Optimality of deadline contracts and dynamic moral hazard Tuomas Laiho∗ November 24, 2016

Abstract This paper analyzes dynamic moral hazard with limited liability in a model where a principal hires an agent to complete a project. We first focus on moral hazard with regards to effort and show that if the agent is patient enough the optimal contract takes the form of a deadline contract. At the deadline the principal either fires the agent or lets him work part time depending on how impatient the agent is. We then extend the model to include moral hazard with regards to quality by considering an imperfectly observed quality choice. Contracts that implement high quality are of ’efficiency wages’ type, i.e the principal leaves enough rents to the agent to incentivize high quality. The main result of the paper is that when the agent is impatient enough, moral hazard with regards to effort goes away but with regards to quality it can actually become worse.

JEL classification: D86 Keywords: Dynamic contracts, moral hazard

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Introduction

Contracts between firms for products or services often stipulate a ’time is of the essence’ clause, which requires the parties to perform their contractual duties within a prescribed time frame. This paper studies the optimality of deadline contracts in a dynamic moral hazard framework, where a firm is outsourcing a project to another firm. It is easy to see why deadlines might be useful for the principal in the moral hazard framework: it is a way to incentivize the agent to put effort into a project because he is fired at the deadline. To provide concrete examples, the principal could be a firm employing a headhunter or outsourcing an R&D project. We analyze both effort and quality moral hazard to see how ∗ Aalto

University, [email protected]. I thank Pauli Murto and Juuso Välimäki for support throughout this project and seminar audiences at HECER for comments.

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the principal resolves the tradeoffs between quality, quick completion of the project and rents. The paper builds a stylized and tractable model of dynamic moral hazard to analyze the problem of outsourcing a project to a subcontractor. The baseline model is the simplest case: binary actions and outcomes. We also assume the the agent is risk neutral and protected by limited liability. The agent either chooses to work or to shirk, which determines the how fast the project is completed. The principal decides on a bonus it uses to incentivize the agent and a hiring probability so that she can scale the agent’s effort between zero and one. We then extend the model to include an element of static moral hazard: the agent has an additional option to choose a lower quality project. The principal cannot write the contract directly on the quality and observes it imperfectly. A low quality project results in a verifiable failure with a positive probability. Otherwise it cannot be told apart from a high quality one. We focus in particular on the role intertemporal preferences play for the dynamic moral hazard problem. The main result of the paper is that the effort and quality dimensions of moral hazard behave very differently as the agent becomes more impatient. As the agent grows more impatient the effort dimension of moral hazard goes away: the principal implements the first best contract. When there is moral hazard with regards to both effort and quality the opposite can happen: the principal implements low quality as the agent gets impatient enough. In the model, the agent decides whether to exert effort at each point of time. He thus faces an intertemporal trade-off between exerting effort today versus exerting it tomorrow. Intertemporal preferences play naturally a large role here, since the more impatient the agent is the less value he puts on working tomorrow. Imposing a deadline for the project restricts the agent’s opportunities to shirk and thus enhances the principal’s payoff. When we add a quality dimension to the model, the agent faces an additional tradeoff between completing a high quality project and low quality project, which can completed more quickly. If the punishment from getting caught is too small, the agent prefers to do the low quality project. Thus in order to implement high quality, the monopolist has to offer enough rents to the agent so that she is unwilling to risk the loss from completing the project with low quality and getting caught. Quality moral hazard thus leads to ’efficiency wages’ type of contracts, since the agent needs to ensured large enough rents. We solve the full commitment solution of the principal, which we call the optimal contract. We show that the optimal contract front-loads effort and has a finite deadline TD > 0 until which the agent works full time. At the deadline the principal imposes a punishment on the agent from not completing the project. If there is moral hazard only with regards to effort and the principal and the agent are equally patient, it is optimal for the principal to fire the agent. If the agent is moderately more impatient than the principal, the principal punishes the agent by scaling down the project at the deadline. This scale is the larger the more 2

impatient the agent is. Finally, when the agent is very impatient relative to the principal, the optimal contract is the same as the first best so that the moral hazard problem essentially goes away and deadlines are not needed anymore. The optimal contract that we derive has the characteristics of a ’bang-bang’ solution but with an interesting twist: there is a nonstationary part that is ’bang’ and a stationary part that specifies an interior project scale. The solution is ’bang-bang’ if and only if the agent is as patient as the principal. Thus the way the discount rates enter the optimal contract show especially in the stationary part. One interpretation for this is that the more impatient the agent is the smaller the punishment needs to be to give him the same incentives to work today. Thus when the agent is impatient the principal does not need to punish the agent as harshly to get the same incentive effect. In a sense, controlling the size of the punishment is the principal’s way of using the agent’s impatience to her advantage. Thought this way the result with regards to firing the agent only when the principal and the agent are equally patient is intuitive. The structure of the optimal contract stays the same when we add moral hazard with regards to quality, but it is possible that principal wants to implement low quality even if it is not the first best solution. We derive conditions under which the principal wants to implement high quality. In essence we measure how restrictive the incentive compatibility constraint for quality is for the principal’s problem. Simply put, the payoff from high quality needs to be larger that the extra rents needed to implement it. If there is only moral hazard with regards to effort, the trade-off between efficiency and rents disappears from the model completely as the agent becomes more and more myopic. That is the principal does need to punish the agent at all. We show that this happens already at a finite discount rate not only at the limit. When we add imperfectly observable quality to the model, under a relatively mild restriction for the profitability of the low quality project we are able to show an exact opposite result: there is a cutoff for the discount rate of the agent above which the principal always implements low quality. This is because the quality dimension of moral hazard is unaffected by the discount rate: the agent still needs to be offered the same amount of rents to implement high quality. As a final part of the analysis, we also consider whether it is possible that as quality becomes imperfectly observable it actually improves the efficiency of the contract. We show that this is true for all contracts for which the principal implements high quality. Furthermore, we show that it can lead to strictly greater total surplus. This illustrates that introducing moral hazard with regards to quality changes the problem nontrivially. It might result in a overall lower effiency (low quality is implemented) or it might in fact increase the efficiency of the contract. Previous work on the topic includes Mason and Välimäki (2015), Hörner and Samuelson (2013) and Varas (2015). All three papers are closely connected to the model we study. Mason and Välimäki derive the full commitment and 3

sequentially rational solution to the principal’s problem, when the agent can choose his effort from a continuum and faces a convex cost. In contrast to our result with regards to effort, they show that in such a setting it is never optimal for the principal to stop the project in finite time. Their optimal contract also depends on the discount rates and they show that the arrival rate of the project converges to zero if and only if the principal and the agent are equally patient. Otherwise it converges to some positive value. Unlike this paper, they do not analyze what happens, when there is moral hazard with regards to quality. Hörner and Samuelson (2013) analyze a model that is similar to ours but with learning about project quality included. Their paper in turn builds on work by Bergemann and Hege (1998, 2005). While most of the paper focuses on a bandit problem with moral hazard, they also include a section on pure moral hazard in their online appendix. The key difference is that we include moral hazard with regards to quality. They also assume equal discount rates for the agent and the principal unlike in this paper. Their solution to the bandit problem is similar in spirit to the optimal contract we derive in that the principal front-loads the effort and then switches to the worst possible equilibrium. Varas (2015) studies the same question as we do with moral hazard both with regards to effort and quality of a project. However, his model includes learning about the project quality once it is completed. He shows that the optimal payment structure is such that the payment to the agent happens only if bad news about the project quality is not observed before some deadline. The optimal contract thus, unsurprisingly, takes form that is similar to the optimal contract in this paper. There are key differences, however, between his and our approaches. The first main difference is obviously learning about the project quality, which changes the optimal payment scheme at the end. Secondly, he assumes that the low quality technology is such that it completes the project immediately and results in a negative payoff for the principal. Thus very differently to our model there is no trade-off between quality and rents, because the principal never wants to implement low quality. The paper is organized as follows. Section 2 derives the optimal contract, when there is moral hazard only with regards to effort. Section 3 extends this analysis to include moral hazard with regards to quality as well.

2 2.1

Perfectly observable project quality The model

There is a principal (she), who has to hire an agent (he) to complete a project. The project yields π > 0 to the principal, when completed. Time is continuous and starts at t = 0 at which the principal offers the agent a contract that she can fully commit to. The principal and the agent discount the future with rates 0 < rP ≤ rA , so that the agent is at most as patient as the principal.

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The contract that the principal proposes to the agent at t = 0 consists of the probability of hiring the agent at each point of time, h(t), and a bonus, b(t), that is paid upon completing the project at t. In addition, the principal has to pay a flow cost of cP > 0 for hiring the agent. Both the agent and the principal are risk neutral and punishments to the agent are constrained by limited liability (b(t) ≥ 0). Given the contract the principal designs, the agent decides on each point of time whether to work or shirk. Let at denote the agent’s action at time t so that it is chosen from A = {a, 0}. Agent’s effort level determines the probability of completing the project at each point of time and if the agent works we let the arrival rate of the project simply be denoted by a > 0. If the agent shirks, she gets a flow benefit cA > 0 that she saves from not working. For simplicity, we assume that the agent’s outside value is zero. The model we have described above is very close to Bergemann and Hege (2005) and Hörner and Samuelson (2013) but without learning. As Hörner and Samuelson point out it can be described as the simplest case for moral hazard as there are only two actions and two outcomes with a limited liability constraint. One can think that the binary actions come from a linear cost of effort so that the agent either chooses full effort or no effort at all. The principal’s static problem is linear so that we can think of the hiring probability h(t) simply as the scale of the project. Because we analyze the continuous time limit we can think of it as the share of time the agent works. The principal can use it to achieve any arrival rate between [0, a] for the project. With this interpretation, the model is close to Mason and Välimäki (2015), who allow the agent to take actions from a continuum that determines the arrival rate. Interpreting the agent’s shirking benefit, cA , as the amount of cash he can divert to himself from the project is also in line with Hörner and Samuelson (2014) and links the model to the literature on cash diversion (e.g. DeMarzo and Sannikov 2006). In this vein, it is fairly natural to interpret cA as portion of the flow that principal pays (cP ) that the agent can appropriate to himself. If the shirking benefit is equal to zero there is no moral hazard problem. In relation to cash diversion, we can also think that the limited liability in this model consists of two separate issues: the agent is cash-constrained in i) being able to operate (cP > 0) and ii) in payments to the principal (b(t) ≥ 0). In other words, limited liability not only restricts the principal from selling the project to the agent, but it also makes her to pay the running costs of the project. Thus the moral hazard problem stems from the combined effect of this limited liability and the agent’s ability to divert cash from the project to his own use. One might note that we have already written the bonus conditional on completing the project into the model, but this is without a loss of generality in our setting. It is easy to see that the principal never wants to pay the bonus or any

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sort of compensation except the flow cost before the completion of the project (this would only have adverse effect on incentives). Also, since rP ≤ rA , there is no reason for the principal to delay the payment to the agent. If the agent is more patient, rP > rA , infinitely backloading the payment is optimal.

