Competitive Contracting and Employment Dynamics Dayanand Manoli and Yuliy Sannikov∗ April 26, 2005
Abstract How are contracts written in competitive settings? What effects does competition have on efficiency and the movement of agents between firms? To address these questions, we study optimal contracts and employment dynamics in a setting where multiple firms with different production technologies compete for identical agents. In our dynamic, continuous-time model, an agent’s hidden effort affects the expected output of his current employer. We fully characterize optimal contracts and derive properties of the allocation of workers between firms. In equilibrium the agent’s wages depend not only on the agent’s performance in the current firm, but also on his performance with previous employers, and his past outside opportunities. To some extent the agent carries his “continuation value” between firms in equilibrium, i.e. future employers take care of the agent’s past incentives. Relative to binding contracts with a single firm, some inefficiency appears in two forms: first, potential outside options reduce a firm’s ability to provide incentives and second, the agent is not always employed with the most productive firm.
1
Introduction.
We study dynamic employment contracts in an environment with competition between firms and informational frictions. We focus on the moral hazard problem. In our settings all agents have observable skills and all firms have observable production techologies. In particular, when a firm and an agent consider a potential employment, they know the quality of the match between them. However, the agent’s effort once employment starts is unobservable. The firm sees only the output, which stochastically depends on the agent’s effort. In our setting employment involves long-term interaction between a firm and an agent. As time advances the output unfolds gradually. Employment is complicated by the fact that ∗
Department of Economics, University of California at Berkeley. E-mail:
[email protected] and
[email protected] We are thankful to seminar participants at Caltech for helpful comments.
1
from time to time the agent encounters outside opportunities with other firms who would like to employ him. As a result, the competition between firms may produce complex employment dynamics. An outside employment opportunity can affect the agent in a number of ways. For example, the agent may quit the current firm to work for a competitor. Also, the current firm may improve the terms of the agent’s employment to retain the agent. We fully characterize optimal equilibrium contracts. Our characterization allows us to answer to many interesting questions. For example, consider the following questions, to which the naive tempting answer is “yes.” Is the agent always happy to encounter an outside opportunity? If the agent encounters a firm that is a better match, does he always move? If the agent encounters a firm that is a worse match, is it true that he never moves? Does the agent always end up with the most productive firm in the long run? Surprisingly, the correct answer to all of the above questions is “no.” The reasons lie in the optimal provision of incentives subject to the constraints of competition. Naturally, the competition between firms makes the provision of incentives more difficult. A firm cannot bind the agent for good. A firm’s ability to punish the agent after undesirable performance is limited, because the agent may quit when he encounters the first outside opportunity. Therefore, competition makes the total social surplus lower relative to the setting with a single firm and a single agent (see Phelan and Townsend (1991) or Sannikov (2004) for a single-firm setting). Yet, competition still leaves a lot of room for incentive provision. First, a firm can provide incentives by performance-dependent pay during the agent’s employment. Second, by its bidding strategy a firm can influence the terms of the agent’s employment with competitor firms when the agent encounters outside opportunities. If the agent performs well, the current firm will bid tough to match outside opportunities. If the agent performs badly, the current firm will fire the agent as soon as he encounters an outside opportunity, letting him accept the worst terms that an outside firm is willing to offer. As a result, the agent’s current wage reflects not only his recent performance with the current employer but also his performance with all past employers. In our model this is a result of dynamic moral hazard, not adverse selection. In our continuous-time setting the output is a diffusion process with the drift determined by the agent’s effort. The agent is compensated with a flow of wages from the firm. The agent consumes his wages immediately, that is, he cannot save or borrow in our model. The contract specifies how the payments depend on output, as well as the agent’s encounters with outside firms. We assume that the agent becomes matched with potential outside employers at Poisson arrival times. Whenever the agent encounters an outside employer, the current firm and the outsider bid for the agent. As in related models with a single firm, we find that an optimal contract is described in terms of a single state variable, the agent’s continuation value. The continuation value completely determines the agent’s pay at any moment of time. In a setting without competition, the evolution of the agent’s continuation value would be completely determined by the current output. Due to competition the agent’s continuation value also changes discontinuously every time when the agent encounters an outside opportunity. The contract is written in such a way that the firm does not need to remember the details of the agent’s 2
past performance or his outside offers. The firm only needs to keep track of the current continuation value. The same continuation value could arise purely due to the agent’s performance, or from a combination of performance and outside offers. Let us see some details about the effect of competition on the contracts and employment dynamics. When an agent encounters an outside opportunity, his decision whether to stay is determined by the terms of employment that the current employer and the competitor are willing to offer. In our setting, the competitor’s strategy is determined by two numbers: the minimum offer (measured in terms of the agent’s payoff) and the ceiling up to which the competitor is willing to bid. If the agent’s continuation value with the current employer is sufficiently high, then the current employer does not alter the terms of the agent’s employment, and the agent ignores the outside opportunity. This situation can happen even if the quality of the match between the agent and the outside employer would be better than with the current employer. For an intermediate range of the agent’s continuation values, in equilibrium the firms bid until the outsider’s ceiling. The agent moves if and only if the competitor firm is a better match. In this case, the current firm bids for the agent in order to reward him for past performance, even though it knows that the agent will leave. If the agent’s current employer is punishing him with a very low continuation value, it will always fire the agent an let him accept the minimum from a competitor. This happens even if the new firm provides an inferior match. The fact that the current employer does not bid is a part of an incentive mechanism. Interestingly, as a result of such an outside encounter, the agent may end up worse off than before the encounter. Our paper makes contributions in the areas of dynamic contracting, contracting with competition between firms and continuous-time contracts. Key papers in the literature on dynamic contracting in discrete time include work by Rogerson (1985), Spear and Srivastava (1987) and Phelan and Townsend (1991). Rogerson (1985) considers a multi-period, principal-agent setting and shows that in general, payments to the agent will depend on the entire history of output produced. Spear and Srivastava (1987) follow this work and show that complex history-dependence in optimal contracts has a simple representation using the agent’s continuation value as a state variable.1 Phelan and Townsend (1991) develop an algorithm to compute the form of such optimal dynamic contracts in discrete time. In our setting, agents’ payments exhibit similar history dependence in that an agent’s payments from an employing principal depends on the past output produced with that principal. We also use the agent’s continuation value as a state variable. In an optimal contract in our setting, the agent’s continuation value summarizes not only the agent’s past performance, but also the history of the agent’s past outside options. We develop a simple iterative algorithm to compute such optimal contracts by solving ODEs that characterize firms’ profits. Phelan (1995) presents a discrete-time setting with competition between identical principals to contract with identical agents. In his model, agents can break contracts at any time. As a result, equilibrium consists of principals offering identical contracts that give 1
Abreu, Pearce and Stacchetti (1986 and 1990) illustrate how players’ continuation values can be used as state variables in more general repeated games settings.
