Commission Sharing among Agents Simple Storage Service
Commission Sharing among Agents∗ Zhiyun (Frances) Xu University of Hong Kong
October 17, 2010
Abstract Commission sharing enables agents to trade the outcomes of their efforts. One incentive to do that is to gain from a better match of the outcomes to the principals. The possibility of such interaction affect the agents’ incentive to exert effort and also the principals’ choice of contracts. I exhibit that there are three forces that determine the choices of the commissions by the principals in response to this possibility of interagent interaction. One is the usual incentive to use the commission to motivate effort in one’s own agent, one is the incentive to use the commission to influence other agents who can potentially trade with one’s own agent, and the other is the incentive to use commission to influence the transfers among the agents. I study the change in the equilibrium outcomes, and show that the equilibrium effort under commission sharing would be lower, while the change in the commission is in general dependant on the degree of alignment between the principals’ taste. I also show that the efficiency gain from inter-agent interaction can also be due to a more efficient effort level. I demonstrate that whether principals can contract with multiple agents or not plays a role in the efficiency of the equilibrium. ∗
I thank participants of seminars at HKUST, CUHK, HKU, Sinica, Lingnan University, 2010 Midwest Theory Summer Conference, 2010 International IO Conference. I am grateful for helpful comments from Rongzhu Ke, Jin Li, Yuk-fai Fong, Johannes H¨ orner, Marco Ottaviani, and Wing Suen. All errors remain my own. I thank the support from Research Grants Council of Hong Kong. This project began when I was affiliated with Cornerstone Research.
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1
Introduction
Principals typically hire a limited number of agents. Even when a principal hires only one agent, it is not hard to see that there are interactions among agents that cause a principal to “indirectly” hire other agents. For example, real estate agents typically share commissions, so if one agent cannot find a suitable outcome for his own principal, he could get an outcome from another agent and share the commissions with that agent. Similar practices happen among headhunters and subcontractors. When principals are “connected” in such a way, questions arise regarding what externality the interaction creates among the principals, how such interaction affect the equilibrium efforts by the agents, the principals’ contracts, payoffs and the overall efficiency. In this paper, I present a rather specific model for understanding the phenomenon of inter-agent interaction. In particular, I consider the case when principals are heterogenous in their taste and the fit between the outcomes created by the agents and the preferences of the principals is not entirely determined by the agents’ effort, but is subject to some uncertainty. The agents here not only have the task of exerting efforts, but also the task of “trading” with other agents, where they might choose to divert outcomes to other agents for a payment or to pay other agents to receive outcomes. These post-effort trading actions are typically hard to observe or to contract on, so I assume that the principal can only contract on the value of the outcome she receives. In the discussion, I also consider the contracting environment where the principals can partially contract on the post-effort trading actions by being able to identify the source of the results delivered. I study a symmetric two-principal-two-agent model that can be generalized to more than two principals and two agents. I model the effort of an agent as being able to increase the probability of a good fit with a principal. Naturally, when principals’ taste are different, effort targeted at one principal decreases the probability of this outcome being also a good fit for the other. To understand the externality introduced by the possibility of inter-agent interaction, our first benchmark is a model where the agents do not have the option to interact. I also solve for the fully efficient efforts given efficient trading as a second benchmark. Our comparisons with the benchmarks are all based on the symmetric equilibrium. A crucial parameter of the model is the level of alignment between the two principals’ taste. For example, two principals looking for a house to buy may have the following tastes. One wants a big traditional house and the other wants a small modern house. In this case, effort that is invested in looking for a good fit with one principal necessarily makes the outcome unlikely to fit the other. However, another possible situation is that one principal wants a big traditional house, while the other wants a small traditional house, then effort targeted at one might create a fit for the other as well, with a probability higher than in the 2
previous situation. The other extreme case is when both principals want a big traditional house each, then effort for one is equally valuable for the other. I parameterize this level of alignment between the principals and study its effect on equilibrium contract and payoffs. One may guess that once the agents can trade post-effort, they tend to exert less targeted effort because of the chance of serving the interest of another principal makes it less meaningful to target a particular taste. This conjecture would definitely be correct if the principals’ commission levels do not change when interaction among agents is introduced, however, it is not a priori clear whether the principals will increase or decrease their commissions/piece rates in response to the possibility of post-effort trading. In the absence of the possibility of trading, the optimal/equilibrium contract is simply a sell-out contract with 100% commission, which is just the piece rate in a two part contract. I show that there are two countervailing forces that determine how the principal’s choice of the commission would change once trading is allowed. First, the fact that the principal cannot directly contract with the other agent means that some of her commission will be leaked to the other agent through trading and this principal has no ability to extract it back through a participation constraint. This creates an incentive to lower the commission. Second, under trading each principal’s commission has an influence on the other agent and can de-motivate the other agent to exert effort targeted at the other principal. This creates an incentive to raise the commission. Therefore, whether the principals offer higher or lower commission compared to the no trading benchmark depends on the relative strength of these two forces. The strength of the first force decreases when the principals are less aligned in their preference, because in that case there is a smaller chance that an outcome will be diverted and a commission will be leaked outside. The strength of the second force however increases when the principals are less aligned in their preference, because each principal’s commission has a stronger de-motivating effect on the other agent. Therefore, in general the equilibrium commission can be higher or lower depending on the alignment of the principals’ preference. However, under quadratic cost function, the leakage effect is stronger than the cross-motivating effect and the commission is always lower under trading than under no trading regardless of the alignment of taste. Even though the direction of change for the commission is parameter-dependent in the above sense when I do not impose quadratic cost function, the effort level comparison is not parameter dependent. Nor is it cost function dependent. That is, the effort under trading is always lower than that under no trading. This is because, on top of the effects of trading on the choice of commission, each symmetric pair of commission becomes weaker in motivating efforts as they start to counteract each other in the agents’ incentives when trading is allowed. I also compare both the equilibrium effort under trading and under no trading with the fully efficient level of effort. The efficient level of effort (under efficient trading) is always 3
lower than the no trading equilibrium effort. This is because the efficient trading takes care of the matching of outcome to principals, and the effort is only valuable from a social point of view if it moves the probability from an outcome that fits nobody to an outcome that fits somebody, and thus targeting one principal at the expense of the other is a socially wasteful activity. Since the equilibrium effort under trading is lower than the equilibrium effort under no trading, trading allows the problem of over-investment in targeted effort to be mitigated. However, the equilibrium effort under trading can be overly low in the sense that it can fall below the efficient effort level. When this happens, the efficiency comparison depends on how important the efficiency gain from better matching is. I show that, with quadratic cost function for efforts, the gain from better matching dominates, so the efficiency is always improved when the agents can trade. The increase in surplus implies that if the principal has the ability to impose an exclusivity condition that bans interaction into the contract offered, then there exist an equilibrium where both do not choose to impose exclusivity. I also study the impact on the principals’ payoff, which is closely related to the efficiency comparison. This is because the gain in efficiency would accrue to the principals. However, while trading unambiguously improves efficiency under a quadratic cost function, trading also improves the agents’ outside and this hurts the principal. When agents cannot interact with each other, the principals can push the agents to an outside option equaling unemployment. However, now an agent who refuses a contract can also work indirectly if the other agent accepts a contract. I show that, with quadratic cost function, this effect based on the outside option is dominated and the principals’ equilibrium payoff is higher when there is agent-interaction. The legal literature has expressed concern that inter-agent interaction, such as Multiple Listing Service among the real estate brokers, makes it easy for agents to collude, because an agent can carry out punishment against another agent who has accepted a low commission.1 While this paper is far from offering a model specifically for the market of real estate agency, it can be viewed as pointing out a potentially important aspect of inter-agent interaction: it may improve efficiency over and above the gain from better matches, because it promotes a lower powered contract that leads to less socially wasteful targeted efforts. From a more theoretical point of view, this paper adds to the multiple principal and multiple agent literature by considering a setting where agents can interact directly among each other, while imposing the restriction that each principal can only offer contract to one agent.2 In this paper, since a principal only contracts with one agent and thus does not internalize the other agent/principal’s utility, the efficient outcome cannot be achieved even 1
See White (2006). See Bernheim and Whinston (1986), Bernheim and Whinston (1985), Prat and Rustichini (2003) for models where each principal can offer contract to multiple agents. 2
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though the collusive outcome would be efficient. I also show that efficiency can be restored in this model if each principal can offer two contracts, one to each agent (shown in the section of Discussions), echoes the result in Segal (1999), which has shown that, when a principal can offer multiple bilateral public offers, if there is no externality on agents’ reservation utilities, the equilibrium is efficient. My assumption that each principal can only contract with one agent is a way of saying that it is very costly for a principal to contract with more than one agent. I didn’t explicitly model the cost of contracting, but one can imagine that agents are professionals who repeatedly perform similar tasks and are thus informed about what other agents are in the market, so they can “trade” with other agents at minimal additional cost after they have paid the fixed cost of getting to know the field. However, the principals are typically only in the market once and it is therefore not worthwhile for them to invest in learning about other principals or agents, so a principal simply does not have the information about the identities of other agents and principals and thus has no way to contracting with them directly. In the same spirit, I also assumed that a principal cannot condition her contract on the acceptance decision of the other agent (unlike in the common agency literature). The restriction of only one agent per principal of course seems extreme, but this is only way of restricting the number of agents that a principal can contract with to be less than the number of the agents whose effort is potentially useful in a twoprincipal-two-agent model. Viewed in such light, my result simply presents the case when contracting cost is high enough, while the multiple-principal-multiple-agent literature with full directions of contracting is simply the case when contracting cost is low enough, then it is not surprising that efficiency is affected in my model. There is also a literature that focuses on agents that are hired to do search. Lewis and Ottaviani (2008) studied a dynamic single-principal and single-agent model where the agent can gain private benefits over the cause of searching, and their focus was on the interaction of the agent’s incentives to exert search effort with agent’s incentives to report the private information the agent acquires during the search process. In comparison, this paper is static, and looks at the agents’ incentives to exert targeted search effort when they can trade according to their private interest. There were studies of referrals, another form of interaction among agents. See Garicano and Santos (2004) and Pauly (1979). “Referral” refers to the possibility that an agent can pass the client to another agent before exerting the effort. Garicano and Santos (2004) focused on matching opportunities with agents’ talent, rather than on matching created outcomes with principals’ taste. The rest of the paper is organized as follows. Section 2 presents the model. Next, section 3 presents the analysis of the two benchmarks. Section 4 shows the main model and contrasts its results with the benchmarks. Section 5 discusses the assumptions, and extends the model
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in different directions, including a n-principal-n-agent model. Section 6 concludes.
2
Model
There are two principals, 1 and 2. Since each principal can only contract with one agent, without loss of generality, principal 1 can only contract with agent 1; and principal 2 can only contract with agent 2. I will refer to an agent as him and a principal as her. The principals differ in their taste, so there are only three possibilities for the outcome an agent can create. If it fits principal 1, then I call it a “1-fit”. If it fits principal 2, I call it a “2-fit”. The last possibility is when it fits neither of the principals, in which case I call it a “no-fit”. After being hired by a principal, the agent learns about the specific taste of the principal and can exert targeted effort ei ≥ 0. In the following table, I show the probabilities of agent 1 getting each types of the outcome as a function of agent 1’s targeted effort e1 and the corresponding value of the outcome to each principal. Probability “1-fit” q + e1 = q + λe1 + (1 − λ)e1 “2-fit” q − λe1 “no-fit” (1 − 2q) − (1 − λ)e1
Value to principal 1 d 0 0
Value to principal 2 0 d 0
The assumption is d > 0, q ∈ (0, 31 ), λ ∈ [0, 1]. This simple structure says the following. If agent 1 exerts no effort, then there is an equal chance of getting a 1-fit and a 2-fit. The effort of agent 1 moves probability from getting a 2-fit to getting a 1-fit by the amount of λe1 , and also moves probability from getting a no-fit to getting a 1-fit by the amount of (1 − λ)e1 . Such effort is called “targeted effort”, because it is targeting the taste of principal 1 specifically. 3 The parameter λ is a measurement of the alignment of taste between the two principals. If the two principals’ taste are quite aligned, then effort targeted at creating a good fit for one does not diminish the chance of getting a good fit for the other by much. On the other hand, if the two principals’ taste are very opposite, then effort targeted at one principal will lead to very low probability of this outcome that fitting the other. One can refer to the 3
Naturally, there is another distinct type of effort where the effort increase the value of the outcome to both principals. This can be termed “un-targeted effort”. The main intuition with un-targeted effort would be the same as the targeted effort, while the implication for efficiency would be different, as “un-targeted effort” does not have a socially wasteful component.