2.2 2.2.1

Dynamic incentives Incentive compatibility

We can characterize the agent’s incentive compatibility with a constraint that tells us that the expected payoff from working needs to greater than the expected payoff from shirking. The principal has two ways of influencing the agent’s incentives: the bonus and the path of the hiring probability. We will use the incentive compatibility constraint to pin down the bonus so that we can write the principal’s problem in terms of finding the optimal path of the hiring probability. Let W denote the agent’s continuation value, we can then write the incentive compatibility constraint in discrete time (up to dt2 terms) as ah(t)b(t)dt + (1 − ah(t)dt)(1 − rA dt)Wt+dt ≥ h(t)cA dt + (1 − rA dt)Wt+dt The left-hand side is the value of working and the right-hand side is the value of shirking. Note that since working results in the project being completed with probability ah(t)dt (think h(t) as a share of time the agent works) the continuation value is scaled down by (1−ah(t)dt), if the agent works. In addition to losing the shirking benefit, this is the trade-off that the agent faces: working today forgoes the possibility of working tomorrow. In the limit as dt → 0, we can write the agent’s IC-constraint as (see appendix A for details) ˙ (t) ≥ h(t)cA − rA W (t) + W ˙ (t) ah(t)b(t) − (ah(t) + rA )W (t) + W

(1)

From above, we can see clearly that the role of h is just to scale down the agent’s effort between a and 0. The underlying Poisson production technology allows us to characterize incentive compatibility in a very simple way, since the principal wants to pay the bonus that leaves the agent indifferent between working and shirking. Setting the above equation to hold with equality yields the following. Lemma 1. The minimal bonus that induces the agent to work equals bM (t) = cA /a + W (t). Proof. Letting (1) to hold with equality and solving for b yields the minimal bonus. The minimal work inducing bonus consists of a static part (cA /rA ) that corresponds to the loss of the shirking benefit at that instance of time and a 6

dynamic part W that corresponds to the agent losing the continuation value, if the project is completed. The bonus measures the cost of providing incentives to work at given the agent’s value. The way this cost of incentives is determined already points out to the key dynamic in the model: the higher the value the principal offers to agent, the higher the bonus that the agent will demand. The principal then has to trade-off the probability of completing the project with having to offer the agent an increasing share of the surplus. The principal never has a reason to offer more than bM (t) due to the binary nature of the agent’s effort and thus this will be the bonus the principal uses. Our assumption on the binary effort for the agent thus allows us to abstract away from intensity of effort questions with regards to the incentive compatibility constraint. Given that the principal implements the minimal work inducing bonus we can substitute it back to find that the agent’s value, W (t), is governed by: ˙ (t) = rA W (t) − h(t)cA W

(2)

The agent’s value is pinned down by the path of the hiring probability h, so that future opportunities for work determine his value today. The source of the agent’s rents is the shirking value, cA , since without it the agent’s value will be independent of h(t) and thus the principal can give him a value of zero. Thus cA is a central part of the model and generally speaking measures how severe the moral hazard problem will be for the principal. The discount rate, rA , will also be crucial as the agent’s value will be proportional to it. We will discuss its role in more detail later. Note also that from this we see that the agent’s value will never be greater than W 1 = cA /rA , i.e. when he is offered the contract with constant scale equal to one. 2.2.2

Principal’s value

Let’s turn to the principal’s value. We can write it with the agent’s value as a state variable to account for history dependency. The reasons why this works are the standard ones in the literature (see for example Spear and Sristava 1987). In essence, the agent’s continuation value can be used to summarize the past history and future promises. The value function tells the principal’s value for the agent’s value that the principal delivers. For a contract that specifies a scale h(t) and a bonus b(t) we can write the principal’s Bellman equation in discrete time (up to dt2 terms) as V (Wt ) = max {hadt(π − b) − hcP dt + (1 − hadt)(1 − rP dt)V (Wt+dt )} h,b

subject to the incentive compatibility constraint of the agent. Dependency on W is dropped from the right-hand sie for notational convenience. The first part is the principal’s flow payoff given the bonus and the hiring probability and the latter part naturally her expected continuation value. Taking the limit 7

dt → 0 and using our expression for the minimal bonus, the Hamilton-JacobiBellman (HJB) equation is (see appendix A for details) rP V (W ) = max {h(aπ − cA − cP − a(W + V )) + VW (rA W − hcA )} , h

where we have used the fact that the time differential for the agent’s value is (2). The structure of the HJB-equation again highlights why it is natural to think h as the scale of the project. It scales the flow payoff that consists of a static part, aπ − cA − cP , and a dynamic part, a(W + V ), up and down. We have written the value function that solves the HJB-equation as a function of the agent’s value alone as the time dimension of the principal’s problem is fully captured by W given our incentive compatibility constraint (minimal bonus). Thus, we can interpret the value function that solves the HJB-equation as the expected value of the principal for the contract that optimally delivers the agent value W . And since the agent’s value fully summarizes all relevant information, we can let the optimal h depend on W and then from the path of W (IC-constraint) infer h for each t. Ultimately, we are interest in the time path of h, so we will speak of h often as being implemented at a given time even though the HJB-equation above is defined in terms of W . A central feature of the problem is the trade-off the principal faces in terms of providing static versus dynamic incentives. The principal’s problem is essentially to decide whether to implement the project at full scale today or tomorrow, and she faces the trade-off that implementing it full scale tomorrow will always scale down his flow payoff today. The principal thus has an incentive to constrain the value she gives to the agent by scaling down the project at some point. This in turn implies that the drift of the agent’s value must be negative so that his value is decreasing over time. We return to this issue when solving for the optimal contract. We focus on the commitment solution to HJB-equation, that is the principal maximizes her t = 0 payoff. The main difference to the sequential rational solution is that the punishments to the agent are restricted (see e.g. Mason and Välimäki 2015). One can imagine different ways how the principal can attain commitment power such as budgeting her resources so that she cannot make decisions after the project has started or by making promises to some outside party that she cannot break so that the value of the project is zero after the deadline. The optimal contract consists of two parts: the path for the scale of the project, h, which we will devote the next section, and the minimal work inducing bonus, bM (t) = cA /a + W (t), which we have already solved. What is left then is to solve the optimal path of h and then the optimal W ∗ that the principal wants to implement in the beginning of the contract. The first part of the problem amounts to solving the solution to the HJB-equation and the second part to finding its maximum value.

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2.3

First best

A natural place to start the analysis of the optimal contract is to look what happens in the first best. When the principal can observe the agent’s effort, she can threaten to fire the agent if he does not work and thus keep the agent to his outside option. That is the agent is employed until the project is completed and the principal just pays the flow cost cP and keeps the surplus from the project to herself. Proposition 1. In the first best, the principal keeps the agent working until the project is completed. The principal gives the agent the value of his outside option, so that bF B = 0. The principal’s expected payoff equals V F B = (aπ − cP )/rE , where rE = a + rP is the principal’s effective discount rate, when the project is done at full scale. Proof. The proposition follows immediately from the fact that it is optimal to hire the agent (aπ − cP > 0) and to pay a bonus of zero as the principal gets less otherwise. We can solve the optimal contract in the case the agent is completely myopic in similar vein. If we let the agent’s discount rate approach infinity, it must be the case that the agent’s value equals zero, since otherwise he places no weight on future earnings. Then the minimal bonus that we derived earlier says that bM = cA /a and thus the principal has no reason to worry about the dynamic aspect of hiring the agent and simply hires him until the project is completed. This will give the principal an expected payoff of V M = (aπ − cA − cP )/rE . To ensure that the myopic contract has a positive payoff we need to assume that aπ − cA − cP > 0. We will call projects that satisfy this constraint feasible and will assume it throughout. The above discussion makes it clear that the moral hazard problem is a lot more severe, when the agent is forward looking, since the dynamic trade-off goes away for the agent. Indeed, the problem with a myopic agent resembles closely a static moral hazard problem. Both in the first best, when effort is observable, and when the agent is myopic, there is no efficiency loss, because the principal has no reason to manage the agent’s rents.

2.4

Stationary contracts

Let’s next try to think about the possible values the principal can attain, when there is moral hazard with regards to effort. Let’s first work out what the first best contract gives to the principal. In this contract the principal sets h = 1 for all t. We call this contract the full-scale stationary contract, since the contract stays unchanged over time and it naturally implements the full-scale for the project. We will see that it plays a special role in the analysis, because all other contracts are proportional to it. We have already argued that it will yield the agent value W 1 = cA /rA . 9

Lemma 2. The full-scale stationary contract specifies a bonus bS1 = cA /a + cA /rA and the agent works until the project is completed. The principal’s expected payoff equals (aπ − cA − cP − a rcA ) A V1 = , rE where rE = a + rP is the principal’s effective discount rate. Proof. Plugging in h = 1 for all t to the principal’s HJB-equation yields the value. Proposition 1 and lemma 2 establish bounds for the principal’s value from the optimal contract. The first best payoff must be the upper bound and the stationary contract payoff must be the lower bound, because then the agent gets the highest possible value (W 1 ) for himself. Thus, V ∗ ∈ [V 1 , V F B ]. We can generalize the notion of stationary contracts to all contracts that set a constant h ∈ [0, 1] for all t. These are contracts for which the system of the ˙ = 0. In principal’s and the agent’s values, V and W , can be at rest so that W other words, these are the steady states of the system. If we write the agent’s value as a function of h, W = h(cA /rA ), we can write V as a function of W alone. This leads to the next lemma. Lemma 3. A stationary contract specifies a constant h ∈ [0, 1] for all t so that the agent works at a constant scale until the project is completed. The agent gets value W = h(cA /rA ) so that the work inducing bonus is bSh = cA /a + h(cA /rA ). Inverting h as a function of W , we can write the principal’s expected payoff as a function W and it is defined for W ∈ [0, W 1 ] as V S (W ) =

aπ − cA − cP − aW rA W, rP cA + arA W

Proof. Plugging in h(t) = t for all t to the principal’s HJB-equation yields the value function, which can be expressed as a function of W using W = h(cA /rA ). Plugging in W = h(cA /rA ) into the expression in lemma 3 leads to principal’s value as a function over h. The expression over W is more convenient, however, since we are solving our optimal contract over W and as one might guess it will converge to a steady state, i.e. a point on V S (W ). Note that V S (W ) is concave in W so that if we optimize over h we would find the best stationary contract for the principal. Our optimal contract will, obviously, yield at least this and generally more, because as we will see front-loading effort brings gains for the principal both in terms of incentives and discounting. Spreading the effort evenly as the stationary contract does is inefficient. We devote the next section for deriving the optimal contract for the principal.

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2.5

Optimal contract

We derived earlier that the principal’s HJB-equation is rP V (W ) = max {h(aπ − cA − cP − a(W + V )) + VW (rA W − hcA )} , h

(3)

Subject to the following constraints 0 ≤ W (t) ≤

cA rA

0 ≤ h(t) ≤ 1 In addition, the principal’s value also has to be nonnegative, but this will be guaranteed by our solution as long as the project is feasible. We also have from the incentive compatibility constraint that the agent’s value is governed by (2), which we will use later. We derive the solution to HJB-equation by first conjecturing some properties of solution and then later verifying these properties. To begin our analysis, let’s first take the first order condition with regards h, which equals    > 0 ⇒ h = 1 aπ − cA − cP − a(V + W ) − VW cA

= 0 ⇒ h ∈ [0, 1]   < 0 ⇒ h = 0

(4)

Note that since the HJB-equation is linear in the scale of the project, we have corner solutions, which are captured by the right-hand side conditions for h above. The first order condition tells us the marginal productivity of hiring the agent given the agent’s value W . Naturally, if it is above zero the agent is hired for full time and if below zero the agent is not hired at all. The first order condition includes derivative of value function with regards to the state variable, VW , which captures the effect that hiring the agent today has on the cost of incentives (the bonus) in the preceding periods. We can use the first order condition to describe the evolution of the optimal scale of the project. First of all, at t = 0 (or W = W ∗ ), when the principal is designing the contract, if an optimal solution exists, it must be the case that the derivative with regards to the agent’s value is zero. From (4) we then have that the scale is one in the beginning if aπ − cA − cP > V ∗ + W∗ a The left-hand side is the maximum total surplus when rP = 0 and on the right-hand side we have the total surplus for the optimal contract. Thus this must be always true when rP > 0 like we do. Thus h = 1 at W = W ∗ . In fact, this condition shows that h = 1 for W > W ∗ as well, since then VW < 0. Naturally, we are assuming here that such W ∗ exists, which we verify later.

