Overinvestment and fraud

Available online at www.sciencedirect.com

Journal of Mathematical Economics 44 (2008) 484–512

Overinvestment and fraud

Marcelo Pinheiro ∗Belk College of Business, Department of Finance and Business Law, UNC at Charlotte, 9201,

University City Boulevard, Charlotte, NC 28223-0001, USA

Received 15 November 2006; received in revised form 24 September 2007; accepted 2 October 2007Available online 19 November 2007

Abstract

We analyze the interactions between two managerial tasks: investing and revealing information. We assume that a manager caninvest influencing the firm’s quality, then he reports this quality to investors. Whenever truthful reporting is not an equilibrium,the manager has incentives to overinvest relative to shareholders. Therefore, the potential for market manipulation is the key inunderstanding investment policy; it is the desire to manipulate prices that leads to inefficient investment. Also, more manipulationoccurs when the manager is in control, so prices are less informative. Finally, we show that the manager is better off with anexogenous reporting policy.© 2007 Elsevier B.V. All rights reserved.

JEL Classification: D82; D84; G14; G31

Keywords: Executive compensation; Investment policy; Effort; Market manipulation; Short-termism; Relative performance

1. Introduction

Corporate scandals have permeated the news in the past few years with an ensuing debate over the pros and cons ofexisting executive compensation packages. The governance failures at Enron, Tyco and WorldCom, among others, havecast a shadow over the corporate governance system in the US. More importantly, it cast doubt over the effectiveness ofusing stocks and options as part of executives’ pay packages with the intent of aligning their interests with shareholders’.

In the center of this discussion is the idea that short-termist behavior has been a major negative consequence ofpackages that are highly sensitive to stock price performance. At a first glance providing stock-based compensationmay align incentives as both managers and shareholders now have a common component in their objective function,but it may also create countervailing incentives as these agents may have different horizons. Extra sensitivity to stockprice may lead managers to engage in activities that maximize short-run value and neglect the long-run value of thefirm since managers are not guaranteed to be around for the long run.

We analyze these issues under a new framework modified to let the potential for information manipulation havespillover effects and influence investment policy. Investment is inefficient when the potential and temptation to manip-ulate information is greatest. More precisely, we show how managers end up investing over and above the shareholders’

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M. Pinheiro / Journal of Mathematical Economics 44 (2008) 484–512 485

optimal level.1 The driving force behind this result comes from two components: first, as the novel part of the model,the manager chooses not only how to invest but also how to reveal information to the market. Second, agents’ expectedhorizons differ. The result that we wish to emphasize here is the ensuing interaction between information revelation(or manipulation) and investment distortion. Inefficient investment takes place because of the desire to inflate pricesthrough information manipulation.

Our model has three types of agents: the manager, existing shareholders and investors (or new shareholders). Theexisting shareholders are present at the time the manager is hired but may need to sell their shares and exit in the nextperiod. On the other hand, the manager either sells his shares (cash in his pay package) or just quits the firm and takesan outside option; he is never around for the long haul.2 Finally, investors come into the picture as participants in themarket for the stock; they become the new shareholders that hold the firm until its final value is realized. Therefore,we see that managers have the shortest horizon, followed by the “existing” shareholders, and finally the investors. As aconsequence, stock price maximization is essentially the only objective of the manager while investors only care aboutfinal value and shareholders’ objectives are in-between. One can view short-termism in the current paper as a greaterconcern with stock price than is optimal for shareholders. It is this excessive weight put on stock prices by the managerthat leads to the incongruence of incentives between the managers and other agents. Alternatively, this could be viewedas a model of mere “fraud”, but, naming it short-termism is natural because it is the manager’s shorter horizon thatleads to the desire for fraud.

We focus on the comparison between the manager’s behavior and the behavior that would arise if shareholders wererunning the firm. To complete the analysis, we also compare these with value-maximizing action and the action thatmaximizes investors’ utility. Throughout the analysis we assume that investment is costly.3 In the model, the investorsare risk averse while the manager and the shareholders are risk-neutral.4 The manager has his compensation packagetied to the stock price performance and he faces a two-dimensional decision problem. First, he decides how much toinvest. The amount invested stochastically influences the firm’s quality.5 After the investment takes place, the qualityof the company is privately revealed to the manager (and to the shareholders). Then, he has to report the observedquality to investors. At this point, he may try to manipulate the market by issuing misleading reports. We assume thatif a misleading report is issued the SEC may find out and “punish” (sue) the manager for improper behavior. Hence,this strategy can be seen as costly lies. After the manager’s report has been issued, investors rationally update theirbeliefs and submit their demand schedule. A portion of the investors has relative performance objectives.6,7 They careabout their performance relative to a benchmark. Market clearing then determines prices. In the following period, thecompany’s payoff is realized and fully paid out to shareholders.

Surprisingly, depending on the strength of the investors’ relative performance objective, the manager may wantto invest MORE than it is optimal for the shareholders and reveal less information to the market. We see this as astriking result since most of the agency problems literature would predict shirking on part of the manager, whichtranslates into under-investment in our model’s language.8 Therefore, the potential for market manipulation (throughmisleading information) is crucial in determining managers’ investment policy. We also show that for low strength

1 As explained below, we can interpret the investment policy as an effort choice. However, we adhere to the “investment” terminology since it ismore in line with our motivating idea.

2 The model could be modified, at the expense of tractability, presentation and clarity, to incorporate the fact that the manager may not need tosell all his shares, or he may not quit. More precisely, we could allow for a situation where the manager is hit by a shock with some probability andin this case needs to decide whether to sell his shares or quit. With complementary probability, he stays on board. Qualitatively all results wouldfollow as long as the probability of this shock was higher than the probability of the shock that hits shareholders.

