Irreversible Investment under Dynamic Agency

Transcription

1 Irreversible Investment under Dynamic Agency Steven R. Grenadier, Jianjun Miao, and Neng Wang December 2004 Abstract This paper provides a recursive contracting model to analyze the irreversible investment decision for a decentralized firm when the manager has private information about the investment cash flows. We show that compared to the full information benchmark, investment is delayed under asymmetric information. We also show that managerial risk aversion lowers the option value of waiting and increases information rents. The net effect of risk aversion is to delay investment. The model also predicts that according to the optimal contract, the owner punishes the manager by lowering his compensation over time when the manager does not invest, and rewards the manager when he makes the investment. The reward increases with the profits upon investing. Keywords: irreversible investment, information asymmetry, dynamic agency, recursive contracts, real options. We thank seminar participants in Boston University for helpful comments, especially, Simon Gilchrist, Larry Kotlikoff, Kevin Lang, Rasmus Lents, Bart Lipman, and Michael Manove. Graduate School of Business, Stanford University, Stanford, CA and National Bureau of Economic Research, Cambridge, MA, USA. sgren@stanford.edu. Tel.: Department of Economics, Boston University, 270 Bay State Road, Boston, MA miaoj@bu.edu. Tel.: Columbia Business School, 3022 Broadway, Uris Hall 812, New York, NY nw2128@columbia.edu; Tel.:

2 1 Introduction One of the most important topics in corporate finance and macroeconomics is the formulation of the optimal investment strategies of firms. The investment decision has two components: how much to invest and when to invest. The first is the capital allocation decision, and the second is the investment timing decision. The standard textbook prescription for the capital allocation decision is that firms should invest in an amount such that the marginal product of capital is equal to the user cost of capital. The standard theory for the investment timing decision is that firms should invest in projects when their net present values (NPVs) are positive. Recently, the alternative real options approach has been widely accepted. The real options approach posits that the opportunity to invest in a project is analogous to an American call option on the investment project, and the timing of investment is economically equivalent to the optimal exercise decision for an option. This real options approach is well summarized in Dixit and Pindyck (1994) and Trigeorgis (1996). Both the user cost theory and the standard real options approach fail to account for the presence of conflicts of interests induced by information asymmetries between the owner and the delegated manager. In most modern corporations, shareholders delegate the investment decision to managers, using managers special skills and expertise. In such decentralized settings, there are likely to be information asymmetries. In general, managers are better informed than owners about project cash flows. A number of papers in the literature provide models of capital budgeting under asymmetric information and agency. 1 The focus of this literature is on the first element of the investment decision: the amount of capital allocated to managers for investment. Thus, this literature provides predictions on whether firms over- or under-invest relative to the first-best no-agency benchmark. The focus of this paper is on the second element of the investment decision: the timing of investment. We extend the real options framework to account for the issues of information and agency in a decentralized firm. Analogous to the notions of over- or under-investment, our paper provides results on hurried or delayed investment. No agency conflicts arise in the standard real options paradigm since it is assumed that the option s owner makes the exercise decision. 2 By contrast, in this paper, a risk-neutral 1 See Stein (2003) for a summary. 2 While our paper focuses on the agency issues that arise from the conflict of interests between owners and managers, similar issues exist between stockholders and bondholders. 1

3 owner delegates the option exercise decision to a risk averse manager. The owner s problem is to design an optimal contract under asymmetric information. Absent any mechanism that induces the manager to reveal his private information voluntarily, the manager could have an incentive to lie about the true level of the project cash flows and divert cash flows for his private interests. For example, the manager could divert privately observed cash flows by consuming excessive perquisites or building empires. To overcome these problems, an optimal contract selects an investment rule and designs a compensation scheme to the manger so as to maximize the owner s value such that the manager follows voluntarily this investment rule and his behavior is incentive compatible. Since the manager is risk averse, the owner will provide partial consumption insurance to the manager in order to facilitate the manager to smooth his intertemporal consumption. An optimal contact must trade off insurance against incentives. Briefly summarized, unlike traditional real options models, our model incorporates at least two important real-world features into dynamic real option models: managerial risk aversion and informational asymmetry, using the optimal contracting approach. We show that these two features jointly generate new and interesting implications for the investment timing decision and the dynamics of managerial compensation. Our model implies that investment behavior differs substantially from the one implied by the standard real options approach with no agency problems. We show that the manager displays greater inertia in their investment behavior, in that they invest later than implied by the firstbest solution. This is because of the agency cost induced by informational asymmetry. The agency cost reflects the fact that the manager captures some rents from his private information about the investment project. By waiting longer, the owner saves more information rents to the manager. Importantly, managerial risk aversion affects the investment timing decision in two opposite ways. First, a larger degree of risk aversion lowers the manager s option value of waiting due to partial insurance or market incompleteness. 3 This result has been obtained by Miao and Wang (2004) in a dynamic incomplete-markets real options model without agency issues. Second, the information rents captured by the manager increases with risk aversion. Our extensive numerical exercises indicate that the latter agency cost effect dominates the former. 3 In the standard real options approach, capital markets are assumed to be complete. By the standard arbitrage argument, the option value and investment timing are independent of preferences (see Dixit and Pindyck (1994)). In the conracting framework, this result also holds true if the owner fully insures the manager, as shown in Proposition 2. 2