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The fact that principal wants to set h = 1 in the beginning is a central feature of the optimal contract, and it makes intuitive sense that the principal wants to implement a full scale phase at first in order to complete project quickly. Bear in mind that the principal is unlikely to set h = 1 throughout unless the agency problem is especially mild, because this will give the agent the maximum rents. Thus the problem of finding the optimal contract is really one of finding the optimal boundary value for the family of curves that are characterized by the HJB-equation and h = 1. This stems from the linearity of the static problem and naturally simplifies the analysis as we do not have to solve a differential equation for h(t) as it will at most get two values. When does the principal want slow down the project? From the first order condition we see that the principal wants to set h lower than one if and only if the first order condition is equal to zero, that is when the agent’s value is such that W = W P ∈ [0, W 1 ) (arbitrary at this point) so that VW is large enough. The principal’s value is then V (W P ) = V P . In the time dimension this happens at some deadline TD ∈ (0, ∞). From (4) we have that at W = W P VW =

aπ − cA − cP − a(V P + W P ) cA

(5)

This is the condition that the marginal productivity of hiring the agent is equal to zero due to the cost of providing incentives having risen high enough. After this point having the agent working at full scale will decrease the principal’s value, since she then has to give the agent too much rent to induce him to work. What does the principal want to do when (5) is true? The condition by itself does not pin down the value of the optimal project scale, since our HJB-equation is linear in h. We have already argued that the principal’s problem corresponds to find the optimal boundary and we are now ready to see why. To understand the nature of the solution, let’s first use (5) to solve the HJB-equation. Plugging in the condition to the HJB-equation (3), we can solve for the principal’s value, which equals V S (W ) =

aπ − cA − cP − aW rA W, rP cA + arA W

(6)

We have encountered this expression before since it describes the principal’s value given that she willing to put h ∈ [0, 1] for all t. That is, it is the stationary contract curve for which the scale of the contract stays constant. Thus the boundary for the HJB-equation lies on the stationary contract curve. This makes sense, if we remember that for these V and W the change in W can be zero, so what we are really saying here that at W = W P , for which (5) is true, we are in a steady state. We now know that at W = W P the value equals (6) and the derivative of the optimal value function equals (5), but to pin down W P we need an additional 12

S condition: smooth pasting at W = W P so that VW (W ) equals (5). We show in appendix D that smooth pasting holds for all value functions such that set h = 1 in the beginning (for W > W P ) and have a boundary point on the stationary curve. Thus this must also hold for the solution to the HJB-equation. This smooth pasting property of the optimal contract determines the value of W P uniquely as the next lemma shows.

Lemma 4. The punishment value W P is the value the principal gives to the S agent after a deadline TD > 0, and it is the W for which VW (W ) equals (6). Let P A = aπ − cA − cP , then the punishment value W solves the following quadratic equation 2

(rA − rP )a2 (W P ) + (arP cA + arA A + (rA − rP )a(A − − (rA − rP )A

rP cA )W P rA

rP cA = 0 rA

Proof: appendix B. With the restriction for the discount rates, rP ≤ rA , the quadratic equation only has one positive root and thus the point at which the value function is tangent to the stationary solution, W P , is uniquely determined. We call the agent’s value W P as the punishment value, because it is the value that the principal imposes on the agent to punish him for not completing the project before the deadline TD . It balances the principal’s goal of wanting to finish the project even after TD with her having to give more rents to the agent in order to do so. The question whether the principal wants to stop the project at some point is then really a question when the principal wants to give the agent a punishment value of zero. Proposition 2. The principal stops the project at t = TD if and only if rP = rA .

Proof. Solving the equation in lemma 4 yields that W P = 0 if and only if rA = rP . Proposition 2 says that the principal will stop the project in finite time, if the agent is at least as patient as the principal. This is because the optimal punishment value, W P , is zero if and only if rA = rP as we can see from the previous lemma. The result is contrary to Mason and Välimäki (2015) where the principal never stops in finite time, but makes intuitive sense in our setting. The incentive for the principal to try to slow down the scale of the project is to exploit the agent’s relative impatience, so that by slowing down the agent loses relatively more than the principal. As the agent becomes as patient as the principal, this incentive naturally disappears, because then waiting is as costly for the both of them. Or put it in other terms, the more impatient the agent

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is the smaller the punishment value can be to achieve the same reduction in incentive costs (the bonus). Now that we have found the optimal boundary W P , from the HJB-equation and h = 1 we see that our task is simply to solve the following ordinary differential equation for W ∈ [W P , W 1 ]: rP V (W ) = aH π − cA − cP − aH (V + W ) + VW (rA W − cA ) Before going further, let’s first try to graphically understand the solutions to this differential equation.

Figure 1: The solution to the principal’s HJB-equation, when rP < rA . Figure 1 depicts different values the principal and the agent attain from different types of contracts. The upper most (green) line is the value from the optimal contract and that is our solution to the HJB-equation. As we can see from the figure, it is tangent to the family of stationary contracts (red line) precisely at our W P as we have defined. The difference between the two contracts is the amount the principal gains from using a nonstationary contract. The third line (blue), which crosses the curve for the stationary contracts is the pure deadline contract for which the principal stops the project at t = TD . The fact that it crosses the curve of the stationary contracts already highlights that it cannot be optimal for the principal, since surely the principal can gain by not letting W go to zero. She can e.g. attain a higher payoff by stopping at the point where the two contracts cross. This illustrates the idea that the

14

optimal contract must be stop at the stationary curve and is as the figure shows indeed tangent to it. Let’s next solve the details that we still do not know about the optimal contract. We now know that the agent’s value from the optimal contract consists of two phases: the phase up to hitherto arbitrary TD at which the principal implements the project at full scale and a stationary phase at which the agent receives W P . To back out TD we will need to use both the incentive compatibility constraint and the principal’s value function (W ∗ ). To find out the scale at which the principal implements the project from TD onwards, we can simply set h = hP for all t and use the agent’s incentive compatibility constraint to get hP = W P

rA cA

This is the project scale that the principal wants to implement at t = TD . The punishment scale naturally depends on the punishment value and thus crucially on the difference between the discount rates of the principal and the agent. One can show that the punishment scale is monotonically increasing in rA , so that as the agent gets more and more impatient the punishment part of the contract becomes more efficient. Using the punishment value, we can solve for the agent’s value using the incentive compatibility constraint. Lemma 5. The agent’s value from the optimal contract is (
+ e−rA (TD −t) W P for t ≤ TD 1 − e−rA (TD −t) rcA A W (t) = W P for t ≥ TD Proof: appendix B. Note that TD is still arbitrary as it is pin down by the optimality condition of the principal. Before solving it we need to first solve the HJB-equation with the condition that V (W P ) = V P , where V P is the value that the principal gets from a stationary contract with agent’s value W P and scale hP . Lemma 6. The principal’s value function for W ∈ [W P , W 1 ] equals  rE  V (W ) = 1 − δ(W ) rA V 1    rE  a cA P rA + δ(W ) − δ(W ) −W rD rA rE

+ δ(W ) rA V P ,   −rA W where δ(W ) = ccAA−r and rD = a + rP − rA . P W A Proof: appendix B. This is the solution to the HJB-equation (3), which we specified in the principal’s problem. One can immediately note that as W → W 1 the principal’s value tends to V 1 and similarly as W → W P then V → V P . 15

Maximizing the principal’s value function with regards to W gives the optimal W , i.e. the value the principal wants to give to the agent in the beginning. We can do this by taking the first order condition. Rather than expressing this in terms of the agent’s value, we can use the incentive compatibility constraint and the agent’s value to solve for the optimal deadline, TD . Lemma 7. The optimal time to slow down the project is   1 rE rE (V 1 − V P )rD TD = ln + rD rA a(cA − rA W P )

Proof: appendix B. This is the deadline that will maximize the principal’s payoff at t = 0 and determines the value the agent gets from the optimal contract, W ∗ . At t = TD the principal’s value will equal V P and agent’s value will equal W P . This naturally begs the question that when the deadline is well specified, that is when can we find such a W ∈ [W P , W 1 ] that VW = 0. To verify this we show that under a condition for the agent’s discount rate the value function is concave and there is an interior solution to the principal’s problem. Lemma 8. Given that rA < rE the principal’s value function is concave, VW (W P ) > 0 and VW (W 1 ) < 0 so that there is an interior optimum W ∗ ∈ [W P , W 1 ] such that VW (W ∗ ) = 0. Proof: appendix D. The lemma just says that the condition under which there is an interior solution to the principal’s problem is that rA < rE . What happens when rA > rE ? Since the principal’s value is increasing in W throughout the interval [W P , W 1 ], it must be the case that it is optimal for her to set W = W 1 .1 Thus the trade-off between efficiency and rents disappears as the agent becomes impatient enough and the principal implements the first best (full scale) contract. We are finally ready to state the form of the optimal contract. Proposition 3 establishes the main result of the paper. Proposition 3. For rP = rA , the optimal contract is such that the principal implements the project at full scale until some t = TD and then fires the agent. For rP < rA < rE , the optimal contract is such that the principal implements the project at full scale, h(t) = 1, until time t = TD and then slows down the project to hP ≤ 1for t ≥ TD . For rA > rE the principal implements the full-scale stationary contract. 1 We can think of this in terms of the principal first solving an unconstrained problem for which W ∈ R. We show in the appendix that the value function is concave up to some ∗ >r ∗ rA E so if rA < rA that there still is a root for the derivative. Since the value function is ∗ increasing up to W it must be that in the constrained problem the principal sets W = W 1 . ∗ the value function is convex and the principal wants to set W = W 1 (independent If rA > rA of the form of the contract).

16

Proof. This follows directly from lemma 4, proposition 2, lemma 7 and lemma 8.

Proposition 3 summarizes the previous discussion: we get two kinds of deadlines out of the model. When the agent and the principal are equally patient the contract is a ’pure deadline’ contract and the agent is fired at TD . If the agent is moderately impatient, the optimal contract is a mixture between a deadline contract and a stationary contract in which the agent keeps working part time after TD . If the agent is very impatient, the problem goes away and the optimal contract is simply the first best contract in which the agent works until the project is finished. One way to think about this is that as the agent becomes more impatient the optimal contract becomes ’more stationary’ in the sense that punishment part becomes less severe (in terms of hP ) and then disappears completely as the agent is impatient enough.