3 Investment is costly because it takes effort to choose the right policy. So, we model this as a private cost. We are abstracting from the “real” costof investing, the cost of the project. Results would be identical if we additionally assumed a fixed real cost to invest, and would be strengthened ifwe modeled projects with a higher likelihood of success as more expensive.

4 These investors are funds controlled by risk-averse portfolio managers.5 One should think about the investment policy as a choice among projects with differing likelihood of success, with “better” projects being more

costly.6 They can be thought of as representing institutional investors (mutual or hedge funds).7 Later we further generalize the model by allowing for the presence of so-called arbitrageurs, i.e., investors that are fully informed about the

state of the world. Since this seemingly simple change to the model creates a lot of difficulty in obtaining closed form solutions and interpreting theresults, it is left for the Appendix E. We show that the results are in essence the same as the ones presented in the main text.

8 The investment policy can be interpreted as costly effort choice.

486 M. Pinheiro / Journal of Mathematical Economics 44 (2008) 484–512

of the relative performance objective the traditional result obtains, i.e., managers would like to under-invest (shirk)relative to shareholders optimal investment level. So, the interesting and novel part of the model is the interactionbetween relative performance objectives, information revelation (market manipulation) and investment policy.

Furthermore, we show that the manager himself is worse off with a discretionary reporting policy, when comparedwith a situation where the reporting policy is exogenous (truthful revelation of the state of the world). And, for agiven investment level in the first stage, the probability of observing lies in equilibrium is increasing in the relativeperformance parameter.

It is important to stress that the results of this paper hold under both voluntary and mandatory disclosure rules. Undermandatory disclosure we have the exact current model: the manager must disclose his information, but can lie. Undervoluntary disclosure, the model works much in the same way because, as long as investors know that the manager hasprivate information, a non-disclosure event is interpreted as a bad signal, hence such event never happens. When themanager wants to report a good signal he must disclose (make a report) and when he wants to report a bad signal he isindifferent between disclosure and no-disclosure.

It is clear that, since we assume that both the manager and the shareholders are risk-neutral, the interest divergenceproblem is not generated by the need to share risk. The payoff asymmetry that delivers our result emanates from thefact that shareholders have a potentially longer horizon than the manager.9 We do not address the question of what isthe optimal compensation package for the manager but rather analyze the effects of existing packages that tie executivecompensation to stock price performance.10

As mentioned above, part of the investors, to be termed institutional, has a relative-performance-type objectivefunction. This objective captures the fact that mutual or hedge funds are compensated based on a comparison toa benchmark. Palomino (1999) shows that these objectives obtain if investors compare performance across fundswhen deciding how much to invest in each fund. We show how the relative performance objective helps determinethe strength of demand and how this influences the optimal reporting policy and investment choice. Earlier empiricalevidence provides us with support for this assumption by showing that fund managers may have this type of objective;11

because either the fund uses a “fulcrum fee”12 and managers get a portion of it, or the fund managers themselves haverelative performance contracts.13

Alternatively one could give a different interpretation to the model under which these investors would be employeeswith keeping up with the Joneses utility functions. That is, the employees would have utility functions that dependon a reference consumption level as well as their own level, where the relevant reference group would be the otheremployees in this company. For examples, applications and empirical support of such utility functions see, for instance,Gali (1994); Shore and White (2003) and Pinheiro (2007a) and references therein.

We also lay out some of the empirical implications of our model and some supportive evidence from previous work.Finally, the appendices present a further generalization of the model that allows for the presence of investors that arefully informed about the state of the world. Throughout the paper we focus on the “investment” interpretation, but thereader is free to think of it, interchangeably, as “effort”.

The rest of the paper is organized as follows: Section 2 summarizes the related literature and places our paperin context, stressing its main contributions. Section 3 describes the model in greater detail. Section 4 analyzes thecommunication stage of the game. We fix an investment level chosen by the agent in charge and analyze the ensuing

9 As it will become clear when the model is presented, even when the shareholders are choosing the investment level we still have a manager incharge of the reporting policy. And, the manager does not incur any cost. The idea behind this assumption is that the task itself is not costly, butchoosing its optimal level is; the decision making process is costly. This assumption is made to maintain manager’s and shareholders’ payoffs assymmetric as possible.10 Investment directly translates into the ex-ante probability of the company being of a good type. In order for investors to be able to update their

beliefs, they need to know this probability, hence the investment level. In this case, the design of a contract that implements shareholders’ firstbest would be trivial. If we were to address the optimal contract design, we could additionally assume that investment, although observable, isnon-contractible. However, this is not the objective of the paper.11 See, among others, Elton et al. (2003); Brown et al. (1996) and Palomino (1999).12 A fulcrum fee is an incentive fee that is centered on a benchmark. The fee is higher for performances above this benchmark, and lower for poorer

performances, with gains and losses being symmetric.13 There is also an extensive analysis of the theoretical issues involved in relative performance contracts. For instance, Eichberger et al. (1999)

analyzes the benefits (and potential shortfalls) for fund owners of rewarding their fund managers based on absolute and relative performancemeasures.


equilibrium in the market for the company stock. In other words, we analyze how much information is revealedto the market, how the investors respond and what the equilibrium price is. In Section 5, we solve for the optimalinvestment policy, given our solution for the communication stage, under different assumptions concerning who isin charge. We look at the problem from four different viewpoints: manager, shareholders, investor and an outsideentity that maximizes the firm’s final value. In Section 6, we put the two stages together in an equilibrium result,point out the potential for multiplicity of equilibria and discuss the main results concerning investment choice, theassociated information manipulation and the manager’s well-being under different reporting structures. We also discussthe empirical implications of the model. Section 7 concludes.