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Introduction section for examples of different levels of alignment of taste. This parameter λ also controls how socially wasteful the targeted effort is. Moving probability from a 1-fit to a 2-fit is socially wasteful, while moving the probability from a no-fit to a 1-fit is not wasteful. I maintain symmetry in our assumptions with respect to principal 1 and principal 2, therefore, the corresponding table for agent 2’s effort, e2 ≥ 0, is the following: Probability “2-fit” q + e2 = q + λe2 + (1 − λ)e2 “1-fit” q − λe2 “no-fit” (1 − 2q) − (1 − λ)e2
Value to principal 1 0 d 0
Value to principal 2 d 0 0
I assumed that efforts are non-negative, which means that each agent can only exert effort targeted to their own principals, that is, they cannot secretly exerting effort targeted at the other principal. This assumption is justified as follows. Even though an agent after being hired knows the taste of one’s own principal and knows how misaligned the two tastes are, there are many ways that taste can be misaligned and an agent is still not sure about the taste of the other principal and thus has no way of targeting it. Without this assumption, the analysis would be very similar, however, for some parameter case I will not have symmetric pure strategy equilibrium. In the same spirit, I assume that if an agent is not hired, he can only choose e = 0, because he does not know what taste to target. Each agent can only create one result, however, a principal can derive value from multiple units of results.4 I restrict the effort choice set by ei ≤ q so that the probabilities of each type of outcomes are well defined. The two agents have the same cost function of effort, C(e). For e > 0, I assume C 0 (e) > 0, C 00 (e) > 0, C 000 (e) ≥ 0, and C(0) = 0, C 0 (0) = 0, C 000 (e) ≥ 0 for any e ∈ [0, q]. I assume C 0 (q) > d, which ensures interior effort choices by the agents in the no trading benchmark. To get some results, I would further assume that C is quadratic, i.e., C 00 (e) is a constant and then C 0 (q) > d is equivalent to Cd00 < q. 4
This assumption removes the complication of multiple effort equilibria given a pair of contracts accepted by the agents (thus the need to arbitrarily select a subgame equilibrium), as what would happen in a coordination game. Suppose each principal values only one outcome. In this case, an agent will have a bigger chance of supplying an outcome to another principal if the other principal’s agent does not work hard to create a good fit. This gives him incentive not to work hard in creating a good fit with his own principal either. On the other hand, if the other agent works hard, this agent also wants to work hard because it is less likely that an outcome will be supplied to the other principal and a good fit with his own principal is more important.
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After exerting effort and after observing the realization of the result, an agent can choose from one of the two actions: 1. Divert the result to the other agent, who will in turn deliver it to the other principal. 2. Not divert the result, i.e., deliver it to his own principal.
2.1
Timing and Information
The timing of the game is as follows. 1. Principals 1 and 2 simultaneously make take-it-or-leave-it public contract offers (to be specified later) to their respective agents. 2. Agents 1 and 2 simultaneously decide whether to accept the offers or not, after observing both contracts offered. 3. Agents 1 and 2, if hired, simultaneously exert efforts after observing the hiring outcomes. Efforts are not observable to others. 4. Project values are realized and are observed only by the agents that created them. 5. Agents decide on whether to divert or not divert the result; an agent who is not hired can only choose to divert the result to an agent that is hired. 6. A diverter and a intended receiver Nash Bargain with equal bargaining power to determine the transfer between them under common knowledge of the value of the results being diverted.5 7. Each principal pays the promised payments to her agent. The effort is not observable or contractible. The value of the outcome delivered is observable and contractible. I will assume that each principal cannot tell whether the outcome delivered to her is created by the agent himself or by another agent she does not contract 5
I allow the diverting and receiving agent to have different bargaining power in the Discussion. Alternatively, I can allow the agents Nash Bargain on a way to split the commission before they exert the effort, as the analysis would reach the same conclusion. The typical norm among real estate brokers and headhunters are to split any commission half-half.
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with.6 Since the value to a principal is binary, I assume the contract has two parts: a piece rate for an outcome created ki ∈ [0, ∞), and a fixed fee Fi ∈ (−∞, ∞).7 To be realistic, I also assume that a principal cannot condition her contract on the acceptance decision of the other agent (unlike in the common agency literature), which is consistent with the spirit of the assumption that it is costly for a principal to contract with more than one agent, an assumption that is discussed in more detail in the Introduction.
2.2
Equilibrium concept
All players are risk-neutral and they maximize their expected payoff. An agent’s payoff is the payment received minus the cost of effort. A principal’s payoff is the value of the project received minus the total payment to the agent. I assume there is no discounting. If an agent is not paid and he does not exert any effort, then his payoff is zero. Principals’ strategies are simply the contracts they offer. The agents’ behavioral strategies are, 1) the acceptance decision given the two contracts offered, 2) effort choices given what contracts are accepted, and 3) the choice between diverting and not diverting given the realized outcomes. Despite that there is incomplete information in this model, since the principals only act once at the beginning of the game by offering contracts, their beliefs will not play a role. Also, when deciding whether to divert or not, the agents have all the information that can affect their payoffs. Therefore, like in any pure screening model, the set of Subgame Perfect Nash Equilibrium is identical to the sets of strategy profiles in Weak Perfect Bayesian Equilibrium or Sequential Equilibrium, so I will use the concept of Subgame Perfect Nash Equilibrium.
2.3
Benchmarks
I will consider the following two benchmarks 8 : 6
Holmstr¨ om and Milgrom (1988) made the similar assumption that when multiple principals contract with a single agent, each principal only observe the agent’s performance in their own relationship. I will relax this assumption in the Discussions. 7 Alternatively, I can allow the principal to observe which type of outcome it is, i.e., 1-fit, 2-fit or no-fit. The equilibrium payment for 2-fit and no-fit would be exactly the same, so it will boil down to a two part contract as well. The restriction of ki ≥ 0 is WLOG because ki < 0 is dominated. In the discussion, I will consider a different contracting environment when the principal can tell whether the outcome delivered to her is created by the agent himself or by another agent she does not contract with. 8 There is naturally a third benchmark: allowing trading but no fee-sharing, that is, I allow the agent to divert the result to the other agent, but does not allow them to make any transfers to each other. It turns out to be identical to the “no trading benchmark” in terms of contracts and efforts on the equilibrium (though potentially different efficiency implications depending on how agents break a tie), so I omit it here.
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• No trading benchmark. This is where I exogenously shut-down all interactions among the agents. • Efficient benchmark. This is where I solve for socially optimal pair of efforts, assuming that the agents divert the results in a socially optimal manner. Then I solve for the pair of two-part contracts that can implement this efficient outcome when the agents are strategic. I will adopt the following notations for the symmetric equilibrium outcomes for the benchmarks and the main model:
No trading benchmark Efficient benchmark Allowing interaction
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Effort eˆ e∗ e˙
Contract kˆ k∗ k˙
Welfare ˆ W W∗ ˙ W
Benchmarks
3.1
No trading benchmark
There is no connection between two principal-agent pairs, so I can just study each pair in isolation. The analysis is very standard: a sell-out contract is optimal for the principal. The contract implements the effort level that is efficient from the point of view of only one pair of principal and agent, which is the solution to:
max (q + e)d − C(e) ⇒ d − C 0 (ˆ e) = 0 e
Lemma 1. In the no trading benchmark, the unique equilibrium outcome is the following: kˆ = 1 and eˆ = (C 0 )−1 (d). ˆ = (q + eˆ)d − C(ˆ The equilibrium payoff of a principal is Π e).
3.2
Efficient benchmark
Under efficient trading, from a society’s point of view an outcome that fits principal 1 is just as good as an outcome that fits principal 2, so any effort exerted to move probability from 10
1-fit to 2-fit or vice versa is a social waste and the only socially desirable part of the effort is to move the probability from a no-fit outcome to one that fits one of the two principals, so the efficient effort level solves:
max (1 − λ)ed − C(e) ⇒ (1 − λ)d − C 0 (e∗ ) = 0 e
Therefore, e∗ = (C 0 )−1 ((1 − λ)d). This shows that in the no trading benchmark, efforts are overly high. The parameter λ controls how socially inefficient is the effort, so the higher λ is, the lower is the efficient level of effort. Next, I solve for the symmetric pair of contracts that can implement this efficient level of effort given strategic agents if they are accepted. First, observe that if both principals offer the same contracts with strictly positive piece rate k, and they are both accepted, then the agents will trade efficiently given any realized results. When agent 1 diverts a result, in additional to what he would have got if he has not diverted it, he would receive a payment equal to the half of the surplus created by diverting. It is half because of the nature of Nash bargaining with equal bargaining power. The amount of surplus is proportional to the piece rate of principal 2. When agent 1 received a result, in additional to what he would have got if he has not received it, he gets a reward from his principal for a good fit, but he also has to pay out to agent 2 an amount equal to the half of the surplus created by agent 2’s diverting, which is proportional to the piece rate of principal 1. When an agent gets a no-fit, there is no change in the agent’s payoff as no surplus is created. Therefore, fixing a symmetric piece rate k, agent 1 maximizes the following objective function:
k k max (q + e1 )dk + (q − λe1 )d + (q − λe2 )d(1 − ) − C(e1 ) e1 2 2 The second term is the additional expected payoff created by agent 1 diverting out an outcome and the third term is the additional payoff created by agent 2 diverting in an outcome. Since C 0 (0) = 0 and C 0 (q) > d, for any k ∈ (0, 1−1 λ , the solution is interior and satisfies: 2
λ d(1 − )k − C 0 (e1 ) = 0 2 11
If k =
1−λ , 1− λ 2
then e1 = e∗ solves the problem because:
λ d(1 − )k − C 0 (e∗ ) = d(1 − λ) − C 0 (e∗ ) = 0 2 1−λ This implies k ∗ = 1− λ can implement the efficient effort level if agents can strategically 2 trade. Note that this is not an equilibrium piece rate because I didn’t allow the principals to be strategic in choosing the piece-rate, but this is an interesting benchmark piece rate.
