrial organization, vol 1. Edward Elgar, London, pp 453–485
MachoStadler I, PérezCastrillo D, Porteiro N (2014) Coexistence of longterm and shortterm contracts. Game Econ Behav 86:145–164
MachoStadler I, PérezCastrillo D (2020) Agency theory meets matching theory. Barcelona GSE wp 1140
Mirrlees J (1974) Notes on welfare economics, information and uncertainty. In: Balch M, McFadden D, Wu S (eds) Essays in economic behavior in uncertainty. NorthHolland, Amsterdam
Mirrlees J (1975) The theory of moral hazard and unobservable behavior—part I. Oxford University Press, Oxford
Mukherjee A, Vasconcelos L (2018) On the tradeoff between job assignment and efficiency in turnover: the role of breakup fees. J Law Econ Organ 34(2):230–271
Prendergast C (2002) The tenuous tradeoff between risk and incentives. J Polit Econ 110(5):1071–110
Ross S (1973) The economic theory of agency: the principal’s problem. Am Econ Rev 63:134–139
Roth A, Sotomayor M (1990) Twosided matching. A study in gametheoretic modeling and analysis, vol 18. Econometric society monograph series. Cambridge University Press, London
Sattinger M (1975) Comparative advantage and the distributions of earnings and abilities. Econometrica 43(3):455–468
Serfes K (2005) Risk sharing versus incentives: contract design under twosided heterogeneity. Econ Lett 88(3):343–349
Serfes K (2008) Endogenous matching in a market with heterogeneous principals and agents. Int J Game Theory 36(3–4):587–619
Shapley L, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1:111–130
Sørensen M (2006) How smart is smart money? An empirical twosided matching model of venture capital. J Financ 62:2725–2762
Sotomayor M (1992) The multiple partners game. In: Majumdar M (ed) Equilibrium and dynamics. The MacMillan Press LTD, London
Sotomayor M (2007) Connecting the cooperative and competitive structures of the multiplepartners assignment game. J Econ Theory 134:155–74
Talman D, Yang Z (2011) A model of partnership formation. J Math Econ 47:206–212
von LilienfeldToal U, Mookherjee D (2016) A general equilibrium analysis of personal bankruptcy law. Economica 83(329):31–58
Waldman M (1984) Job assignments, signalling, and efficiency. RAND J Econ 15(2):255–267
Wright DJ (2004) The risk and incentives tradeoff in the presence of heterogeneous managers. J Econ 83(3):209–223
Funding
This study was funded by the Ministerio de Ciencia y Tecnología and FEDER (PGC2018094348BI00) and Severo Ochoa Programme (SEV20150563), and Generalitat de Catalunya (2017 SGR 711 and ICREA under the ICREA Academia Programme).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Inés MachoStadler declares that she has no conflict of interest. David PérezCastrillo declares that he has no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This paper corresponds to the Presidential Address delivered by the second author at the 44th Simposio de la Asociación Española de Economía in Alicante. We gratefully acknowledge comments received from Kaniska Dam and two anonymous referees. We also acknowledge financial support from the Ministerio de Ciencia y Tecnología and FEDER (PGC2018094348BI00), Severo Ochoa Programme (SEV20150563) and Generalitat de Catalunya (2017 SGR 711, and ICREA under the ICREA Academia Programme).
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
MachoStadler, I., PérezCastrillo, D. Agency theory meets matching theory. SERIEs 12, 1–33 (2021). https://doi.org/10.1007/s13209020002153
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13209020002153
Keywords
 Incentives
 Contracts
 Matching
 Moral hazard
JEL Classification
 D86
 D03
 C78