2. Related literature

Our main result can be seen as one of overinvestment or excessive effort: the manager is investing (through effortexertion) in projects that attach too high a probability to the good state of the world, from shareholders’ perspective. Hedoes not incur in the optimal trade-off between cost and return. More importantly, lies/misreports are the motivationbehind this overinvestment, and, furthermore, there is a direct connection between these results and the institutionalinvestors’ objectives as relative performance maximizers. Viewed in this way the paper can easily be compared anddifferentiated from the main strand of literature on investment.

Stein (2003) provides a superb and condensed view of the relevant progress made so far in the investment liter-ature as it relates to agency and information asymmetry problems. The main divergence between this literature andour paper is that in our paper we allow for information manipulation to interact with the objectives of investors andmanagers’ investment, showing that overinvestment is more pervasive during misreporting periods due to the pres-ence of relative performance. Our conclusion is in stark contrast with the initial idea of the existing literature, thatis, our model predicts excess effort/investment and not insufficient effort/investment, originating from the desire tolater manipulate prices. The literature on empire building also predicts overinvestment but for completely differentreasons, and as astutely pointed out by Stein: “...it would be wrong to conclude that empire building tendencies nec-essarily leads to an empirical prediction of overinvestment on average.” Finally, as he points out overconfidence canalso lead to a form of overinvestment, but again, the mechanism is distinct from the one discussed here, resorts tobehavioral explanations and the typical result is that, for overconfident managers, investment is more responsive tocash-flow, so there is overinvestment only when there is excess cash-flow. So, again, the results are not of unconditionaloverinvestment.14

Bebchuk and Stole (1993) argue that if managers are concerned with short-run stock price and the marketcan observe investment but not its productivity, overinvestment occurs. We can draw a direct analogy with ourpaper: here, investment is observable, but the signal concerning its productivity is privately observed by man-agers and this leads to overinvestment. The main qualitative difference is that in the current paper the managermay also manipulate the stock price through misleading information revelation, not only through his investmentpolicy. The mechanism driving the results is also different. In Bebchuk and Stole (1993), the manager overin-vests to signal that the project is highly profitable. Here, he overinvests because it facilitates market manipulation.Similarly, Stein (1989) shows that the desire to achieve high stock price induces managers to behave myopi-cally, inflating current earnings at the expense of future earnings (through forsaken/liquidated good investments)and, therefore, hurting the firm in the long run. It is essentially a model of costly earnings manipulation wherethe investment decision is the mechanism that is used to attempt manipulation. And, it is actually a result ofunder-investment (as good investments are liquidated or not undertaken). Here, manipulation is tried via fraudu-lent information that is not directly tied to investment. Stein’s results are not of overinvestment but of short-termist(and actually under-) investment so he does not analyze the links between fraud, overinvestment and relative perfor-mance.

In our model, relative performance affects prices much in the same way as would biased variance forecasts (orreduced risk aversion parameter). Prices are still expected discounted values, but the discount rate must be cleverlychosen to incorporate relative performance objectives. If one interprets this effect as a form of market inefficiency, thenthis model draws a relation between investment policy, inefficiencies and manager’s horizon, much as in Stein (1996).

14 See Heaton (2002) and Roll (1986) for the theory and Malmendier and Tate (2005) for the empirical application.


In Stein’s paper, the inefficiency comes from the fact that investors have biased forecasts concerning the expectedvalue of the firm, but higher-order moments are formed rationally. The converse happens here as the effect on prices isthrough high order moments (variance). However, the effect of relative performance is a rational result of the structureof contracts faced by fund managers. Therefore, we could see our result as a rational way of modeling a phenomenonthat Stein (1996) obtained by means of irrationality/inefficiency. Also, Stein (1996) does not allow for any form ofmarket manipulation through information revelation.

This paper can also be related to Faure-Grimaud and Gromb (2001). They show the importance of giving incentives,via price informativeness, to a large shareholder to exert value-increasing effort. Our results point out to a caveat insuch analysis: the sensitivity of prices to effort increases incentives to exert effort (invest), however it does it so much asto lead to an effort choice above optimal-overinvestment. This result is due to the fact that the price informativeness isunder manager’s control via his reporting policy. And, under some policies, company stock prices may be too sensitiveto investment/effort choice.

Pinheiro (2007b) uses a similar model but concentrates on the conflicts of interest of an informed trader whenhe has to reveal information, that is, this paper builds on that by generalizing the model to allow for an interactionbetween information and investment in an equilibrium framework and by allowing (see Appendix E) for the presenceof arbitrageurs.

Finally, building on these papers, the current paper can be seen as showing how the presence of relative performancecontracts for portfolio managers may exacerbate the problem of distorted investment policy when firm managers areconcerned with stock prices.

To the best of the author’s knowledge, this is the first paper to address investment policy in conjunction with a“reporting policy choice” when market participants have relative performance objectives. The feedback between thetwo choices proves to be of importance to understand short-termist behavior, and more so when fund managers haverelative performance contracts—the presence of relative performance contracts is what makes the interaction betweenreporting policy and investment non-trivial.

3. The general framework

In this section, we start with a brief overview of the timing in the model and then we present it in more detail.We focus mainly on two different situations concerning the control over the investment choice: the manager or theshareholders choosing the projects. Therefore, we sometimes refer to the agent responsible for the choice genericallyas the insider.

