Dynamic Agency with Persistent Exogenous Shocks

Transcription

1 Dynamic Agency with Persistent Exogenous Shocks Rui Li University of Wisconsin-Madison (Job Market Paper) Abstract Several empirical studies have documented the phenomenon of pay for luck a CEO s compensation may depend on factors completely outside his control. I present a continuous-time repeated moral hazard model which explains this phenomenon, and I develop a theoretical method to characterize the optimal contracts. In my model, an investor hires a manager to administrate a firm by offering a long term contract. In addition to the idiosyncratic disturbance influencing the manager s performance, I introduce exogenous uncontrollable shocks impacting the firm s profitability, which are publicly observable, unpredictable and persistent. I show the optimal way to pay the manager for luck (the factors beyond his control) and his performance in the optimal contracts. An optimal contract determines whether the manager should be rewarded or punished upon the arrival of a good or bad impact to the firm, with the reward-punishment pattern being history-dependent. My result provides an explanation for why the bonuses of many highly ranked CEOs soared in 2009, the year following the financial crisis which damaged the firms under their administration. Beyond this application, I develop a relatively general method to completely characterize the efficient frontier of the payoffs to the participants and the optimal history-dependent contracts in models with persistent exogenous shocks. Keywords: Dynamic agency, pay for luck, executive compensation. 1 Introduction Firms are always exposed to external perturbations. For example, the fluctuation of input and output prices, arrivals of severe natural disasters, large accidents, financial crises, and changes in government policies. In many situations, these outside shocks are publicly observable, verifiable, and can be written into a contract. Therefore, there is no reason to assume that a contract maker Department of Economics, University of Wisconsin-Madison, 1180 Observatory Dr., Madison WI rli2@wisc.edu. Phone: I am grateful to Noah Williams for his guidance and encouragement. I also thank Yuliy Sannikov, Neng Wang, Daniel Quint, Bill Sandholm, Ricardo Serrano-Padial and Marzena Rostek for their invaluable advices. This project benefited from discussions with Marek Weretka, Raymond Deneckere, Lones Smith, Bo Chen, Ming Li, Chao He, Yu Zhu, Chang-Koo Chi and Michael Rapp. All remaining errors are mine. 1

2 should ignore the relevant exogenous factors. The more a contract relies on relevant information, the more efficient it will be, especially when exogenous shocks have persistent 1 effect (for instance, a permanent drop in the price of an input). The results of this paper show that optimal contracts indeed depend on exogenous disturbances. In earlier empirical research on agency models, some researchers, including Bertrand and Mullainathan [2001], Garvey and Milbourn [2006], and Hermalin and Hubbard [2000], have documented the phenomenon of pay for luck in various industries. This means that a CEO s compensation depends on luck, the observable shocks which are not controllable. For example, by examining the compensation packages of the CEOs of 51 large American oil companies between 1977 and 1994, Bertrand and Mullainathan [2001] found that the compensation of the CEOs were significantly correlated with the fluctuations of the crude oil price, average industry performance, and exchange rates. Obviously, all of these three factors are beyond the control of a single American CEO. However, traditional static moral hazard models, for instance Hölmstrom [1979], cannot explain why executives are paid for luck. Tying their pay to uncontrollable disturbances cannot provide additional incentives, but only makes the contracts more risky and inefficient. More features need to be added into traditional models to explain this observation 2. In this paper, I investigate pay for luck in a theoretical dynamic moral hazard model. I develop a continuous-time model, with a persistent exogenous disturbance, in which a risk neutral investor hires a risk averse manager to operate a firm. The effort exerted by the manager at each instant is a hidden action which is imperfectly monitored by the investor through the income of the firm. In addition to the idiosyncratic disturbance in the income flow which generates the moral hazard problem, I introduce an exogenous disturbance with persistent effect into the model. This disturbance influences the firm s profitability, is publicly observable, and is completely beyond the control of the investor and the manager. In spite of its necessity, persistent exogenous disturbances in dynamic agency models have been relatively unexplored. The main reason is that adding a state variable indicating a persistent disturbance into a traditional discrete-time dynamic agency model makes the model complicated so that history-dependent contracts cannot be characterized completely. The recently developed continuous-time dynamic agency models enable us to accomplish this research with the help of stochastic control tools. This paper contributes a method to characterize the efficient frontiers of the expected payoffs to the investor and the manager in a relatively general setting with persistent exogenous shocks. The frontiers are characterized by value functions which help us to obtain 1 Here persistence means that the current state of the disturbance influence the probability distribution of the future state. For example, the price tomorrow is correlated with today s price. 2 Some researchers explain pay for luck by adding new features to the basic model. For example, Bertrand and Mullainathan [2001] argue that CEOs have power to control their own payments and they are able to pay themselves more in good time because shareholders scrutinize their firms less carefully in good time; Garvey and Milbourn [2006] argue that CEOs are able to influence exogenous benchmarks to their performance ex-post when the shareholders decide their payment; Hermalin and Hubbard [2000] argue that the CEO s human capital varies as the industry circumstance changes and their payments match their outside option. 2