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zero profit in expectation. Two key elements of our environment are heterogeneity of principals and stochastic match times. We show that heterogeneous principals compete in equilibrium by offering different contracts. Further, because agents can only break contracts at stochastic times, firms may get strictly positive profits. These elements of our model lead to interesting employment dynamics in equilibrium. Continuous-time contracts have also been examined in the past. Holmstrom and Milgrom (1987) present a continuous-time, principal-agent setting in which the agent controls the drift of the principal’s output diffusion process. In their model, the agent receives payments after a fixed interval of time. In their setting they show that the agent’s payments depend on aggregate output produced up to the time of payment and additionally that the payments are linear in aggregate output. Williams (2004) presents a continuous-time, finite-horizon, principal-agent setting in which the agent controls the drift of the output process and receives continuous payments. He characterizes the optimal contract in terms of several state variables, a partial differential equation and stochastic differential equations. In our setting, agents control the drift of principals’ output processes and receive continuous payments in an infinite-time horizon. DeMarzo and Sannikov (2004) and Sannikov (2004) also characterize optimal contracts in such continuous-time settings. Our model of competition is based on the single-firm model of Sannikov (2004). In a single-firm setting, the optimal principal-agent contract uses the agent’s continuation value as a state variable, the principal’s payoff is computed by solving an ODE. We extend the stochastic calculus methodology developed by Sannikov (2004) to characterize optimal contracts in our competitive setting with multiple principals and agents. This methodology allows us to cleanly illustrate the intuition behind the dynamic contracts, easily compute optimal contracts, and characterize exciting employment dynamics. This paper is organized as follows. We present the model in Section 2. In Section 3, we describe the properties of our equilibrium and illustrate them on several computed examples. The primary purpose of Section 3 is exposition: to convey all the main intuition behind optimal contracts and employment dynamics. We leave the intensity of the formal derivation to Section 4. Section 5 summarizes the formal characterization of the equilibrium. Section 6 describes our computational techniques. Section 7 presents extensions, and Section 8 concludes.
2
The Model
First, let us paint a general picture of our setting. There are many firms and many agents ¯ according to a known in the market. The firms’ productivities θ are distributed on [0, θ] distribution Φ. When an agent works for one of the firms, with intensity λ he is randomly matched with another firm from distribution Φ. The new firm has an opportunity to steal the agent from the current employer. Every time this happens, the current employer observes the type of the competing firm and may revise the terms of the agent’s current employment. After that, the new firm has an opportunity to make an offer, and the agent
4
decides whether to accept it or not.2 According to our definition of equilibrium, the agent accepts the outside offer if his expected payoff from accepting it exceeds his expected payoff from staying with the current employer. If the inside and outside offers match, then the agent may accept either offer. An offer is a contract that specifies the agent’s payments as a function of the agent’s performance and his past outside opportunities. Now let us lay out the details. First, we explain the production technologies that the firms own. Second, we define the payoffs of each agent and each firm. Lastly, we explain the range of contracts that we allow and define strategies and a contracting equilibrium. At all times t from the beginning of employment until termination the agent puts effort at ∈ [0, a ¯] into the firm’s production. It is important to keep in mind that at is determined not only by the time t, but also the agent’s contract and the agent’s history within the firm. The agent’s effort affects the firm’s profits through the output process. The path of ¯ is given by output of a firm with productivity3 θ ∈ [0, θ] θ dXt = θ(at dt + dZt ) where at is the agent’s effort and Z is a standard Brownian motion. The firm observes output and but cannot observe the agent’s effort. For the duration of employment, the agent gets a flow of consumption payments ct ∈ [0, c¯] from the firm, which he consumes immediately. These history-dependent payments are specified by the contract. The agent enjoys consumption, dislikes effort and discounts utility at a rate r. If we normalize the time when employment starts to 0, the agent’s expected payoff is given by Z ∞ −rt W0 = E e (u(ct ) − v(at )) dt , 0
where consumption and effort after the termination time are associated with future employers. The function u : [0, c¯] → [0, ∞) with u0 > 0 and u00 < 0 captures the agent’s flow of utility from consumption and v : [0, a¯] → [0, ∞) with v 0 > 0 and v 00 > 0 captures the agent’s flow of disutility from effort. The firm has the same discount rate r, and its expected payoff from a given agent is given by Z τ −rt e (θ dXt − ct dt) , E 0
where τ denotes the termination time. 2
It may be more natural to think that the outsider moves first to make an offer to the agent, and then the current employer decides about a revised contract. Because in our setting the current employer’s reaction is contractually specified, we change the order of moves to avoid the situation where the current employer commits to a very aggressive reaction that deters all outside offers. We believe this situation to be unnatural. In a setting where the outsider moves first, perhaps such threats can be ruled out by a perturbation instead of changing the order of moves. 3 Alternatively, we can interpret θ as the quality of the match between the firm and the agent.
5
2.1
Contracts, Strategies and the Equilibrium.
We now define contracts, strategies, and the equilibrium in our competitive contracting environment. We allow fully history-dependent contracts, in which a history summarizes what the firm observed from the time when the contract was signed. Let us define contracts formally. For simplicity, we normalize the time when the agent begins employment with a given firm to 0. If the agent has not quit the firm until time t, an agent’s employment history specifies everything that the firm has observed from the beginning of employment until time t. That is, a history ht consists of the path of output produced at the employing firm (Xs )s∈[0,t] and a history of matches {(θ1 , τ1 ), . . . (θn , τn )} with outside firms. A history until time t collects all information that determines the payment from the principal to the agent at time t in a given contract. We assume that the firm does not observe the details of outside offers that the agent receives, so we do not include their characteristics in employment histories. Denote the set of all finite histories by H. A contract ψ specifies a payment flow that the agent receives ψ : H → [0, ∞) as a function of history. Technically, we require process ct = ψ(ht ) to be progressively measurable under any contract. Our definition of employment histories allows firms to specify how the terms of employment are revised when the agent encounters outside opportunities. Denote the set of contracts by Ψ, with a typical element ψ. We assume that a firm is able to commit to any contract. Note that after any history ht ∈ H the terms of employment that the agent is facing is also an element of Ψ. The continuation of contract ψ after history ht is defined by ψht (ht0 ) = ψ(ht ht0 ), where ht ht0 denotes a history until time t + t0 in which ht is followed by ht0 . We also refer to the continuation contract ψht as the current terms of employment. Next, we define a firm’s strategy. When an outside firm encounters an agent at time τ, we assume that it does not see the agent’s past employment history. However, the new firm can see the terms ψhτ that the agent is currently facing, where ψ denotes the contract that was signed originally. We define a strategy of a firm to specify an offer ψ 0 ∈ Ψ to the agent as a function of the agent’s current terms of employment. Formally, a strategy is a function σ : Ψ → Ψ. Finally, let us define an agent’s strategy. For simplicity, we define an agent’s strategy to be a pair of functions a : Ψ × H → A and b : Ψ × Ψ → {0, 1}. Function a(ψ, ht ) specifies an effort level that the agent takes when he is employed under contract ψ after history h t . Function b(ψ, ψ 0 ) specifies whether the agent rejects (0) or accepts (1) an outside offer ψ 0 when the current terms of employment are given by ψ. Note that we intentionally restrict an agent’s possible strategies. If we wanted to be more general, we would define the agent’s full histories to contain all the information about all of the agent’s all past employers. Then the agent’s strategy would specify effort and accept/reject decisions after all full histories. We believe that this level of generality would lead to the same result, but it would unnecessarily complicate the analysis to a great degree. Let us define a contracting equilibrium. Definition. Consider a vector of strategies [a, b], σ = (σθ )θ∈[0,θ]¯ , where σθ denotes 6
the strategy of a firm of type θ. This vector of strategies is a contracting equilibrium if 1. The agent’s strategy [a, b] is optimal in all situations, given the firms’ strategies, i.e. • For any contract ψ ∈ Ψ currently in place, the agent’s strategy maximizes his expected payoff given the strategies of all firms. • For any pair of contracts (ψ, ψ 0 ) between which the agent is deciding, his strategy maximizes his expected payoff given the strategies of all firms. 2. The strategy of each type of a firm θ is optimal in all situations given the strategies of the agent and all firms. Specifically, for any contract ψ currently in place with another firm, the strategy σθ maximizes the payoff of firm θ at the point when it offers the contract to the agent. These conditions have to hold even for those contracts ψ and ψ 0 that are never offered by any firm. In the next section we explain the phenomena that happen in a unique contracting equilibrium in this setting. We state many facts without a justification, and only focus on the most interesting conclusion. In particular, we omit many micro details about the form of history-dependence exhibited by the contracts. We focus on macro details about the form of the competition: what happens when the agent encounters a potential outside employer, how contracts are revised, what determines whether or not the agent moves. We also present several computed examples, and defer the formal derivation of the equilibrium until Section 4.