3.1. Timing and interpretations

We now briefly describe the timing of the model. At t = 0, a manager (M) accepts employment at a firm that giveshim a reward function solely based on the price that will prevail in the market. Existing shareholders of the firm havea potentially longer horizon than the manager. With probability η a shareholder is hit by a liquidity shock, in whichcase he sells the shares in the market, otherwise he waits to collect the final payoff of the company. For simplicity,we assume that the manager is given a share on the company and that each shareholder has one share as well. Wefurther assume that the manager may quit the firm and pursue an outside option valued by him at L. Both manager andshareholders are assumed to be risk-neutral.

The company is ex-ante extremely profitable, but its quality is uncertain. It can be either good or bad. If it is bad,it has zero expected value and positive variance. If it is good, it has positive expected value and lower variance. Themanager and shareholders have privileged information about the quality of the company’s stock. On top of that, theyhave the ability to affect the probability of the good state by choosing the optimal investment project. Investment takesplace at t = 1. As mentioned, investment is costly because it takes effort to choose the optimal project and manage it.So, we model this as a private cost. Results would be qualitatively the same if we additionally assumed a “real” costto invest, as long as the firm can borrow freely to finance it.

At t = 2, the manager has the option to quit the firm or to sell his share in the stock market. The idea behind thisframework is that of vesting of shares (or options) granted to executives coupled with a liquidity need. He needs moneyat t = 2, so he either quits or sells his share; he cannot stay aboard and retain his share until the final value of thecompany is paid out. If he quits, he is not entitled to the shares since they would not have vested. Results would be


qualitatively the same if we did not allow for an outside option and/or if we allowed the manager to stay on board forthe long run.15

Before any trading takes place, the manager has to issue a public statement about the quality of the company. Hefaces conflicting interests: if he lies to pump-up the stock price, he may incur a cost (e.g., lawsuit costs from SEC orreputational costs); but he wants to get the maximum profit from the sale of his shares. Agents are not irrational, so theyanticipate this and form beliefs about the relevance of his report accordingly. Given their beliefs, they submit limit ordersand market clearing determines the stock price. We assume that, with some given probability p and independently ofall other random variables, the manager is caught lying. In this case, he incurs a monetary punishment of K. In orderto simplify matters, we only use C = pK, the expected costs of lying. The company pays off its final value in the lastperiod, t = 3.

3.2. The model

Let V denote the company’s final value. Its distribution depends on the realization of a binary random variable. Thecompany is good with prior probability of ν0 > 1/2 and bad with the complementary probability.16 If the company isgood, it has expected value E[VH ](=: E1) and variance σ2

H .17 If the company is bad, then the first and (uncentered)

second moments areE[V L] = 0 < E1 andE[(VH )2] < E[(V L)

2].18 The interpretation for these distributional assump-

tions has to do with the format of investors’ preferences, assumed to be of mean-variance type. With mean-varianceutility, a random variable with higher variance and lower expected value is inexorably worse.

We define the ex-ante variance by σ20 , which we derive next. First note that:

E[V ] = ν0E[VH ] + (1 − ν0)E[V L] = ν0E1,

E[V 2] = ν0E[(VH )2] + (1 − ν0)E[(V L)

2].

So

σ20 = E[V 2] − (E[V ])2 = ν0E[(VH )

2] + (1 − ν0)E[(V L)

2] − (ν0E1)2,

now we add and subtract ν0E21 to get

σ20 = ν0{E[(VH )

2] − (E1)2} + (1 − ν0)E[(V L)

2] + ν0(E1)2 − (ν0E1)2

= ν0σ2H + (1 − ν0)σ2

L + ν0(1 − ν0)(E1)2 (1)

Note that the derivative of the ex-ante variance with respect to the prior probabilities is negative.19 Also, notice thatσ2

0 > σ2H . This property will be useful later and is valid for any ex-ante probability.

The manager (or shareholders) may actually change the probability of the good state, by choosing differentinvestment projects. If he wants the company to have a probability ν of being good, he must incur a (private) cost

Q(ν) = κ(ν − ν0)2, (2)

15 As long as we maintained the assumption that the manager is more likely to be hit by a shock than the shareholders. We need him to have ashorter expected horizon than shareholders.16 As we will see later, the restriction on the possible values of ν0 is needed in order for prices to be increasing in ν0 (a desirable characteristic,

given that ν0 stands for the probability of the good state of the world). More generally, we need the demand functions of the investors, who are ofmean-variance type, to be increasing in the probability of the good state.17 All values are per-share.18 Note that σ2

L > σ2H iff E[(V L)

2] − E[V L]

2> E[(V H )

2] − (E1)2, i.e., E[(V L)

2] > E[(V H )

2] − (E1)2, so E[(V H )

2] < E[(V L)

2] is sufficient for

σ2L > σ2

H .19 It is equal to

σ2H − σ2

L + (1 − 2ν0)(E1)2 < 0,

where the inequality follows since ν0 > (1/2) and σ2H < σ2

L. This holds for any ex-ante probability bigger than (1/2).


to be seen as the cost of analyzing all existing projects, picking and managing the appropriate one. The cost alsoincreases with the quality of the project picked. This assumption is meant to incorporate the idea that better projectsmay be harder to implement and manage.

At t = 2, the manager releases a report: “good” (g) or “bad” (b).20 Technically, there will not be any b reports inequilibrium since the manager will quit in that instance. At this point the manager might be caught lying and he takesinto account the resulting expected cost, C, when he makes a decision. By lying he potentially increases the price forwhich he sells his share, however, he expects to incur costs.

At t = 3, the value of the asset is realized. Unless the manager is caught lying, the true state of the world is neverknown with certainty by the public. To make this assumption internally consistent, we restrict the support of thedistributions (support of V L) ⊆ (support of VH ).