4 Thus, we find that investment is delayed and is delayed more if the manager is more risk averse. In addition to generating predictions on investment decisions, our model also provides predictions on the dynamic evolution of managerial compensation. We show that managerial compensation decreases over time before investment is made and increases when the manager invests. Moreover, the amount of increase in managerial compensation depends on the cash flows generated by the project. This increase in managerial compensation upon investment provides an agency based explanation to the observation that the manager is rewarded with bonus upon completion of successful projects. Finally, our model also predicts that firm value jumps upwards when the manager invests. This is consistent with the empirical finding that the announcements of unexpected increases in investment lead to increases in stock prices, as documented by McConnell and Muscarella (1985). Since our model allows for repeated interactions between the manager and the owner in a long-term relationship, the contracting problem becomes inherently dynamic and the optimal contract potentially depends on the history of the manager s reported cash flows. In order to keep the model analytically tractable, we use the techniques developed in the recursive contracting literature. 4 The key insight of the recursive contracting methodology is to capture the history dependence of dynamic relationship between the owner and the manager by using a state variable so that one can formulate the history-dependent optimal contacting problem as a recursive one. This important state variable is the promised value by the owner to the manager. The promised value describes the manager s lifetime utility from the consumption (or compensation) stream delivered by the owner. It is not only a forward looking variable, but also summarizes the histories relevant for contracting purposes. The paper closely related to ours is Grenadier and Wang (2004). They also analyze how informational asymmetry and agency conflicts distort investment timing decisions. Grenadier and Wang (2004) assume that the owner pays the manager only at the time of investment, and specify managerial compensation contingent upon the manager s announced cash flows of 4 See Abreu, Pearce and Stacchetti (1990), Green (1987), Spear and Srivastava (1987), Thomas and Worrall (1988, 1990) for seminal contributions. The recursive contract approach has been applied widely in macroeconomics recently, e.g., unemployment insurance (Hopenhayn and Nicolini (1997) and Shimer and Werning (2003)), taxation (Kocherlakota (2004)), and social insurance (Atkeson and Lucas (1992)). Albuquerque and Hopenhayn (2004) and Clementi and Hopenhayn (2002) study the optimal lending contracts by analyzing the conflict of interests between borrowers and lenders. See Ljungquist and Sargent (2004) for a textbook treatment on recursive contracts. 3

5 the project at the time of investment. In our setting, the owner and the manager interact repeatedly in a long-term relationship. The optimal contract requires the owner to pay the manager even if the manager does not invest, since the manager is risk averse and compensating the manager before his investment smooths his intertemporal consumption. Thus, the key predictions of our model rely heavily on managerial preferences for intertemporal consumption smoothing, induced by risk aversion. This leads to the next key distinction between our model and Grenadier and Wang (2004), who assume that managers are risk neutral and are protected by limited liability. Finally, unlike Grenadier and Wang (2004) who analyze the effects of both informational asymmetry and moral hazard on the investment timing decision, we focus on the role of informational asymmetry only. Our work also relates to the capital budgeting literature, particularly to Harris and Raviv (1996, 1998) and Bernardo, Cai, and Luo (2001). 5 Harris and Raviv (1996) apply a static costly state verification model to examine capital budgeting processes in a single-division firm. Harris and Raviv (1998) generalize that model to a multi-division firm. Both papers assume that all parties are risk neutral and that managerial compensation is exogenous. Bernardo, Cai, and Luo (2001) consider the capital allocation decision for a decentralized firm under both asymmetric information and moral hazard in a static model. They use the optimal contracting approach to jointly derive the optimal investment and compensation policies. Unlike the preceding papers, we analyze the (irreversible) investment timing decision in a dynamic framework. As in Bernardo, Cai, and Luo (2001), we derive managerial compensation as part of the optimal contract. However, our dynamic model has important implications for the dynamics of managerial compensation. The remainder of the paper is organized as follows. Section 2 presents and solves the optimal contracting problem in a static model. Section 3 introduces the dynamic setting and solves for the optimal contract using recursive methods. Section 4 analyzes the model s implications on investment decision and wage dynamics. Section 5 concludes. Proofs are relegated to an appendix. 5 See Harris et al. (1982), Antle and Eppen (1985), and Holmstrom and Ricart i Costa (1986) for related early work. 4