3 clean and complete solutions of the optimal history-dependent contracts. Using my methods, a wide variety of applications can be addressed. In particular, the results of this paper show the optimality of paying for luck in the contracts, and depict the optimal way to do so in a long term principal-agent relationship. Based on the optimal contracts, I characterize the adjustments of the expected continuation utility of the manager, the utility he will experience in the rest of his career, when an exogenous good or bad impact arrives. As compensation packages typically include promotion and retirement plans, shares or options in firm stocks, as well as salary, the promised continuation utility captures the manager s payoffs in long term. The reward-punishment pattern is history-dependent. A manager with a long period of good performance, for example, making the firm s productivity high for a long time and being promoted to a high rank, is more likely to be rewarded when the firm is impacted by a bad shock, and punished when the firm is impacted by a good shock. Symmetrically, a manager with a long period of poor performance is more likely to be punished when the firm is impacted by a bad shock and rewarded when the firm is impacted by a good shock. This reward-punishment pattern provides an explanation for why some highly ranked executives receive rewards when their firms incur bad shocks. For example, in 2009, the year following the financial crisis, the average bonus for executives of FTSE 100 firms jumped from 95% to 98%, and their bonus hurdles were set very low so that the executives were able to get their bonuses easily. However, the firms under their administration were significantly damaged by this financial crisis 3. Similar evidence can be found in the data sets in Bertrand and Mullainathan [2001] and Garvey and Milbourn [2006]. The explanation of pay for luck in my model depends on the interaction between the wealth effect and incentives. Due to the wealth effect, a rich executive with high rank is hard to motivate and less sensitive to incentives, as compensating his effort cost is expensive. Note that before the arrival of a bad shock, the firm s profitability is relatively high. Then, the owner of a firm should try to make the CEO easier to motivate, so that more incentives can be provided and the full potential of the firm can be realized before a bad shock hits. To do so, the owner of the firm should be tough on the CEO, by making promotion harder and bonuses difficult, and lower the CEO s wealth level during the good times. But the owner needs to compensate the CEO for his hard work, and so should promise a reward upon the arrival of a bad shock. Although this reward makes the CEO hard to motivate after the impact, the owner is still better off because more effort is enforced in the high profitability times and the manager s effort is not as valuable in the bad times. In some extreme situations, my results suggest that the owner should reward the CEO with a bonus but then close the firm permanently when a bad shock comes. The explanation of the other cases in the reward-punishment pattern is given in Section The statistics and analysis are provided by Deloitte Touche Tohmatsu Limited. See the news from Management Today: you-win-some-win-some-ftse-executive-bonuses-soar/. 3

4 I now provide more detail on the theoretical model. Based on the continuous-time repeated moral hazard model introduced by Sannikov [2008], I assume that an investor delegates the operation of a firm to a manager over a long period of time. The instantaneous income of the firm at each moment is determined by the investor s action (for instance, the choice of the production scale or the investment of the input material), the effort level exerted by the manager, and an idiosyncratic disturbance. The moral hazard problem in the model is due to the inability of the investor to monitor the effort of the manager. She observes the income flow of the firm, which is a noisy signal of the manager s effort. I introduce a persistent exogenous disturbance into the model which is a jump process 4 affecting the firm s profitability. For example, it could be the productivity of the firm, the price of the product, or an input material, determined by the market. The investor offers a contract at the beginning, which specifies the investor s action, the amount of consumption delivered to the manager, and suggests an effort level at each instant according to the entire history of the firm performance and the exogenous disturbance. The investor s action, the income of the firm and the exogenous disturbance are publicly observable and verifiable. My model is the first repeated moral hazard model, in which large (or jump) persistent exogenous risks are considered, with relatively general preferences and production technology. In this paper, I pay special attention to the case in which the investor chooses the amount of production input and the exogenous disturbance is the fluctuation of the input price. I investigate the following four types of processes 5 : simple jump processes with a certain jump, which model the arrival of an unpredictable 6 large event with a known impact to the firm; simple jump processes with an uncertain jump, which model the arrival of an unpredictable large event with an unknown impact to the firm; multiple jump processes with finitely many jumps, which model arrivals of several unpredictable large events; and two-state Markov switching processes, which model cyclical changes in the profitability of the firm. As in previous literature on dynamic contracts, the continuation utility of the manager plays an important role in characterization of the optimal contracts in my model. All relevant information about the history of the performance and the exogenous disturbance is encoded in it. It acts as a key state variable, determining the behavior of the participants in the contract, and its dependence on the performance creates the incentives of the manager 7. Now I introduce some basic features of the optimal contracts in my model. Since wealth effects exist, as the utility level of the manager increases, the marginal utility of consumption decreases. Thus more consumption is needed to compensate the same expense of effort. Therefore, 4 Another way to model persistent effects is to assume the disturbance to be a diffusion process as in Williams [2009a,b]. However, with this assumption, the state variable indicating the disturbance has infinite and uncountable possible values, which makes the model hard to solve in general. 5 See Section 3 for detailed explanation. 6 Here, the unpredictability of an event means that the event will occur for sure sooner or later, but nobody can anticipate its arrival time. 7 This form of history dependence also appears in the constructions of sequential equilibria in repeated game models. See Abreu et al. [1990] for example. 4