To implement the efficient effort level, the fixed payment does not affect efficiency as long as it is high enough for both agents to accept. The optimal contracts for the principals keep the agent at the participation constraint. If an agent rejects a contract, he cannot target effort at any principal, but he can still hope to earn payoff through an outcome that fits the other principal. Suppose agent 1 rejects the contract offer, if he is lucky to get an outcome that fits principal 2, then agent 2 will leave him half of the surplus, so agent1’s expected ∗ dq and this is the amount payoff after rejecting principal 1’s contract is U0 (k ∗ ) ≡ q k2d = 1−λ 2−λ of payoff the contract has to leave to an agent for the agent to accept given that the other agent accepts. Therefore, I have the following lemma: Lemma 2. The efficient effort is e∗ = (C 0 )−1 ((1 − λ)d). If both principals offer a piece rate 1−λ of k ∗ = 1− λ and a sufficiently high fixed payment, then in the subgame after these contract 2 offers, there exists a subgame equilibrium where both agents accept the contracts and exerts the efficient effort. The payoff of each principal under a pair of symmetric contracts that implements the 1−λ efficient efforts satisfies Π∗ ≤ (2q + (1 − λ)e∗ )d − C(e∗ ) − 2−λ dq. Let’s call the contract that gives the principal the above payoff the “optimal-efficient” contract, denoted by (k ∗ , F ∗ ). Notice that k ∗ is lower than the sell-out piece rate kˆ = 1. This is determined by two forces. First, the efficient effort level e∗ is lower than eˆ, so implementing the efficient effort level calls for a smaller piece rate. Second when one principal’s piece rate motivates effort, the other principal’s piece rate de-motivates effort, so given the same pair of common piece rate, the strength of the pair to motivate effort is lower. This calls for a higher piece rate. Since the Nash bargaining process dilutes the de-motivating effect, the ˆ second force is weaker, so overall k ∗ < k. The efficiency level under the efficient k ∗ is unambiguously higher than that in the no trading benchmark because 1) the effort level is more efficient and 2) there is efficient matching of outcomes to principals. The difference of the half of the total welfare, i.e., welfare per principal-agent pair, is: 12
ˆ = max{(q + e)d + (q − λe)d − C(e)} − max{(q + e)d − C(e)} > 0 W ∗ (λ) − W e
e
Despite this efficiency gain, the possibility of trading also raises the outside option of the agents, so the comparison of the profits of the principals depend on the comparison between ˆ ≥ W ∗ (1) − W ˆ =. ˆ and 1 qd. The former is decreasing in λ, so W ∗ (λ) − W W ∗ (λ) − W 2 qd The latter is decreasing in λ as well, and the maximum is 2 . Therefore, the “optimalefficient” contract under trading gives the principals a better payoff (though not a part of an equilibrium) than the no trading equilibrium contract.
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Main Model: allowing trading
4.1
Agent’s problem
When agent 1 diverts a result, in additional to what he would have got if he has not diverted it, he would receive a payment equal to the half of the surplus created by diverting due to the nature of Nash bargaining with equal bargaining power. The amount of surplus is proportional to the piece rate of principal 2. When agent 1 received a result, in additional to what he would have got if he has not received it, he receives a reward from his principal for a good fit, but he has to pay out to agent 2 an amount equal to the half of the surplus created, which is proportional to the piece rate of principal 1. When an agent gets a no-fit outcome, there is no change in the surplus and the agent’s payoff. Therefore, the objective functions are: Agent 1 solves: max
e1 ∈[0,q]
(q + e1 )dk1 + (q − λe1 )d
k1 k2 + (q − λe2 )d(1 − ) − C(e1 ) 2 2
Agent 2 solves: max
e2 ∈[0,q]
(q + e2 )dk2 + (q − λe2 )d
k1 k2 + (q − λe1 )d(1 − ) − C(e2 ) 2 2
Given two k1 and k2 , if the solutions (denoted by e˜1 and e˜2 ) are interior, then they are determined by the first order conditions: 13
k2 − C 0 (e1 ) = 0 2 k1 dk2 − λd − C 0 (e2 ) = 0 2
dk1 − λd
This easily gives us the partial derivatives of the subgame equilibrium efforts with respect to piece rates:
d ∂˜ ei = 00 >0 ∂ki C (˜ ei ) ∂˜ ei λd 1, this effect is negative because motivating more effort is payoff decreasing for the principal. At k1 = 1, this effect is second order due to the Envelope Theorem. Cross-motivating effect Third, there is an effect of k1 through agent 2’s effort, e˜2 . When k1 < 2, principal 1 wants a lower e˜2 to increase the chance of a fit with agent 2’ result. This pushes up k1 . When k1 > 2, principal 1 wants to discourage receiving result from agent 2 because too much commission (more than 100%) would be paid to agent 2 in that case. ∂˜ e2 ), which is first positive, then negative, The effect is represented by the term λd(1 − k21 )(− ∂k 1 as k1 increases, with a cutoff of k1 = 2. I first look at the necessary conditions for an equilibrium with symmetric piece-rates, and then compare the piece-rate and the corresponding effort levels to those of the benchmarks. Let k denote a value for the common piece rate, and let e denote the corresponding effort. Define a function f (k) as follows:
f (k) ≡
∂Π1 |k =k =k ∂k1 1 2
Applying a symmetric k to the agents’ problem, from the first order condition, I have:
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kd =
C 0 (e) 1 − λ2
Therefore, using (1), I have:
1 d k λd f (k) = − (q − λe)d + d(1 − k) 00 + λd(1 − ) 00 2 C (e) 2 2C (e)
(2)
If there is an symmetric equilibrium where the common piece-rate is positive, then the ˆ first order condition of the principal holds at k:
˙ =0 f (k) Lemma 3. If there exists a symmetric equilibrium with a positive effort, i.e., e˙ > 0, then it q . Also, it must is the only symmetric equilibrium with a positive effort and C 00d(0) > λ2 2(1+
be that k˙ ≤
2 1+ λ2 2 1+ λ4
2
)
ˆ Regardless of λ, e˙ > eˆ. Also, . Moreover, when λ is small enough, k˙ > k.
ˆ ∈ (0, 1) such that if λ < λ ˆ then e˙ is less than the efficient level, and there exists a cutoff λ ˆ then e˙ is higher than the efficient level. if λ > λ Proof. From the agent’s problem, e˙ > 0 implies k˙ > 0. First, I show that k˙ ≤ Suppose not, k˙ >
2
1+ λ2
2
1+ λ4
.
2
1+ λ2
2
1+ λ4
. Denote the corresponding effort level by e. ˙ Then,
d λ2 λ2 ˙ 1 ˙ = − 1 (q − λe)d f (k) ˙ + d 00 ((1 + ) − (1 + )k) < − (q − λe)d ˙ 2 C (e) ˙ 2 4 2 This forms a contradiction. 2
Next, I show that f (k) is strictly decreasing in k for k < 16
1+ λ2
2
1+ λ4
.
∂e λd λ2 λ2 d2 λ2 d2 000 [ − d((1 + ) − (1 + )k) 00 (1 + ) C (e)] − ∂k 2 2 4 (C (e))2 C 00 (e) 4 ∂e λ(1 − λ) − 2 λ2 λ2 C 000 (e) =d [ − d((1 + ) − (1 + )k) 00 ] 0, then I have f (0) > 0.
qd d2 λ2 d q + 00 (1 + ) > 0 ⇒ 00 > 2 C (0) 2 C 2(1 + λ2
1+ Also kˆ = 1 ∈ [0, λ22 ] and k∗ = 1+
4
2
1−λ 1− λ 2
∈ [0,
1+ λ2
2 1+ λ4
λ2 ) 2
], the comparison with kˆ and k ∗ is entirely
determined by the sign of f (k) evaluated at kˆ and k ∗ . 1. Comparison with piece-rate in the no trading benchmark. At k1 = k2 = kˆ = 1, the second term of the first order derivative condition in (2) disappears (i.e., the self-motivating effect is zero): λ 1 1 1 f (1) = − (q − λ(C 0 )−1 ((1 − )d)d + λ2 d2 00 0 −1 2 2 4 C ((C ) ((1 − λ2 )d)) Here one can see that there are two competing incentives: the leakage effect tries to pull the piece rate below one, and the cross-motivating effect tries pull the piece rate to be above one. When λ goes to zero, the cross motivating effect vanishes, but the leakage effect is still present because q > 0, so f (1) < 0 for small enough λ. 2. Comparison with effort in the no trading benchmark. From Lemma 3, I know that k˙ ≤
2
1+ λ2
2 1+ λ4
. To implement the same effort level as in the no
trading benchmark, now I need a higher piece rate kˇ = d(1 − λ2 )kˆ − C 0 (ˆ e) = 0. Therefore, kˇ >
2
1+ λ2
2
1+ λ4
17
1 , 1− λ 2
because kˆ =
1 1− λ 2
, which implies kˇ > k˙ and eˆ > e. ˙
implies
This shows that the equilibrium with agent interaction (if exists) lowers the equilibrium effort level from the overly-high level of no trading benchmark. 3. Comparison with piece rate and effort in the efficient benchmark. At k1 = k2 = k ∗ =
1−λ , 1− λ 2
the effort level are efficient, i.e., at e∗ :
1 λd d λd λd f (k ∗ ) = − (q − λe∗ )d + + 00 ∗ 2 2 − λ C (e ) 2 − λ 2C 00 (e∗ ) When λ is close to 0, f (k ∗ ) < 0, so k˙ < k ∗ and e˙ < e∗ . When λ is close to 1, e∗ → 0. Since e˙ > 0, I have e˙ > e∗ .
Next, I impose quadratic cost function to get existence, comparative statics and sharper comparisons with the benchmarks. Proposition 1. With quadratic cost function, if
d C 00
∈(
q 2 , q), 2(1+ λ2 )
then there exists a sym-
metric equilibrium that is the unique symmetric equilibrium and it has an interior effort e˙ ∈ (0, q). The equilibrium piece rate is increasing in λ. The equilibrium effort is increasing in λ when λ is small. Both the piece rate and the effort level are below those of the no trading ˆ e˙ < eˆ. benchmark, i.e., k˙ < k, Note that the result that e˙ < eˆ, is not dependant on the quadratic assumption. The intuition is that to implement an effort level as high as eˆ when agents are being de-motivated by other principals, both principals have to offer a very high piece rate: kˇ = 1−1 λ . At such 2 a high piece rate, each principal has the incentive to deviate to a lower piece rate, because the negative self-motivating effect alone dominates the positive cross-motivating effect. The self motivating effect is negative because kˇ > 1. In addition to that, there is the negative leakage effect pulling down the piece rate:
2
1 + λ4 d2 λ2 ˇ = − 1 (q − λˆ f (k) e)d + 00 ((1 + ) − ) 0. Since
d C 00
< q, k˙
0,
Therefore, e˙ ∈ (0, q). Also, there is a unique
2
solution to f (k) = 0 within k ∈ (0,
1+ λ2
2
1+ λ4
).
The assumption that the cost function is quadratic ensures that the objective function 2 for a principal given another principal’s positive piece rate is quadratic. Notice that ∂∂ke˜2i = 0 for i = 1, 2. Also,
∂˜ e2 ∂k1
< 0 and
∂˜ e1 ∂k1
1
> 0, therefore:
∂ 2 Π1 λd ∂˜ e2 ∂˜ e1 λd ∂˜ e2 = −d − (− ) 0.