6 2 A Static Model We first solve for the optimal contract in a static setting, which provides intuition for the nature of dynamic contract analyzed in Section 3. Consider a decentralized firm. The risk-neutral owner of the firm has an investment project. The project costs I and generates a stochastic project value x, which is randomly drawn from a given distribution function F (x) over the interval [a, b]. Assume that F has a positive continuously differentiable density f > 0 and I (a, b). 6 The owner delegates the investment decision to the manager because the manager has special human capital. The manager decides whether to invest in the project or not. The manager is risk averse with an increasing, strictly concave and continuously differentiable utility function u. The manager and the owner sign a contract. The key assumption is that the manager has private information about the project value, and thus has incentives to lie in order to hide project value from the owner. An optimal contract induces the manager to truthfully reveal his privately observed project value. Moreover, the contract also ought to provide some insurance to the manager since the manager is risk averse and the owner is risk neutral. These two goals often come into conflicts. For example, a fully insured manager has no incentives to reveal his project value truthfully, when he has a high project value. We will show later that the optimal contract trades off incentive elicitation against insurance provision. Intuitively, one should expect that managerial compensation must be tied to the manager s investment decision and his reported project value, while partially help the manager smooth his consumption across states. The optimal contracting problem can be described formally as a mechanism design problem or a message game. By the revelation principle, we can restrict to the direct revelation mechanism in which the message space is the set of possible project values. The manager s investment decision is binary. The contract specifies an investment rule with a threshold value x such that the investment is made if and only if the reported project value is above the threshold x. Suppose that the manager observes the project value is x. If he reports a high value x x to the owner, the owner pays the manager a wage y A ( x) and the manager must make the investment 6 It will be clear below that the assumption on f ensures the existence of a solution under asymmetric information. The assumption on investment cost I rules out the uninteresting case where the project is either invested immediately or never invested. 5

7 and give the owner the reported project value. If the manager reports a low value x < x, then the owner pays him a wage y A ( x) and the manager does not invest. An optimal contract maximizes the owner s value (or firm value) subject to meeting the manager s participation and truth-telling incentive constraints. Formally, we formulate the contracting problem as subject to and max (y A,y B,x) v = x a x a y B (x)df (x) + u ( y B (x) ) df (x) + b x b x [ x y A (x) I ] df (x) (1) u ( y A (x) ) df (x), (2) u ( y A (x) ) u ( y A ( x) + x x ), x, x x, (3) u ( y A (x) ) u ( y B ( x) ), x x > x, (4) u ( y B (x) ) u ( y A ( x) + x x ), x x > x, (5) u ( y B (x) ) u ( y B ( x) ), x > x, x. (6) Equation (2) is the individual rationality constraint which describes that the participating manager has a reservation utility level v. Inequalities (3)-(6) are incentive constraints. For example, the left side of inequality (3) states that the manager with project value x x, will invest and receive compensation y A (x). The right side of (3) gives the utility for the manager if he lies and reports x x. Under such a scenario, the manager can hide the part of project value (x x) and receive compensation y A ( x). Incentive constraint (3) ensures that the manager has no incentive to lie to any x x. The other three incentive constraints rule out the manager s incentives to lie in other situations. For notational convenience, let d A (x) = x y A (x) denote the dividend payment to the owner, when the manager invests. The following lemma simplifies the incentive constraints. Lemma 1 If the manager does not invest, then he receives constant compensation y B, which is independent of the reported project value, in that y B (x) = y B ( x) = y B, for x, x < x. If the manager invests, then the owner receives constant dividend d A, which is independent of the reported project value x, in that d A (x) = d A ( x) = d A, for x, x x. Moreover, y B = y A ( x) = x d A. 6

8 The intuition behind the above lemma is as follows. Since the owner cannot observe the project value, the optimal contract must be designed in such a way that the owner s payoff may only depend on the manager s observable and verifiable investment/no-investment decision, and cannot respond to the manager s private information beyond what is conveyed by the managerial investment decision. If the manager invests, the payoff for the owner is d A (x) = x y A (x). If the manager does not invest, then the owner pays the manager compensation y B (x). Lemma 1 shows that both y B and d A are constant, consistent with our intuition. It is worth noting that the constancy of y B and d A does not hinge upon the manager s risk aversion and solely derives from incentive compatibility conditions. Finally, when the project value is at the threshold x, the manager is indifferent between investing and not investing, and hence receives the same compensation y B = y A ( x) = x d A. Using Lemma 1, we can simplify the optimal contracting problem as follows: max ( x d A) F ( x) + ( d A I ) (1 F ( x)) (7) (d A,x) subject to v = u ( x d A) F ( x) + b x u ( x d A) df (x). (8) Proposition 1 In the static model with asymmetric information, the optimal contract is as follows. (i) The optimal trigger x and the dividend payment d A satisfy the equation and the participation constraint (2), where x I = F ( x ) [ λu ( x f ( x d A) 1 ], (9) ) λ 1 = u ( x d A) b F ( x) + u ( x d A) df (x). (10) x (ii) For x < x, the manager does not invest, and receives constant compensation y B = x d A. For x x, the manager invests and receives compensation y A (x) = x d A. Before we explain the optimal contract described in this proposition, we first outline the efficient investment rule under symmetric information. Under symmetric information, there are no incentive constraints (3)-(6). One can easily show that the investment trigger is equal to the investment cost I, which is also the trigger value under the standard NPV rule. Moreover, 7