5 the investor stops motivating the manager when his continuation utility is very high. On the other hand, because of limited liability, negative consumption cannot be delivered. So motivation and production are terminated when the manager s continuation utility is so low that he cannot be punished. Therefore, as in Spear and Cheng [2005], the manager retires when he is too rich to be motivated or too poor to be punished. Here, retirement means receiving constant consumption flow and not exerting effort. With the persistent exogenous disturbances, in some situations it is optimal to shutdown the firm temporarily, namely, not to motivate the manager to take effort or produce until the arrival of the next exogenous shock. Typically, shutdown happens when the current circumstance is bad, but good times are expected in the future. One reason for shutdown is that, in the good times, the investor can bear larger enforcement costs with the same promised utility level. Therefore, when the manager s continuation utility increases to a high level in the bad times, the investor should close the firm and reopen it when profitability of the firm improves. In addition, to motivate the manager, the manager s continuation utility must be tied to his performance. So the continuation utility may be driven into the region where the manager should retire and the firm should be closed permanently. If that happens in the bad times, the investor will lose the chance to produce in the good times. So closing the firm in the bad times and waiting for the good times is optimal when the continuation utility is close to the retirement region, for example, point zero. In a shutdown phase, the manager s continuation utility as well as the amount of consumption are constant 8 and the effort and output level are zero. The remainder of the paper is organized as follows: in Section 2, I review related literature; in Section 3, I introduce the basic model and assumptions; in Section 4, I give an overview of the constructions of the value function and the optimal contracts in case of simple jump process with a certain jump; in Section 5, I apply the method to the special case of input price disturbance, show two numerical examples and demonstrate pay for luck in the optimal contracts; Section 6 fills in detail on the constructions in Section 4; in Sections 7, 8 and 9, I demonstrate the constructions in the cases of simple jump processes with an uncertain jump, multiple jump processes and two-state Markov switching processes respectively; the conclusion is in Section 10. A separate Appendix contains proofs omitted from the text and available On-line 9. 2 Literature Review My work is closely related to the contemporaneously-developed work by Hoffmann and Pfeil [2010], who investigate optimal reward for luck in a continuous-time dynamic moral hazard model. In their model, both the principal and the agent are risk neutral, the production technology is linear, and 8 Note that, in the optimal unemployment insurance contracts derived by Hopenhayn and Nicolini [1997], where the worker s effort of searching affects the arrival of a job offer, the worker s continuation utility and consumption decreases as the unemployment spell gets longer. 9 The URL link is 5

6 the principal is more patient than the agent (so that the optimal contracts can be solved uniquely). They adopt a simple jump process to model the disturbance in productivity and show that the agent is always rewarded when a good shock comes and punished when a bad shock comes 10. However, this result is inconsistent with the findings in empirical studies that some executives get significant rewards when their firms incur a bad impact. My results are more general, as the agent is risk averse and the production technology is non-linear. In addition, I consider a wider vary of shock processes. Therefore, my model is more flexible and may have broader applications. As discussed, this flexibility is key to explain the empirical findings that reward-punishment patterns for executives are of ambiguous sign. This paper belongs to the growing literature on continuous-time dynamic agency models, which stem from the traditional discrete-time models, for instance the models introduced by Spear and Srivastava [1987] and Phelan and Townsend [1991]. I classify them into four categories. The first category is hidden action models, namely repeated moral hazard models, which were initially put forward by Holmstrom and Milgrom [1987]. In their setting, the agent has a CARA preference, the principal is risk neutral and the idiosyncratic disturbance making the agent s action unobservable is a white noise and does not have persistent effect. Schättler and Sung [1993] extend their model and provide the partial differential equations characterizing the value function and the optimal contracts. Sannikov [2008] develops an algorithm to completely compute the value functions and the optimal contracts in a continuous-time hidden action model with the agent s preference being additively separable in consumption and effort. The second category is hidden information models, in which the agent privately observes a cash flow and reports what he observed to the principal at each moment. The first and most widely utilized model was introduced by DeMarzo and Sannikov [2006] where hidden information is not persistent and both the principal and the agent are risk neutral. Researchers have adapted their model to study hidden information problems with firm size changes (He [2009]) and capital adjustments (Biais et al. [2010], DeMarzo et al. [2009], Philippon and Sannikov [2007]). Williams [2009a] studies a model with risk aversion and persistent private information, which is characterized by a diffusion process. Zhang [2009] and Tchistyi [2006] also investigate persistent hidden information by assuming that the privately observed cash flow is a Markov switching process. The third category is hidden information models with persistent exogenous disturbances. This category includes Piskorski and Tchistyi [2007, 2008] who investigate the optimal house mortgage design with an unobservable income flow of the borrower and an uncontrollable exogenous stochastic house appreciation, and DeMarzo et al. [2009] who study a capital investment model with hidden information and stochastic changes in capital prices. The fourth category is hidden action models with persistent exogenous disturbances, to which Hoffmann and Pfeil [2010] and my model belong. In addition to these, Williams [2009b] puts forward a highly general model with both the idiosyncratic and exogenous disturbances being persistent 10 As we will see later, the key reason for this result is that the linear preference does not generate a wealth effect. 6