3
Contracting Equilibrium: A Discussion.
Dynamic contracts, heterogeneous firms and competition are exciting ingredients of our model. They let us address several important and interesting questions. What determines the attractiveness of a contract to the agent and the profitability of a contract? How do contracts motivate agents? How do firms compete? How are contracts designed to deal with competition? What effect does competition have on the movement of agents between the firms? How does competition affect the agents’ welfare? In this section, we illustrate the key features of competitive contracting and employment dynamics in our environment and present several computed examples to answer these questions. This section is meant to be expository while the next section formally justifies all the results. In this section we pay particular attention to the major features of contracts, competition and employment dynamics, without looking at the fine details of stochastic calculus. The two most important characteristics of any contract are its profitability to the firm and its attractiveness to the agent. These characteristics have been explored extensively both in discrete-time and in continuous-time literatures in a setting without competition. The level of attractiveness of a contract to the agent is measured by his future expected 7
payoff from a contract, a.k.a. continuation value4 denoted by W. When the agent is hired, it is optimal for a firm to choose the most profitable contracts among all those that deliver to the agent the same continuation value. Figure 1 shows the firm’s profit from the best contract that delivers any value to the agent in a setting with a single firm. The lowest value that any contract can have is 0, because the agent can guarantee the utility of 0 by taking zero effort. The profit function is concave and peaks at a value R > 0. Above R, the profit function decreases indefinitely. Profit F 0
Value, W
M F0
PSfrag replacements
Figure 1: The profit function with a single firm. Let us discuss how contracts provide incentives, why the profit function has this shape and how we expect this function to change with firm heterogeneity and with competition. A profitable contract must motivate the agent to put effort, which is unobservable. The incentives to put effort are created by linking the agent’s continuation value, and thus payments, with output and hence indirectly to effort. The firm’s profit function peaks at a strictly positive value because successful incentive provision requires both rewards and punishments. In a setting with a single firm, the only way to deliver to the agent value 0 is with payments of 0 at all times, which leave no room for incentives. Competition generally hurts the firm’s profits, but the basic form of the profit function remains concave. The profits tend to decrease with competition because the availability of outside options makes it more difficult for a firm to punish the agent.5 In our model λ = 0 corresponds to the absence of competition and the degree of competition is increasing with 4
In our setting any contract has a unique continuation value because all agents are identical, i.e. they do not have hidden wealth, hidden skills, or any other hidden information. 5 In our model competition may improve the profit of the low-productivity firms for a range of values because it allows them to reward the agents by affecting the terms of their employment when they go to the high-productivity firms.
8
λ. The left panel of Figure 2 shows how the profit function changes with λ. Note that the minimal value that a firm can deliver to the agent increases with competition. This value is delivered by “firing” the agent: making 0 payments until he has an opportunity to switch to a different firm. With larger λ the agent encounters this outside opportunity sooner. The minimal value that any firm can deliver is independent of the type of the firm, but more productive firms earn higher profits, as shown in the right panel of Figure 2.
g replacements Profit
Profit R(θH ) Fθ R(θL ) λ=0
0
λ = .5
Fθ
L
H
E(θH ) E(θL )
F0
λ=1
¯ H) R(θ ¯ L) R(θ
Value, W
0 λ = 10
Figure 2: Profit with competition for different values of λ and θ. Let us discuss what happens when an agent working for an incumbent firm encounters a competitor of type θ. The situation of the competitor is more straightforward than that of the current employer. The strategy of the competitor can be deduced from the form of its profit function. Two points on the profit function of the competitor have special significance: point M (θ) that maximizes the firm’s profit, and point E(θ) > M (θ) at which the firm breaks even (see the right panel of Figure 2) Ideally the competitor would like to hire a worker at M (θ). This value could entice a worker if his value with his employer is low. However, if necessary the competitor is willing to bid more aggressively for an agent. E(θ) is the most aggressive bid that type θ will make. Any higher bid leaves the firm with negative profit if it attracts the agent and any lower bid leaves the firm with 0 profit if it is outbid for the worker. Thus, for a type θ firm, M (θ) and E(θ) exactly capture the bidding strategy to hire a worker from another firm. The situation of the current employer is much more complicated. In our setting the reaction of the current employer, which is contractually specified, must take into account not 9
Value, W
only profitability, but also past incentives and past performance. To understand incentives in greater detail, let us take a detour to discuss the structure of optimal contracts. As in a setting with a single firm, optimal contracts with competition are recursive in the agent’s continuation value, which uniquely determines the agent’s payments. During the course of employment the firm adjusts the agent’s continuation value depending on performance to provide incentives, while keeping track of the value that the agent has consumed and the value that the firm owes. Optimal contracts are recursive because after any history, it is best for the firm to deliver the value that it owes to the agent in a unique manner that maximizes the firm’s profits. Therefore, the continuation value uniquely determines the continuation of a contract after any history. With competition, the continuation value takes into account not only the payments and effort during current employment, but also the value of future employment opportunities. Therefore, the reaction of the current employer to competition must depend on the agent’s current value. When the competitor is of type θ, the current employer has the following options: to fire the agent and let him get value M (θ), to bid for the agent and let him accept any value in the interval (M (θ), E(θ)], or to retain the agent with value E(θ) or higher. Not surprisingly, in equilibrium the first option is used with the agent’s continuation value is low (typically when the principal’s profit function is increasing in the agent’s value). The last option is used when the agent’s continuation value is very high. If the current firm bids for the agent and lets him accept an outside offer (the middle option), it is optimal to bid all the way to E(θ), because bidding is a costless way to deliver value to the agent. Whether or not the middle option is used depends on the relative productivities of the current employer and the outsider. If the current firm is more productive than the outsider, it will retain the agent because it can make positive profit at value E(θ). Mechanically, the current employer “bids” by revising the agent’s continuation value to a new level. To see more details, Figure 3 illustrates the responses of a current employer to competititors, depending on whether the current employer or the competitor is more productive. Let us describe how the reaction of the current employer is determined geometrically. When a competitor enters the picture, the profit function of the current employer collapses: values below M (θ) become unattainable and the profit on the interval [M (θ), E(θ)) becomes 0. Geometrically, if the agent’s value W and the firm’s profit F before the competitor’s appearance maximized λW + F for some λ ∈ t. Then Vt is a martingale with respect to the filtration F X generated by X, because Z ∞ −rt X Vt = E e (u(ct ) − v(at )) dt | Ft 0
By the martingale representation theorem, there is a process β ψ such that e(r+λ)t dVt = βtψ (dXt − aψt dt) 13
(3)
Differentiating (2) with respect to t we obtain ˜ t dt + (u(cψt ) − v(aψt )) dt − (r + λ)Wt dt + dWt e(r+λ)t dVt = λW Combining (3) and (4), we obtain ψ ψ ˜ dWt = (r + λ)Wt − u(ct ) + v(at ) − λWt dt + βtψ (dXt − aψt dt),
(4)
(5)
as desired.