9 the owner fully insures the manager. That is, the owner offers a constant identical wage to the manager no matter whether he makes the investment. We now turn to the optimal contract under asymmetric information described in Proposition 1. Part (i) demonstrates that the investment trigger and compensation are jointly determined by equations (9) and (2). To better understand equation (9), we rewrite it as f ( x ) ( x I) = F ( x ) [ λu ( x d A) 1 ]. (11) The left side of this equation represents the marginal cost from an additional unit increase in the investment threshold. The cost arises because low type managers x < x do not invest at project values below x and thus the NPV ( x I) is lost. It is multiplied by f ( x ) which represents the probability of this cost. The right side of the preceding equation represents the marginal benefit. The benefit arises because the owner saves information rents λu ( x d A) 1 to each type x below x. This happens with probability F ( x ). Note that λ is the Lagrange multiplier associated with the participation constraint (8). It represents the shadow price of the reservation utility. Given the above discussion, we interpret the right side of (9) as information rents or agency costs due to asymmetric information. Managerial risk aversion (u (x) < 0) implies that these agency costs are positive, and thus the investment trigger is higher than the cost of investment ( x > I). Consequently, the investment threshold under asymmetry information is higher than that in the full information benchmark. This means that asymmetric information leads to underinvestment. The intuition is the following. When designing an optimal contract, the owner faces the trade off between providing insurance and eliciting incentives. He must offer information rents to the privately informed manager such that he will not lie. This generates agency costs and distorts investment efficiency. Note that if the manager is risk neutral, then investment will be efficient because the manager is equally capable of bearing risk as the owner does. Part (ii) of Proposition 1 demonstrates that the owner offers a constant wage y B to the manager if he does not make the investment. However, the manger is rewarded if he makes the investment. Moreover, the wage y A is positively related to the project value. Thus, wage provides incentives to the manager such that he tells the owner the true project value and makes the investment. In addition, the wage must also provide insurance to the risk averse manager. 8

10 Because of incentive problems, the manager does not obtain constant full insurance wage. We now turn to the question as to how managerial risk aversion influences investment decision. Unfortunately, we cannot derive a general result analytically for general utility functions. The constant absolute risk aversion (CARA) utility u (c) = e γc /γ is an exception, as shown in the following Corollary. Before presenting it, we define the certainty equivalent η ( x; γ) for a lottery offering the payoff of max {x x,0}, where x is drawn from the distribution F. 7 For CARA utility, it is immediate to show that { [ 1 γ η ( x; γ) = log F ( x) + ] b x e γ(x x) df (x), γ > 0, b x (x x)df(x), γ = 0. (12) Corollary 1 Suppose u (c) = e γc /γ. (i) If [1 F (x) + η (x; γ)]/f (x) is decreasing in x [a, b], then the optimal trigger value x is the unique solution to the equation x I = 1 F ( x ) + η ( x ; γ) f ( x. (13) ) Moreover, x is increasing in γ. (ii)the optimal compensation policy is given by y B = 1 γ log ( vγ) η ( x ; γ) for x < x, (14) y A (x) = x x + y B for x x. (15) Notice that for CARA utility, the investment trigger is determined by a single equation, independent of compensation and reservation utility. This is because CARA utility has no wealth effect. This feature is more important when we turn to the dynamic contract in Section 3. Thus, in most of analysis below we will adopt this utility specification. The intuition behind Corollary 1 is the following. When risk aversion is higher, the manager prefers to have smoother consumption. This insurance objective is in conflict with the truthtelling incentives. In order to elicit incentives, the owner must offer more information rents to the manager. Thus, agency costs are higher; that is, the term on the right-hand side of (13) is higher. This implies investment is distorted more in the sense that the investment trigger is higher. This intuition carries over for the constant relative risk aversion utility specification u (c) = c 1 γ / (1 γ), γ 0. We illustrate this point by numerical simulations. We choose a standard 7 Use u(η) = E [u (max {x x, 0})], where u is CARA utility. 9

11 uniform distribution over [0,1] and set the investment cost I = 0.5. We also set the reservation utility v = γ / (1 γ). That is, the manager receives a consumption level of 0.1 if he does not participate in the contract. 8 Figure 1 plots the investment threshold as a function of the risk aversion parameter. This figure indicates that the investment threshold increases with the risk aversion parameter. [Insert Figure 1 Here] In the next section, we turn to the dynamic setting. When the manager and the owner live in a dynamic environment with repeated relationship, two important distinctions from the static setting arise. First, there is an option value of waiting even in the absence of agency issues. Second, the owner can use an intertemporal punishment and rewarding compensation scheme to elicit incentives. We show that the interaction between the option value of waiting and the intertemporal incentive scheme gives interesting implications for investment timing. 3 Optimal Recursive Contract We begin this section by first describing the dynamic setting and formulating the problem in a recursive manner. We then provide the solution to the first-best full-information investment problem, as a benchmark for further analysis. Finally, we derive the solution to the recursive contracting problem under asymmetric information. 3.1 Dynamic Setting Unlike the static setting, the manager privately observes the project s cash flows x t in each period t = 0, 1,.... The project cash flows are independently and identically drawn from a distribution function F over the interval [a, b]. 9 If the manager invests at the cash flow value x in some period τ, the firm pays the cost I immediately, and the project generates an equal cash flow x in period τ and each period thereafter. This assumption simplifies the contracting 8 Note that we do not choose a fixed utility level, say 0, as the reservation value. Otherwise, the manager will not participate in the contract when his risk aversion is bigger than 1 since in any contract his utility is always less than 0 given u (c) = c 1 γ / (1 γ) specification. We rule out this situation in our numerical example. 9 Assume that F has a positive continuously differentiable density f > 0 and I (a/ (1 β), b/ (1 β)). It will be clear below that the assumption on f ensures the existence of a solution under asymmetric information. The assumption on I rules out the uninteresting case where the project is either invested immediately or never invested. The IID assumption is important to formulate the optimal contract recursively. 10