3 3.1

Imperfectly observable quality The model

What happens if the quality of the project is not fully observable? There are many reasons why the principal might cannot observe the project quality immediately, but rather learn it gradually over time. For example, in the case of headhunting an employee his productivity becomes known only when he has worked long enough for the company. Next we extend the model of the previous section to account for imperfectly observable quality to see how the results change due to the multidimensionality of the moral hazard problem. We do this simply allowing the agent’s action to affect not only the arrival rate but also the payoff of the project. Let the action set be of the agent be A = {aL , aH , 0}, so that there is a low quality action aL in addition to the high quality action aH available to the agent. For the agent, the difference between high and low quality is that the low quality action allows the agent to complete the project more quickly, so that we have aL > aH . For the principal the payoff from the project also changes. The high quality action yields the principal πH while the low quality action yields πL < πH . We interpret this as the standards for the project that the agent provides is lower for the low quality action and thus it yields less and is completed more quickly. For example, we might think that there is always a chance that the project yields zero and that this risk is affected by the agent’s action. Thus we can equivalently think aH as the low risk action and aL as the high risk action. To keep the problem interesting, we must have that the agent faces a possible punishment from completing the project with the wrong action. Otherwise, he always chooses aL as it completes the project faster. To keep our analysis as close to the previous section as possible, we take an imperfect contracting 17

stance on the possible punishments for the agent: if the project is completed with the wrong action with probability 0 < 1 − σ < 1 there is enough evidence for the principal to punish the agent and deny him the bonus. Otherwise the agent receives the ’normal’ bonus for completion. Thus the principal is able to condition the payment to the agent on the project quality, but this happens only with some probability less than one. Our interpretation is that the principal is liable to submit evidence e.g. to a court or a mediator before she can deny paying the agent the bonus, if the project is completed. This assumption also means that we can rule out the principal possibly having beliefs about the project quality based on the arrival rates. We also assume that the game ends when the project is completed. Our approach is in contrast to Varas (2014), who allows learning about the project quality after it is completed and conditioning the payment on this learning. What is essential for the principal’s problem is that the agent gets punished with probability 1 − σ and that the punishment gives him zero value. This means that we must scale the arrival rate for the bonus with σ. In order for the quality aspect to be a problem we need three things that we assume here: aH πH > aL πL (high quality is first best), 0 < σ < 1 and aL σ > aH . The last condition ensures that the agent prefers aL to aH for some value of W even if punished. If the last two conditions are not true we are back to the case with observable quality. Note that even when the conditions hold, the principal can always implement the low quality action similarly as in section 2 simply by not punishing the agent. Our way of modelling moral hazard with regards to quality in essence amounts allowing the agent to cheat in terms of the project that he passes on to the principal. Another way to model this would be to make quality stochastic but observable so that the agent could produce with a positive high quality with low quality action and vice versa. This is a very different problem, however, since then the principal only needs make sure that the bonus for high quality project is high enough with regards to the bonus for the low quality project.

3.2

Incentive compatibility

How can we characterize incentive compatibility when there is moral hazard with regards to effort and quality? Let’s assume that the principal wants to implement aH . We then have two constraints: one that ensures that the agent wants to choose aH instead of aL and another that ensures that agent wants to choose aH instead of not working. Writing these out in discrete time (up to dt2 ), we first have that the high quality must yield more than the low quality for the agent:

aH ht bt dt + (1 − aH ht dt)(1 − rA dt)Wt+dt ≥ aL σbt ht dt + (1 − aL ht dt)(1 − rA dt)Wt+dt 18

Secondly, the high quality action must yield more than shirking (choosing a = 0):

aH ht bt dt + (1 − aH ht dt)(1 − rA dt)Wt+dt ≥ ht cA dt + (1 − rA dt)Wt+dt We have two variables with which to satisfy these constraints: the bonus and the agent’s value. The first condition leads to a constraint for the agent’s value and the second a constraint for the bonus b(t) similarly as before. That is, if the agent’s value is low enough he will always choose aL and if it is high enough he will always choose aH . Why? Here the chance that the principal might observe agent’s cheating plays a key role: if the agent cheats (implements aL ) he loses his continuation value (W ) for nothing with probability 1 − σ and thus there has to be a W high enough so that the agent chooses aH instead of aL . Taking dt → 0 we can solve from the first constraint a constraint for the agent’s value W (t) ≥

aL σ − aH b(t) aL − aH

(7)

This tells what the agent’s value needs to be relative to the bonus so that he will choose the high quality action. The condition describes exactly what we argued above: principal needs to offer high enough dynamic incentives (continuation value) relative to the static incentives, which are described by the downward scaled bonus on the right-hand side. The bonus is what the agent gets by completing the project today, the continuation value is what he receives by simply waiting. It is fairly intuitive that if the bonus is high enough relative to the continuation value, then the agent wants to take the gamble and use the low quality action. The second incentive constraint is exactly as we had for effort before and so the work inducing bonus will be the same as before (lemma 1). We can now characterize incentive compatibility with regards to implementing aH in the limit as dt → 0 with the following lemma. Lemma 9. Given the minimal work inducing bonus, the agent will choose aH over aL if and only if W (t) ≥ W H =

aL σ − aH cA (1 − σ)aL aH

Proof: appendix E. The lemma simply summarizes what we already discussed. If the agent’s value is above W H , then the agent will choose action aH and if it is below he will choose aL . At W H the agent is indifferent between the two and we assume that he chooses the action that the principal wants to implement. The reason why the principal can use the continuation value of the agent to police 19

his actions is the same as for any efficiency wage type contract: in order for the punishment (firing) to be effective, the agent needs to get enough rents in the first place. If W H ≤ W 1 the agent’s value is determined the same way as in section 2, but there obviously is no guarantee that W H will satisfy this constraint. If the quality of the project is hard to observe (high σ), then it might well happen that W H > W 1 . The principal then has to offer the agent more value than what is given by the full scale stationary contract. She can accomplish this by paying an extra unconditional flow payoff to the agent. Note that simply increasing the bonus will not work, because of the inequality (7) we derived earlier. Increasing the bonus also increases the incentives to cheat and thus the principal will always want to pay just the minimal work inducing bonus. Let’s denote the extra flow payment by cE > 0 (depends possibly on W ), when W H > W 1 . The agent’s value is then determined by the same differential equation as before, but now with a constant term of cA + cE instead of cA . We will return to the issue of optimal cE later, when we derive the optimal contract. Note that the static part of the bonus changes, when the action changes to take into account that the two qualities arrive at different rates. This does not matter for the principal’s problem, however, as this change is offset by the change in the arrival rate for the project and thus the flow payoff to the principal is defined the same way as we had before. As a final point concerning incentive compatibility: would the principal want to set a more lenient punishment than firing the agent? Clearly not, since if she wants to implement aH , she can ensure the lowest W H with the most severe punishment. Furthermore, in the equilibrium there will be no mistakes and the principal knows the quality she is implementing. Thus she has no reason to have a more lenient punishment. The analysis in this section shows that introducing imperfectly observable quality to the model adds an incentive compatibility constraint for the principal’s problem, if she wants to implement high quality. Thus the principal now has to resolve a trade-off between rents and quality as well as between rents and effort.

3.3

Optimal contract with imperfectly observable quality

Let’s start our analysis of the optimal contract with the first best. When the principal observes the action the agent chooses, she can fire the agent if he chooses the wrong action and thus keep the agent to his outside value, which we have defined as zero. Thus the principal will implement high quality, aH , if aH πH − cP aL πL − cP > , rEH rEL where rEH = aH + rP and rEL = aL + rP . A sufficient condition is that aH πH > aL πL , which we have already assumed. 20

Proposition 4. In the first best, the principal implements high quality and keeps the agent working until the project is finished. The agent is kept to his outside value so that bF B = 0. The principal’s expected payoff equals V F B = (aH πH − cP )/rEH . As previously, we assume that the principal has full commitment power. And similarly to the previous section we also assume unless stated otherwise that aL πL − cA − cP = AL > 0, i.e. that the low quality project is feasible, which also then implies that the high quality project is feasible. In addition to the scale of the project, the principal now has to decide on quality of the project she wants to implement and whether she also wants to switch the quality if the project is not completed up to some point. It is fairly natural to think that if she changes the quality she will first implement the high quality action and then switch to low quality. This will indeed be the case. Why? If the principal would first implement aL and then aH there would have ˆ , W H ] at which the agent would not work, because it would be to be a region [W the impossible to satisfy the incentive constraint for effort. Thus implementing ˆ , W H ] and, since aH aH first results in two gains: the agent would work on [W yields more than aL , the principal also gains in discounting. Given that the principal always implements the high quality action first, there are five different forms the optimal contract might take: a stationary (full scale) contract with action aH , a nonstationary contract with action aH , a nonstationary contract first with action aH and then with action aL , a nonstationary contract with action aL and a stationary contract with aL . These are all possible and are all implemented depending on the parameter values of the project. Let’s first analyze the situation when the disciplining value is greater than the full scale stationary value for the agent, that is W H > W 1 . We cannot analyze this case with the same HJB-equation as before, because if the principal wants to implement aH , she has to set h = 1 and pay the extra flow cost cE so that W ≥ W H . First note that the principal never wants to pay more than W H , since it will only cost her more and she will receive no benefit from increasing cE . On the other hand, the agent has get at least W H otherwise he will not choose aH . Therefore, if the if the principal implements aH she has to do it with a stationary contract that gives the agent value W H and yields the principal V H. The principal also has the alternative of implementing aL with the optimal contract from section 2, which will yield her V L (W ∗ ). The principal’s problem is thus to choose max{V H , V L (W ∗ )}. We therefore arrive to the following proposition. Proposition 5. Given that W H > W 1 the optimal contract with imperfectly observable project quality implements high quality if and only if V H > V L (W ∗ ), where V L is the value from the optimal contract with observable quality and action aL and V H is the value of a stationary contract that gives the agent W H 21

and is defined as VH =

aH πH − cA − cP − cE − aH W H , aH + rP

where cE > 0 is the extra flow payment to the agent defined as cE = W H rA −cA . Proof: Appendix E. We still need max{V H , V L (W ∗ )} > 0 otherwise the project will obviously not implemented at all. This is guaranteed by our assumption that AL > 0. All the previous results from the section with observable quality regarding the optimal contract for aL hold, so that V L might be nonstationary or the full scale stationary contract. Let’s now move on to the case, when W H < W 1 . We can then analyze the problem with a familiar looking HJB-equation that also takes into account optimization over the action of the agent. Writing out the principal’s problem: rP V (W ) = max{h(aπa − cA − cP − a(V + W )) + VW (rA W − hcA )} h,a

(8)

This is subject to the incentive compatibility constraints we derived earlier, 0 ≤ h ≤ 1 and 0 ≤ W ≤ W 1 . Naturally, a ∈ {0, aL , aH }. Note that the structure of the problem is identical to section 2 with exception of optimization over the action. Thus, we are again solving for the optimal boundary point, but now taking into account that it might lie either on the stationary curve for aL or aH . Optimization over the action is straightforward, since we have assumed that the flow from the high quality action is higher for a given W . This follows from our assumption that aH πH > aL πL . Thus, if the principal can implement aH at some W , she wants always to do it. Therefore for W > W H the principal wants to implement aH and for W < W H , she implements aL as the agent cannot be incentivized to do high quality. The critical point is obviously what happens at W H , but we return to this a little later. The first order condition with regards to h stays the same:    > 0 ⇒ h = 1 aπ − cA − cP − a(V + W ) − VW cA

= 0 ⇒ h ∈ [0, 1]   < 0 ⇒ h = 0

(9)

In the beginning of the contract, when the agent value is optimal, VW = 0, the principal thus wants to set h = 1 for the same reasons as before. Thus the point where the principal stops the agent’s value depends whether W H > W P or vice versa. If W H < W P then the optimal contract stays the same as in the observable quality case with aH implemented throughout, and h = 1 until the deadline TD . Thus imperfectly observable quality does not affect the problem at all if W H is low enough. 22

Lemma 10. If W H < W P , then the principal always implements high quality with the optimal contract from section 2. Proof. If W H < W P , then the value constraint does not bind and the solution to the principal’s problem stays unchanged. In what follows we will assume that W H > W P , so that the moral hazard problem also has a quality dimension. We have already solved optimization over the action except for the point W = W H , at which the principal has to decide whether to implement aL and let W go below W H or the stationary contract that gives the agent W H . The principal naturally implements the stationary contract, if it yields more than the contract that lets W decrease below W H and implements aL . Thus the principal implements aH if V H ≥ V L = V L (W H ), where V L (W H ) is the optimal contract implementing aL at W = W H , and otherwise she implements aL . Our analysis can then broken down into to parts: value from W ∈ [W P , W H ] and value from W ∈ [W H , W 1 ]. For the first part, the value function, V L (W ), is a solution to (8) with a = aL and h = 1 until the boundary W P . Solving for this yields the following lemma. Lemma 11. Given W H < W 1 , the principal implements aL for W ∈ [W P , W H ] and the value function is   rE L V L (W ) = 1 − δL (W ) rA V 1L    rE L cA aL + δL (W ) − δL (W ) rA − WP rA rDL + δL (W )

rE L rA

V P,

−rA W P where δL (W ) = ccAA−r is the punishment value P , rEL = aL + rP − rA and V AW defined as in section 2.