We say that a choice rule b selects optimally if ( 0 b(ψ, ψ 0 ) = 0 when W0ψ > W0ψ 0 b(ψ, ψ 0 ) = 1 when W0ψ < W0ψ
and
Proposition 2. (Incentive Compatibility) An agent’s strategy (a, b) is optimal for any current contract ψ and choice between any ψ and ψ 0 if and only if the following conditions hold 1. for all contracts ψ and output histories, aψt maximizes aβtψ − v(a)
(6)
2. choice rule b selects optimally Proof. Clearly, if b does not select optimally, then the agent’s strategy (a, b) is not optimal. Therefore, let us assume that b selects optimally, and prove that (a, b) is optimal if and only if condition 1 holds. Consider an alternative effort strategy a ˆ in contract ψ. If the agent follows aˆ from time 0 until time t or the first matching time τ, whichever comes sooner, and then switches back to (a, b), then he gets payoff (YS: need to find a better interpretation of this expression) Z t Z t −(r+λ)s ˜ Vt = λe Ws ds + e−(r+λ)s (u(cψs ) − v(ˆ as )) ds + e−(r+λ)t Wt . 0
0
We can use (5) to derive that at ) + βtψ (ˆ at − aψt )) dt + βtψ dZt . dVt = e−(r+λ)t (v(aψt ) − v(ˆ
Therefore, the expected payoff from putting in effort aˆ until time τ is Z ∞ Z ∞ ψ ψ ψ ψ −(r+λ)t E[V∞ ] = V0 + E dVt = V0 + E e (βt aˆt − v(ˆ at ) − (βt at − v(at ))) dt , 0
0
(7)
14
where V0 is the agent’s payoff from following aψ until time τ. Suppose condition 1 fails. Then let a ˆ be the effort strategy such that a ˆt maximizes aβtψ − v(a) for all t ≤ τ. Then for all t v(aψt ) − βtψ aψt ≤ v(ˆ at ) − βtψ aˆt with strict inequality on a set of strictly positive measure. Therefore, (7) implies that E[V∞ ] > V0 , so the agent’s strategy (a, b) is suboptimal. Proving the converse (that stategy (a, b) is optimal if condition 1 holds) is more challenging. The reason is that even if E[V∞ ] ≤ V0 for all alternative effort strategies aˆ, we cannot conclude that the agent cannot improve his payoff by deviating both before and after time τ. Therefore, we need to come up with a new trick to save the day. ¯ 0ψ the be the agent’s payoff when he faces Suppose that condition 1 holds. Denote by W a contract ψ ∈ Ψ at time 0 and follows an optimal strategy. Let ¯ ψ − W ψ ). K = sup(W 0 0 ψ∈Ψ
To show that (a, b) is optimal, we need to prove that K = 0. ¯ 0ψ − W0ψ > λK . Suppose that Suppose not, i.e. K > 0. Pick a contract ψ such that W r+λ ψ ¯ W0 is attained by following effort strategy aˆ until time τ, when the agent chooses between contracts ψτθ and σθ (ψτθ ). From his optimal choice, the agent obtains value ψ˜θ σθ (ψτθ ) ψτθ ψτθ ¯ σθ (ψτθ ) ¯ + K = W0 τ + K. ≤ max W0 , W0 max W0 , W0 As a result, the agent’s expected continuation value conditional upon encountering an outside option at time τ when he follows an optimal strategy is ˜¯ = W t
Z
θ¯
max
0
Therefore,
¯ 0ψ = E W E
Z
∞
λe
−(r+λ)s
0
Z
¯ 0ψτθ , W ¯ 0σθ (ψτθ ) W
∞
λe
−(r+λ)s
dΦ(θ) ≤
˜¯ ds + W s
0
˜ s + K) ds + (W
Z
Z
Z
∞ 0
a contradiction.
Z
θ¯ 0
ψ˜θ
˜ t + K. (W0 τ + K) dΦ(θ) = W
e
−(r+λ)s
(u(cψs )
0
∞
e
∞
−(r+λ)s
0
V0 +E |{z} W0ψ
(u(cψs )
− v(ˆ as )) ds ≤
− v(ˆ as )) ds = E[V∞ ] +
λK = r+λ
λK λK e−(r+λ)t (βtψ a ˆt − v(ˆ at ) − (βtψ aψt − v(aψt ))) dt + ≤ W0ψ + , | {z } r+λ r+λ ≤0
15
4.2
The Optimality of The Firm’s Strategy.
¯ Recall that the problem is to find Now we consider the problem of a firm of type θˆ ∈ [0, θ]. a profit maximizing offer (if there is one), given the agent currently faces terms ψ 0 with another firm. Denote by "Z # τ (ψ) ˆ ψ F θ,ψ = E e−rt (θˆ dXt − ct ds) | ψ 0
the profit of firm θˆ if it currently employs the agent under contract ψ, where τ (ψ) denotes the termination time. Then optimal strategy of firm θˆ can be found by solving the problem ˆ
maxψ F θ,ψ 0 s.t. W0ψ ≥ W0ψ .