12 problem after investment. In Section 4.4, we will relax this assumption. We also assume that investment is irreversible in the sense that if the manager invests, he does not make any investment decision again and forgoes all future opportunities to draw high values of cash flows. As in the static setting, the owner delegates the investment decision to the manager. The risk averse manager has a time-additive expected utility given by E [ t=0 βt u (c t ) ], where β [0, 1) is the subjective discount factor. The owner is risk neutral and discounts cash flows according to the discount factor β. The owner does not observe the project cash flows, which are the manager s private information. However, the owner observes whether or not the manager has made the investment in the previous periods. As is standard in the contracting literature, we assume that at time zero, the owner makes a take-it-or-leave-it offer. The terms of the dynamic contract are contingent on histories of the manager s reports and investment status. Let s t = 0 if the investment is not made in period t and s t = 1 if the investment is made in period t. Let x t = (x 0, x 1,..., x t ) and s t = (s 0, s 1,..., s t ) denote the history of cash flows and the history of investment status, respectively. Since the investment is irreversible, s t satisfies the property that s τ = 1 for all t τ t if t is the minimum τ such that s τ = 1. By the revelation principle, we can restrict the manager s reporting strategy to be a function of the form x = { x t ( x t, s t)} t=0. The dynamic contract specifies an investment rule and a managerial compensation scheme contingent on the histories of reports and investment status h t = ( x 0, s 0 ; x 1, s 1 ;...; x t, s t ). 10 The optimal contract maximizes the owner s expected discounted payoffs subject to the manager s intertemporal incentive and participation constraints. As shown by Rogerson (1985) in a related two-period moral hazard model, an optimal dynamic contract depends on the history of reports. This history dependence makes the analysis complicated. In order to tackle this problem, we follow the recursive contract literature and create a state variable that summarizes the history of reports. 11 Then, we may convert the history dependent optimization problem into a recursive (Markovian) one. The newly created state variable is the manager s utility promised by the owner. Intuitively, the promised continuation utility captures what the manager cares for his future. After all, the manager s objective is to maximize his life-time utility. Thus, using current compensation and promised continuation 10 We will offer more detailed comments on this contract form in the next subsection. 11 See Spear and Srivastava (1987), Thomas and Worrall (1990) and Abreu et al (1990) for early contribution to this approach. 11

13 utility, the owner is able to deliver the manager s reservation value. 3.2 Recursive Formulation We now describe the dynamic contract in a recursive manner. Specifically, consider any date when investment has not been made before. Suppose at that date the manager is promised an expected lifetime utility level v. The dynamic contract is described as follows: 1. The manager observes a value of cash flows x drawn from the distribution F ( ) and then makes a report x to the owner. 2. The owner observes the report x and recommends the manager to follow a trigger investment rule. Specifically, there is a threshold value x, such that the investment is undertaken when the manager reports a high value x x; and the investment is not undertaken, otherwise. 3. If the investment is not undertaken, the owner offers the manager compensation y B ( x) and a promised continuation value w B ( x) for the next period. If the investment is undertaken, the manager obtains the true value of cash flows x and hands over the reported cash flows x to the owner in each period after investment. Moreover, the owner offers compensation yn A ( x) in the n th period after investment, n 0. Two comments on the contract form are in order. First, as in the static model described in Section 2, the dynamic contract specifies an investment threshold such that the investment is made if the reported cash flows are higher than this threshold and if there is no investment before. This investment rule is intuitive and related to the trigger policy in the standard investment model without agency issues, e.g., McDonald and Siegel (1986) or its discrete time variant described in the proof of Proposition 2 in the appendix. Thus, we are able to compare the investment policy in our model with that in the standard model without agency issues. Note that the investment trigger x may depend on the state variable promised utility v. In Section 3.4, we will show that it is a constant independent of v under the CARA specification. In Section 4.3 we will also consider the case where the investment policy is not written in the contract and follows a simple NPV rule often recommended in practice. We will contrast the resulting solution with the optimal contract. Second, after investment the owner knows that 12