7 (characterized by diffusion processes). state, for example, a hidden saving or investment account. Furthermore, the agent has a privately observed hidden 3 The Basic Model In this section I introduce the basic model. Suppose that an investor hires a manager to administrate a firm over a time horizon [0, ). The cumulative income process of the firm is denoted by the process {y t } 11 t [0, ) which satisfies the following stochastic differential equation: dy t = g(b t, a t, p t )dt + σ(b t, p t )dz t and y 0 = 0. (1) Here b t is an action chosen by the investor relevant to the production of the firm at t. For example, this could be the input level or the production scale. Without loss of generality, b t [0, b] with b R+ 12 for t [0, ). The effort level taken by the manager to administrate the firm is denoted by a t A, where A is the set of optional effort levels. I allow A to be either binary, {0, ā}, or a continuum, [0, ā], for some ā R ++. The idiosyncratic disturbance of the profitability of the firm is characterized by the process {z t } t [0, ), a standard Brownian Motion. The exogenous disturbance influencing the profitability is characterized by the process {p t } t [0, ) with p t P, a finite set in R. P is a jump process, which will be introduced in detail shortly below, and its motion has a persistent effect. According to (1), given the current values of b t, a t and p t, g(b t, a t, p t ) is the expected instantaneous income of the firm, σ(b t, p t ) is the instantaneous volatility, and dy t is the realized profit at t. The functions g(,, ) : [0, b] A P R and σ(, ) : [0, b] P R satisfy the following assumptions. Assumption 1. For each p P: (a) g(,, p) is twice differentiable with g a (b, a, p) > 0, g aa (b, a, p) 0, g bb (b, a, p) 0, g ab (b, a, p) = g ba (b, a, p) > 0 for all (b, a) [0, b] [0, ā] 13 ; (b) max b [0, b] g( b, 0, p) = 0; (c) σ(, p) is differentiable with σ(, p) > 0 for all b [0, b]. Let σ = min b [0, b], p P σ( b, p). Remark 3.1. Part (b) of Assumption 1 means that, without the effort of the manager, the investor cannot obtain positive profit. However, she can prevent herself from incurring loss by taking some safe action, for example, closing the firm. In part (c), by assuming σ(, ) to be positive, I prevent the investor from recognizing the manager s effort by observing the process Y. I do not allow the 11 In the remaining of this paper, after the first definition of a process, it will be denoted by the corresponded upper case letter. For example, Y refers to process {y t } t [0, ) hereafter. 12 If b = 0, the investor does not have any decision to the production procedure. 13 In this paper, the derivatives on the boundary of the closed domain of a function are given by its one-sided derivatives. Or we can extend the domain to a open neighborhood and assume that the function in consideration is differentiable in the extended domain. 7

8 manager s effort level to affect the volatility of the idiosyncratic disturbance. Otherwise, since we have a Brownian information structure, the investor would be able to directly observe the manager s action by computing the quadratic variation of the cumulative income process, Y. In this paper, I pay special attention to the case where p t is an input price and b t is the input invested by the investor, p t is the price of input at time t, with P R ++, and (1) simplifies to: dy t = (f(b t, a t ) p t b t )dt + σ(b t )dz t and y 0 = 0. (2) The price of output is normalized to 1 dollar per unit and the expected instantaneous income is f(b t, a t ) p t b t. Here f(, ) : [0, b] A R + is the production function of the firm which satisfies the following assumption. Assumption 2. f(b, 0) = 0 for all b [0, b], f(, ) is twice differentiable, concave and f(b, a) 0, f a (b, a) > 0, f b (b, a) 0, f aa (b, a) 0, f bb (b, a) 0, f ab (b, a) = f ba (b, a) > 0 for all (b, a) [0, b] [0, ā]. Remark 3.2. Assumption 2 is fairly common for a production function. With minimal scale one example is f(b, a) = (b + b) α 1 a α 2 with b > 0, α 1 (0, 1] and α 2 (0, 1]. Note that, Assumption 2 implies that the firm can generate positive income with positive effort level of the manager when the input is zero. This point can be explained by a required minimal input amount needed to run the firm. With this fixed input amount the firm generates positive expected income if the manager takes effort. However, if the manager does not take effort, the expected output level is zero. Note this special case satisfies the assumptions of the general case given above. Let {F Y t } t [0, ) be the filtration generated by Y. The jump process P has a right continuous trajectory and generates filtration {F P t } t [0, ). I assume that P and Z are independently distributed and let F t = F Y t F P t for t [0, ). So filtration {F t } t [0, ) indicates the joint history of Y and P. I assume that both Y and P are publicly observable and verifiable. But the history of the process A is the manager s private information. The investor is not able to determine whether a large income increment is due to the manager s high effort a t or to a shock dz t. Therefore, a moral hazard problem emerges and the investor has to delegate the operation of the firm to the manager by offering a contract at the beginning. A contract is represented by a triple of processes ({a t } t [0, ), {b t } t [0, ), {c t } t [0, ) ) with a t being the suggested effort level, b t the action taken by the investor (or the amount of input), and c t the amount of consumption delivered to the manager. All these three terms depend on the history of processes Y and P realized up to time t. Then (A, B, C) is progressively measurable with 8