˙ This is, the higher λ is, the higher is the equilibrium effort level k. From the first order condition,
e˙ =
d (1 C 00
+
λ2 ) 2
−
2
1+ λ4
1− λ 2
−
λ 2
q 2
=
d (1 C 00
+
1+
λ2 ) − 2q 2 λ2 2−λ
Take derivative with respect to λ. Let ∼ denote the same sign. 20
∂ e˙ d λ(4 − λ) d q λ2 λ2 ∼ 00 (1 + )− )− ) ( (1 + 2 00 ∂λ C 2−λ (2 − λ) C 2 2 λ d = ( 00 λ(6 − 2λ + λ2 ) + q(4 − λ)) 2 2(2 − λ) C d ∼ 00 λ(−6 + 2λ − λ2 ) + q(4 − λ) C When Cd00 is close to 2q from above, the expression is close to q(λ(−3 + λ − 12 λ2 ) + 4 − λ)], which is positive for any λ ∈ [0, 1], so the effort is increasing in λ. When Cd00 is close to q from below, the expression is close to q(λ(−6 + 2λ − λ2 ) + 4 − λ), which is positive for small λ, but negative for large λ, so the effort is first increasing in λ and then decreasing in λ. Comparison with benchmarks Lemma 3 already shows that the comparison of k˙ and kˆ depends on the sign of f (k) ˆ It I impose quadratic cost function, I get: evaluated at k. d λ 1 1 1 f (1) = − (q − λ 00 ((1 − ))d + λ2 d2 00 2 C 2 4 C 1 d λ2 d λ2 ∼ − (q − 00 (λ − )) + 00 2 C 2 C 4 d 1 = (−q + 00 ) 2 C e∗ = (1 − λ) Ce00 , since e˙ < eˆ, efficiency is always higher under trading because e˙ is in between eˆ and e∗ and the efficiency level of effort is quadratic in effort. Moreover, (1 − λ) 2Cd 00 ≤ e∗ , since φ(e) ˙ is decreasing for e˙ > (1 − λ) 2Cd 00 , φ(e) ˙ > 0 for 23
e˙ ≥ (1−λ) 2Cd 00 . The lowest value of e˙ is achieved by λ = 0, at which e˙ = Cd00 − 2q . Since φ(e) ˙ is q d 1 1 d2 d C 00 2 increasing for e˙ < (1−λ) 2C 00 , for this range of e, ˙ φ(e) ˙ ≥ φ( C 00 − 2 )|λ=0 = 2 qd+ 4 C 00 − 4 q > 0. q The inequity is due to Cd00 ≥ . λ2 2(1+
2
)
˙ >W ˆ. This implies W Case 2. e˙ = 0. ˙ −W ˆ = qd − (ˆ Then W ed − C(ˆ e)) = d(q −
d ) 2C 00
> 0.
Even though trading creates more total surplus, the principal also has to leave the agents a bigger share because of their better outside options under trading, so these two countervailing effects determines the comparison of payoffs with the no trading benchmark. Corollary 2. (Payoff Comparison) With quadratic cost function, the principal’s payoff is ˙ > Π. ˆ higher under trading than when trading is not allowed, i.e., Π Proof. The difference in the payoff equals to the difference in the efficiency level plus the difference in outside option: qC 00 d ˙ = (q − λe)d ˙ −Π ˆ = (W ˙ −W ˆ ) − U0 (k) ˙ 2− e˙ Π ˙ − C 00 ( 00 − e) 2C 2 − λ d 1 d 2 d q 00 2 = C (q 00 − ( 00 ) − (e) ˙ + (1 − λ) 00 − e) ˙ C 4 C C 2−λ 2 Define φ(λ) ≡ q Cd00 − 41 ( Cd00 )2 − (e(λ)) ˙ + (1 − λ) Cd00 − determines the comparison.
q 2−λ
e(λ). ˙ The sign of this function
Case 1. e˙ > 0. This implies Cd00 ∈ ( 2q , q), then φ(λ) is increasing and then decreasing in λ. Next I show that φ(0) > 0 and φ(1) > 0.
φ(0) =
d 1 d (q − )>0 00 C 4 C 00
5 d 13 d 3 d d φ(1) = q 00 − ( 00 )2 + q 2 ∼ (10q − 3 00 )(q − 00 ) > 0 8 C 16 C 16 C C 24
Therefore, φ(λ) > 0. Case 2. e˙ = 0. ˙ −W ˆ − 0 = d(q − Then W
d ) 2C 00
> 0.
˙ > Π. ˆ Therefore, Π
5
Discussion
5.1
Knowing the source of the outcome
If a principal knows the source of the outcome and it is verifiable and contractible, then she will pay positive but arbitrarily small piece rate for an outcome diverted in from the outside to minimize the leakage. Assuming that the agent still diverts the project to its bestfitting principal even when the gain of doing that is zero (i.e., when they are indifferent), then there exist an equilibrium where both principals offer Fˆ , kˆ when there is trading, i.e. the same contract as in the no trading benchmark. However, because there is trading the efficiency level is higher. Also, since now the agent’s outside option under trading is zero, the principal’s payoff is higher under trading as well because all the efficiency gain goes to the principals.
5.2
The restriction of one agent to contract with
In the paper, I made the following assumption. Assumption: each principal can only contract with one agent. This assumption is important for the comparison with the fully efficient benchmark. To see that, I consider a model where I relax this assumption. Let each principal simultaneously offer one contract to each agent and let the contract offer also specify that it is void unless the other agent also accept the contract offered by the other principal (such as in Bernheim and Whinston (1985)). I maintain the assumption that the source of the outcome is unknown to make the problem non-trivial.9 9
If the source of the outcome is known then there is an easy equilibrium that achieves efficient, where
25
I can denote a contract offered to agent 1 by principal 1 as (k1 , F1 ), one offered to agent 2 by principal 1 as (k˜2 , F˜2 ), one offered to agent 1 by principal 2 as (k˜1 , F˜1 ), and one offered to agent 2 by principal 2 as (k2 , F2 ). Recall that e∗ denotes the efficient effort level. Proposition 3. There exists a efficient symmetric equilibrium, where each principal pays 1 λ piece-rate k1 = k2 = 1+λ to her own agent and piece-rate k˜1 = k˜2 = 1+λ to the other agent. The equilibrium effort level is e∗ . The proof is in the Appendix. Here principal 1 not only pays agent 1 more when she gets a good fit, she also pays agent 2 more, not knowing who created the good fit. Efficiency is achieved because the principal internalizes both agents’ payoff. Then, when each principal is maximizing by picking piecerate, she is maximizing the total payoff of four players. Notice that k1 + k˜2 = 1, so each principal is effectively selling out the whole project to the agents, not to only one agents, but to two agents together. Despite the desirability of such an equilibrium, there are many real world applications where it is costly for the principals to offer contracts to many agents. As discussed in the Introduction, a model where principals can contract with as many agents as she wants can be viewed as the result of a more general model that explicitly model the contracting cost, when the cost of contracting with more agents is low enough; and a model where principals can only contract with one agent can be viewed as the result of this same more general model when the contracting cost is high enough.
5.3
Un-targeted effort
The paper modeled targeted effort, efforts that is useful to only one principal. Similar analysis can be applied to efforts that are “un-targeted”, i.e., efforts that are valuable to both principals. One example of such effort is efforts that just increase the vertical value of the outcome. The “leakage” problem is still there and it depresses the piece-rates and the efforts compared to the no trading benchmark. Since such “un-targeted” effort does not have a socially wasteful component, the no trading benchmark gives the efficient level of effort. Therefore introducing inter-agent interaction lowers the equilibrium level of effort from the efficient level, and is thus welfare-decreasing. each principal commits to pay the sell-out commission to the source.
26
5.4
Agents can choose whether to exert targeted or un-targeted effort
The interesting parameter case is when the productivity of targeted effort is higher than the productivity of un-targeted effort, but not twice as high in a two-principal-two-agent setup. In the no trading benchmark, the agents are only focused on targeted effort. Once trading is allowed, the other principal’s piece-rate dampens the incentive to exert targeted effort and the agent become focused only on un-targeted effort, which is welfare-improving However, the leakage effect causes the level of un-targeted effort to be below the fully efficient level.
5.5
More than two principals
The extension to n principals and n agents with n > 2 is straightforward if we maintain that each principal can only contract with one agent. The model is symmetric, so what I describe for principal 1 and agent 1 applies to all other agents with the appropriate substitution of index. Suppose N principals have different taste, so if agent 1 exerts effort in the direction 1 and of principal 1, the effect of his effort is the following, with the assumption that λ < n−1 1 10 q < n+1 . Probability “1-fit” q + e1 “i-fit” with i 6= 1 q − λe1 “no-fit” (1 − nq) − (1 − (n − 1)λ)e1
Value to principal 1 0 d 0
Value to principals i with i 6= 1 d 0 0
Given such a symmetric setup, the analysis of n principals and n agents is almost a word to word replica of the case n = 2. If agent 1 exerts effort e1 in the direction of principal 1, then his result has probability (q + e1 ) to fit principal 1 and probability (q − λe1 ) to fit any other principals. Agent 1’s problem, given contracts (k1 , k2 , . . . , kn ), becomes:
max (q + e1 )k1 d + ((n − 1)q − λΣn2 ei ) e1
Σn ki k1 d + (q − λe1 ) 2 d − C(e1 ) 2 2
10
It is arguable that when the number of principals and agents increase, the gap between a good fit and a bad fit, d, should shrink as well. Since the magnitude of d does not play a role in the analysis, any assumption regarding d can be easily accommodated by the model.
27
The effect of a piece-rate on the subgame equilibrium effort level is: ∂˜ e1 d = 00 ∂k1 C (e1 ) ∂˜ ei λd =− for all i 6= 1 ∂k1 2C 00 (ei ) Therefore, the principal 1 maximizes the following payoff:
U1 = (q + e˜1 )d + ((n − 1)q − λΣn2 e˜i )(d −
k1 Σn ki d) + (q − λ˜ e1 ) 2 d − C(˜ e1 ) 2 2
∂˜ e1 1 k1 ∂ e˜i ∂U1 = − ((n − 1)q − λΣn2 e˜i )d + (1 − k1 )d + (1 − )λd(−Σn2 ) ∂k1 2 ∂k1 2 ∂k1 At a symmetric equilibrium, the first order derivative is:
U1 d k λd 1 f (k) ≡ + (n − 1)(1 − )λd 00 = − (n − 1)(q − e)d + (1 − k)d 00 k1 k1 =k2 =...=kn =k 2 C (e) 2 2C (e) with
kd 2
= C 0 (e).
Here one can see that the impact of looking at n > 2 instead of 2 is that both the leakage effect and the cross-motivating effect are stronger here. The more other principals and agents there are, the more likely it is for the principal to receive a result from outside. The analysis is essentially the same as for the case n = 2. Therefore, this model has the following versions of Proposition 1 and 2 under quadratic cost function: Proposition 4. If
d C 00
∈(
q 2 , q), 1 2( n−1 + λ2 )
then there exists a symmetric equilibrium that is the
unique symmetric equilibrium and it has an interior effort e˙ ∈ (0, q). The equilibrium piece rate is increasing in λ. The equilibrium effort is increasing in λ when λ is small. Both the ˆ e˙ < eˆ. piece rate and the effort level are below those of the no trading benchmark, i.e., k˙ < k, Proposition 5. If Cd00 ≤ (n − 1) 2q , there exists a symmetric equilibrium with kˆ = 0. If there exists a symmetric equilibrium with kˆ = 0, then Cd00 ≤ (n − 1) 2q . 28
5.6
Bargaining with unequal bargaining power
The agents do not have to have equal bargaining power. As long as the contributor of the result gets strictly positive bargaining power, the results of the model go through. To keep the model symmetric, one can specify that the agent who diverts a result gets r share of the inter-agent surplus and the agent who receives it and delivers it to his principal gets 1 − r, with r ∈ (0, 1). The major expressions become:
Π1 = (q + e˜1 )d + (q − λ˜ e1 )rk2 + (q − λ˜ e2 )d(1 − rk1 ) − C(˜ e1 )
f (k) = −(q − λe)dr + d(1 − k)
λrd d + λd(1 − rk) 00 00 C C
Therefore,
f (0) = −qdr +
d2 λ2 rd2 d qr + > 0 ⇒ 00 > 00 00 C C C 1 + λ2 r
1+λ2 r2
(1 − λ)r + 1−λr C 00 q d 1 + λ2 r2 d 2 f( ) = −q(1 − λ)r + 00 (1 + λ r) − q < 0 ⇒ 00 < q d(1 − λr) C 1 − λr C 1 + λ2 r 2 2
and
r (1−λ)r+ 1+λ 1−λr 2 1+λ r
> 1.