14 the project cash flows are locked in at the reported value upon investment, even though he does not know the true value. 12 Thus, any non-flat reports after investment are irrelevant to the owner. Consequently, the owner will offer wages in every period after investment, contingent on the reported value x at the time of investment only. We solve the contracting problem by dynamic programming. Let P (v) denote the value function of the owner or firm value. The optimal contract solves the following problem: P (v) = x max (y B,w B,yn A, x) a + [ y B (x) + βp ( w B (x) )] df (x) (16) b x [ β n ( x yn A (x) ) ] I df (x) n=0 subject to v = x [ u ( y B (x) ) + βw B (x) ] df (x) + b a x n=0 β n u ( yn A (x) ) df (x) (17) and β n u ( yn A (x) ) β n u ( x x + yn A ( x) ), x, x x, (18) n=0 n=0 β n u ( yn A (x) ) u ( y B ( x) ) + βw B ( x), x x > x, (19) n=0 u ( y B (x) ) + βw B (x) β n u ( x x + yn A ( x) ), x x > x, (20) n=0 u ( y B (x) ) + βw B (x) u ( y B ( x) ) + βw B ( x), x > x, x. (21) The interpretation of the participation and incentive constraints (17)-(21) is similar to that given in the previous section. When x < x, the manager does not invest. The owner offers the manager compensation y B (x) in the current period and continuation utility w B (x). The 12 As an example, suppose that the contracted trigger value is 5. The manager s draw of cash flow value has been below the trigger level 5 from the initial period to period 9. Thus, there is no investment before (and including) period 9. Suppose that the manager draws a high value 7 of cash flows in period 10. However, the manager reports the cash flow value to be 6 in period 10. According to the contract, the manager invests in period 10, and hands over the reported cash flow value 6 in period 10 and each period thereafter. But he obtains the true value 7 in period 10 and each period thereafter. The owner does not observe the true value, which is 7. If the manager decides to change his reported cash flow to another false level, say he reports 5 in period 12, this new report in period 12 is payoff irrelevant to the owner. The wage in any period after investment is contingent on the report 6. 13

15 owner does not receive any cash flow, but obtains continuation firm value P ( w B (x) ). When x x, the manager invests. The owner offers the manager compensation yn A (x) in the n th period after investment. The owner pays the investment cost I at the time of investment and obtains truthfully reported cash flows x in each period thereafter. By the principle of optimality, summing up the owner s value under both investment and no-investment regions and integrating with respect to the distribution F give the owner s current value function P (v). 3.3 Full Information Benchmark As a benchmark, we analyze the case in which the manager and the owner have symmetric information about the cash flows. The optimal contract is the solution to the optimization problem (16) subject to the promise-keeping constraint (17). The following proposition summarizes the optimal contract. Proposition 2 The optimal contract under full information is characterized as follows: (i) The investment threshold x FB is the unique solution to the following equation: x FB (1 β)i = β b ( x x FB ) df (x). (22) 1 β x FB (ii) The managerial compensations, y B (x) and y A n (x), before and after investment are equal and given by y FB = u 1 ((1 β)v) for all x and n. (iii) The continuation value if investment is not made is given by w B (x) = v. Firm value is given by P (v) = u 1 (v (1 β)) 1 β b x (x (1 β)i)df (x) + FB (1 β)[1 βf (x FB. (23) )] The interpretation of (22) is as follows. The left side of equation (22) represents the marginal cost of waiting when increasing the threshold value x FB by one unit. This cost is the forgone NPV of the project, measured in the flow sense. The right side of (22) represents the option value of waiting for one more period to receive better draws x x FB. Equation (22) prescribes the firm to invest at the threshold value x FB such that the marginal benefit from waiting for one more period equals the marginal cost of waiting. Note that under full information, the manager is fully insured and obtains constant wages. Thus, he behaves in a risk neutral way and the investment threshold is independent of preferences. In fact, in the appendix we show that x FB is equal to the investment trigger under risk neutrality. 14

16 For much of the remainder, we use the CARA utility. With such a utility specification, the managerial compensation under first-best benchmark is given by y FB = log ( vγ (1 β))/γ. 3.4 Model Solution We now turn to the optimal contract under asymmetric information. We first simplify the incentive constraints and derive some general properties of the contract. Lemma 2 (i) The incentive constraints (19)-(21) can be replaced with the condition u ( y B (x) ) + βw B (x) = β n u ( yn A ( x) ) for all x < x. (24) n=0 (ii) Let d A n (x) = x yn A (x). Suppose u (c) = e γc /γ. If P ( ) is strictly concave and differentiable, then at the optimum y B (x) and w B (x) are independent of x for all x < x, and d A n (x) is independent of n and x for all x x, and n 0. The intuition behind this lemma is similar to that behind Lemma 1. Before investment, the manager does not receive any cash flows. Thus, offering a constant wage and a constant continuation value to the manager can induce his truth-telling incentives. The resulting lifetime utility of the manager is the same as that upon investment at the threshold value x, as shown in (24). Moreover, under CARA specification the incentive constraint (18) is redundant. This is because for the CARA utility the manager type variable x can be cancelled out in the incentive constraint (18). Consequently, an optimal contract cannot distinguish the manager upon investment and hence requires all types of manager to hand over identical dividends to the owner. By this lemma, without risk of confusion we may simply use d A, y B, and w B to denote d A n (x), y B (x), and w B (x), respectively. Let w A (x) = n=0 βn u ( y A n (x) ) denote the continuation value after investment. We are now ready to present our main result. Proposition 3 Suppose the assumptions in Corollary 1 hold. Then the optimal contract under asymmetric information is characterized as follows: (i) The trigger value x is the unique solution to the equation x (1 β)i = β 1 β η ( x ; γ) + [1 F ( x ) + η ( x ; γ)][1 βf( x )] (1 β)f ( x. (25) ) 15