9 respect to {F t }. For technical reasons, I require A and B to be {F t }-predictable 14. To interpret this requirement, I assume that, at each moment, the suggested effort and the investor s action are given before an exogenous shock is realized 15. Denote the set of all contracts by C. Given processes A and B, let µ B,A be the joint probability distribution of Y and P. Since P is exogenous, A and B only affect the conditional distribution of Y. Let E B,A be the expectation with respect to this measure. A policy of the manager is an {F t }-predictable effort level process  with value in A. The set of all policies is denoted by A. After accepting a contract (A, B, C), the manager solves the following problem: max re B,Â[  A ˆ 0 e rt (u(c t ) h(â t ))dt]. (3) Here r > 0 is the common discount rate of the manager and the investor, u( ) : R + R + is the instantaneous utility function, and h( ) : R + R + is the instantaneous cost function of exerting effort. I make the following assumption about these two functions. Assumption 3. u( ) is twice differentiable with u(0) = 0, u, u > 0 and lim c u (c) = 0; h( ) is twice differentiable with h(0) = 0, h, h > 0 and h (0) = h for some real number h > 0. If A = {0, ā}, let h = h(ā) h(0). A contract (A, B, C) is incentive compatible if the suggested effort process, A, is the optimal policy that maximizes the manager s expected utility in (3), and that A is implemented by B and C. The investor chooses among incentive compatible contracts to maximize her expected profit. Namely, she solves max (A,B,C) C reb,a [ ˆ 0 e rt (dy t c t dt)] = max (A,B,C) C reb,a [ ˆ 0 e rt (g(b t, a t, p t ) c t )dt], (4) such that A is implemented by B and C, (5) and re B,A [ ˆ 0 e rt (u(c t ) h(a t ))] w. (6) The equality in the objective function is due to the fact that dy t g(b t, a t, p t )dt is a white noise. Condition (5) is the incentive compatibility constraint and (6) is the individual rationality constraint. Here w is the reservation utility level of the manager. Note that, the manager can always guarantee himself a non-negative expected utility by not taking effort, therefore I assume w 0. In this paper, I assume that the enforcement of a signed contract is complete. That is to say 14 See Elliott [1982] Chapter 5 for the definition of predictability of a process and Chapter 13 for Girsanov theorem. 15 In fact, in my model, the requirement of predictability matters at most countable many time points which has zero measure and does not affect the expected payoffs to the investor and the manager I define below. Because of the limited liability of the manager, I require c t 0. 9

10 that, neither the investor nor the manager can quit after they commit to the contract. Furthermore, there is no renegotiation after the commitment. The main theoretical contribution of this paper is that I develop an algorithm to completely characterize the efficient frontier of the expected payoffs to the participants and optimal historydependent contracts. My methods are adapted to different assumptions on the exogenous process P. I start with a simple case and extend progressively to more general processes. With the general setup of the model, the algorithm applies to three types of jump processes of P. The first type is of simple jump processes with a certain jump. Specifically, P possesses one jump. Participants know the initial value of the process and the new value after the jump, but they do not know the jump time. This case models the arrival of a large event with a certain impact to the firm. The second case is a simple jump process with an uncertain jump, this is like the previous case except that the new value P jumps to is unknown before the jump and drawn from a given probability distribution supported by a subset of P. This case models the arrival of the a large event with an uncertain impact to the firm. The third type is a multiple jump processes. Namely, P has finite many jumps. The participants know the initial value and the the probability distribution of new value after each random jump. This models the arrival of several large events with uncertain impacts on the firm. When I specialize to consider the disturbances in the input price, the method can handle two-state Markov switching processes. Specifically, the input price can be high, in bad times, or low, in good times. This allows me to model cyclical changes of the economy with uncertain switching times between good and bad times. 4 Simple Jump Processes with a Certain Jump An Overview In this section, I provide an overview the constructions of the value function and the optimal contracts. Detailed discussion is given in Section The Setup and Notation I assume that P is a simple jump process with p t {p, p} and p > p for t [0, ). This allows me to introduce the notations and methods in a simple setting, and provide much of the insight for more general cases. The initial value at time 0 is p 0 {p, p} and p t = p 0 until the process jumps to p 1 {p, p} \ {p 0 }. The jump time is denoted by a random time T, which is drawn from an exponential distribution with rate parameter λ (0, ). Once P jumps to the new value, it will not change any more. Therefore, at any time t such that p t = p 0, we have Pr{p t+ t p 0 p t = p 0 } = t λ + o( t ) for all t > 0. The motion of the exogenous disturbance process has the following features: first, the probability that the jump occurs at time 0 is zero; second, the probability that jump does not occur is zero; 10