ˇ = −(q − λˆ f (k) e)dr +
e˙ =
d (1 + λ2 r) − C 00 1+λ2 r2 − λr 1−λr
d2 1 + λ2 r 2 2 ((1 + λ r) − )
October 17, 2010
Abstract Commission sharing enables agents to trade the outcomes of their efforts. One incentive to do that is to gain from a better match of the outcomes to the principals. The possibility of such interaction affect the agents’ incentive to exert effort and also the principals’ choice of contracts. I exhibit that there are three forces that determine the choices of the commissions by the principals in response to this possibility of interagent interaction. One is the usual incentive to use the commission to motivate effort in one’s own agent, one is the incentive to use the commission to influence other agents who can potentially trade with one’s own agent, and the other is the incentive to use commission to influence the transfers among the agents. I study the change in the equilibrium outcomes, and show that the equilibrium effort under commission sharing would be lower, while the change in the commission is in general dependant on the degree of alignment between the principals’ taste. I also show that the efficiency gain from inter-agent interaction can also be due to a more efficient effort level. I demonstrate that whether principals can contract with multiple agents or not plays a role in the efficiency of the equilibrium. ∗
I thank participants of seminars at HKUST, CUHK, HKU, Sinica, Lingnan University, 2010 Midwest Theory Summer Conference, 2010 International IO Conference. I am grateful for helpful comments from Rongzhu Ke, Jin Li, Yuk-fai Fong, Johannes H¨ orner, Marco Ottaviani, and Wing Suen. All errors remain my own. I thank the support from Research Grants Council of Hong Kong. This project began when I was affiliated with Cornerstone Research.
1
1
Introduction
Principals typically hire a limited number of agents. Even when a principal hires only one agent, it is not hard to see that there are interactions among agents that cause a principal to “indirectly” hire other agents. For example, real estate agents typically share commissions, so if one agent cannot find a suitable outcome for his own principal, he could get an outcome from another agent and share the commissions with that agent. Similar practices happen among headhunters and subcontractors. When principals are “connected” in such a way, questions arise regarding what externality the interaction creates among the principals, how such interaction affect the equilibrium efforts by the agents, the principals’ contracts, payoffs and the overall efficiency. In this paper, I present a rather specific model for understanding the phenomenon of inter-agent interaction. In particular, I consider the case when principals are heterogenous in their taste and the fit between the outcomes created by the agents and the preferences of the principals is not entirely determined by the agents’ effort, but is subject to some uncertainty. The agents here not only have the task of exerting efforts, but also the task of “trading” with other agents, where they might choose to divert outcomes to other agents for a payment or to pay other agents to receive outcomes. These post-effort trading actions are typically hard to observe or to contract on, so I assume that the principal can only contract on the value of the outcome she receives. In the discussion, I also consider the contracting environment where the principals can partially contract on the post-effort trading actions by being able to identify the source of the results delivered. I study a symmetric two-principal-two-agent model that can be generalized to more than two principals and two agents. I model the effort of an agent as being able to increase the probability of a good fit with a principal. Naturally, when principals’ taste are different, effort targeted at one principal decreases the probability of this outcome being also a good fit for the other. To understand the externality introduced by the possibility of inter-agent interaction, our first benchmark is a model where the agents do not have the option to interact. I also solve for the fully efficient efforts given efficient trading as a second benchmark. Our comparisons with the benchmarks are all based on the symmetric equilibrium. A crucial parameter of the model is the level of alignment between the two principals’ taste. For example, two principals looking for a house to buy may have the following tastes. One wants a big traditional house and the other wants a small modern house. In this case, effort that is invested in looking for a good fit with one principal necessarily makes the outcome unlikely to fit the other. However, another possible situation is that one principal wants a big traditional house, while the other wants a small traditional house, then effort targeted at one might create a fit for the other as well, with a probability higher than in the 2
previous situation. The other extreme case is when both principals want a big traditional house each, then effort for one is equally valuable for the other. I parameterize this level of alignment between the principals and study its effect on equilibrium contract and payoffs. One may guess that once the agents can trade post-effort, they tend to exert less targeted effort because of the chance of serving the interest of another principal makes it less meaningful to target a particular taste. This conjecture would definitely be correct if the principals’ commission levels do not change when interaction among agents is introduced, however, it is not a priori clear whether the principals will increase or decrease their commissions/piece rates in response to the possibility of post-effort trading. In the absence of the possibility of trading, the optimal/equilibrium contract is simply a sell-out contract with 100% commission, which is just the piece rate in a two part contract. I show that there are two countervailing forces that determine how the principal’s choice of the commission would change once trading is allowed. First, the fact that the principal cannot directly contract with the other agent means that some of her commission will be leaked to the other agent through trading and this principal has no ability to extract it back through a participation constraint. This creates an incentive to lower the commission. Second, under trading each principal’s commission has an influence on the other agent and can de-motivate the other agent to exert effort targeted at the other principal. This creates an incentive to raise the commission. Therefore, whether the principals offer higher or lower commission compared to the no trading benchmark depends on the relative strength of these two forces. The strength of the first force decreases when the principals are less aligned in their preference, because in that case there is a smaller chance that an outcome will be diverted and a commission will be leaked outside. The strength of the second force however increases when the principals are less aligned in their preference, because each principal’s commission has a stronger de-motivating effect on the other agent. Therefore, in general the equilibrium commission can be higher or lower depending on the alignment of the principals’ preference. However, under quadratic cost function, the leakage effect is stronger than the cross-motivating effect and the commission is always lower under trading than under no trading regardless of the alignment of taste. Even though the direction of change for the commission is parameter-dependent in the above sense when I do not impose quadratic cost function, the effort level comparison is not parameter dependent. Nor is it cost function dependent. That is, the effort under trading is always lower than that under no trading. This is because, on top of the effects of trading on the choice of commission, each symmetric pair of commission becomes weaker in motivating efforts as they start to counteract each other in the agents’ incentives when trading is allowed. I also compare both the equilibrium effort under trading and under no trading with the fully efficient level of effort. The efficient level of effort (under efficient trading) is always 3
lower than the no trading equilibrium effort. This is because the efficient trading takes care of the matching of outcome to principals, and the effort is only valuable from a social point of view if it moves the probability from an outcome that fits nobody to an outcome that fits somebody, and thus targeting one principal at the expense of the other is a socially wasteful activity. Since the equilibrium effort under trading is lower than the equilibrium effort under no trading, trading allows the problem of over-investment in targeted effort to be mitigated. However, the equilibrium effort under trading can be overly low in the sense that it can fall below the efficient effort level. When this happens, the efficiency comparison depends on how important the efficiency gain from better matching is. I show that, with quadratic cost function for efforts, the gain from better matching dominates, so the efficiency is always improved when the agents can trade. The increase in surplus implies that if the principal has the ability to impose an exclusivity condition that bans interaction into the contract offered, then there exist an equilibrium where both do not choose to impose exclusivity. I also study the impact on the principals’ payoff, which is closely related to the efficiency comparison. This is because the gain in efficiency would accrue to the principals. However, while trading unambiguously improves efficiency under a quadratic cost function, trading also improves the agents’ outside and this hurts the principal. When agents cannot interact with each other, the principals can push the agents to an outside option equaling unemployment. However, now an agent who refuses a contract can also work indirectly if the other agent accepts a contract. I show that, with quadratic cost function, this effect based on the outside option is dominated and the principals’ equilibrium payoff is higher when there is agent-interaction. The legal literature has expressed concern that inter-agent interaction, such as Multiple Listing Service among the real estate brokers, makes it easy for agents to collude, because an agent can carry out punishment against another agent who has accepted a low commission.1 While this paper is far from offering a model specifically for the market of real estate agency, it can be viewed as pointing out a potentially important aspect of inter-agent interaction: it may improve efficiency over and above the gain from better matches, because it promotes a lower powered contract that leads to less socially wasteful targeted efforts. From a more theoretical point of view, this paper adds to the multiple principal and multiple agent literature by considering a setting where agents can interact directly among each other, while imposing the restriction that each principal can only offer contract to one agent.2 In this paper, since a principal only contracts with one agent and thus does not internalize the other agent/principal’s utility, the efficient outcome cannot be achieved even 1
See White (2006). See Bernheim and Whinston (1986), Bernheim and Whinston (1985), Prat and Rustichini (2003) for models where each principal can offer contract to multiple agents. 2
4
though the collusive outcome would be efficient. I also show that efficiency can be restored in this model if each principal can offer two contracts, one to each agent (shown in the section of Discussions), echoes the result in Segal (1999), which has shown that, when a principal can offer multiple bilateral public offers, if there is no externality on agents’ reservation utilities, the equilibrium is efficient. My assumption that each principal can only contract with one agent is a way of saying that it is very costly for a principal to contract with more than one agent. I didn’t explicitly model the cost of contracting, but one can imagine that agents are professionals who repeatedly perform similar tasks and are thus informed about what other agents are in the market, so they can “trade” with other agents at minimal additional cost after they have paid the fixed cost of getting to know the field. However, the principals are typically only in the market once and it is therefore not worthwhile for them to invest in learning about other principals or agents, so a principal simply does not have the information about the identities of other agents and principals and thus has no way to contracting with them directly. In the same spirit, I also assumed that a principal cannot condition her contract on the acceptance decision of the other agent (unlike in the common agency literature). The restriction of only one agent per principal of course seems extreme, but this is only way of restricting the number of agents that a principal can contract with to be less than the number of the agents whose effort is potentially useful in a twoprincipal-two-agent model. Viewed in such light, my result simply presents the case when contracting cost is high enough, while the multiple-principal-multiple-agent literature with full directions of contracting is simply the case when contracting cost is low enough, then it is not surprising that efficiency is affected in my model. There is also a literature that focuses on agents that are hired to do search. Lewis and Ottaviani (2008) studied a dynamic single-principal and single-agent model where the agent can gain private benefits over the cause of searching, and their focus was on the interaction of the agent’s incentives to exert search effort with agent’s incentives to report the private information the agent acquires during the search process. In comparison, this paper is static, and looks at the agents’ incentives to exert targeted search effort when they can trade according to their private interest. There were studies of referrals, another form of interaction among agents. See Garicano and Santos (2004) and Pauly (1979). “Referral” refers to the possibility that an agent can pass the client to another agent before exerting the effort. Garicano and Santos (2004) focused on matching opportunities with agents’ talent, rather than on matching created outcomes with principals’ taste. The rest of the paper is organized as follows. Section 2 presents the model. Next, section 3 presents the analysis of the two benchmarks. Section 4 shows the main model and contrasts its results with the benchmarks. Section 5 discusses the assumptions, and extends the model
5
in different directions, including a n-principal-n-agent model. Section 6 concludes.
2
Model
There are two principals, 1 and 2. Since each principal can only contract with one agent, without loss of generality, principal 1 can only contract with agent 1; and principal 2 can only contract with agent 2. I will refer to an agent as him and a principal as her. The principals differ in their taste, so there are only three possibilities for the outcome an agent can create. If it fits principal 1, then I call it a “1-fit”. If it fits principal 2, I call it a “2-fit”. The last possibility is when it fits neither of the principals, in which case I call it a “no-fit”. After being hired by a principal, the agent learns about the specific taste of the principal and can exert targeted effort ei ≥ 0. In the following table, I show the probabilities of agent 1 getting each types of the outcome as a function of agent 1’s targeted effort e1 and the corresponding value of the outcome to each principal. Probability “1-fit” q + e1 = q + λe1 + (1 − λ)e1 “2-fit” q − λe1 “no-fit” (1 − 2q) − (1 − λ)e1
Value to principal 1 d 0 0
Value to principal 2 0 d 0
The assumption is d > 0, q ∈ (0, 31 ), λ ∈ [0, 1]. This simple structure says the following. If agent 1 exerts no effort, then there is an equal chance of getting a 1-fit and a 2-fit. The effort of agent 1 moves probability from getting a 2-fit to getting a 1-fit by the amount of λe1 , and also moves probability from getting a no-fit to getting a 1-fit by the amount of (1 − λ)e1 . Such effort is called “targeted effort”, because it is targeting the taste of principal 1 specifically. 3 The parameter λ is a measurement of the alignment of taste between the two principals. If the two principals’ taste are quite aligned, then effort targeted at creating a good fit for one does not diminish the chance of getting a good fit for the other by much. On the other hand, if the two principals’ taste are very opposite, then effort targeted at one principal will lead to very low probability of this outcome that fitting the other. One can refer to the 3
Naturally, there is another distinct type of effort where the effort increase the value of the outcome to both principals. This can be termed “un-targeted effort”. The main intuition with un-targeted effort would be the same as the targeted effort, while the implication for efficiency would be different, as “un-targeted effort” does not have a socially wasteful component.