Proof: Appendix E. Using this we can solve for the principal’s value for W ∈ [W H , W 1 ], which obviously depends which action the principal wants to implement at W = W H . Unfortunately, the condition V H > V L does not yield a simple restriction on the parameters due to the nonlinear nature of V L (W ). The next lemma establishes conditions under which we can be sure that the principal does not want to switch the quality of the project. L Lemma 12. Suppose that VW (W H ) > 0 and implements aH at W = W H .

aH +rP aL +rP

≤ hP . Then the principal

Proof: Appendix E. The condition in lemma is intuitive, because it says that if the principal’s value is still increasing at W H for low quality, it must be the case that the implementing high quality with the stationary contract yields 23

more than switching to low quality. The second condition in the lemma is need purely for technical reasons. Note that the condition max{V H , V L } determines whether the principal lets the value to fall below W H , if she starts from somewhere above. So if V H > V L the principal stops at W H if she starts from above W H and at W P if she starts from below. As before, both these stopping points lie on the corresponding stationary curves for the actions. Let’s next solve for the value function for W ∈ [W H , W 1 ]. It is a solution to (8) with a = aH and h = 1 with boundary at W H . Lemma 13. Given that W H < W 1 , the principal implements aH for W ∈ [W H , W 1 ] and the value function is   rE H H rA V (W ) = 1 − δH (W ) V 1H    rE H cA aH H rA + δH (W ) − δH (W ) −W rA rDH + δH (W )

rE H rA

max{V H , V L },

−rA W SH where δH (W ) = ccAA−r is the full-scale stationary H , rEH = aH +rP −rA , V AW H H contract with action a , V is the stationary contract that gives the agent W H and V L = V L (W H ) is the value defined in lemma 14.

Proof: Appendix E. Figure 3 depicts the solution to the value function in the case when the principal does not switch quality at W H . The value function is thus discontinuous at W H . It approaches the stationary curve for aH above W H and the stationary curve for aL below W H . The figure nicely illustrates that when W H > W P , the incentive compatibility constraint for high quality essentially limits the point where the principal can stop W as the stationary curve is cut in half. Despite this, it is still clear that in this example the principal wants to implement high quality. It is also clear that if we start moving W H towards the right at some point implementing the low quality contract becomes optimal for the principal. The case when the principal switches the action from high quality to low quality at W H , would essentially look the same that except the value function would cross W H somewhere above the stationary curve for aH . While the value function is then continuous, in general it is not differentiable at W H , since its left and right derivatives do not agree. This follows directly from the form of the value function. There is also no guarantee that the value function will be concave throughout [W P , W 1 ] even if the value functions for both aL and aH are concave. Thus optimization over W is now more convoluted than in the observable quality case. The way we solve optimization over W is by doing it in a piecewise manner as our value function is also defined piecewise. To find where the principal starts the contract, we need to compare the maximum values the principal can attain 24

Figure 2: The solution to the principal’s HJB-equation, when V H > V L . from the two regions [W P , W H ] and [W H , W 1 ]. We first need to find WL∗ that is ∗ optimal for [W P , W H ] and WH that is optimal for [W H , W 1 ] and then compare ∗ the payoffs these yield, VL and VH∗ . Let’s first characterize the maximum value the principal can get from [W P , W H ]. Lemma 14. Given rA < rEH , V L (W ) defined in lemma 14 is concave and has L an interior optimum at WL∗ ∈ [W P , W H ) if and only if VW (W H ) < 0. Thus ∗ L ∗ L H ∗ L H VL = V (W ) if VW (W ) < 0 and VL = V (W ) otherwise. Proof: Appendix E. This lemma simply tells us that with our familiar restriction for the agent’s discount rate, the value function is concave and the maximum will be in the interior if the derivative at the upper boundary is negative. We can characterize the maximum value from [W H , W 1 ] similarly. Lemma 15. Given rA < rEH , V H (W ) defined in lemma 16 is concave and has ∗ H an interior optimum at WH ∈ [W P , W H ) if and only if VW (W H ) > 0. Thus ∗ H ∗ H H ∗ H VH = V (W ) if VW (W ) > 0 and VH = V otherwise. Proof: Appendix E. This lemma just tells us that if the derivative is positive at the lower boundary for high quality, the principal wants to give the agent more than W H . Thus given the restriction for the agent’s discount rate there is also an interior optimum. The principal implements high quality if and only if VH∗ > VL∗ . Note that when we are talking about implementing high quality here, it still might be 25

optimal for the principal to switch quality at W H unless e.g. we impose the condition given in lemma 12. The next lemma establishes a few conditions under which the principal wants to implement high quality. Lemma 16. Let W H < W 1 , we then have that i) suppose that AH − aH W H < 0. The principal then never implements high quality. L ii) suppose that AH − aH W H > 0 and VW (W H ) > 0. The principal then implements high quality. iii) suppose that there is an interior optimum for V L (W ) and V H (W ). Then the principal implements high quality if ∗ aH πH − aL πL > WH − WL∗ Proof. Appendix E. The first part of the lemma follows from the fact that if AH − aH W H < 0 then implementing high quality will yield a negative payoff, but given our assumptions there is always a contract that implements low quality that yields a positive payoff. The second part is intuitive as well, since if the principal still at W = W H finds it optimal to give more value to the agent, it must be that she wants to implement high quality. One way to think about this is that whenever the value for low quality is still increasing, W H is not too much of an restriction for the principal, since she still is on the increasing part of the curve. Note that if the derivative is negative then the solution that implements low quality is the same one as it would be if the agent only had access to the low quality technology. The third condition says that the difference in payoffs needs to be greater than the difference in rents. This exactly the principal’s trade-off as high quality must be more valuable to her than the extra rents she has to offer for the agent. Note that we can always find a πL that makes this true e.g. by setting it zero. The right-hand side clearly still depends on the parameters on the left-hand side, but unfortunately expanding it does not yield an easier characterization in terms of the parameters. Once we have figured out the W ∗ the principal wants to implement, we can use the agent’s incentive compatibility constraint to pin down the deadlines. Then if W ∗ > W H we have a deadline TH at which the principal either scales down the project to hH < 1 or switches the action to aL . If the action changes, there is an another deadline TP at which the principal scales down the project to the punishment scale hP . Details on how to derive these and the expressions for the deadlines are in appendix E. We are now almost ready to state the form of the optimal contract, but we still need to analyze the case when the agent is very impatient, that is when rA > rEH . It seems intuitive that we should have that the principal wants give the agent the maximum value as we had in section 2. The next lemma establishes this. Lemma 17. Suppose that rA > rEH and W H < W 1 . Then it is optimal for 26

the principal to set W = W 1 . Proof: Appendix E. The lemma follows because now our assumption that W H > W P is no longer constraining our optimization. Note that W H < W 1 is very important here. Proposition 5 establishes the form of the optimal contract, when W H < W 1 and project quality is imperfectly observable. Proposition 6. Suppose that W H < W 1 . Then if rA < rEH , the principal implements high quality if and only if VH∗ > VL∗ . If V H > V L , she implements aH with h = 1 until TH and then with h = hH < 1. If V H < V L , she implements aH with h = 1 until TH and then switches to aL with h = 1 until TP at which she sets h = hP < 1. If VH∗ < VL∗ the principal implements low quality with the optimal contract. If rA > rEH then the principal implements high quality with the full-scale stationary contract. Proof. The proof follows directly from the previous lemmas. The proposition effectively sums up our discussion so far in laying the conditions under which the principal implements high quality. Together with proposition 5 it summarizes the optimal contract in the case when there is moral hazard both with regards to effort and quality. Comparing proposition 5 and proposition 6 we see how central the assumption that W H < W 1 is. Essentially, when W H > W 1 , the effort dimension of the problem goes away as the principal always uses the full scale stationary contract for high quality. Compared to section 2, we have that ’pure deadline’ contracts are rarer when quality is imperfectly observable. This is because we now need not only that the discount rates agree, rP = rA , but in addition either that the principal implements low quality or that she implements a contract that switches the quality at W H . This is intuitive, because our incentive compatibility constraint with regards to the quality of the action just essentially limits the punishment the principal can impose on the agent. Also note that when rP = rA , the quality aspect always has bite, since by our assumptions W H > 0. Given the optimal contract, we are now ready to analyze of how the problem changes, when the agent grows more impatient. Central here is that W H does not change as rA changes. That is the incentive constraint for quality does not become easier to satisfy as the agent becomes more impatient. However, we do know from the previous section that the rents for effort do become smaller as rA grows. This means that using the contract that implements the low quality action becomes more attractive for the principal. Proposition 7 establishes the main result of this section. Proposition 7. Suppose that πL − πH + W H > 0 and that at rA = rP the principal implements high quality. Then there is a r˜A such that for rA > r˜A the principal implements low quality instead of high quality.

27

Proof: Appendix E. The proposition 7 says that the moral hazard problem may become worse with regards to quality as the agent becomes more impatient. Therefore agent’s impatience affects the effort and quality dimensions of the moral hazard problem in opposing ways: it makes the inefficiency in the effort dimension go away, but it makes the inefficiency in the quality dimension worse. As a final point we analyze how agent’s ability to cheat the principal, as measured by σ, affects the efficiency of the optimal contract in the effort dimension. How does the total surplus change when σ = 0 compared to when σ is such that W H > W P ? In other words, can the agent’s ability to cheat be sometimes socially beneficial? Remember that when σ = 0, the principal wants to implement the optimal contract from section 2 for aH . We want to compare this to the optimal contract from propositions 5 and 6 in terms of the total surplus. First remember that we can write the principal’s value from the optimal contract in the beginning (VW = 0) as V∗ =

ai πi − cA − cP − ai W ∗ ai + rP

So that the principal’s value is decreasing in W ∗ . Then we can recall that the total surplus from the contract is simply the sum of the principal’s and the agent’s payoffs and with above this gives V ∗ + W∗ =

aπ − cA − cP + rP W ∗ a + rP

This is increasing in the agent’s value. This together with the above leads to the following proposition. Proposition 8. Suppose that the principal finds it optimal to implement aH . Then the total surplus will be at least as large when the agent is able to cheat than when he is not. Proof. In the problem with σ > 0 we have constrained the agents value, say WC , to [W H , max{W 1 , W H }] (aH is implemented) where as in the problem with σ = 0 the agents value, WN C , belongs to [0, W 1 ]. Furthermore, W ∗ fully characterizes which contract yields more to the principal so that a lower W ∗ means that the principal’s value is larger. It must be that in the less constrained problem the principal gets at least as much as in the constrained problem and thus WC∗ ≥ WN∗ C . From the expression we have for the total surplus this then implies that VC∗ + WC∗ ≥ VN∗ C + WN∗ C . The intuition for the proposition is straightforward: if the agent can cheat, it must be the case that when high quality is implemented he gets a larger value than otherwise. One could strengthen the proposition by imposing a condition on payoff from low quality, such as πL = 0, so that under this condition the 28

ability to cheat always leads to larger total surplus as long as the payoff to the principal is larger than zero. This is also a stark result. One might also wonder whether a similar proposition might hold when the principal implements low quality, but this makes comparing the total surpluses tricky due to the quality issue and the fact that the optimal value for the agent is nonmonotonic in the arrival rate. Note that inequality can be strict, so that the agent’s ability to cheat actually increases efficiency. This happens e.g. whenever W H > W 1 .