11 third, before the jump of P, the investor and the manager are not able to anticipate the jump time T or to update their beliefs on this random time according to the history they observed. According to the features of P, the setting of the model here matches situations in practice in which an unpredictable large event impacts the firm permanently. For example, a potential new production technology inside the firm is under development, which can raise the productivity significantly. But the time of the success of the new innovation is not predictable. To model the information contained in the history of P, it is convenient to us to define a state indicator process, {s t } t [0, ), where s t = 0 if p t = p 0 and s t = 1 if p t = 1. Obviously, S starts with s 0 = 0, possesses a right continuous, piecewise constant trajectory and a single jump of 1 with rate λ. The compensated jump martingale, {m t } t [0, ), associated with S is given by m t = s t (T t)λ, for t [0, ). Then M is a {Ft P } adapted martingale under the probability measure of P 16. Note that dm t = s t 1({t < T })λdt, for t [0, ). Thus M conveys the information about the process P. 4.2 The Sketch of the Construction The value function indicates the efficient frontier of the payoffs to the participants with the information constraint and the optimal contracts in case of simple jump processes. There are two state variables-the continuation utility promised to the manager, w t, and the current the state indicator, s t. To define w t, suppose that under a contract (A, B, C) C the manager chooses the suggested policy A. I define the continuation utility of the manager at time t as: w t = re B,A [ ˆ t e r(τ t) (u(c τ ) h(a τ ))dτ F t ]. Then, w t is the conditionally expected utility that the manager can experience in the rest of his career from time t on under the contract. I define the value function as J(, ) : [0, u( )) {0, 1} R. For each pair (w, s), J(w, s) is the greatest expected profit that the investor can obtain, with current state indicator s, by an incentive compatible contract which promises utility w to the manager. The construction of the value function is in a backward manner and contains four steps. I start from the value function of the period after the jump of P. First, I compute the retirement 16 See Boel et al. [1975] Proposition

12 value function J R ( ), which indicates the maximal profit generated by the contracts in which the investor provides constant consumption and the manager exerts no effort. Of course J R ( ) may not be optimal because the investor has the option to provide incentives for the manager to work. Therefore, in the second step I find the improvements generated by motivating the manager. These are characterized by a set of concave curves, satisfying a Hamilton Jacobi Bellman (HJB here after) ordinary differential equation and boundary conditions relating to J R ( ). Once we find all improvements, we have the post-jump value function J(, 1). Then, I turn to the period before the jump of P. Given J(, 1), I compute the shutdown value function J S ( ) in the third step. This gives the maximal expected profit of the investor when she is not allowed to motivate the manager before the arrival of the exogenous shock, but after the shock there is no restriction on the contract. Again, this may not be optimal. So in the fourth step, I find the improvements in profit obtained by motivating the manager before P jumps. Again they are characterized by a set of concave curves, satisfying another HJB equation and boundary conditions relating to J S ( ). Hence, with the improvements to J S ( ), we have the value function J(, 0). To provide a sense of how the contracts work, note that the motion of W is driven by the processes Y and P, and governs the incentive provision and the production adjustment in response to the exogenous shock. Given the evolution of W and S and the value function, the optimal contract can be characterized by the maximizers in the objective function in the associated HJB equation at each instant. The investor s action, the manager s effort level and the amount of consumption delivered are functions of the state variables. Before the exogenous shock, suppose that w t is in the interior of an interval where J(, 0) > J S ( ), then the optimal expected profit given by the value function is achieved by motivating the manager to exert effort. To motivate the manager, his continuation utility must be tied to his performance, the process Y. Therefore, before P jumps, W is a diffusion process inside the interval, and the continuation payoff vector of the participants moves along J(, 0). The manager s effort level and the output level are positive, until W hits one of the boundaries where J(, 0) = J S ( ). Then the contract switches to the shutdown contract promising the utility level on the boundary. If w is not in the interior of such an interval, the investor offers an shutdown contract promising w. Similarly, after the exogenous shock, if w t is in the interior of an interval where J(, 1) > J R ( ), then incentive provision is needed to achieve the expected profit J(w t, 1). Therefore, W is bound with Y and a diffusion process inside this interval. The effort and output level are positive until one of the boundaries of the interval is reached, when the manager retires with promised utility level on the reached boundary. When P jumps, the continuation utility is adjusted according to the following rule: either the slope of J(, 1) at the adjusted utility level equals the slop of J(, 0) before the adjustment, or the continuation utility changes to zero if the slope of J(, 0) at the pre-jump utility level is too large. 12

13 5 Input Price Disturbances and Pay for Luck Before entering the detailed discussion of the method to construct the value function and the optimal contracts, I show an application of the method and demonstrate some numerical examples. I specialize to the input disturbance case as in (2). While previous construction applies to many models, here I illustrate pay for luck with a good or bad shock. Here b t is the input invested by the investor and p t is the price of input for t [0, ). If p 0 = p, the firm is in a good state initially and incurs a bad shock at T with the input becoming more expensive; if p 0 = p, the firm is in a bad state initially and incurs a good shock at T, with the input becoming cheaper. 5.1 Setup and Numerical Examples Note that in the specific setting, the exogenous disturbance does not affect the monitoring structure and the incentive of the manager, so the following result is straightforward. Proposition 5.1. For any w [0, u( )), if p 0 = p, then J(w, 0) J(w, 1); if p 0 = p, then J(w, 0) J(w, 1). J(w, 1) for all w [0, u( )). Consequently, either J(w, 0) J(w, 1) for all w [0, u( )) or J(w, 0) To show this result, suppose that the price jumps upward, for example. Since the price does not affect the incentive of the manager, any post-jump contract is implementable before the jump. If the promised expected utility of the manager is w, by implementing the post-jump contract promising this utility level, the investor can earn no less than J(w, 1). Compared with the expected profit in the environment where the price is always p, the investor has the same output level and pays less for the input before the price jumps up. By definition, J(w, 0) is generated by the optimal contract, then we have J(w, 0) J(w, 1). The case of downward jumps is discussed in Appendix C.4. According to the construction of J(, 0), Proposition 5.1 and Proposition 6.6, we have the following result. Corollary 5.1. Let {[ w k L, wk R ]} k N and {[w k L, wk R ]} k N be the post-jump active intervals and the active intervals 17 respectively. Then: (a) if p 0 = p, k N [ w k L, wk R ] k N[w k L, wk R ]; (b) If p0 = p, k N [w k L, wk R ] k N[ w k L, wk R ]. Corollary 5.1 implies that, in case of an upward jump, computing J(, 0) only requires the retirement value function J R ( ), and we do not need to compute J S ( ). Furthermore, according to the overview in the previous section, in this case, shutting down the firm before the price change is equivalent to permanent retirement. That is, when the shock will be bad, there is no option value to temporarily shutting down. Now, I provide two numerical examples. One is of upward price jump, a bad shock, and the other is of downward price jump, a good shock. 17 The definitions of post-jump active intervals and active intervals are given in Section 6. In the interior of a post-jump active intervals J(, 1) > J R ( ), and in the interior of an active intervals J(, 0) > J S ( ). 13