17 (ii) The optimal compensation policy is given by y B = 1 γ log ( vγ (1 β)) η ( x ; γ) for x < x, (26) y A (x) = x x + y B for x x. (27) (iii) The continuation values are given by w B = v exp(γη ( x ; γ)) for x < x, (28) w A (x) = v exp(γη ( x ; γ) γ (x x)) for x x. (29) (iv) The owner s value function is given by P (v) = log ( vγ (1 β)) γ (1 β) + ( x I (1 β)) (1 F ( x )) + η ( x ; γ) (1 βf( x )) (1 β). (30) In general, an optimal solution to the contracting problem (16)-(21) is of the following form: x = x (v), w B (x) = w B (x, v), y B (x) = y B (x, v), y A n (x) = y A n (x, v), n 0. Note that the promised utility is a state variable, which summarizes the history of reports. The contract dynamics can be described in terms of the evolution of the promised utility. Specifically, suppose there is no investment before date t and the promised utility level is v t at the beginning of date t. Then the investment trigger at date t is given by x (v t ). If the manager truthfully reports a high value of cash flows x t x (v t ), then the manager invests and obtains cash flows y A n (x t, v t ), at the n th period after date t. If the manager truthfully reports a low value of cash flows x t < x (v t ), then the manager does not invest, and obtains compensation y B (x t, v t ) and continuation value w B (x t, v t ). In date t + 1, the starting promised utility level is given by v t+1 = w B (x t, v t ). The contract then has the same form as that described earlier for date t. Proposition 3 demonstrates under the CARA utility specification, the above contract form can be simplified significantly. First, the investment trigger is a constant independent of the state variable promised utility. This is similar to Corollary 1 due to the lack of wealth effect. Second, the compensation y B and continuation utility w B before investment are independent of cash flows x. Moreover, the dividend d A = x y A after investment is also independent of x. We next turn to the discussion of model implications. 4 Model Implications We first analyze the model s implication for investment timing. We next turn to the implication for the dynamics of managerial compensation. After that, we contrast our model with one in 16

18 which investment is assumed to follow the NPV rule. We finally outline some extensions. 4.1 Investment Timing Investment timing is determined by equation (25), which characterizes the investment threshold x. Its interpretation is similar to that for (9). The left side of (25) is the marginal cost from waiting (measured in the current period flow payoff). The right side of (25) contains two types of marginal benefit from waiting. The first term reflects the option value of waiting as in (22). 13 Unlike the full information model, risk aversion impacts the option value. From the classical utility theory, we know that the certainty equivalent η ( x ; γ) decreases in risk aversion γ (Mas- Colell et al. (1995)). Thus, risk aversion lowers the first term on the right side of (25), the option value of waiting, ceteris paribus. The second term on the right side of (25) is analogous to the term on the right side of (13). It reflects information rents saved by the owner from waiting. One can show that this information rents term is positive. These information rents represent agency costs due to asymmetric information. The owner rationally incorporates both the option value of waiting and managerial information rents in designing the optimal contract. Consequently, the optimal investment rule trades off the cost from waiting against the option value of waiting plus saved agency costs. Importantly, risk aversion affects both components of the benefit. As a reference for comparison, we first consider the special case in which the manager is risk neutral. When γ = 0, the owner does need to take risk sharing motive into account when he designs the contract. Therefore, even in the presence of asymmetric information, the first-best efficient investment decision is achieved 14. When the manager is risk averse, investment timing decisions reflect both the owner s motives to meet the manager s incentive constraint and to offer consumption smoothing insurance. We may show that Proposition 4 (i) Holding investment trigger x fixed, the option value β 1 β η ( x ; γ) is decreasing in γ and the agency cost term [1 F ( x ) + η ( x ; γ)][1 βf( x )] (1 β)f ( x ) (31) 13 We call this term the option value component because it vanishes in the static model in (13). 14 Mathematically, the certainty equivalent η is simply given by η( x ; 0) = b (x x) df(x) and the derivative x of η with respect to the investment trigger is given by η ( x ; 0) = (1 F( x )). Thus, the second term in (25) vanishes and the investment trigger under risk neutrality is the same as the value maximizing trigger under perfect information. 17

19 is increasing in γ. (ii) If E [x y x y] < 1 βf(y) βf(y) for all y [a, b], then there is underinvestment under asymmetric information when risk aversion is high enough; that is, x > x FB for γ large enough. Part (i) shows that risk aversion has two opposing effects. It lowers the option value of waiting, thereby speeding up investment. However, it also increases agency costs, thereby delaying investment. The overall impact of risk aversion and asymmetric information depends on which effect dominates. Part (ii) demonstrates that when risk aversion is high enough, the agency cost effect may dominate. This happens when a condition on distribution F is satisfied. One can verify that this condition is satisfied for exponential and uniform distributions. For general distributions and risk aversion parameter values, we cannot analytically characterize the effect of risk aversion and asymmetric information on investment timing. We thus turn to numerical analysis. We choose three specifications for the distribution F: normal distribution with mean µ and variance σ 2, uniform distribution over [0, b], and exponential distribution F(x) = 1 e αx over [0, ). Throughout numerical calculations in this section, we choose the subjective discount factor β = 0.98 and fix the investment cost at unity (I = 1). [Insert Figures 2-4 here.] Figure 2 graphs both the left side and the right side of (25), for a standard normal distribution. The left side (x (1 β)i) measures the net benefit from investment. The right side captures both the option value of waiting and agency costs. We plot the right side of (25) for various levels of risk aversion coefficient γ. The intersection in Figure 2 gives the investment trigger. Figure 2 shows that the investment trigger increases with risk aversion coefficient γ. This is because the agency cost effect dominates the option value of waiting. Similarly, Figures 3 and 4 graph the left side and the right side of (25) for exponential distribution and uniform distribution, respectively. Both figures also show that investment trigger increases with risk aversion. While there are two opposing effects of risk aversion on investment timing, our extensive numerical exercises show that the agency cost effect dominates the option effect. As a result, the investment trigger increases in the degree of risk aversion. We next turn to the effects of volatility on the investment timing decision. Consider normal distribution with mean zero and 18