As mentioned before, we have two types of investors: institutional investors (mutual and hedge funds), I, andothers, O.21 They both update their beliefs after the manager’s report.22 They only differ in their objectives, namely,institutional investors are those that we mentioned would have relative performance objectives. We denote by EI andEO the expected value of the company’s share for I and O, respectively, after they observe the report. Similarly, σ2

I andσ2

O stand for the updated variances. Given these beliefs, we are able to characterize the investors’ demands. We firstpostulate their utility functions:

UI = EI [wI3 − γW3] − 12ρVarI [wI3 − γW3]; (3)

UO = EO[wO3] − 12ρVarO[wO3], (4)

where wi3 denotes final wealth, γ ∈ [0, 1) is the parameter measuring the strength of relative performance objectivesand W3 is a benchmark of comparison (measuring the performance of other institutional investors). We define thebenchmark as

W3 =[

1

N

∑i ∈ I

αi

](V − PR) =: α(V − PR). (5)

We use N as the cardinality of the institutional investors’ set, αi as the amount invested in the stock, P to representthe price of the stock and R as the gross interest rate. The parameter γ is meant to capture the idea that fund managerscare about their relative performance. Because either the fund uses a “fulcrum fee” and they get a portion of it, or, moredirectly, they have a contract that pays depending on their relative performance. Even without any such mechanisms,if they are rewarded based on the size of their fund and consumers invest more in funds that perform better, a similarobjective obtains.23 Therefore, their variable of interest is not final wealth (final value of their investment fund), butfinal wealth relative to other funds’ final wealth. In essence, this reflects the idea that they are only rewarded if theydo better than their peers. As mentioned in the introduction, an alternative formulation would model these investors asemployees of the company, and this utility function specification would reflect peer-group effects akin to a keeping upwith the Joneses story.

We assume that there are only two assets available, the risk-free and the company stock.24 Hence, final wealth canbe written as

wi3 = (wi1 − αiP)R + αiV , for i ∈ {I, O}, (6)

where wi1 stands for initial wealth.

20 Lowercase denotes messages and uppercase the true state of the world.21 In Appendix E, we allow for a third type: fully informed arbitrageurs.22 In the present setting, price will only reflect information already possessed by agents. Hence, there is no need to further condition on price when

updating beliefs. However, in Appendix E, when we allow for arbitrageurs, we will need to make an extra assumption to maintain the tractability ofthe model.23 See for instance Palomino (1999).24 This is basically an immaterial assumption made only for convenience. Qualitatively, all results would follow even if we allowed for multiple

risky assets.


With the assumed utility functions, we get the following demand functions for the risky asset:

αI = 1

1 − γ

[EI − PR]

ρσ2I

; αO = [EO − PR]

ρσ2O

. (7)

Note that γ ends up being directly related to the strength of the institutional investors’ demand, and hence, may beseen as a measure of their trading aggressiveness. In total, there is a supply of m shares in the market25 and institutionalinvestors represent a measure 1 − λ of the market while the rest constitutes a measure λ. Prices are determined by themarket clearing condition:

λαO + (1 − λ)αI = m. (8)

Alternatively, we can, more realistically, assume that institutional investors affect prices and take this effect intoaccount when maximizing their utility function; while still assuming that the rest of the investors behave competitively.Under this modified framework one can show that most of the structure of the model and results are, qualitatively,unaltered, however more algebraically involved. Hence, we omit this from the paper.26

We could have allowed for a degree of asymmetry across institutional investors’ pay packages as well. Moreprecisely, we can assume that each institutional investor has a different relative performance parameter, γi. One caneasily show that the demand for each institutional investor would then be

αIi = [EI − PR]

ρσ2I

+ γi

1 − γ

[EI − PR]

ρσ2I

, (9)

with γ =∑Ni=1γi/N. So, their per-capita demand becomes

αI = 1

N

N∑i=1

αIi = 1

1 − γ

[EI − PR]

ρσ2I

. (10)

With this change, we would observe asymmetric behavior among institutional investors, in line with what is observedin reality, without losing any of the model’s results. All results to be discussed would follow in the same fashion withthe slight modification that we should replace the parameter γ with the average measure of relative performance,γ = (

∑Ni=1γi)/N.

Next, we postulate the main assumptions on the parameters of the model. The following one deals with theprofitability of quitting the job. The outside option value is assumed to satisfy

L + C > P(0,E1, σ2H ) = 1

R[E1 − mρσ2

H ], (11)

L + C < limγ→1

P(γ, ν0E1, σ2I ) = 1

Rν0E1. (12)

The interpretation of these conditions is intuitive. The first inequality says that when agents have optimistic beliefsand there are no relative performance objectives, if someone tries to sell m shares of the company for L + C, themarket will not clear (there will be excess supply). Note that this would happen to be the case if the beliefs wereex-ante beliefs. Alternatively, it simply means that, in this situation, it would be better to quit the firm than to lie andsell the shares running the risk of being sued. The second part of the assumption guarantees that as the degree ofrelative performance becomes higher and the institutional investors have beliefs equal to ex-ante beliefs, there wouldbe excess demand for the company stock if the price charged was L + C. This would clearly be true if the beliefs werethe ones in the good state. Alternatively, it says that, in this case, quitting the job is not optimal even with law suitcosts.

25 As mentioned before, the manager holds 1 share, and we assume that there are m − 1 other shares outstanding. Without loss of generality, weabstract from the shares that will be sold by liquidity-strained shareholders. We further discuss this simplification at the end of the section.26 A proof of the statement is available from the author upon request.