4

Conclusions

The project setting with Poisson technology allows us to analyze the dynamic incentive problem in a transparent way: the principal has to trade-off the probability of success with rents to the agent, which in the full commitment solution results in a front-loaded effort scheme. In our case with binary effort this is especially stark, as the principal might stop the project altogether or at least scales it down discretely at some finite point in time. The setting also makes it easy to identify the agent’s discount rate as a key determinant of the dynamic moral hazard problem. Indeed, we have illustrated two extreme cases. Above some finite cutoff for the discount rate, the dynamic moral hazard problem disappears and the agent is hired until the project is completed. On the contrary, if the the agent is as patient as the principal, he is fired if he does not complete the project before a finite deadline. For intermediate cases, the optimal contract is a combination of a deadline and and some stationary part with less than full scale. Extending the model to allow unobservable quality for the project results either in an ’efficiency wages’ type of contract or then one where the project quality changes if the project is not done until some deadline. This is intuitive, because to implement high quality the principal needs to offer the agent enough rents so that he is not willing to take the risk of cheating and getting caught. The optimal contract nevertheless has the same structure as in the case with observable quality the difference being that the punishments the principal can implement are restricted. The agent’s discount rate is crucial also for the moral hazard problem with regards to quality as it is with regards to effort, but then we get an exact opposite result to: given that implementing low quality is profitable enough, there is always a cutoff for the discount rate of the agent such that for higher discount rates the principal always implements low quality.

29

References Bergemann, Dirk and Ulrich Hege, “Venture capital financing, moral hazard, and learning,” Journal of Banking & Finance, 1998, 22 (6–8), 703 – 735. Bergemann, Ulrich Hege Dirk, “The Financing of Innovation: Learning and Stopping,” The RAND Journal of Economics, 2005, 36 (4), 719–752. DeMarzo, Peter M. and Yuliy Sannikov, “Optimal Security Design and Dynamic Capital Structure in a Continuous-Time Agency Model,” The Journal of Finance, 2006, 61 (6), 2681–2724. Hörner, Johannes and Larry Samuelson, “Incentives for experimenting agents,” The RAND Journal of Economics, 2013, 44 (4), 632–663. Mason, Robin and Juuso Välimäki, “Getting it done: dynamic incentives to complete a project,” Journal of the European Economic Association, 2015, 13 (1), 62–97. Spear, Stephen E. and Sanjay Srivastava, “On Repeated Moral Hazard with Discounting,” The Review of Economic Studies, 1987, 54 (4), 599–617. Varas, Felipe, “Contracting Timely Delivery with Hard to Verify Quality,” Mimeo, Duke University, 2015.

30

Appendices Appendix A: Derivation of the agent’s IC-constraint and the principal’s problem Proof of lemma 1: Let’s start by defining the agent’s value in the case when the principal hires him with scale ht at each period. Value from working in discrete time (up to dt2 terms) is determined by Wt = ah(t)b(t)dt + (1 − aht dt)(1 − rA dt)Wt+dt , And value from shirking is equivalently determined by Wt = h(t)cA dt + (1 − rA dt)Wt+dt , Reorganizing terms and taking the limit as dt → 0, these can be written as ˙ = rA W (t) − ah(t)b(t) W ˙ = rA W (t) − h(t)cA W The incentive compatibility constraint is then that working must yield at least as much as shirking ah(t)b(t)dt + (1 − aht dt)(1 − rA dt)Wt+dt ≥ h(t)cA dt + (1 − rA dt)Wt+dt Taking the limit as dt → 0 and setting the IC-constraint to hold with equality, the agent’s IC-constraint becomes ah(t)b(t) − (ah(t) + rA )W (t) + W 0 (t) = h(t)cA − rA W (t) + W 0 (t) = 0 So that the work inducing bonus equals (from the IC-constraint) b(t) =

cA + Wˆ(t) a

The agent’s value is governed thus by ˙ = rA W (t) − h(t)cA W Deriving the principals value We can write the principal’s value as a function of the value that she delivers to agent at each period, W . In discrete time, we have that the principal’s value equals (up to dt2 terms) V (Wt ) = max {h(a(π − bt ) − cP )dt + (1 − ahdt − rP dt)V (Wt+dt )} h

31

This is subject to the agent’s IC-constraint above. Reorganizing terms, dividing with dt and letting dt → 0 yields rP V (W ) = max {h(a(π − b) − cP ) − ahV (W ) + VW (W )(rA W − hcA )} , h

Plugging in the minimal work inducing bonus to the principal’s problem yields the Hamilton-Jacobi-Bellman (HJB) equation in the text rP V (W ) = max {h(a(π − cA − cP − a(V + W )) + VW (W )(rA W − hcA )} , h

where dependency on W is suppressed from the right-hand side for notational convenience. Note that there is a constraint on the scale of the project 0 ≤ h ≤ 1 and the agent’s value 0 ≤ W ≤ cA /rA . The latter follows from nonnegativity and the fact that the most the agent can get is an infinite stream of cA . Proof of proposition 1: The first part of follows from simply observing that if the principal sees the agent’s actions sufficient incentives are provided by paying the flow cost of working and threathening to fire the agent if he does not work. The threat of firing is enough to incentivize the agent (he is indifferent) and we can set the bonus on completion equal to zero bF B = 0. The flow payoff is then thus (aπ − cP ), which gives the expression for the expected payoff in the proposition. Proof of lemma 2: We can find the agent’s bonus from the IC-constraint by solving the differential ˙ = 0. This yields W 1 = cA /rA (the equation, when h(t) = 1 for all t and thus W agent gets an infinite stream of cA ) and thus b1 = cA /a+cA /rA . The principal’s expected payoff is then substituting the into the HJB-equation. Proof of lemma 3: The value for the contract that specifies a constant h ∈ [0, 1] and pays the minimal work inducing bonus can be found from the HJB-equation V (h) = h

aπ − cA − cP − ah rcA A ah + rP

A gives the expression in the lemma. Plugging in h = W rcA

Appendix B: Solving the HJB-equation Proof of lemma 4 Let’s first rewrite the principal’s problem for convenience: rP V (W ) = max {h(aπ − cA − cP − aW ) − ahV (W ) + VW (rA W − hcA )} , h

(10) Subject to 0 ≤ W ≤

cA rA

and 0 ≤ h ≤ 1. In addition, we have the agent’s

32

IC-constraint. The first order condition with regards to h is    > 0 ⇒ h = 1 aπ − cA − cP − a(V + W ) − VW cA

= 0 ⇒ h ∈ [0, 1]   < 0 ⇒ h = 0

(11)

The conditions for h follow from the fact that our choice of h has to maximize the HJB-equation above. From the arguments presented in the text we know that h = 1 in the beginning as VW = 0, if such W exists. We now come to the proof of lemma 4. From the first order condition we see that h = 1 as long as (11) is greater than zero, that is until VW =

aπ − cA − cP − a(V P + W P ) , cA

(12)

where V P = V (W P ). Substituting this to the HJB equation gives us the principal’s value as a function of W P : V S (W P ) =

aπ − cA − cP − aW P rA W P , rP cA + arA W P

(13)

Letting W be arbitary this is the stationary contract curve, i.e. contracts ˙ = 0. that set a constant h ∈ [0, 1] for all t and for which it W We know that V (W P ) = V S (W P ). Now we want use the smooth pasting 0 condition to find W P , i.e. set V 0 (W P ) = V S (W P ). Consult appendix C for details why this holds. Taking the derivative of (13) with regards to W gives VW =

arA W aπ − cA − cP − aW 2 aπ − cA − cP − aW rA − − arA W rP cA + arA W rP cA + arA W (rP cA + arA W )2

Setting this equal to (12) yields aπ − cA − cP − aW P arA W P aπ − cA − cP − aW P 2 P rA − − arA W = P P rP cA + arA W rP cA + arA W (rP cA + arA W P )2 aπ − cA − cP − a(V P + W P ) cA Simplifying this expression yields and plugging in our expression for V P from (13) yields a quadratic equation for W P , and letting A = aπ − cA − cP we have lemma 4: rP 2 (rA − rP )a2 (W P ) + (arP cA + arA A + (rA − rP )a(A − cA ))W P rA rP (14) − (rA − rP )A cA = 0 rA There are two roots to this equation, one positive and one negative, which we can disregard, since W ≥ 0. W P is thus fully pinned down by the optimality (12) condition and the smooth pasting condition. 33

We can also solve for the punishment value from (14), with a little manipulation the positive root equals   r  2  rA A A A + 1 + + 4 cAA rrPA − 1 + rA −rP 1 + cA 1 + rAr−r cA P P W = 2 crAAraP Proof of proposition 2 This is a direct corollary to lemma 3, since with our feasibility assumption, A > 0, so that W P will be zero if and only if rA = rP . Then (14) has a negative root and a zero root. If W = 0, then naturally h(t) = 0 for all t > TD . Proof lemma 5 To express W P in terms of the path of h we need to use the IC-constraint to back out the agent’s value in terms of h. Since W P has to satisfy the equation ˙ = 0, we have that W P = hP cA in lemma 3 for all t ≥ TD and must have that W rA A for some hP ∈ [0, 1]. Thus the scale of the project will simply be hP = W P rcA . To sum up the previous analysis, our solution to the HJB-equation implements ( 1 for W > W P or t < TD h(t) = hP for W = W P or t ≥ TD which in turn implies for the agent’s value that ( + e−rA (TD −t) W P for t < TD (1 − e−rA (TD −t) ) rcA A W (t) = W P for t ≥ TD This follows directly from the differential equation for the agent’s value. Proof lemma 6 We still need to solve the HJB-equation for t < TD and the deadline TD . Plugging in h = 1 and writing out the HJB-equation gives rP V (W ) = aπ − cA − cP − a(V + W ) + VW (W )(rA W − cA ) Or more conveniently rE V (W ) + (1 − W

rA )cA VW (W ) = (aπ − cA − cP − aW ), cA

(15)

where rE = a+rP . This is a first order linear ODE and we have the boundary condition that V (W P ) = V P , where V P equals to the stationary contract with scale hP : VP =

hP (aπ − cA − cP − aW P ) ahP + rP

34

Or equivalently plugging in W P to (13). A solution to the ODE that satisfies the boundary condition is:    rE rE  cA a 1 P rA rA V (W ) =(1 − δ(W ) )V + δ(W ) − δ(W ) −W rA rD rE

+ δ(W ) rA V P

(16)

−rA W where δg (W ) = ccAA−r P and rD = rE − rA To check that this indeed is a AW solution, first take the derivative with regards to W :    rE rE rE 0 rE 0 cA a 1 0 P rA −1 rA −1 V + δ (W ) − VW = − δ (W )δ(W ) δ (W )δ(W ) −W rA rA rA rD rE rE 0 −1 + δ (W )δ(W ) rA V P , rA

Now note that     a cA a rA rA S − rE V (W ) + (1 − W )cA VW (W ) = rE V + 1 − W cA cA rD rA rD Plugging this in to the LHS of (15) and simplifying will yield that LHS equals RHS so that (16) indeed solves (15). Proof lemma 7 Finally, we want to solve for the optimal W that principal wants to give to agent in the beginning. This must be when VW (W ) = 0 so that    rE rE rE 0 rE 0 cA a 1 0 P rA −1 rA −1 − δ (W )δ(W ) V + δ (W ) − δ (W )δ(W ) −W rA rA rA rD rE rE 0 −1 + δ (W )δ(W ) rA V P = 0 rA Simplifying and plugging in δ(W ) = cA − rA W = cA − rA W P



cA −rA W cA −rA W P

yields

rE rE (V 1 − V P )rD + rA a(cA − rA W P )

− rrA

D

Plugging in our expression for W gives cA − (1 − e−rA TD )cA − e−rA TD rA W P = cA − rA W P



rE rE (V 1 − V P )rD + rA a(cA − rA W P )

− rrA

D

Solving for TD yields the optimal deadline at which the principal will slow down the project:   rE rE (V 1 − V P )rD 1 ln + TD = rD rA a(cA − rA W P )

35

Appendix C: Smooth-pasting property of the value function We can write our earlier solution for the value function for an arbitary stopping value W S that lies on the stationary contract curve:  rE  V (W, W S ) = 1 − δ(W, W S ) rA V1    rE  cA a + δ(W, W S ) − δ(W, W S ) rA − WS rA rD + δ(W, W S )V S (W S ), −rA W where δ(W ) = ccAA−r S , V1 is the value from a stationary contract with AW S S h = 1 and V (W ) is the value from a stationary contract that gives the agent value W S .

