We now go back to consider a principal–agent market. The set of heterogeneous, risk-neutral 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
$$\begin{aligned} U_{j}(w,e)=-\exp \left[ -r_{j}\left( w-\frac{1}{2}v_{j}e^{2}\right) \right] \text {.} \end{aligned}$$
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:
$$\begin{aligned} x=\alpha +e+\sigma _{ij}\varepsilon \end{aligned}$$
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 one-dimensional 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 cross-partial 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 cross-partial 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,
$$\begin{aligned} S^{\mathrm{SI}}\left( \sigma _{i}^{2},r_{j}\right) =\alpha +\frac{1}{2v},\qquad \text { and}\qquad \frac{\partial ^{2}S^{\mathrm{SI}}\left( \sigma _{i}^{2},r_{j}\right) }{ \partial \sigma _{i}^{2}\partial r_{j}}=0; \end{aligned}$$
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,
$$\begin{aligned} S^{\mathrm{SI}}\left( \sigma _{i}^{2},v_{j}\right) =\alpha +\frac{1}{2v_{j}}. \end{aligned}$$
Also in this case, the cross-partial 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 cross-partial 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:
$$\begin{aligned} S^{\mathrm{MH}}\left( \sigma _{i}^{2},r_{j}\right) =\alpha +\frac{1}{2v}\left( \frac{1 }{1+r_{j}v\sigma _{i}^{2}}\right) . \end{aligned}$$
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 low-volatility project, and a “good” agent has a low degree of risk aversion. Moreover:
$$\begin{aligned} \frac{\partial ^{2}S^{\mathrm{MH}}\left( \sigma _{i}^{2},r_{j}\right) }{\partial \sigma _{i}^{2}\partial r_{j}}=-\frac{1}{2}\frac{\left( 1-r_{j}v\sigma _{i}^{2}\right) }{\left( 1+r_{j}v\sigma _{i}^{2}\right) ^{3}}. \end{aligned}$$
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 risk-averse 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:
$$\begin{aligned} s^{\mathrm{MH}}\left( \sigma _{i}^{2},r_{j}(\sigma _{i}^{2})\right) =\frac{1}{ 1+r_{j}(\sigma _{i}^{2})v\sigma _{i}^{2}}. \end{aligned}$$
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
$$\begin{aligned} S^{\mathrm{MH}}\left( \sigma _{i}^{2},v_{j}\right) =\alpha +\frac{1}{2v_{j}}\left( \frac{1}{1+rv_{j}\sigma _{i}^{2}}\right) , \end{aligned}$$
which cross-partial 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,
$$\begin{aligned} \frac{\partial \Pi _{i}^{\mathrm{SI}}}{\partial c_{i}}=-\frac{\partial \Pi _{i}^{\mathrm{SI}}}{ \partial \sigma _{i}^{2}}=0. \end{aligned}$$
However, an agent with higher ability obtains a higher level of utility:Footnote 13
$$\begin{aligned} \frac{\partial U_{j}^{\mathrm{SI}}}{\partial c_{j}}=-\frac{\partial U_{j}^{\mathrm{SI}}}{ \partial v_{j}}=-\frac{\partial S^{\mathrm{SI}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) }{\partial v_{j}}=\frac{1}{2v_{j}^{2}}. \end{aligned}$$
Similarly, under moral hazard, we obtain:
$$\begin{aligned} \frac{\partial \Pi _{i}^{\mathrm{MH}}}{\partial c_{i}}= & {} -\frac{\partial \Pi _{i}^{\mathrm{MH}}}{\partial \sigma _{i}^{2}}=-\frac{\partial S^{\mathrm{MH}}\left( \sigma _{i}^{2},v_{\mu (i)}\right) }{\partial \sigma _{i}^{2}}=\frac{1}{2}\frac{r}{ \left( 1+rv_{\mu (i)}\sigma _{i}^{2}\right) ^{2}}\text {, and} \\ \frac{\partial U_{j}^{\mathrm{MH}}}{\partial c_{j}}= & {} -\frac{\partial U_{j}^{\mathrm{MH}}}{ \partial v_{j}}=-\frac{\partial S^{\mathrm{MH}}\left( \sigma _{\mu (j)}^{2},v_{j}\right) }{\partial v_{j}}=\frac{1}{2v_{j}^{2}}\frac{ 1+2rv_{j}\sigma _{i}^{2}}{\left( 1+rv_{j}\sigma _{i}^{2}\right) ^{2}}. \end{aligned}$$
Therefore,
$$\begin{aligned} \frac{\partial \Pi _{i}^{\mathrm{MH}}}{\partial c_{i}}>\frac{\partial \Pi _{i}^{\mathrm{SI}}}{ \partial c_{i}}=0,\qquad \text {and}\qquad \frac{\partial U_{j}^{\mathrm{SI}}}{ \partial c_{j}}>\frac{\partial U_{j}^{\mathrm{MH}}}{\partial c_{j}}>0 \end{aligned}$$
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
$$\begin{aligned} S^{\mathrm{MH}}\left( \sigma _{i}^{2},\sigma _{j}^{2}\right) =\alpha +\frac{1}{2v} \left( \frac{1}{1+rv\left( \sigma _{i}^{2}+\sigma _{j}^{2}\right) }\right) . \end{aligned}$$
Therefore, the cross-partial 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