We also assume that the cost parameter κ satisfies

κ > max

{(E1)2ρm(1 − γ)

R(1 − γλ),C + (E1)/(R) − (ρm(1 − γ))/(R(1 − γλ))(σ2

H − σ2L − (E1)2)

2(1 − ν0),

(η(L + C))/(ν0) + ((1 − η)E1)/(R)

2(1 − ν0)

}. (13)

This guarantees that solutions to the investment project choice problem are interior, i.e., the probability of the goodstate is set above ν0 and below 1.27

As mentioned before, shareholders have objectives that are different from those of the manager. They are alsorisk-neutral but with probability η a shareholder is hit by a liquidity shock, in which case he sells the shares in themarket, otherwise he waits to collect the final payoff of the company. We assume that if the shareholder was to choosethe project, he would face the same cost as the manager. Also, it is important to note that, in order to simplify matters,we abstract from the effect that the liquidity shock has on the supply of the company’s share. Ideally, one should noticethat if a shareholder is hit by a liquidity shock the supply increases to m + 1. This simplifying assumption can be seenas an approximation for the case where m is large enough, so that an extra share has basically no effect on prices, orthe case where the probability of a liquidity shock is small enough. However, to incorporate this effect would poseno major analytical problems. Since shareholders and the manager are risk-neutral, they would only care about theexpected value of the price when deciding on investment and reporting policy. So, the supply that would matter isthe expected one, m + ηs, where s represents the number of shareholders, assuming liquidity shocks are i.i.d. acrossshareholders. Then, since we assumed that the liquidity shock is independent of the asset’s return distribution, all theanalysis would follow just by redefining the supply of the asset as m1 := m + ηs.28

The exact format of manager’s and shareholders’ payoffs depend on the equilibrium in the communication stagegame. We solve the model by first characterizing this stage, and then, given this solution, we go back to solve for theoptimal investment policy. Since there is a non-trivial feedback between the two decisions, we need to carefully puteverything together in a “general” equilibrium. We do this in the next-to-last section.

4. Communication stage game

This section is similar to the work developed by Pinheiro (2007) with the exception that here the manager determinesthe profitability of the firm, so we generalize that work by endogenizing the profitability of the firm allowing for aninteraction between investment and information revelation and by adding aribtrageurs to the model (see Appendix Efor the latter). We assume that γ is common knowledge, so there is no reason for investors to hold different beliefs(EI = EO =: E and σ2

I = σ2O =: σ2). However, we have not characterized how beliefs are updated. This clearly

depends on how M reports his information. Assume that his strategy and investors’ updating leads them to believe thatPr(G|g) = ν. In this case, prices are given by

Pν(γ) := 1

R

[νE1 − mρσ2

ν

(1 − γ)

1 − λγ

], (14)

where σ2ν is the ex-ante variance if beliefs attach probability ν to the good state, that is,

σ2ν = νσ2

H + (1 − ν)σ2L + ν(1 − ν)(E1)2. (15)

We characterize the actions of the manager depending on the relative performance parameter. In what follows leadingup to Proposition 1, we show that there exists a region of the γ-parameter space such that, for any γ in this region,it is ex-post optimal to lie in the bad state. The model described here can be seen as an example of a signaling gameplayed by the informed trader and the investors in periods t = 1 and t = 2. The manager is the sender (or leader) andthe investors are the receivers (or followers). The quality of the company is to be interpreted as the type of the sender,

27 Given the assumptions of the model, the second order conditions are always satisfied.28 Obviously, the realized and expected prices would differ so the actual payoffs would not be as expected. But, when the payoff realizes they have

no further action to be taken, so the analysis is unaltered.


even though it really concerns the company and not the informed agent. So, the sender has private information abouthis type and sends a message to the receivers, who must update their beliefs upon observing this message and thentake actions. The message space is limited to g or b. Since we have multiple receivers, we concentrate on symmetricequilibria. The equilibrium concept to be used here is the standard Perfect Bayesian Equilibrium, which guaranteesthat the receivers’ strategies are optimal responses to the sender’s action (message) given their beliefs. Their beliefsare updated using Bayes’ rule and the equilibrium strategy of the sender. Finally, the sender’s message maximizes hisutility given the responses of the receivers.29

For Proposition 1 we fix the initial beliefs, which will be investors’ priors, at ν. As we clarify in Proposition 2, ν

actually depends on the action taken by the agent choosing the projets. The choice of project in turn depends on whichregion of the parameter space we are. Due to this endogeneity, the equilibrium is more complicated. For now, we justgive the main result to motivate the next section. Later, when we present the two choices together, we take care of theendogeneity.

Proposition 1. For any investment project choice ν ∈ (ν0, 1) in the first stage of the game, the second stage (commu-nication) has a unique Perfect Bayesian Equilibrium. The equilibrium is determined by two thresholds, γ1 < 1 andγ2 ≥ 0, with

Pν(γ1) = L + C; P1(γ2) = L + C. (16)

(i) If γ ≥ γ1, M always issues a g report for the company stock. Beliefs are unchanged at ν. (Pooling.)(ii) If γ2 < γ < γ1, there are no incentives to always lie. However, there are not enough incentives to be always

truthful either. Hence, the manager uses a mixed strategy. (Semi-separating.) More precisely, in the case of goodnews he always says g, and in the case of bad news he says g with probability p1, defined as

p1(γ) = ν

1 − ν

1 − ν(γ)

ν(γ), (17)

where ν(γ) is such that

Pν(γ)(γ) = L + C, ∀γ ∈ [γ2, γ1]. (18)

In this region, we have P(G|g) = ν(γ), and if there is a bad report then agents attach probability one to thebad state, P(G|b) = 0.

(iii) Finally, if γ ≤ γ2, the manager always tells the truth and is fully believed by investors. (Separating.)