6
Introduction section for examples of different levels of alignment of taste. This parameter λ also controls how socially wasteful the targeted effort is. Moving probability from a 1-fit to a 2-fit is socially wasteful, while moving the probability from a no-fit to a 1-fit is not wasteful. I maintain symmetry in our assumptions with respect to principal 1 and principal 2, therefore, the corresponding table for agent 2’s effort, e2 ≥ 0, is the following: Probability “2-fit” q + e2 = q + λe2 + (1 − λ)e2 “1-fit” q − λe2 “no-fit” (1 − 2q) − (1 − λ)e2
Value to principal 1 0 d 0
Value to principal 2 d 0 0
I assumed that efforts are non-negative, which means that each agent can only exert effort targeted to their own principals, that is, they cannot secretly exerting effort targeted at the other principal. This assumption is justified as follows. Even though an agent after being hired knows the taste of one’s own principal and knows how misaligned the two tastes are, there are many ways that taste can be misaligned and an agent is still not sure about the taste of the other principal and thus has no way of targeting it. Without this assumption, the analysis would be very similar, however, for some parameter case I will not have symmetric pure strategy equilibrium. In the same spirit, I assume that if an agent is not hired, he can only choose e = 0, because he does not know what taste to target. Each agent can only create one result, however, a principal can derive value from multiple units of results.4 I restrict the effort choice set by ei ≤ q so that the probabilities of each type of outcomes are well defined. The two agents have the same cost function of effort, C(e). For e > 0, I assume C 0 (e) > 0, C 00 (e) > 0, C 000 (e) ≥ 0, and C(0) = 0, C 0 (0) = 0, C 000 (e) ≥ 0 for any e ∈ [0, q]. I assume C 0 (q) > d, which ensures interior effort choices by the agents in the no trading benchmark. To get some results, I would further assume that C is quadratic, i.e., C 00 (e) is a constant and then C 0 (q) > d is equivalent to Cd00 < q. 4
This assumption removes the complication of multiple effort equilibria given a pair of contracts accepted by the agents (thus the need to arbitrarily select a subgame equilibrium), as what would happen in a coordination game. Suppose each principal values only one outcome. In this case, an agent will have a bigger chance of supplying an outcome to another principal if the other principal’s agent does not work hard to create a good fit. This gives him incentive not to work hard in creating a good fit with his own principal either. On the other hand, if the other agent works hard, this agent also wants to work hard because it is less likely that an outcome will be supplied to the other principal and a good fit with his own principal is more important.
7
After exerting effort and after observing the realization of the result, an agent can choose from one of the two actions: 1. Divert the result to the other agent, who will in turn deliver it to the other principal. 2. Not divert the result, i.e., deliver it to his own principal.
2.1
Timing and Information
The timing of the game is as follows. 1. Principals 1 and 2 simultaneously make take-it-or-leave-it public contract offers (to be specified later) to their respective agents. 2. Agents 1 and 2 simultaneously decide whether to accept the offers or not, after observing both contracts offered. 3. Agents 1 and 2, if hired, simultaneously exert efforts after observing the hiring outcomes. Efforts are not observable to others. 4. Project values are realized and are observed only by the agents that created them. 5. Agents decide on whether to divert or not divert the result; an agent who is not hired can only choose to divert the result to an agent that is hired. 6. A diverter and a intended receiver Nash Bargain with equal bargaining power to determine the transfer between them under common knowledge of the value of the results being diverted.5 7. Each principal pays the promised payments to her agent. The effort is not observable or contractible. The value of the outcome delivered is observable and contractible. I will assume that each principal cannot tell whether the outcome delivered to her is created by the agent himself or by another agent she does not contract 5
I allow the diverting and receiving agent to have different bargaining power in the Discussion. Alternatively, I can allow the agents Nash Bargain on a way to split the commission before they exert the effort, as the analysis would reach the same conclusion. The typical norm among real estate brokers and headhunters are to split any commission half-half.
8
with.6 Since the value to a principal is binary, I assume the contract has two parts: a piece rate for an outcome created ki ∈ [0, ∞), and a fixed fee Fi ∈ (−∞, ∞).7 To be realistic, I also assume that a principal cannot condition her contract on the acceptance decision of the other agent (unlike in the common agency literature), which is consistent with the spirit of the assumption that it is costly for a principal to contract with more than one agent, an assumption that is discussed in more detail in the Introduction.
2.2
Equilibrium concept
All players are risk-neutral and they maximize their expected payoff. An agent’s payoff is the payment received minus the cost of effort. A principal’s payoff is the value of the project received minus the total payment to the agent. I assume there is no discounting. If an agent is not paid and he does not exert any effort, then his payoff is zero. Principals’ strategies are simply the contracts they offer. The agents’ behavioral strategies are, 1) the acceptance decision given the two contracts offered, 2) effort choices given what contracts are accepted, and 3) the choice between diverting and not diverting given the realized outcomes. Despite that there is incomplete information in this model, since the principals only act once at the beginning of the game by offering contracts, their beliefs will not play a role. Also, when deciding whether to divert or not, the agents have all the information that can affect their payoffs. Therefore, like in any pure screening model, the set of Subgame Perfect Nash Equilibrium is identical to the sets of strategy profiles in Weak Perfect Bayesian Equilibrium or Sequential Equilibrium, so I will use the concept of Subgame Perfect Nash Equilibrium.
2.3
Benchmarks
I will consider the following two benchmarks 8 : 6
Holmstr¨ om and Milgrom (1988) made the similar assumption that when multiple principals contract with a single agent, each principal only observe the agent’s performance in their own relationship. I will relax this assumption in the Discussions. 7 Alternatively, I can allow the principal to observe which type of outcome it is, i.e., 1-fit, 2-fit or no-fit. The equilibrium payment for 2-fit and no-fit would be exactly the same, so it will boil down to a two part contract as well. The restriction of ki ≥ 0 is WLOG because ki < 0 is dominated. In the discussion, I will consider a different contracting environment when the principal can tell whether the outcome delivered to her is created by the agent himself or by another agent she does not contract with. 8 There is naturally a third benchmark: allowing trading but no fee-sharing, that is, I allow the agent to divert the result to the other agent, but does not allow them to make any transfers to each other. It turns out to be identical to the “no trading benchmark” in terms of contracts and efforts on the equilibrium (though potentially different efficiency implications depending on how agents break a tie), so I omit it here.
9
• No trading benchmark. This is where I exogenously shut-down all interactions among the agents. • Efficient benchmark. This is where I solve for socially optimal pair of efforts, assuming that the agents divert the results in a socially optimal manner. Then I solve for the pair of two-part contracts that can implement this efficient outcome when the agents are strategic. I will adopt the following notations for the symmetric equilibrium outcomes for the benchmarks and the main model:
No trading benchmark Efficient benchmark Allowing interaction
3
Effort eˆ e∗ e˙
Contract kˆ k∗ k˙
Welfare ˆ W W∗ ˙ W
Benchmarks
3.1
No trading benchmark
There is no connection between two principal-agent pairs, so I can just study each pair in isolation. The analysis is very standard: a sell-out contract is optimal for the principal. The contract implements the effort level that is efficient from the point of view of only one pair of principal and agent, which is the solution to:
max (q + e)d − C(e) ⇒ d − C 0 (ˆ e) = 0 e
Lemma 1. In the no trading benchmark, the unique equilibrium outcome is the following: kˆ = 1 and eˆ = (C 0 )−1 (d). ˆ = (q + eˆ)d − C(ˆ The equilibrium payoff of a principal is Π e).
3.2
Efficient benchmark
Under efficient trading, from a society’s point of view an outcome that fits principal 1 is just as good as an outcome that fits principal 2, so any effort exerted to move probability from 10
1-fit to 2-fit or vice versa is a social waste and the only socially desirable part of the effort is to move the probability from a no-fit outcome to one that fits one of the two principals, so the efficient effort level solves:
max (1 − λ)ed − C(e) ⇒ (1 − λ)d − C 0 (e∗ ) = 0 e
Therefore, e∗ = (C 0 )−1 ((1 − λ)d). This shows that in the no trading benchmark, efforts are overly high. The parameter λ controls how socially inefficient is the effort, so the higher λ is, the lower is the efficient level of effort. Next, I solve for the symmetric pair of contracts that can implement this efficient level of effort given strategic agents if they are accepted. First, observe that if both principals offer the same contracts with strictly positive piece rate k, and they are both accepted, then the agents will trade efficiently given any realized results. When agent 1 diverts a result, in additional to what he would have got if he has not diverted it, he would receive a payment equal to the half of the surplus created by diverting. It is half because of the nature of Nash bargaining with equal bargaining power. The amount of surplus is proportional to the piece rate of principal 2. When agent 1 received a result, in additional to what he would have got if he has not received it, he gets a reward from his principal for a good fit, but he also has to pay out to agent 2 an amount equal to the half of the surplus created by agent 2’s diverting, which is proportional to the piece rate of principal 1. When an agent gets a no-fit, there is no change in the agent’s payoff as no surplus is created. Therefore, fixing a symmetric piece rate k, agent 1 maximizes the following objective function:
k k max (q + e1 )dk + (q − λe1 )d + (q − λe2 )d(1 − ) − C(e1 ) e1 2 2 The second term is the additional expected payoff created by agent 1 diverting out an outcome and the third term is the additional payoff created by agent 2 diverting in an outcome. Since C 0 (0) = 0 and C 0 (q) > d, for any k ∈ (0, 1−1 λ , the solution is interior and satisfies: 2
λ d(1 − )k − C 0 (e1 ) = 0 2 11
If k =
1−λ , 1− λ 2
then e1 = e∗ solves the problem because:
λ d(1 − )k − C 0 (e∗ ) = d(1 − λ) − C 0 (e∗ ) = 0 2 1−λ This implies k ∗ = 1− λ can implement the efficient effort level if agents can strategically 2 trade. Note that this is not an equilibrium piece rate because I didn’t allow the principals to be strategic in choosing the piece-rate, but this is an interesting benchmark piece rate.