(8)
Assuming that the agent selects optimally, the strategy of firm θˆ is optimal if σθˆ(ψ 0 ) solves (8) when the objective is nonnegative and σθˆ(ψ 0 ) is a contract that the agent rejects when the objective is nonpositive. Instead of solving (8) directly, we consider ˆ
F (W ) = maxψ F θ,ψ s.t. W0ψ = W
(9)
Then the solution of problem (8) is given by maxW ∈[W ψ0 ,∞] F (W ). 0 Proposition 2 from the previous subsection will help us find optimal contracts that solve (9). The incentive compatibility condition (6) links the sensitivity βtψ of the agent’s continuation value toward the output X with the agent’s effort, and thus the firm’s profits. However, one big challenge still stands in the way of finding optimal contracts and characterizing the firms’ strategies. The jump of the agent’s continuation value from Wτψ to ψˆθ W0 τ at a match time τ depends on the strategy of an outside firm θ, which has not been characterized yet. Because of this difficulty, we must characterize an optimal contract of a firm of type θˆ for an arbitrary collection of outside firms’ strategies (σθ )θ∈[0,θ]¯ . As we mentioned above, a contract ψ of a firm of type θˆ must specify the agent’s consumption before the matching time, and a revised contract at the matching time. Let us introduce several definitions about the revision possibilities at a match time τ when the outside competitor is of type θ, for an arbitrary collection of strategies (σθ )θ∈[0,θ]¯ of outside firms. When the current employer of type θˆ revises the terms to a new level ψ 0 (let us not forget that this revision is contractually specified) the competitor firm offers the contract σθ0 (ψ 0 ). Let us define the set of continuation possibilities of the original firm by 0
ˆ
σ (ψ 0 )
G(θ) = {(W0ψ , F θ,ψ ) | b(ψ 0 , σθ (ψ 0 )) = 0} ∪ {(W0 θ 0
16
, 0) | b(ψ 0 , σθ (ψ 0 )) = 1}.
This set, schematically pictured in Figure 5, conveniently summarizes the complexity of the strategies of outside firms. Profit (W µ,θ , F µ,θ )
(W µ,θ , F µ,θ ) Value
G(θ)
PSfrag replacements
Figure 5: Schematic form of the set G(θ). The set of continuation possibilities gives the pairs of the agent’s continuation value and current employer’s profits achievable by revising the terms of employment at the match time. It is intuitive that the optimal contract must use only some pairs from the set G(θ). In fact, we will show that only those pairs that solve max
(w,f )∈G(θ)
µw + f.
(10)
for µ ∈ [−∞, ∞] are used in an optimal contract. It is convenient to denote such pairs by (W µ,θ , F λ,θ ). Denote by ψ µ,θ the contract that corresponds to this value pair. The following proposition characterizes the profit of a firm of type θˆ and the optimal contracts for an arbitrary form of the set G(θ). We supress the index θˆ on F for simplicity. Proposition 3. A solution to problem (9) exists on the range [R, u(¯ c)/r] with R defined by Z θ¯ λ R= W −∞,θ : dΦ(θ). r+λ 0
The solution is given by a concave function F that solves an ordinary differential equation ˆ −c+λ (r + λ)F (W ) = max θa a,c
17
Z
0
θ¯
F −F
0 (W ),θ
: dΦ(θ)+
(11)
F 0 (W ) (r + λ)W − u(c) + v(a) − λ
Z
θ¯
W
−F 0 (W ),θ
: dΦ(θ)
0
!
+
v 0 (a)2 00 F (W ) 2
with boundary conditions λ F (R) = r+λ
Z
θ¯
F −∞,θ : dΦ(θ),
0
and F (u(¯ c)/r) = −¯ c/r.
(12)
c)/r] → [0, a¯] and c : [R, u(¯ c)/r] → [0, c¯] be the maximizers in (11). Then Let a : [R, u(¯ the contract that achieves profit F (W ) and delivers to the agent value W is defined as follows. Starting from the value W0 = W, the firm computes Wt as a solution to the SDE ˜ t : dt + v 0 (a(Wt )) : (dXt − a(Wt ) : dt) dWt = (r + λ)Wt − u(c(Wt )) + v(a(Wt )) − λW with
˜t = W
Z
θ¯
W −F
0 (W
t ),θ
dΦ(θ)
0
until time τ when the agent is matched with an outside firm for the first time. Before time τ the contract specifies a payment to the agent of c(Wt ). At time τ, the contract is revised 0 to a new contract ψ F (Wt ),θ , where θ denotes the type of the outside firm.
Proof. First, let us justify the boundary conditions (12). We start with R. Note that W −∞,θ = min(w,f )∈G(θ) w is a lower bound on the continuation value that the agent gets when he encounters an outside firm of type θ. By putting zero effort from time 0 until the matching time τ, the agent can guarantee himself the utility of at least # "Z Z Z ¯ τ
θ
e−rt u(cψt ) dt + e−rτ
E
0
τ
e−rt u(cψt ) dt + R.
W −∞,θ dΦ(θ) = E
0
0
We see that the firm cannot deliver to the agent any value W < R, so for those values problem (9) does not have a solution. The minimal value that the firm can deliver is W = R. To deliver value R, the firm must provide zero consumption until the first match, when it can offer the contract ψ −∞,θ to maximize profit. As a result, " # Z θ¯ Z θ¯ λ −rτ −∞,θ F (R) = E e F dΦ(θ) = F −∞,θ dΦ(θ). r+λ 0 0 For the boundary condition at u(¯ c)/c, note that u(¯ c) is the maximal utility flow that the agent can possibly get in our model. The only way to get it is by consuming c¯ and
18
putting effort 0 at all times. To deliver this value, the firm must pay c¯ to the agent forever. Therefore, F (u(¯ c)/r) = −¯ c/r. Note that the firm cannot force any other firm provide c¯ after the match time in equilibrium, since this would give negative profit to the other firm. Similarly, if the current firm promises a stream of c¯ forever, nobody can steal the agent from the current firm in equilibrium. We conclude that the boundary conditions (12) must be valid, and that problem (9) is c)/r]. unsolvable outside the range [R, u(¯ Lemma 1 in the Appendix demonstrates that function F that solves (11) is concave. We proceed to the main part of the proof. We must show that the function F defined by (11) gives the principal’s profit under an opitmal contract, and that the contract layed out at the end of the proposition achieves profit F. Consider an arbitrary contract ψ, defined by consumption cψt for each output history before time τ and a revised contract ψtθ for each output history that ends with type θ encountering the agent at time t = τ. Define the process R t −rs e (θˆ dX − cψs ) ds + e−rt F (Wtψ ) when t < τ Gt = R0τ −rs ˆ s e (θ dXs − cψs ) ds + e−rτ F˜τ (θ) when t ≥ τ 0
where
F˜τ (θ) =
ˆ
θ
F θ,ψτ when b(ψτθ , σθ (ψτθ )) = 0 0 when b(ψτθ , σθ (ψτθ )) = 1
Let us show that G is a supermartingale for an arbitrary contract ψ and a martingale for the contract defined at the end of the proposition. By Proposition (1) before time τ the agent’s continuation value evolves as follows
dWtψ =
(r + λ)Wtψ − u(cψt ) + v(aψt ) − λ
where ˜ t (θ) = W
Z
θ¯
˜ t (θ) dΦ(θ) W
0
!
dt + βtψ (dXt − aψt dt),
θ
W ψt when b(ψτθ , σθ (ψτθ )) = 0 θ W σθ (ψt ) when b(ψτθ , σθ (ψτθ )) = 1
Using Ito’s formula with jumps, the drift of G is given by ˆ ψ − cψ + F 0 (Wtψ ) (r + λ)Wtψ − u(cψt ) + v(aψt ) − λ Dt = e−rt θa s s (β ψ )2 + t F 00 (Wtψ ) − rF (Wtψ ) + λ 2 19
Z
θ¯ 0
Z
θ¯
˜ t (θ) dΦ(θ) W
0
ψ F˜t (θ) dΦ(θ) − Ft (Wt− )
!!