20 variance σ 2. Figure 5 plots the investment trigger as a function of the volatility parameter σ and the risk aversion coefficient γ. It reveals that a higher volatility leads to a larger investment trigger for any value of risk aversion parameter. Moreover, the impact of volatility on investment is larger for more risk averse managers. This is because our numerical simulations indicate that both the agency cost component and the option value component increase with volatility, reinforcing each other and leading to greater incentives to wait. This effect is stronger for more risk averse managers. [Insert Figure 5 here.] 4.2 Wage Dynamics Unlike the static model described in Section 2, in a dynamic setting the owner can choose an intertemporal incentive scheme to punish or reward the manger such that the managerial incentives are aligned with the owner. We now describe the wage dynamics implied by the optimal contract. Let the initial promised utility to the manager be v 0 < 0 in period 0 (the utility function is negative exponential). According to the optimal contract described in Proposition 3, when the manager reports truthfully the initial value of cash flows x 0 x, the manager makes the investment and hands over x 0 to the owner. It follows from (27) that the owner pays the manager compensation y0 A (x 0) = x 0 x + y0 B, which is higher than yb 0, the compensation if the manager does not invest. Notice that managerial compensation y0 A depends on the project cash flow x 0. The manager who invests in a higher cash-flow project captures more information rents, ceteris paribus. The owner also promises the manager with a continuation utility value w0 A (x 0 ) = v 0 exp( γ (x 0 x η ( x ; γ))) = w0 B e γ(x 0 x ), x 0 x. (32) It is immediate to note that w0 A is higher than wb 0. Thus, the owner rewards the manager for his investment both in terms of current compensation and his future value (via continuation utility). In period 1, the contract starts with an initial utility level v 1 = w0 A. In all periods t 1, the owner offers a completely flat consumption profile to the manager in that yt A (x 0 ) = y0 A (x 0). When the manager reports truthfully a low value of cash flows in period 0, x 0 < x, the manger will rationally choose not to make investment according to the optimal contract. The owner pays the manager a constant compensation in the amount of y0 B = yfb (v 0 ) η ( x ; γ), 19

21 which is lower than y FB (v 0 ), the compensation under full information to yield the manager with utility level v 0. Moreover, to induce the manager to stay in the long-term contract, the owner also promises the manager a continuation utility given by w B 0 = v 0 exp(γη ( x ; γ)). Since the initial utility v 0 < 0, it is immediate to see that w B 0 < v 0. That is, upon a no-investment decision, the owner punishes the manager by lowering not only his current compensation but also his continuation utility. In period 1, the contract repeats with the manager s initial utility level v 1 = w0 B, which is promised by the owner to the manager in period 0. This reflects the recursive nature of the optimal contract: The promised utility in period 0 is the manager s starting utility level at the beginning of period 1. We next turn to period 1 when there is no investment in period 0. If the manager reports truthfully a value of cash flows x 1 < x in period 1, the optimal contract instructs the manager not to make investment. By (26), the owner pays the manager y B 1 = log ( v 1γ (1 β)) γ η ( x ; γ) = log ( w0 B γ (1 β)) η ( x ; γ) = y0 B η ( x ; γ), (33) γ which is lower than y0 B. Moreover, the owner promises a continuation value to the manager w B 1 = v 1 exp(γη ( x ; γ)) = w B 0 exp(γη ( x ; γ)), (34) which is lower than w0 B. Thus, the owner punishes the manager by lowering both managerial compensation and continuation utility over time during the period when the manger does not make investment. If there is no investment in period 0 and the manager reports truthfully a value of cash flows x 1 x in period 1, then the optimal contract instructs the manager to make investment. The owner pays the manager y1 A (x 1) = x 1 x + y1 B, and promises him utility w A 1 (x 1 ) = v 1 exp( γ (x 1 x η ( x ; γ))) = w B 0 exp( γ (x 1 x η ( x ; γ))). (35) Thus, the owner gives a current period reward x 1 x and promises a higher continuation utility to the manager if he makes the investment. In all periods t 2, the owner offers a fat compensation yt A (x 1 ) = y1 A (x 1). While we have extensively discussed the contract dynamics for periods 0 and 1, the preceding analysis applies for any period. We now summarize the dynamics of compensation and continuation utility. Let T be the first time at which the manager invests. Then, for all t < T, 20