14 Example 5.1. Let p 0 = and p 1 = 0.5 with jump rate λ = 0.5; r = ; u(c) = c; f(b, a) = a b and σ(b) = 2.5(b ) for b [0, 4]; A = {0, 1} with h(0) = 0 and h(1) = 1. The value functions are depicted in Figure 1. There is one post-jump active interval [ w 1 L, w1 R ] = [0, 0.33] and one active interval [wl 1, w1 R ] = [0, 0.47]. To the right of w1 R = 0.47, J(, 0) coincides with J S ( ) and J R ( ); to the right of w R 1 = 0.33, J(, 1) coincides with J R ( ). The reason is that when the utility level is very high, the manager is not sensitive to incentives due to the low marginal utility and the high cost to compensate his effort expense. Therefore, it is optimal to stop motivating the manager and producing if the investor needs to promise a high continuation utility. On the other hand, J(, 0) and J(, 1) pass the original point O. The only way to promise a zero utility is to deliver zero consumption constantly. If the investor provided positive amounts of consumption with positive probability, the manager would be able to obtain positive utility by not exerting effort. The optimal contracts can be read from Figure 2, which shows the policy functions. Except in the upper-left chart, the solid curves are the pre-jump period policy functions and the dashed curves are the post-jump period policy functions. Not shown is the effort level policy. The upper-left chart demonstrates the adjustment of the continuation utility of the manager when P jumps, given the utility promise current. Since the slopes of the value functions are equal before and after the jump, the adjustment is zero if the continuation utility is zero or greater than w 1 R = 0.47 because, in this region, J(, 0) = J(, 1) = J R ( ). In addition, there is a threshold level in (0, 0.47) at which the slope of the pre-jump and the post-jump value function are equal and therefore the adjustment is zero. To the left of this utility level the manager is punished and the to the right of this level he is rewarded when the bad shock comes. This demonstrates the pay for luck-with a low utility promise the manager is punished on the arrival of a bad shock while with a high utility promise he is rewarded on the arrival of a bad shock. The upper-right chart show how the continuation utility determines the input amount. Obviously, with a cheaper price, the investor tends to purchase more. When the continuation utility is higher than wr 1 in the pre-jump period or higher than w1 R in the post-jump period, the production is terminated and the input is zero. The effort level is 1 if the utility level is in (0, wr 1 ) before the jump or in (0, w1 R ) after the jump. Otherwise, the manager does not take effort. The lower-left chart demonstrates the consumption delivered to the manager given the promised continuation utility. Consumption is increasing in the continuation utility. As shown in the next section, it is completely determined by the slope of the current value function. The solid and dashed curves intersect at the threshold level mentioned previously, at which the the slopes of pre-jump and post-jump value functions are equal. Before P jumps, if the utility level is higher than w 1 R, or, after P jumps, if the the utility level is higher than w R 1, the retirement contract with the corresponding utility promise is offered. So the solid and dashed curves are given by u 1 ( ) to the right of w 1 R and w1 R respectively. The lower-right chart demonstrates the drift, namely the expected instantaneous increment of 14

15 W at each utility level. In this example, it is positive, and zero if w = 0, or w > wr 1 before the exogenous shock, or w > w R 1 after the exogenous shock because in this region the manager retires and the continuation utility is constant. Therefore, over time, we expect the manager s utility to increase and then consumption to increase O w 1 R w 1 R w J(w,0) J S (w) J(w,1) J R (w) Figure 1: The Value Functions of Example 5.1. Example 5.2. Let p 0 = 0.5 and p 1 = with jump rate λ = 0.2; r = ; u(c) = c; f(b, a) = a b and σ(b) = 2.5(b ) for b [0, 4]; A = {0, 1} with h(0) = 0 and h(1) = 1. The value functions in Example 5.2 are depicted in Figure 3. There is one post-jump active interval [ w L 1, w1 R ] = [0, 0.54] and one active interval [w1 L, w1 R ] = [0.0032, 9]. Note that, different from the previous example, in Example 5.2, the investor shuts down the firm before P jumps when the promised expected utility is very close to zero. If the investor were to enforce positive effort level, the continuation utility would evolve as a diffusion process, as in equation (25). The probability that it hits zero is therefore very large when it is nearby. The only way to enforce zero expected utility is by retiring the manager, and in that case the investor would lose the chance to produce in the future good times. Hence, it is optimal to stop producing and wait for the good times. The optimal contracts can be read from Figure 4, which shows the policy functions in the similar way to the previous example. In this good shock case, the reward-punishment pattern given by 15