Proof. See Appendix A. �The basic idea of Proposition 1 is immediate. We have assumed that, for γ close to zero, quitting is good when

compared to the price given with optimistic beliefs. But, this is exactly the price that would emerge if the managerwere to lie and be believed. Hence, for small γ , it is optimal to collect the outside option in the bad state implyingalways being truthful. As γ approaches 1, quitting starts to look unprofitable, so there are incentives to lie.

For very high γ , it is optimal to always lie, even taking into consideration that investors will anticipate this behaviorand will not respond to manager’s statements and that they might be punished. High γ leads to high demand and highprice, enhancing the incentives to lie.

For γ2 < γ < γ1, the manager knows he is not fully believed when he says g, otherwise he would lie. If he wereto always lie, agents would anticipate this. Given beliefs and the potential monetary punishment, he would prefer toquit. But, if he prefers to quit, lying merely gives him extra costs, contradicting the assumption that he lies. Hence, theonly equilibrium is in mixed strategies. When the state is good, he always tells the truth and minimizes the punishmentcosts; and he sometimes lies in the bad state.

For γ < γ2, we obtain a fully revealing equilibrium where agents correctly anticipate that the manager will behonest, and it is indeed the case in equilibrium.

The main point of the proposition is that the presence of “captive demand” by the institutional investors may leadto increased incentives to misrepresent the truth. We name this captive since, for a given level of expected returns, they

29 For a formal definition and further discussion of Perfect Bayesian Equilibrium see, for example, Fudenberg and Tirole (1998).


are more willing to hold the stock than outsiders. This happens because of the relative performance effect present inthe utility function of these investors.

As mentioned in the Introduction, the results of this paper hold under both voluntary and mandatory disclosurerules. Under mandatory disclosure, we have the exact current model: the manager must disclose his information, butcan lie. Under voluntary disclosure, the model works much in the same way with non-disclosure interpreted as a badsignal, hence never happening in equilibrium. When the manager wants to report a good signal he must choose todisclose. And when he wants to report a bad signal he is indifferent between disclosure and non-disclosure, since henever reports a bad signal in the good state, so cannot save on punishment costs by not disclosing.

We are ready to solve the project choice stage. However, first let us present an interesting corollary to Proposition1.

Corollary 1. For any project choice ν ∈ (ν0, 1), the ex-ante probability of observing lies in equilibrium is increasingin γ .

Proof. The ex-ante probability of lies is

(1 − ν)Pr(g|B) =

⎧⎪⎪⎨⎪⎪⎩

(1 − ν), for γ ≥ γ1;

ν1 − ν(γ)

ν(γ), for γ2 < γ < γ1;

0, otherwise

(19)

Notice that ν(γ1) = ν and ν(γ2) = 1. Given that ν(γ) is decreasing in γ ,30 we have that ν(1 − ν(γ))/(ν(γ)) is increasingin γ . �

5. Investment project stage

The choice of which investment project to adopt depends on which equilibrium we are in, hence it depends on γ .But, more importantly, it depends on the reporting policy being used by either the manager or shareholders. From thediscussion above we have an idea of the possibilities: they can be fully honest, follow a mixed strategy, or always liein the bad state. We describe the solution based on these possible states. We start with the case where the reportingis exogenous, i.e., the case where any information is truthfully revealed to investors. This serves as a benchmark ofcomparison to better understand the effects of the reporting policy choice on the first stage investment decision. It alsoserves to check whether without such a choice the model delivers more traditional results. And, clearly, it correspondsto the equilibrium for very low values of γ . In addition, we present two alternative benchmarks: the project choicethat maximizes investors’ discounted utility and the one that maximizes discounted expected firm value. Then, thediscretionary reporting case is presented.

Throughout we make the assumption that [1 − ((E1)2ρm(1 − γ))/(κR(1 − γλ))] > 0, which requires

κ >(E1)2ρm(1 − γ)

R(1 − γλ). (20)

This is the reason for the first term in the restriction stated in (13).

5.1. Benchmarks

In this section, we analyze our benchmark cases:

30 Because Pν(γ) is increasing in ν and γ . Hence, when γ decreases, price decreases. In order to maintain the equality, we need to increase theimplied beliefs, i.e., increase ν. Or, more formally

dν

dγ= − (∂Pν(γ))/(∂γ)

(∂Pν(γ))/(∂ν)< 0.

The inequality follows from the properties of the price function and the fact that (∂Pν(γ))/(∂ν) < 0 for ν > 1/2.


1. The agent choosing the project has no control over the reporting policy and the state of the world is honestly revealedto all investors. We further split this into two sub-cases:(a) Manager chooses investment project;(b) Shareholder chooses investment project.

2. Investors choose the investment level and information is revealed according to the pooling equilibrium describedin Proposition 1.31

3. A third party chooses the investment policy to maximize discounted expected firm value.32

5.1.1. Exogenous reportingWith truthful reporting we can write the manager’s objective function as

νP1(γ) + (1 − ν)L − κ(ν − ν0)2. (21)

The first term stands for the fact that with probability ν he will be able to sell his share at full price. The second termrepresents his outside option. In case of bad news, that are truthfully reported, the company has no market value sincewith risk averse investors and a zero expected value prices would be negative. Hence, the best he can do is abandonhis share and his job, taking the outside option. The last term represents the cost of setting the probability of goodstate to ν, the cost of choosing such a project. The first-order condition for this problem gives us the following projectchoice33:

νM = ν0 + P1(γ) − L

2κ. (22)

We see that the manager picks projects that attach higher probability to the good state of the world as the relativeperformance parameter increases. This is because higher γ implies higher prices, and hence, a better payoff in the goodstate. Accordingly, the manager has more incentives to work hard to “guarantee” the realization of a good state. Onthe other hand, if his outside option has a higher value, he will tend to choose projects that are not as “good”, as is thecase for higher costs of choice.