To implement the efficient effort level, the fixed payment does not affect efficiency as long as it is high enough for both agents to accept. The optimal contracts for the principals keep the agent at the participation constraint. If an agent rejects a contract, he cannot target effort at any principal, but he can still hope to earn payoff through an outcome that fits the other principal. Suppose agent 1 rejects the contract offer, if he is lucky to get an outcome that fits principal 2, then agent 2 will leave him half of the surplus, so agent1’s expected ∗ dq and this is the amount payoff after rejecting principal 1’s contract is U0 (k ∗ ) ≡ q k2d = 1−λ 2−λ of payoff the contract has to leave to an agent for the agent to accept given that the other agent accepts. Therefore, I have the following lemma: Lemma 2. The efficient effort is e∗ = (C 0 )−1 ((1 − λ)d). If both principals offer a piece rate 1−λ of k ∗ = 1− λ and a sufficiently high fixed payment, then in the subgame after these contract 2 offers, there exists a subgame equilibrium where both agents accept the contracts and exerts the efficient effort. The payoff of each principal under a pair of symmetric contracts that implements the 1−λ efficient efforts satisfies Π∗ ≤ (2q + (1 − λ)e∗ )d − C(e∗ ) − 2−λ dq. Let’s call the contract that gives the principal the above payoff the “optimal-efficient” contract, denoted by (k ∗ , F ∗ ). Notice that k ∗ is lower than the sell-out piece rate kˆ = 1. This is determined by two forces. First, the efficient effort level e∗ is lower than eˆ, so implementing the efficient effort level calls for a smaller piece rate. Second when one principal’s piece rate motivates effort, the other principal’s piece rate de-motivates effort, so given the same pair of common piece rate, the strength of the pair to motivate effort is lower. This calls for a higher piece rate. Since the Nash bargaining process dilutes the de-motivating effect, the ˆ second force is weaker, so overall k ∗ < k. The efficiency level under the efficient k ∗ is unambiguously higher than that in the no trading benchmark because 1) the effort level is more efficient and 2) there is efficient matching of outcomes to principals. The difference of the half of the total welfare, i.e., welfare per principal-agent pair, is: 12
ˆ = max{(q + e)d + (q − λe)d − C(e)} − max{(q + e)d − C(e)} > 0 W ∗ (λ) − W e
e
Despite this efficiency gain, the possibility of trading also raises the outside option of the agents, so the comparison of the profits of the principals depend on the comparison between ˆ ≥ W ∗ (1) − W ˆ =. ˆ and 1 qd. The former is decreasing in λ, so W ∗ (λ) − W W ∗ (λ) − W 2 qd The latter is decreasing in λ as well, and the maximum is 2 . Therefore, the “optimalefficient” contract under trading gives the principals a better payoff (though not a part of an equilibrium) than the no trading equilibrium contract.
4
Main Model: allowing trading
4.1
Agent’s problem
When agent 1 diverts a result, in additional to what he would have got if he has not diverted it, he would receive a payment equal to the half of the surplus created by diverting due to the nature of Nash bargaining with equal bargaining power. The amount of surplus is proportional to the piece rate of principal 2. When agent 1 received a result, in additional to what he would have got if he has not received it, he receives a reward from his principal for a good fit, but he has to pay out to agent 2 an amount equal to the half of the surplus created, which is proportional to the piece rate of principal 1. When an agent gets a no-fit outcome, there is no change in the surplus and the agent’s payoff. Therefore, the objective functions are: Agent 1 solves: max
e1 ∈[0,q]
(q + e1 )dk1 + (q − λe1 )d
k1 k2 + (q − λe2 )d(1 − ) − C(e1 ) 2 2
Agent 2 solves: max
e2 ∈[0,q]
(q + e2 )dk2 + (q − λe2 )d
k1 k2 + (q − λe1 )d(1 − ) − C(e2 ) 2 2
Given two k1 and k2 , if the solutions (denoted by e˜1 and e˜2 ) are interior, then they are determined by the first order conditions: 13
k2 − C 0 (e1 ) = 0 2 k1 dk2 − λd − C 0 (e2 ) = 0 2
dk1 − λd
This easily gives us the partial derivatives of the subgame equilibrium efforts with respect to piece rates:
d ∂˜ ei = 00 >0 ∂ki C (˜ ei ) ∂˜ ei λd 1, this effect is negative because motivating more effort is payoff decreasing for the principal. At k1 = 1, this effect is second order due to the Envelope Theorem. Cross-motivating effect Third, there is an effect of k1 through agent 2’s effort, e˜2 . When k1 < 2, principal 1 wants a lower e˜2 to increase the chance of a fit with agent 2’ result. This pushes up k1 . When k1 > 2, principal 1 wants to discourage receiving result from agent 2 because too much commission (more than 100%) would be paid to agent 2 in that case. ∂˜ e2 ), which is first positive, then negative, The effect is represented by the term λd(1 − k21 )(− ∂k 1 as k1 increases, with a cutoff of k1 = 2. I first look at the necessary conditions for an equilibrium with symmetric piece-rates, and then compare the piece-rate and the corresponding effort levels to those of the benchmarks. Let k denote a value for the common piece rate, and let e denote the corresponding effort. Define a function f (k) as follows:
f (k) ≡
∂Π1 |k =k =k ∂k1 1 2
Applying a symmetric k to the agents’ problem, from the first order condition, I have:
15
kd =
C 0 (e) 1 − λ2
Therefore, using (1), I have:
1 d k λd f (k) = − (q − λe)d + d(1 − k) 00 + λd(1 − ) 00 2 C (e) 2 2C (e)
(2)
If there is an symmetric equilibrium where the common piece-rate is positive, then the ˆ first order condition of the principal holds at k:
˙ =0 f (k) Lemma 3. If there exists a symmetric equilibrium with a positive effort, i.e., e˙ > 0, then it q . Also, it must is the only symmetric equilibrium with a positive effort and C 00d(0) > λ2 2(1+
be that k˙ ≤
2 1+ λ2 2 1+ λ4
2
)
ˆ Regardless of λ, e˙ > eˆ. Also, . Moreover, when λ is small enough, k˙ > k.
ˆ ∈ (0, 1) such that if λ < λ ˆ then e˙ is less than the efficient level, and there exists a cutoff λ ˆ then e˙ is higher than the efficient level. if λ > λ Proof. From the agent’s problem, e˙ > 0 implies k˙ > 0. First, I show that k˙ ≤ Suppose not, k˙ >
2
1+ λ2
2
1+ λ4
.
2
1+ λ2
2
1+ λ4
. Denote the corresponding effort level by e. ˙ Then,
d λ2 λ2 ˙ 1 ˙ = − 1 (q − λe)d f (k) ˙ + d 00 ((1 + ) − (1 + )k) < − (q − λe)d ˙ 2 C (e) ˙ 2 4 2 This forms a contradiction. 2
Next, I show that f (k) is strictly decreasing in k for k < 16
1+ λ2
2
1+ λ4
.
∂e λd λ2 λ2 d2 λ2 d2 000 [ − d((1 + ) − (1 + )k) 00 (1 + ) C (e)] − ∂k 2 2 4 (C (e))2 C 00 (e) 4 ∂e λ(1 − λ) − 2 λ2 λ2 C 000 (e) =d [ − d((1 + ) − (1 + )k) 00 ] 0, then I have f (0) > 0.
qd d2 λ2 d q + 00 (1 + ) > 0 ⇒ 00 > 2 C (0) 2 C 2(1 + λ2
1+ Also kˆ = 1 ∈ [0, λ22 ] and k∗ = 1+
4
2
1−λ 1− λ 2
∈ [0,
1+ λ2
2 1+ λ4
λ2 ) 2
], the comparison with kˆ and k ∗ is entirely
determined by the sign of f (k) evaluated at kˆ and k ∗ . 1. Comparison with piece-rate in the no trading benchmark. At k1 = k2 = kˆ = 1, the second term of the first order derivative condition in (2) disappears (i.e., the self-motivating effect is zero): λ 1 1 1 f (1) = − (q − λ(C 0 )−1 ((1 − )d)d + λ2 d2 00 0 −1 2 2 4 C ((C ) ((1 − λ2 )d)) Here one can see that there are two competing incentives: the leakage effect tries to pull the piece rate below one, and the cross-motivating effect tries pull the piece rate to be above one. When λ goes to zero, the cross motivating effect vanishes, but the leakage effect is still present because q > 0, so f (1) < 0 for small enough λ. 2. Comparison with effort in the no trading benchmark. From Lemma 3, I know that k˙ ≤
2
1+ λ2
2 1+ λ4
. To implement the same effort level as in the no
trading benchmark, now I need a higher piece rate kˇ = d(1 − λ2 )kˆ − C 0 (ˆ e) = 0. Therefore, kˇ >
2
1+ λ2
2
1+ λ4
17
1 , 1− λ 2
because kˆ =
1 1− λ 2
, which implies kˇ > k˙ and eˆ > e. ˙
implies
This shows that the equilibrium with agent interaction (if exists) lowers the equilibrium effort level from the overly-high level of no trading benchmark. 3. Comparison with piece rate and effort in the efficient benchmark. At k1 = k2 = k ∗ =
1−λ , 1− λ 2
the effort level are efficient, i.e., at e∗ :
1 λd d λd λd f (k ∗ ) = − (q − λe∗ )d + + 00 ∗ 2 2 − λ C (e ) 2 − λ 2C 00 (e∗ ) When λ is close to 0, f (k ∗ ) < 0, so k˙ < k ∗ and e˙ < e∗ . When λ is close to 1, e∗ → 0. Since e˙ > 0, I have e˙ > e∗ .
Next, I impose quadratic cost function to get existence, comparative statics and sharper comparisons with the benchmarks. Proposition 1. With quadratic cost function, if
d C 00
∈(
q 2 , q), 2(1+ λ2 )
then there exists a sym-
metric equilibrium that is the unique symmetric equilibrium and it has an interior effort e˙ ∈ (0, q). The equilibrium piece rate is increasing in λ. The equilibrium effort is increasing in λ when λ is small. Both the piece rate and the effort level are below those of the no trading ˆ e˙ < eˆ. benchmark, i.e., k˙ < k, Note that the result that e˙ < eˆ, is not dependant on the quadratic assumption. The intuition is that to implement an effort level as high as eˆ when agents are being de-motivated by other principals, both principals have to offer a very high piece rate: kˇ = 1−1 λ . At such 2 a high piece rate, each principal has the incentive to deviate to a lower piece rate, because the negative self-motivating effect alone dominates the positive cross-motivating effect. The self motivating effect is negative because kˇ > 1. In addition to that, there is the negative leakage effect pulling down the piece rate:
2
1 + λ4 d2 λ2 ˇ = − 1 (q − λˆ f (k) e)d + 00 ((1 + ) − ) 0. Since
d C 00
< q, k˙
0,
Therefore, e˙ ∈ (0, q). Also, there is a unique
2
solution to f (k) = 0 within k ∈ (0,
1+ λ2
2
1+ λ4
).
The assumption that the cost function is quadratic ensures that the objective function 2 for a principal given another principal’s positive piece rate is quadratic. Notice that ∂∂ke˜2i = 0 for i = 1, 2. Also,
∂˜ e2 ∂k1
< 0 and
∂˜ e1 ∂k1
1
> 0, therefore:
∂ 2 Π1 λd ∂˜ e2 ∂˜ e1 λd ∂˜ e2 = −d − (− ) 0.