We know that at W = W S the value from the contract will match the value of the stationary contract at W S i.e. we have that V (W S , W S ) = V S (W S ) 0

Do we also have smooth pasting so that VW (W S , W S ) = V S (W S ) at W = W S ? In order to answer this question, let’s first write the value function decomposed into nonstationary and stationary part: V (W, W S ) = V N S (W, W S ) + δ(W, W S )V S (W S ) If we now write the value matching condition we have that at W = W S V N S (W, W S ) + δ(W, W S )V S (W ) = V S (W S ) Totally differentiating both sides gives NS NS S S S VW (W, W S ) + VW (W ) S (W, W ) + δW (W, W )V 0

0

+ δW S (W, W S )V S (W S ) + δ(W, W S )V S (W ) = V S (W S ) Now, at W = W S we have that δ(W S , W S ) = 1 so that the RHS and LHS agree if: NS VW (W S , W S ) + δW (W S , W S )V S (W ) = NS S S S S S − (VW (W S )) S (W , W ) + δW S (W , W )V

Taking the derivative and evaluating it at W = W S , we see that the RHS of this equation equals NS VW (W S , W S ) + δW (W S , W S )V S (W ) =     rA rE a cA rE S V1 + − + −W cA − rA W S cA − rA W S cA − rA W S rD rA rE − V S (W S ) cA − rA W S

36

And the LHS equals: NS S S S S S VW (W S ) = S (W , W ) + δW S (W , W )V     rA rE a cA rE S V − − + − W − 1 cA − rA W S cA − rA W S cA − rA W S rD rA rE V S (W S ) + cA − rA W S

So that we indeed have NS VW (W S , W S ) + δW (W S , W S )V S (W ) = NS S S S S S − (VW (W S )) S (W , W ) + δW S (W , W )V

Thus it must also be that our total derivatives agree, since after cancelling terms RHS equals LHS: 0

0

V S (W ) = V S (W S ) and thus we have smooth pasting at any stopping value W S ∈ [0, W 1 ] that lies on the stationary curve.

Appendix D: Existence of an interior optimum and comparative statics Proof lemma 8 We have earlier derived that the value function equals     rE  rE  cA a 1 P rA rA V + δ(W ) − δ(W ) −W V (W ) = 1 − δ(W ) rA rD + δ(W )V P , This was derived on the assumption that we can set the derivative, VW , equal to zero. That is, we claimed that ∃W ∈ [W P , W 1 ] such that VW (W ) = 0. We will verify the conditions under which this is true by checking that that i) VW (W P ) > 0, ii) VW (W 1 ) ≤ 0 and iii) that the value function is concave (VW W < 0). These conditions also ensure that the optimum, W ∗ , is unique. Let’s first verify i) VW (W P ) > 0. The derivative of the value function with regards to W equals    rE rE rE 0 cA a rE 0 1 0 P rA −1 rA −1 V + δ (W ) − δ (W )δ(W ) −W − δ (W )δ(W ) rA rA rA rD rE rE 0 P rA −1 + δ (W )δ(W ) V , rA Letting W = W P we have that δ(W P ) = (cA − rA W P )/(cA − rA W P ) = 1. Thus the derivative is positive if and only if   rE a rE 1 P (V − V ) + −1 >0 cA − rA W P rD rA 37

A little manipulation then yields (V 1 − V P )rE + a



cA − WP rA

 >0

Since V 1 = (A − a(cA /rA ))/rE and V P = hP (A − aW P )/(ahP + rP ) we get that this is true if and only if A − aW P > 0, where A = aπ − cA − cP This is just the flow at the punishment point, which has to be at least zero by the optimality of W P , because the principal can always stop at W P = 0 and get at least zero. In fact, since we assume A > 0, there is always a h > 0 such that A − ah(cA /rA ) > 0 so the flow must be strictly positive and hence VW (W P ) > 0. Let’s then show ii) VW (W 1 ) ≤ 0. Now, δ(W 1 ) = (cA − rA (cA /rA ))/(cA − rA W P ) = 0, plugging this in yields −

a ≤ 0, rD

which is true if rD = rE − rA > 0. Let’s now show iii) VW W < 0. The second derivative of the value function equals   rE rE rE −2 − − 1 δ 0 (W )2 δ(W ) rA V 1 rA rA     rE cA a rE rE 0 2 P rA −2 − − 1 δ (W ) δ(W ) −W rA rA rA rD   rE rE rE −2 + − 1 δ 0 (W )2 δ(W ) rA V P rA rA This is less than zero if and only if   cA 1 P −rD V − −W a + V P rD < 0 rA We are almost done, since the sign of the above does not depend on W . And since VW (W P ) > 0 and VW (W 1 ) < 0, we must have that VW W < 0 if rA < rE . However, for the sake of completeness we can go a little deeper by exploring the properties of the above inequality. In fact, we can show that VW W < 0 for ∗ ∗ rA < rA with rA > rE , so that rA < rE is not the condition for concavity of the value function but for the existence of an interior optimum in our feasible set for W . To start, let’s simplify the above a little to get (rE − rA )(V 1 − V P )rA + acA − arA W P > 0

38

Then, writing out the difference between the full scale value and the punishment value:   cA hP aπ − cA − cP − ahP rcA aπ − c − c − a A P A rA V1−VP = − a + rP ahP + rP   (1 − hP ) rP (aπ − cA − cP ) − (rP hP + ahP + rP )a rcA A = rE (ahP + rP ) Now noting that acA − arA W P = (1 − hP )acA because W P = hP rcA we get A that   cA (rE − rA ) rP A − (rP hP + ahP + rP )a rA + rE (ahP + rP )acA > 0, rA where A = aπ − cA − cP . Simplifying this further finally yields F (rA ) = (rE − rA )αrA − (rE − rA )rP hP + rA (ahP + rP ) > 0,

(17)

PA where α = rac . This a polynomial in rA with a quadratic term and a square A root term (inside hP ). At rP = rA (our lower bound for rA ) it is positive, but once rA is large enough it will be negative. While we cannot explicitly solve for the roots of the polynomial, we can find a lower and an upper bound for the ∗ cutoff rA above which (17) is negative and thus VW W > 0.

We begin by observing that since F (rA ) is (at least weakly) increasing in hP it must be bounded above by the polynomial for which hP = 1: F (rA ) = (rE − rA )αrA − (rE − rA )rP + rA (a + rP ) 2 = −αrA + (αrE + 2rP + a)rA + (2a − rP )rE

This is a quadratic equation with two roots, one smaller than our lower limit (rP ) and one larger. The larger root equals p αrE + rE + rP + (αrE + rE + rP )2 − 4rE αrP rA = 2α In similar manner F (rA ) must be bounded from below by the polynomial for which hP = 0: F (rA ) = (rE − rA )αrA + rA rP = −αrA + αrE + rP And this will be less than zero (since rA > 0) for rA > rA such that rA = rE +

rP α

∗ Thus VW W < 0 if rA < rA and rA > rE so it must be that rA > rE .

39

Comparative statics Proof of lemma 9: We have that the punishment value equals   r  − 1+

W

P

=

rA rA −rP

1+

A cA

+

1+

rA rA −rP



1+

A cA

2

+ 4 cAA rrPA

2 crAAraP

Now let’s denote the disciminant by ∆(rA ) = B(rA )2 + Γ(rA ), that is   2 rA A A rA ∆(rA ) = 1 + 1+ +4 rA − rP cA cA rP = B(rA )2 + Γ(rA ) Note that ∆ > 1. The project scale in the punishment phase equals p −B(rA ) + ∆(rA ) cA P P h = W = rP rA 2a The derivative of hP with regards to rA equals 1

−B 0 (rA ) + 21 ∆(rA )− 2 (2B 0 (rA ) + Γ0 (rA )) rP h (rA ) = 2a P0

Now Γ0 (rA ) = 4 cAA r1P > 0 and    1 rA A B 0 (rA ) = − 1 + rA − rP (rA − rP )2 cA Or 

0

B (rA ) =

rP − rA − rP

  A 1+ <0 cA

Going back we see that derivative of hP is greater than zero if and only if −B 0 (rA )( Since −B 0 (rA ) > 0 and Γ0 (rA ) > 0 we are done.

p

p

1 ∆(rA ) − 1) + Γ0 (rA ) > 0 2

∆(rA ) > 1 the first part is positive and since

We can also show that W P < W 1 as long as rA < rE : WP <

cA rA

This is true if and only if √ B+

∆<2

40

a rP

This is just 

2

B +Γ<

a B+2 rP

2

= B2 + 4

a a2 B+4 2 rP rP

Simplifying this yields   a A a2 A rA < a + rA 1 + + cA rA − rP cA rP Now

a rA −rP

=

rE −rP rA −rP

< 1 since rA < rE , so it is enough that 0
a a2 rA + , rA − rP rP

which is true always. Thus W P < W 1 and hP < 1 if rA < rE .

Appendix E: Imperfectly observable quality Proof lemma 12 Taking the limit as dt → 0 we can write our incentive compatibility constraints as aH hb(t) − (aH h + rA )W (t) + W 0 (t) ≥ aL σb(t)h − (aL h + rA )W (t) + W 0 (t) And aH hb(t) − (aH h + rA )W (t) + W 0 (t) ≥ hcA − rA W (t) + W 0 (t) We can solve from the first one the what the value of the agent needs to be relative to bonus so that he will choose the high quality action: W (t) ≥

aL σ − aH b(t) aL − aH

Solving the bonus out from the latter one gives b(t) ≥

cA + W (t) aH

Plugging this in and solving for W from the first inequality yields lemma 8: W (t) ≥ W H =

aL σ − aH cA (1 − σ)aL aH

Proof proposition 4 The proposition follows directly from the same arguments as proposition 1. Proof proposition 5

41

The expression for the stationary contract follows directly from having a flow aH πH − cA − cP − cE − aH W H and discounting it with aH + rP . The extra flow cost, cE , is pinned down by the differential equation for the agent’s value, E since we have that W H = cAr+c . A Deriving the principal’s value The principal’s value in discrete time equals (up to dt2 terms) V (Wt ) = max{h(a(πa − bt ) − cP )dt + (1 − ahdt − rP dt)V (Wt+dt )} a,h

We have assumed here that with probability (1 − hdt) the flow value is zero. Taking the limit as dt → 0 yields the expression in the text rP V (W ) = max{h(aπa − cA − cP − a(V + W )) + VW (rA W − hcA )} h,a

Proof lemma 14 Using the fact that h = 1 until W = W P , the principal’s value function can be solved from the ordinary differential equation we have already encountered: rA L rEL V L + (1 − W )cA VW = aL πL − cA − cP − aL W (18) cA with a boundary condition that at W = W P , the punishment value for aL , (derived in appendix B) the value will be V P , which is defined by the stationary contract V P = hP

aL πL − cA − cP − aL hP rcA A aL hP + rP

,

A where hP = W P rcA .