16 rϕ Low Price High Price Jump in Continuation Utility 0.02 O Input 4 0 w O w w 1 R w 1 R Low Price High Price Low Price High Price Consumption Drift of Continuation Utility 0.05 O w w 1 R w 1 R w w 1 R w 1 R Figure 2: The Optimal Contracts of Example 5.1. the upper-left chart is inverted compared with that in Example 5.1. The manager is rewarded if his utility level is lower than a threshold level and punished if it is higher than that level when the good shock comes. These results are described in more detail below. From the solid curve shown in the lower-right chart we can see that inside the active interval, the drift of W is positive. Although in the interval (0, ], the firm is in a shutdown phase and the continuation utility does not move before P jumps, the drift is positive because there is a reward attached to the unpredictable shock. Similarly, in the interval [9, 0.54), the drift is negative in a shutdown phase because the manager is punished upon the arrival of the shock. With the clean and complete solutions, we are able to simulate the optimal contracts. Here, I simulate an optimal contract in Example 5.2 for particular histories. The initial promised expected utility to the manager is 5 and t [0, ]. The idiosyncratic volatility Z is given by the top chart in Figure 5. I consider two histories of the price shock in which price drops at T 1 = 0.02 and T 2 = 0.06 respectively. The trajectories of the continuation utility, input and consumption of the 16

17 0.05 O w 1 L w 1 R w 1 R w J(w,0) J S (w) J(w,1) J R (w) Figure 3: The Value Functions of Example 5.2. two histories are shown in Figure 5 (with the dashed curves being trajectories in the first history and solid curves trajectories in the second history). The effort level is always 1 in the first history. In the second history, the price jumps too late, so that the firm involved in a shutdown phase, the time interval [0.035, 0.6], in which the effort level and input are zero. The idiosyncratic disturbance in this simulation has a small upward trend which helps to boost up the manager s performance. Therefore, W has a similar trend in the two histories if we net out the jump. The continuation utility jumps down (punishment) on impact, since the utility promise is relatively high and the impact is good. 5.2 Pay for Luck The numerical examples illustrate how managers may be rewarded or punished for luck. When a bad shock arrives, the manager is rewarded by a positive adjustment in his continuation utility if his continuation utility before the shock is relatively high, and is punished by a negative adjustment in continuation utility if his continuation utility before the shock is relatively low. Symmetrically, when a good shock arrives, the manager is punished if his continuation utility is relatively high, and is rewarded if his continuation utility is relatively low. To promise a continuation utility level, we have two ways to tie the continuation utility to the 17

18 rϕ High Price Low Price Jump in Continuation Utility 0.02 O Input 0 0 w O w 1 L w 1 R w w 1 R High Price Low Price High Price Low Price Consumption Drift of Continuation Utility 0.05 O w 1 L w 1 R w w 1 R O w 1 L w 1 R w w 1 R Figure 4: The Optimal Contracts of Example 5.2. exogenous shock before it occurs. First, the investor could add in a decreasing trend to W and attaching a reward when the shock arrives; Alternatively, she could add an increasing trend to W and attach a punishment on the arrival of the shock. Suppose that the current state is a low price state and there comes a bad shock. In addition, the manager is promised a high continuation utility. Then the optimal contract should reward the manager for the bad shock. Since the manager already has a high continuation utility and the wealth effect exists, the manager will become not sensitive to incentives easily. Namely, compensating his effort expense becomes more expensive and his continuation utility level will hit the right boundary of the active interval with fewer good signals in performance. Production in these good times must be terminated when that happens. But adding a decreasing trend to the continuation utility makes promotion harder. In other words, improvement in performance induces less reward in wealth. Therefore, before the price jumps the probability that the production terminates soon is lower. Consequently, the expected output level in good times increases. To balance out the 18

19 Z O 0.01 t W w 1 R O 0.01 T1 Shutdown T2 t B O T1 Shutdown T2 t C 0.02 O T1 Shutdown T2 t Figure 5: A Simulation of The Optimal Contract in Example 5.2 (History 1-dashed, History 2-solid). disutility generated by the current tough times, the investor promises a positive adjustment in continuation utility on arrival of the bad shock. Such a reward makes the manager less sensitive, so that with a larger probability the manager retires immediately or soon after the price increases and the production in bad times terminate. As a result, the expected output level after price jumps decreases. But since production in the good times is more profitable than that in the bad times, the benefit dominates the loss. If the continuation utility of the manager is relatively low before the price jumps up, the investor should punish the bad luck. Due to the low utility level, punishment is more problematic before the shock because the manager may become too poor to be punished. By adding an increasing trend to the continuation utility, the probability that the manager retires due to bad signals in performance soon is low and the expected output level in good times increases. The investor needs to punish 19