˙ This is, the higher λ is, the higher is the equilibrium effort level k. From the first order condition,
e˙ =
d (1 C 00
+
λ2 ) 2
−
2
1+ λ4
1− λ 2
−
λ 2
q 2
=
d (1 C 00
+
1+
λ2 ) − 2q 2 λ2 2−λ
Take derivative with respect to λ. Let ∼ denote the same sign. 20
∂ e˙ d λ(4 − λ) d q λ2 λ2 ∼ 00 (1 + )− )− ) ( (1 + 2 00 ∂λ C 2−λ (2 − λ) C 2 2 λ d = ( 00 λ(6 − 2λ + λ2 ) + q(4 − λ)) 2 2(2 − λ) C d ∼ 00 λ(−6 + 2λ − λ2 ) + q(4 − λ) C When Cd00 is close to 2q from above, the expression is close to q(λ(−3 + λ − 12 λ2 ) + 4 − λ)], which is positive for any λ ∈ [0, 1], so the effort is increasing in λ. When Cd00 is close to q from below, the expression is close to q(λ(−6 + 2λ − λ2 ) + 4 − λ), which is positive for small λ, but negative for large λ, so the effort is first increasing in λ and then decreasing in λ. Comparison with benchmarks Lemma 3 already shows that the comparison of k˙ and kˆ depends on the sign of f (k) ˆ It I impose quadratic cost function, I get: evaluated at k. d λ 1 1 1 f (1) = − (q − λ 00 ((1 − ))d + λ2 d2 00 2 C 2 4 C 1 d λ2 d λ2 ∼ − (q − 00 (λ − )) + 00 2 C 2 C 4 d 1 = (−q + 00 ) 2 C e∗ = (1 − λ) Ce00 , since e˙ < eˆ, efficiency is always higher under trading because e˙ is in between eˆ and e∗ and the efficiency level of effort is quadratic in effort. Moreover, (1 − λ) 2Cd 00 ≤ e∗ , since φ(e) ˙ is decreasing for e˙ > (1 − λ) 2Cd 00 , φ(e) ˙ > 0 for 23
e˙ ≥ (1−λ) 2Cd 00 . The lowest value of e˙ is achieved by λ = 0, at which e˙ = Cd00 − 2q . Since φ(e) ˙ is q d 1 1 d2 d C 00 2 increasing for e˙ < (1−λ) 2C 00 , for this range of e, ˙ φ(e) ˙ ≥ φ( C 00 − 2 )|λ=0 = 2 qd+ 4 C 00 − 4 q > 0. q The inequity is due to Cd00 ≥ . λ2 2(1+
2
)
˙ >W ˆ. This implies W Case 2. e˙ = 0. ˙ −W ˆ = qd − (ˆ Then W ed − C(ˆ e)) = d(q −
d ) 2C 00
> 0.
Even though trading creates more total surplus, the principal also has to leave the agents a bigger share because of their better outside options under trading, so these two countervailing effects determines the comparison of payoffs with the no trading benchmark. Corollary 2. (Payoff Comparison) With quadratic cost function, the principal’s payoff is ˙ > Π. ˆ higher under trading than when trading is not allowed, i.e., Π Proof. The difference in the payoff equals to the difference in the efficiency level plus the difference in outside option: qC 00 d ˙ = (q − λe)d ˙ −Π ˆ = (W ˙ −W ˆ ) − U0 (k) ˙ 2− e˙ Π ˙ − C 00 ( 00 − e) 2C 2 − λ d 1 d 2 d q 00 2 = C (q 00 − ( 00 ) − (e) ˙ + (1 − λ) 00 − e) ˙ C 4 C C 2−λ 2 Define φ(λ) ≡ q Cd00 − 41 ( Cd00 )2 − (e(λ)) ˙ + (1 − λ) Cd00 − determines the comparison.
q 2−λ
e(λ). ˙ The sign of this function
Case 1. e˙ > 0. This implies Cd00 ∈ ( 2q , q), then φ(λ) is increasing and then decreasing in λ. Next I show that φ(0) > 0 and φ(1) > 0.
φ(0) =
d 1 d (q − )>0 00 C 4 C 00
5 d 13 d 3 d d φ(1) = q 00 − ( 00 )2 + q 2 ∼ (10q − 3 00 )(q − 00 ) > 0 8 C 16 C 16 C C 24
Therefore, φ(λ) > 0. Case 2. e˙ = 0. ˙ −W ˆ − 0 = d(q − Then W
d ) 2C 00
> 0.
˙ > Π. ˆ Therefore, Π
5
Discussion
5.1
Knowing the source of the outcome
If a principal knows the source of the outcome and it is verifiable and contractible, then she will pay positive but arbitrarily small piece rate for an outcome diverted in from the outside to minimize the leakage. Assuming that the agent still diverts the project to its bestfitting principal even when the gain of doing that is zero (i.e., when they are indifferent), then there exist an equilibrium where both principals offer Fˆ , kˆ when there is trading, i.e. the same contract as in the no trading benchmark. However, because there is trading the efficiency level is higher. Also, since now the agent’s outside option under trading is zero, the principal’s payoff is higher under trading as well because all the efficiency gain goes to the principals.
5.2
The restriction of one agent to contract with
In the paper, I made the following assumption. Assumption: each principal can only contract with one agent. This assumption is important for the comparison with the fully efficient benchmark. To see that, I consider a model where I relax this assumption. Let each principal simultaneously offer one contract to each agent and let the contract offer also specify that it is void unless the other agent also accept the contract offered by the other principal (such as in Bernheim and Whinston (1985)). I maintain the assumption that the source of the outcome is unknown to make the problem non-trivial.9 9
If the source of the outcome is known then there is an easy equilibrium that achieves efficient, where
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I can denote a contract offered to agent 1 by principal 1 as (k1 , F1 ), one offered to agent 2 by principal 1 as (k˜2 , F˜2 ), one offered to agent 1 by principal 2 as (k˜1 , F˜1 ), and one offered to agent 2 by principal 2 as (k2 , F2 ). Recall that e∗ denotes the efficient effort level. Proposition 3. There exists a efficient symmetric equilibrium, where each principal pays 1 λ piece-rate k1 = k2 = 1+λ to her own agent and piece-rate k˜1 = k˜2 = 1+λ to the other agent. The equilibrium effort level is e∗ . The proof is in the Appendix. Here principal 1 not only pays agent 1 more when she gets a good fit, she also pays agent 2 more, not knowing who created the good fit. Efficiency is achieved because the principal internalizes both agents’ payoff. Then, when each principal is maximizing by picking piecerate, she is maximizing the total payoff of four players. Notice that k1 + k˜2 = 1, so each principal is effectively selling out the whole project to the agents, not to only one agents, but to two agents together. Despite the desirability of such an equilibrium, there are many real world applications where it is costly for the principals to offer contracts to many agents. As discussed in the Introduction, a model where principals can contract with as many agents as she wants can be viewed as the result of a more general model that explicitly model the contracting cost, when the cost of contracting with more agents is low enough; and a model where principals can only contract with one agent can be viewed as the result of this same more general model when the contracting cost is high enough.
5.3
Un-targeted effort
The paper modeled targeted effort, efforts that is useful to only one principal. Similar analysis can be applied to efforts that are “un-targeted”, i.e., efforts that are valuable to both principals. One example of such effort is efforts that just increase the vertical value of the outcome. The “leakage” problem is still there and it depresses the piece-rates and the efforts compared to the no trading benchmark. Since such “un-targeted” effort does not have a socially wasteful component, the no trading benchmark gives the efficient level of effort. Therefore introducing inter-agent interaction lowers the equilibrium level of effort from the efficient level, and is thus welfare-decreasing. each principal commits to pay the sell-out commission to the source.
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5.4
Agents can choose whether to exert targeted or un-targeted effort
The interesting parameter case is when the productivity of targeted effort is higher than the productivity of un-targeted effort, but not twice as high in a two-principal-two-agent setup. In the no trading benchmark, the agents are only focused on targeted effort. Once trading is allowed, the other principal’s piece-rate dampens the incentive to exert targeted effort and the agent become focused only on un-targeted effort, which is welfare-improving However, the leakage effect causes the level of un-targeted effort to be below the fully efficient level.
5.5
More than two principals
The extension to n principals and n agents with n > 2 is straightforward if we maintain that each principal can only contract with one agent. The model is symmetric, so what I describe for principal 1 and agent 1 applies to all other agents with the appropriate substitution of index. Suppose N principals have different taste, so if agent 1 exerts effort in the direction 1 and of principal 1, the effect of his effort is the following, with the assumption that λ < n−1 1 10 q < n+1 . Probability “1-fit” q + e1 “i-fit” with i 6= 1 q − λe1 “no-fit” (1 − nq) − (1 − (n − 1)λ)e1
Value to principal 1 0 d 0
Value to principals i with i 6= 1 d 0 0
Given such a symmetric setup, the analysis of n principals and n agents is almost a word to word replica of the case n = 2. If agent 1 exerts effort e1 in the direction of principal 1, then his result has probability (q + e1 ) to fit principal 1 and probability (q − λe1 ) to fit any other principals. Agent 1’s problem, given contracts (k1 , k2 , . . . , kn ), becomes:
max (q + e1 )k1 d + ((n − 1)q − λΣn2 ei ) e1
Σn ki k1 d + (q − λe1 ) 2 d − C(e1 ) 2 2
10
It is arguable that when the number of principals and agents increase, the gap between a good fit and a bad fit, d, should shrink as well. Since the magnitude of d does not play a role in the analysis, any assumption regarding d can be easily accommodated by the model.
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The effect of a piece-rate on the subgame equilibrium effort level is: ∂˜ e1 d = 00 ∂k1 C (e1 ) ∂˜ ei λd =− for all i 6= 1 ∂k1 2C 00 (ei ) Therefore, the principal 1 maximizes the following payoff:
U1 = (q + e˜1 )d + ((n − 1)q − λΣn2 e˜i )(d −
k1 Σn ki d) + (q − λ˜ e1 ) 2 d − C(˜ e1 ) 2 2
∂˜ e1 1 k1 ∂ e˜i ∂U1 = − ((n − 1)q − λΣn2 e˜i )d + (1 − k1 )d + (1 − )λd(−Σn2 ) ∂k1 2 ∂k1 2 ∂k1 At a symmetric equilibrium, the first order derivative is:
U1 d k λd 1 f (k) ≡ + (n − 1)(1 − )λd 00 = − (n − 1)(q − e)d + (1 − k)d 00 k1 k1 =k2 =...=kn =k 2 C (e) 2 2C (e) with
kd 2
= C 0 (e).
Here one can see that the impact of looking at n > 2 instead of 2 is that both the leakage effect and the cross-motivating effect are stronger here. The more other principals and agents there are, the more likely it is for the principal to receive a result from outside. The analysis is essentially the same as for the case n = 2. Therefore, this model has the following versions of Proposition 1 and 2 under quadratic cost function: Proposition 4. If
d C 00
∈(
q 2 , q), 1 2( n−1 + λ2 )
then there exists a symmetric equilibrium that is the
unique symmetric equilibrium and it has an interior effort e˙ ∈ (0, q). The equilibrium piece rate is increasing in λ. The equilibrium effort is increasing in λ when λ is small. Both the ˆ e˙ < eˆ. piece rate and the effort level are below those of the no trading benchmark, i.e., k˙ < k, Proposition 5. If Cd00 ≤ (n − 1) 2q , there exists a symmetric equilibrium with kˆ = 0. If there exists a symmetric equilibrium with kˆ = 0, then Cd00 ≤ (n − 1) 2q . 28
5.6
Bargaining with unequal bargaining power
The agents do not have to have equal bargaining power. As long as the contributor of the result gets strictly positive bargaining power, the results of the model go through. To keep the model symmetric, one can specify that the agent who diverts a result gets r share of the inter-agent surplus and the agent who receives it and delivers it to his principal gets 1 − r, with r ∈ (0, 1). The major expressions become:
Π1 = (q + e˜1 )d + (q − λ˜ e1 )rk2 + (q − λ˜ e2 )d(1 − rk1 ) − C(˜ e1 )
f (k) = −(q − λe)dr + d(1 − k)
λrd d + λd(1 − rk) 00 00 C C
Therefore,
f (0) = −qdr +
d2 λ2 rd2 d qr + > 0 ⇒ 00 > 00 00 C C C 1 + λ2 r
1+λ2 r2
(1 − λ)r + 1−λr C 00 q d 1 + λ2 r2 d 2 f( ) = −q(1 − λ)r + 00 (1 + λ r) − q < 0 ⇒ 00 < q d(1 − λr) C 1 − λr C 1 + λ2 r 2 2
and
r (1−λ)r+ 1+λ 1−λr 2 1+λ r
> 1.
ˇ = −(q − λˆ f (k) e)dr +
e˙ =
d (1 + λ2 r) − C 00 1+λ2 r2 − λr 1−λr
d2 1 + λ2 r 2 2 ((1 + λ r) − )