A solution to (18) is rE L

V L (W ) =(1 − δL (W ) rA )V 1L    rE L cA aL P rA + δL (W ) − δL (W ) −W rA rDL + δL (W )

rE L rA

V P,

−rA W 1L where δL (W ) = ccAA−r is the P , rEL = aL + rP , rDL = rEL − rA and V AW value of the full scale stationary contract with technology aL . One can check that this is indeed a solution the usual way: taking the derivative and plugging it in together with the value function to (18) and checking that LHS and RHS agree.

Proof lemma 15 L We want to show that V H > V L , when VW (W H ) > 0 and L Using the diffential equation for V (18) this implies that

V L (W H ) <

AL − aL W H , rEL 42

aH +rP aL +rP

≤ hP .

L where AL = aL πL − cA − cP . This follows from VW (W H ) > 0. Using this L H H L to evaluate V (W ) we have that V > V if

rEL hH (AH − aH W H ) − rEH (AL − aL W H ) > 0, and simplifying where AH = aH πH − cA − cP . Noting that W H = hH rcA A the above yields   AH rE aL rE AL 2 −hH + rA + H hH − H rA > 0 aH cA rEL aH rEL aH cA This is now a quadratic with two positive roots between which the expression is positive (satisfies our condition). Let the upper bound be h and the lower bound be h. We want to show that h > 1 and h < hP . Plugging in hH = 1 yields   rE aL rE AL AH rA + H − H rA > 0 (19) −1 + aH cA rEL aH rEL aH cA A little manipulation yields rEL AH − rEH AL + (aL − aH )rP > 0 and this is always positive with our assumptions that aL > aH and πH > πL . +rP ≤ hP . Thus it must be that h > 1. Let’s now show that h < hP given aaHL +r P The smaller root to (19) is   r  H

h

=

AH aH cA rA

+

rEH aL rEL aH

−

AH aH cA rA

+

rEH aL rEL aH

2

r

− 4 rEEH L

AL aH cA rA

2

This is less than hP if rEH AL rA < rEL aH cA Now letting

aH +rP aL +rP



AH rE aL rA + H aH cA rEL aH



hP − hP

2

= hP yields aL P 2 2 h − hP > 0 aH

This is always true since

aL aH

> 1.

Proof lemma 16 Plugging in h = 1 and a = aH to the HJB-equation we that for W ∈ [W , W 1 ] the value function will be a solution to the following ordinary differential equation   rA H H rEH V + 1 − W VW = aH πH − cA − cP − aH W (20) cA H

43

with a boundary condition that says that the value at W H equals VL (W H ) = max{V H , V L }, where V L = V L (W H ). A solution to this is:   rE H H rA V (W ) = 1 − δH (W ) V SH    rE H cA aH H rA + δH (W ) − δH (W ) −W rA rDH + δH (W )

rE H rA

max{V H , V L },

−rA W SH where δH (W ) = ccAA−r is the full-scale staH , rEH = aH +rP −rA and V AW H tionary contract with action a . Again, one can verify the solution by plugging it back to (20)

Proof of lemma 17 L L We want to show that VW (W P ) > 0 and that VW W < 0, when rA < rEH . These both follow directly from the same arguments as in appendix B as the only difference now is that our value function has a different upper bound. It is L (W H ) < 0 there is an interior optimum, since the derivative then clear that if VW L (W H ) < 0 gives: changes sign and is continuous. Writing out VW     rE L −1 aL aL cA − rA W P rEL δL (W H ) rA V 1H − V P + − (cA − rA W P ) <0 rA rDL rDL

Whether this is negative or not critically depends how large W H , since we L (W 1 ) < 0. Unfortunately, this inequality does not yield an easier know that VW characterization in terms of the underlying parameters. L (W H ) > 0 then the value To show the last part of the lemma, note that if VW P H function must be increasing throughout [W , W ] (VW W < 0 so the sign cannot change between the two points), and thus the maximum on the interval must L lie at W H . This yields that VL∗ = V L (W ∗ ) if VW (W H ) < 0 and VL∗ = V L (W H ) otherwise.

Proof of lemma 18 H We first want to show that VW (W 1 ) < 0, when rA < rEH . The first part, H VW (W 1 ) < 0, follows directly by taking the derivative with regards to W :

−

rE H −1 rEH 0 δH (W )δH (W ) rA V 1H rA    rE H rEH 0 cA aH 0 H rA −1 + δH (W ) − δ (W )δH (W ) −W rA H rA rDH rE H −1 rE 0 + H δH (W )δH (W ) rA max{V H , V L } rA

and evaluating it at W = W 1 when δ(W 1 ) = 0 gives that is less than zero if −

aH <0 rDH 44

This is true since we have assumed that rA < rEH . Now evaluating the H derivative at W = W H gives that VW (W H ) > 0 if and only if   rEH aH rEH H L 1H − max{V , V }) (V + −1 >0 cA − rA W H rDH rA If V H > V L then the inequality is satisfied if and only if AH − aH W H > 0, where AH = aH πh − cA − cP . If V L > V H , then we must have that VL <

AH − a H W H rEH

H Now suppose VW (W H ) > 0. We prove that the value function is then concave and that there is an interior optimum. Let’s take the second derivative with regards to W :   rE H −2 rEH rEH 0 − V 1H − 1 δH (W )2 δH (W ) rA rA rA     rE H rEH rEH cA aH 0 2 H rA −2 − − 1 δH (W ) δH (W ) −W rA rA rA rDH   rE H −2 rEH rEH 0 + max{V H , V L } − 1 δH (W )2 δH (W ) rA rA rA

This will be less than zero if and only if (rEH − rA )(V 1H − max{V H , V L })rA − (cA − rA W H )aH < 0, H We see that the sign of is independent of W so that either VW W > 0 or H H H 1 < 0 for W . But we have VW (W ) > 0 and VW (W ) < 0, so that it must H be the case that VW W < 0 otherwise there is a contradiction. Also, there must ∗ be an interior optimum WH ∈ [W H , W P ], since the derivative changes sign and is continuous.

H VW W

H (W H ) < 0. Finally, we want to show that the maximum is at W H when VW H We also have that VW (W 1 ) < 0 and we know from previous argument that VW H is monotone (VW W does not change sign), so that the value function must be decreasing throughout [W H , W 1 ] and thus the maximum is at W H .

Proof of lemma 19 Let’s first show i) suppose AH − aH W H < 0 then the principal then never implements high quality. H From above we know that then VW (W H ) < 0 if V H > V L . We know H that then V is the maximum the principal can get from implementing aH . If V H < V L we are done, so suppose V H > V L and that the principal implements high quality and gets V H . Since AH − aH W H < 0, V H < 0. However, we have

45

ˆ < W H such that V L (W ) > 0, assumed that AL > 0, so that there must be W a contradiction. Therefore principal never implements high quality if AH − aH W H < 0. L Then let’s show ii) suppose that AH − aH W H > 0 and VW (W H ) > 0. The principal then implements high quality. L Our assumption VW (W H ) > 0 means that the maximum the principal gets L ∗ from low quality is V (W H ). Now, if V H > V L we are done, since V H (WH )≥ H H L V . So suppose that V < V . Then from the differential equation (20) we can write the derivative of V H (W ) at W = W H as H VW (W H ) =

AH − aH W H − rEH V L (W H ) (cA − rA W H )

Since V H (W H ) = V L (W H ) by our assumption that V H < V L . Then note that from our differential equation (18) we know that V L (W H ) = V L <

AL − aL W H rEL

H (W H ) > 0 if Since the derivative is positive at W = W H . Thus VW

AH − aH W H − AL + aL W H > 0 The inequality follows from aL > aH and is obviously true since AH > AL . ∗ ∗ ) > V H (W H ) = V L (W H ) > W H and thus that V H (WH This implies that WH ∗ by optimality of WH and we are done. Finally, we want show iii) suppose that there is an interior optimum for V (W ) and V H (W ). Then the principal implements high quality if aH πH − ∗ aL πL > WH − WL∗ . L

∗ H L ∗ (WH ) = 0, we see from (18) and VW (WL∗ ) = 0 and at WH Since at WL∗ VW ∗ ∗ (20) that VH > VL if and only if ∗ AH − aH WH AL − aH WL∗ > rEH rEL

And writing this out yields that a sufficient condition for this is that aH πH − ∗ aL πL > WH − WL∗ . Deadlines Let W ∗ > W H . Then the deadline when W = W H follows from −

rE H −1 rEH 0 δH (W )δH (W ) rA V 1H rA    rE H −1 cA rE 0 aH 0 + δH (W ) − H δH (W )δH (W ) rA − WH rA rA rDH rE H −1 rE 0 + H δH (W )δH (W ) rA max{V H , V L } = 0 rA

46

Simplifying and plugging in δH (W ) = cA − rA W = cA − rA W H



cA −rA W cA −rA W H

yields

rEH rE (V 1H − V H )rDH + H rA aH (cA − rA W H )

 − r rA

DH

Now from the incentive compatibility constraint and h = 1 until W H we get that W ∗ = (1 − e−rA TH )(cA /rA ) + e−rA TH W H . Plugging this in and solving for TH yields   1 rEH rEH (V 1H − V H )rDH TH = ln + rDH rA aH (cA − rA W H ) If V H < V L the principal will now stop here, but will continue from W H to W . From the incentive compatibility constraint and h = 1 until W P , we thus have that cA W H = (1 − e−rA TP ) + e−rA TP W H rA P

Solving for TP yields 1 TP = − ln rA



cA − rA W H cA − rA W P



H

AW < 1. If the the principal implements low This is positive since ccAA−r −rA W P quality, then the deadline is the same as in lemma 6.

Proof of lemma 20 We know from section 2 and appendix B that when rA > rEH the solution to the relaxed problem, in which W H = 0, is to set W = W 1 . In order for this solution to be incentive compatible, so that the agent chooses high quality, we only need to check that W H < W 1 , which is true by our assumption. Proof proposition 7 We can separate the analysis into two cases: when r˜A < WcAH and when r˜A > WcAH . Suppose that r˜A > WcAH so that W H > W 1 . Then the principal gets V H , if she implements high quality. We also have that V L (W ∗ ) ≥ V 1L , where V L (W ∗ ) is the payoff for the optimal contract that implements aL . Therefore it is enough to show that there is a rA such that V 1L > V H . Writing out this inequality gives   aL πL − rcA − cA − cP aH (πH − W H ) − cA − cP − cE A > aL + rP aH + rP Simplifying this gives   cA H aL aH πL − πH + W − + (aL − aH )(cA + cP ) + (aL + rP )cE rA     cA H + rP aL π − − aH (π − W ) > 0 rA 47

It is easy to see that it will be enough that πL − πH + W H −

cA > 0, rA

since aL > aH . This will be true for some rA ∈ R, if πL − πH + W H > 0 as we have assumed. Setting this to hold with equality and solving for rA yields r˜A =

cA , πL − πH + W H

which will be an upper bound for r˜A , since we have assumed that r˜A > and used V 1L instead of the optimal V L .

48

cA WH