.
!
Note that from problem (10) ˜ t (θ) + F˜t (θ) ≤ −F 0 (Wtψ )W −F 0 (Wtψ ),θ + F −F 0 (Wtψ ),θ −F 0 (Wtψ )W with equality if ψtθ = ψ −F
0 (W ψ ),θ t
,
0 ψ 2 v 0 (a)2 00 0 ˆ ˆ ψ −cψ +F 0 (W ψ )(v(aψ )−u(cψ ))+ v (at ) F 00 (W ψ ) ≤ max θa−c+F (Wtψ ) (v(a) − u(c))+ F (Wtψ ) θa t s s t t t a,c 2 2
and
v 0 (aψt )2 00 ψ (βtψ )2 00 ψ F (Wt ) ≤ F (Wt ) 2 2
because βtψ enforces effort aψt and F is a concave function by Lemma 1. We conclude that ! Z θ¯ ψ 0 ˆ − c + F 0 (Wtψ ) (r + λ)Wtψ − u(c) + v(a) − λ W −F (Wt ),θ dΦ(θ) Dt ≤ max e−rt θa a,c
0
v 0 (a)2 00 ψ + F (Wt ) − rF (Wtψ ) + λ 2
Z
θ¯
F
−F 0 (Wtψ ),θ
0
dΦ(θ) −
ψ Ft (Wt− )
!!
=0
by (11). We conclude that G is a supermartingale for an arbitrary contract. Also, we have an equality in the expression above for a contract defined in the proposition, so G is a martingale for that contract. We conclude by noting that the firm’s profit from an arbitrary contract ψ is E[Gτ ] ≤ G0 = F (W0ψ ) with equality for the contract defined in the proposition. This completes the proof. ˆ Denote by Now we are ready to characterize the optimal strategy of a firm of type θ. ψθˆ(W ) the contract that delivers to the agent value W and gives the firm of type θˆ profit F (W ). When the firm of type θˆ is bidding for an agent who faces contract ψ 0 with his current employer, its optimal offer comes from the problem maxW ∈[W ψ0 ,u(C)] ¯ F (W ) whenever it has 0 a nonnegative value. If maxW ∈[W ψ0 ,u(C)] ¯ F (W ) < 0, it is optimal to not offer any contract. 0 By Proposition 3 the function F is concave, and the optimal strategy of a firm of type θˆ can be summarized as follows: ˆ be the value that Corollary. Let F be the profit function of a firm of type θˆ and M (θ) 0 ˆ maximizes F. If F (M (θ)) ≤ 0, then the optimal strategy σθˆ(ψ ) must specify no contract ˆ > 0 then σ ˆ(ψ 0 ) depends or a contract that the agent rejects for any ψ 0 ∈ Ψ. If F (M (θ)) θ ˆ > M (θ) ˆ be the solution to F (E(θ)) ˆ = 0. on the contract ψ 0 currently in place. Let E(θ) 20
ˆ when W ψ0 ≤ M (θ), ˆ Then the optimal strategy must offer the contract σθˆ(ψ 0 ) = ψθˆ(M (θ)) 0 0 0 0 ˆ E(θ)] ˆ and no contract when W ψ > E(θ). ˆ σθˆ(ψ 0 ) = ψθ (W0ψ ) when W0ψ ∈ [M (θ), 0 From this result, we can characterize the upper boundary of the continuation possibility set for each firm. For clarity we index the set G(θ) and the profit function F by the type θˆ of the current employer. Then the upper boundary of the set Gθˆ(θ) is [M (θ), E(θ)] × {0} ∪ {(W, Fθˆ(W )) : | : W ≥ E(θ)} Then λ R= r+λ
Z
θ¯
M (θ) : dΦ(θ). 0
The form of the upper boundary of Gθˆ(θ) lets us characterize what happens when an agent is matched with an outside firm. The outcome of the match maximizes λw + f over the upper boundary of the set Gθˆ(θ) for λ = −Fθˆ0 (W ). We rely on the result that Fθˆ(W ) is increasing in θˆ whenever Fθˆ(W ) > 0 (see Lemma 2 in the Appendix) to isolate two cases depending on whether θˆ ≥ θ or not. Figure 6 shows how the revised contract is determined for the two cases from the upper boundary of the set Gθˆ(θ). Fθˆ(E(θ)) . Then at a match When θˆ ≥ θ denote by Wθˆ(θ) the point where Fθˆ0 (W ) = E(θ)−M (θ) time with an outsider of type θ, contracts offered by the incumbent and the entrant, the agent’s decision, and the continuation values of the agent and the incumbent firm: case W ≤ Wθˆ(θ) W ∈ (Wθˆ(θ), E(θ)] W > E(θ)
incumbent’s contract no contract ψθˆ(E(θ)) ψθˆ(W )
competitor’s contract ψθ (M (θ)) no contract no contract
agent’s decision leave stay stay
agent’s value M (θ) E(θ) W
incumbent’s profit 0 Fθˆ(E(θ)) Fθˆ(W )
In the last case the match with an outsider has no effect on the current contract. F (E(θ)) When θˆ < θ denote by Wθˆ(θ) the point where Fθˆ0 (W ) = Wθˆ−E(θ) . Then at a match time with an outsider of type θ, we have case ˆ W ≤ M (θ) ˆ W ∈ (M (θ), Wθˆ(θ)] W > Wθˆ(θ)
incumbent’s contract no contract ψθˆ(E(θ)) ψθˆ(W )
competitor’s contract ψθ (M (θ)) ψθ (E(θ)) no contract
agent’s decision leave leave stay
agent’s value M (θ) E(θ) W
incumbent’s profit 0 0 Fθˆ(W )
Again, in the last case the match with an outsider has no effect on the current contract. It is convenient to define the expectation over all competitor’s types of the agent’s ˜ ˆ(W ) and the incumbent’s profit F˜ˆ(W ). These expressions simplify equation (11). value W θ θ ˜ ˆ(θ, W ) be the agent’s new value and F˜ˆ(θ, W ) be incumbent’s new profit, given Letting W θ θ by the tables above, we have 21
revise to E(θ), keep no revision
fire Profit
g replacements competitor
0
M (θ)
fire revise to E(θ), let go
competitor incumbent M (θ) E(θ)
incumbent
Value, W
E(θ)
no revision
Profit
0
Value, W
collapsed profit collapsed profit
Figure 6: The revised value and contract.
˜ ˆ(W ) = W θ
Z
θ¯ 0
˜ ˆ(θ, W ) dΦ(θ) and F˜ˆ(W ) = W θ θ
Z
θ¯ 0
F˜θˆ(θ, W ) dΦ(θ)
This completes our characterization of the contracting equilibrium.
4.3
The Retirement Profit.
We finish this section by exploring the firm’s choice whether to actively employ the agent by inducing positive effort, or let the agent take effort 0 in an optimal contract. Function ¯ → (−∞, 0] denotes the profit of a firm of productivity 0. This firm does not F0 : [Q, u(C)] employ agents. However, equation (11) implies that if forced to deliver to the agent value W, it would provide a constant stream of payments c such that 1 + u0 (c)F00 (W ) = 0. At the time of the first match τ, the firm’s behavior is described by the same rules as that of any other firm. The following proposition characterizes the choice of a firm of type θˆ > 0 whether to employ the agent or not. ¯ Proposition 4. Let R(θ) = inf{W > R : | : Fθˆ(W ) = F0 (W )}. Then for all θ > 0, 22
¯ Fθ (M (θ)) > 0 and R(θ) > R where λ R= r+λ
Z
θ¯
E(θ) : dΦ(θ). 0
¯ The firm actively employs the agent for Wt ∈ (R, R(θ)) until the “retirement” time when ¯ Wt hits R or R(θ). After the retirement time, the firm behaves as the firm of type 0 would. ¯ Specifically, Wt stays constant and the firm lets the agent take effort 0. For W ∈ (R, R(θ)) function Fθˆ(W ) satisfies the ODE 2 ˆ + c − λF˜ˆ(W )− (r + λ)Fθˆ(W ) − θa θ a>0,c v 0 (a)2 0 ˜ Fθˆ(W ) (r + λ)W − u(c) + v(a) − λWθˆ(W )
Fθˆ00 (W ) = min :
(13)
Profit Fθ L 0 PSfrag replacements
RR
Fθ H ¯ H) R(θ ¯ L) R(θ
F0
Value
Figure 7: Retirement profit. This characterization of the firm’s profit will be very useful for computation. Figure 7 illustrates the retirement profit function F0 together with profit functions of the firms of two different types θL and θH .
23
5
Summary.
In this section, we crystallize the characterization of a contracting equilibrium that follows from the intense derivation of the previous section. In particular, we summarize the features regarding incentive compatibility for the agent, optimal strategies for firms, equilibrium profit functions and equilibrium outcomes. Theorem 1. (Full Characterization of a Contracting Equilibrium) The contract¯ and ing equilibrium is characterized by a family of optimal contracts ψ θ (W ) for θ ∈ [0, θ] ¯ Each contract ψθ (W ) is defined using the agent’s continuation value Wt as W ∈ [R, R(θ)]. a state variable, which is uniquely determined by the starting value W, the output and the past encounters with outside firms. The profit associated with contract ψ θ (W ) is given by ¯ Fθ (W ). Function Fθ (W ) is concave in W and increasing in θ when W ∈ (R, R(θ)). Whenever an outside firm of type θ encounters an agent, its offer is characterized by ¯ two numbers: M (θ) that maximizes Fθ (·) and E(θ) that solves Fθ (W ) = 0 on [M (θ), R(θ)]. Letting W be the agent’s continuation value from the current employer, the competitor’s offer is ψθ (M (θ)) if W ≤ M (θ) ψθ (W ) if W ∈ [M (θ), E(θ)] ∅ if W > E(θ)
The agent works for the firm that offers the best terms. In case of a tie, in equilibrium the agent goes to the more productive firm. ¯ ¯ → [0, c¯] and aθ : [R, R(θ)] → Optimal contracts are defined with functions cθ : [R, R(θ)] [0, a ¯]. The agent earns payments cθ (Wt ) and chooses to put effort aθ (Wt ) when Wt is the agent’s current continuation value in a given contract. In an optimal contract, the firm adjusts the agent’s value depending on the realization of output as follows between match times ˜ dWt = (r + λ)Wt − u(cθ (Wt )) + v(aθ (Wt )) − λWθ (Wt ) : dt+v 0 (aθ (Wt ))(dXt −aθ (Wt )dt), (14) ˜ In this expression Wθ (W ) denotes the expectation of the agent’s value in case he is matched ¯ with a competitor when his value just prior to the encounter was W. If W t hits R or R(θ) before a match time, it is absorbed there. Then the agent gets a stream of constant payments and puts effort 0 (until a match time). If Wt hits R, the agent’s consumption is 0 until the match time, when he accepts the contract ψθ0 (M (θ0 )) from a competitor of type θ 0 . Therefore, R is determined by λ R= r+λ
Z
θ¯
M (θ) : dΦ(θ), 0
Whenever the agent is matched with an outside firm of type θ 0 , the current firm of type θ ˜ θ (θ0 , W ). The new level is defined revises the agent’s continuation value to a new level W from the profit function Fθ of the current firm and the competitor’s values M (θ 0 ) and E(θ0 ) 24
0
Fθ (E(θ )) as follows. If θ ≥ θ 0 , let Wθ (θ0 ) be the point where Fθ0 (W ) = E(θ 0 )−M (θ 0 ) . Then 0 0 ˜ θ (θ , W ) F˜θ (θ , W ) case W 0 W ≤ Wθ (θ ) M (θ0 ) 0 0 0 0 W ∈ (Wθ (θ ), E(θ )] E(θ ) Fθ (E(θ0 )) 0 W > E(θ ) W Fθ (W ) The last column gives the resulting profit for the incumbent firm, which depends also on the agent’s decision to stay or leave. (E(θ 0 )) If θ < θ0 , let Wθ (θ0 ) be the point where Fθ0 (W ) = FWθ −E(θ 0 ) . Then ˜ θ (θ0 , W ) F˜θ (θ0 , W ) W case W ≤ M (θ) M (θ 0 ) 0 W ∈ (M (θ), Wθ (θ0 )] E(θ0 ) 0 0 W > Wθ (θ ) W Fθ (W ) ¯ and the The profit functions Fθ : [R, Wgp (θ)) → < for the firm of each type θ ∈ (0, θ] retirement profit function F0 : [R, u(¯ c)/r] → (−∞, 0] are determined simultaneously by the following conditions. Let
˜ ˆ(W ) = W θ
Z
θ¯ 0
˜ ˆ(θ, W ) : dΦ(θ) and W θ
F˜θˆ(W ) =
Z
0
θ¯
F˜θˆ(θ, W ) : dΦ(θ)
Then the profit function solves the second-order ordinary differential equation 2 Fθ00 (W ) = min 0 2 (r + λ)Fθ (W ) − θa + c − λF˜θ (W )− a>0,c v (a) ˜ θ (W ) Fθ0 (W ) (r + λ)W − u(c) + v(a) − λW
(15)
(16)
with boundary conditions
¯ ¯ ¯ ¯ Fθ (R(θ)) = F0 (R(θ)) and Fθ0 (R(θ)) = F00 (R(θ))
Fθ (R) = 0,
¯ ¯ The functions cθ : [R, R(θ)] → [0, c¯] and aθ : [R, R(θ)] → [0, a¯] are the maximizers in 0 ¯ 0 ¯ ¯ equation (16), with cθ (R) = 0, aθ (R) = 0, F (R(θ))u (cθ (R(θ))) = 1 and aθ (R(θ)) = 0. The retirement profit F0 is defined by ˜ 0 (W ) . (r + λ)F0 (W ) = max −c + λF˜0 (W ) + F00 (W ) (r + λ)W − u(c) − λW c
˜ 0 (W ) are still defined using (15) and the table (this is the In this equation, F˜0 (W ) and W 0 case when θ = 0 < θ ) ˜ 0 (θ0 , W ) F˜0 (θ0 , W ) W case W ∈ [R, W0 (θ0 )] E(θ0 ) 0 0 W > W0 (θ ) W F0 (W ) (E(θ 0 )) with W0 (θ0 ) defined by F00 (W ) = FW0 −E(θ 0) . 25
6
Computation.
We compute the firms’ profit functions under competition using an iterative algorithm. In the first step of iteration we compute the profit function for each firm in a single principal - single agent setting. Then we iterate to compute profits under competition. We compute iteration i + 1 profit using feedback values from iteration i profits. This process gives a sequence of profit functions that ultimately converges to the profit functions under competition. The criterion for convergence is simply based on the distance between break-even values for each firm between consecutive iterations. We follow Sannikov (2004) to describe the profit functions for each firm in a single principal - single agent setting. The profit of a type θ firm without competition is described by the second order differential equation (Fθ0 )00 (W ) = min a,c
2 0 0 0 r(F )(W ) − θa + c − ((r + λ)W − u(c) + v(a)) (F ) (W ) (17) θ θ v 0 (a)2
with boundary conditions Fθ0 (0) = 0,
0 0 0 0 Fθ0 (Wgp ) = F00 (Wgp ) and (Fθ0 )0 (Wgp ) = (F00 )0 (Wgp ),
where the superscript on F 0 refers to the initialization of the iterative procedure, and the initial retirement profit function F00 is given by F00 (u(c)/r) = −c/r. ¯ 0 (θ) such that the solution To solve (17) for each type of a firm θ, we must search for R that starts from the last two boundary conditions also satisfies Fθ0 (0) = 0. The search for ¯ 0 (θ) is illustrated in the figure below.7 R The profit functions in the single principal - single agent setting are used for initial values in computing the profit functions under competition. The profit functions Fθ0 give the initial values for M 0 (θ) and E 0 (θ) for each θ, which are used to find the initial lower boundary R θ¯ 0 R θ¯ 0 λ λ 0 value R0 = r+λ M (θ)dΦ(θ) as well as the initial value R = E (θ)dΦ(θ). Further, r+λ 0 0 we can compute the entire retirement profit function for the next iteration, F01 (W ), using the values of M 0 (θ) and E 0 (θ).8 Once we have the retirement profit function F01 , we proceed to compute the profit functions Fθˆ1 during the next iteration. Let us describe the iterative step that computes the function Fθˆi from functions Fθi−1 ¯ We solve the differential equation for θ ∈ [0, θ]. (Fθˆi )00 (W ) = min : a,c
7
2 (r + λ)(Fθˆi )(W ) − θa + c − λF˜θˆi−1 (W )− v 0 (a)2
In this example, √ we use the following parameter values and functional forms: effort is binary, a ∈ {0, 3}, r = .95, u(c) = 21 c, v(a) = 21 a2 , λ = .25, and θ = 1. 0 8 The retirement profit function F01 (W ) is 0 between Q0 and Q , and for W > Q, F01 (W ) is the convex hull of F00 (W ) and the retirement profit functions when retiring the agent until a match with a firm of ¯ for all θ. type θ ∈ [θ, θ],
26
Computing Profits with No Competition 0.4
Wgp Too High, Decrease Optimal Wgp Wgp Too Low, Increase Retirement Profit
0.3 0.2
Profit
0.1
0 −0.1 −0.2 −0.3 −0.4
PSfrag replacements
−0.5
0
0.5 1 Continuation Value w
1.5
Figure 8: Finding the right value of Wgp . ˜ i−1 (W ))(F ˆi )0 (W ) −((r + λ)W − u(c) + v(a) − λW θ θˆ with boundary conditions Fθˆi (R) = F0i (R) = 0,
ˆ = F i (R ˆ ˆ = (F i )0 (R ˆ ¯ i (θ)) ¯ i (θ)) ¯ i (θ)) ¯ i (θ)). Fθˆi (R and (Fθˆi )0 (R 0 0
F˜θˆi−1 (W ) denotes the expectation of the firm’s profit over all types of outside firms if a match ˜ i−1 (W ) denotes the agent’s expected continuation value from a happens. Analogously W θˆ competitive match when his value just prior to the match is W . In computation we need to resolve the difficulty that both of these values depend on the equilibrium profit functions and are simultaneously necessary to solve the ODEs to determine the profit functions. ˜ i−1 (W ) using M i−1 (θ), E i−1 (θ), and the profit We compute the values F˜θˆi−1 (W ) and W θˆ functions from the previous iteration. The computation is performed using the formulas at the end of subsection 4.2. Thus, using the values from the previous iteration, we can determine all values necessary to solve the ODE. We continue iteration until the distance between break-even values between iterations is sufficiently small. The figure below illustrates the profit functions computed using this iterative method.9 9
In this example, √ we use the following parameter values and functional forms: effort is binary, a ∈ {0, 3}, r = .95, u(c) = 21 c, v(a) = 21 a2 , and λ = .25. We also assume that all firms are identical with θ = 1.
27
Computing Profits Under Competition 0.2
Iteration 1 − No Competition Iteration 2 Iteration 3 Iteration 4 Retireent Profit
0.1
Profit
0 −0.1 −0.2 −0.3 −0.4
PSfrag replacements
−0.5
0
0.5 1 Continuation Value W
1.5
Figure 9: The outcome of iterations.
7
Extensions.
(to be completed)
8
Conclusion.
(to be completed)
Appendix. The following lemma is used in the proof of Proposition 3 and in the characterization of the firms’ optimal bidding strategies. Lemma 1. Equation (11) defines a concave function F (W ).
Proof. (to be completed) Lemma 2. Fθˆ(W ) is increasing in θˆ whenever Fθˆ(W ) > 0. 28
Proof. Consider the contract that achieves profit Fθˆ(W ) > 0 for a firm of type θˆ and delivers to the agent value W. Because this contract achieves positive profit, it must induce positive effort on a set of positive measure. If this contract was offered by a more productive firm, it would induce the same effort and deliver the same value to the agent, but achieve strictly higher profit. Therefore, Fθˆ(W ) must be increasing in θˆ whenever Fθˆ(W ) > 0.
29
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