We now analyze how a shareholder would act if he was the one making the investment decision. His objective thenis

η[νP1(γ) + (1 − ν)0] + (1 − η)νE1

R− κ(ν − ν0)2. (23)

In this case, the first term represents the fact that with probability η he is hit by a liquidity shock and has to sellhis share. In the good state, he gets full price and in the bad state he gets nothing. The second term represents theno-liquidity-shock case, when he can wait to collect the company’s payoff. Hence, his first-order condition gives

νS = ν0 + ηP1(γ) + (1 − η)E1/R

2κ, (24)

where a similar interpretation applies.For the previous analysis to be correct, we need the restriction that νS ∈ [ν0, 1], which translates into

κ >ηP1(γ) + (1 − η)E1/R

2(1 − ν0), (25)

31 We focus on the pooling equilibrium because it is the case that deserves most of our attention and discussion due to the fact that the comparisonbetween manager’s and shareholders’ choice is most interesting.32 Notice that this is independent of the reporting policy. The reporting policy only affects the price at which the company stock is traded, but it

has no effect on the final value of the firm. So, the reporting policy is irrelevant for this benchmark.33 For low values of γ , it may be the case that P1(γ) < L. Then νM = ν0 since he would always quit. Even though we accounted for this case in

the proof of Proposition 1, we abstract from it in what follows. However, results would be almost identical with just a slight change to some of thearguments to come.


This implies κ > (P1(γ) − L)/(2(1 − ν0)) since (E1)/(R) > Pν(γ), assuring νM ∈ [ν0, 1]. However, the second termin restriction (13) guarantees that

κ >C + (E1)/(R) − (ρm(1 − γ))/(R(1 − γλ))(σ2

H − σ2L − (E1)2)

2(1 − ν0)>

(E1)/(R)

2(1 − ν0)>

ηP1(γ) + (1 − η)E1/R

2(1 − ν0).

(26)

In the end, it is clear that, as mentioned before, νS ≥ νM . Therefore, without a reporting policy choice, managerand shareholders have conflicting interests, but the direction of such conflict is as usual—managers are less willing toexpend resources in order to produce profitable outcomes.

5.1.2. Maximizing investors’ utilityHere, we find the project choice that maximizes the investors’ discounted utility,

νi = arg maxv

1

R

[E[w∗

3] − 1

2ρVar[w∗

3]

]− κ(ν − ν0)2, (27)

where w∗3 is the investors’ random final wealth, given optimal behavior in the next period when he determines his

demand for the company stock. We concentrate on type-O investors, however, the analysis is identical for type-Iinvestors. For the sake of comparison, we only analyze the case of a high γ , when there are always lies in the badstate.34 In this case, one can show that

w∗3 =

⎧⎪⎪⎨⎪⎪⎩

[w1 − m(1 − γ)

1 − γλPν(γ)

]R + m(1 − γ)

1 − γλVH, with probability ν;[

w1 − m(1 − γ)

1 − γλPν(γ)

]R + m(1 − γ)

1 − γλVL, with probability 1 − ν

Furthermore, the only interior solution to the investors’ maximization problem is νi = ν0.35 This result may seemsurprising at first, since these investors, being risk-averse, would profit the most from an increased probability ofthe good state. In a partial equilibrium scenario this is certainly true, because higher ν increases expected value anddecreases the variance. However, in a general equilibrium the price reflects the improved characteristics of the assetrendering the change worthless. Given this optimal project choice, the analysis is confirmed to be valid for γ ≥ γi

1,where Pν0 (γi

1) = L + C.

5.1.3. Maximizing firm’s final valueAs a final reference we also find the value maximizing project, i.e., the choice that maximizes discounted expected

value of the firm net of costs. This is simply the solution to the shareholders’ problem when η = 0:

ν∗ = ν0 + E1

R

1

2κ. (28)

5.2. Discretionary reporting and investment policy

Here, we analyze the incentives of the manager and shareholders to choose the right investment projectwhen information revelation is under an agent’s control. We assume that the reporting agent is alwaysthe manager, even when the shareholder is the one choosing the project. This is a simplifying assump-tion meant to capture the fact that usually it is management that “talks” to the market. However, weconsider the effect that the shareholders’ choice has on the reporting policy. We assume that it is the man-ager that does the reporting, so he is the one bearing the potential costs of lies. And, his strategy is

34 As it will be clear soon, the most interesting part of the analysis takes place when γ is high, implying that the pooling equilibrium obtains.35 The proof is omitted, but is available from the author upon request.


determined by the equilibrium of the communication game with ex-ante beliefs given by the shareholders’choice.36

It is important to notice that this section is not an equilibrium analysis, which is left to be covered in the next section.Here, we just describe the incentives and choices given that both the manager and the shareholders are in the same“equilibrium” of the communication game. In the next section, when we put everything together, it will be clear thatthere exist values of γ such that the choice made by the manager and the shareholders might lead to different reportingpolicies. Also, unfortunately, there are areas of the parameter space where multiple equilibria occur. In the case ofmultiplicity of equilibria, we can make the simplifying assumption that both the manager and shareholders are in the“same equilibrium”, and compare their choices given that similar strategies are being played in the reporting stage.This may seem like a restrictive assumption, so when we present the main result of the paper (next-to-last section) wealso discuss the possibility that they are in different equilibria. It is important to notice that the multiplicity of equilibriaactually strengthens our results. If we take into account that the manager and the shareholders may be playing differentequilibria, then it is true that for almost any parameter value (γ) there exists a situation where the manager chooses aproject with higher probability attached to the good state than the shareholders would. We dwell on this issue a littlebit more later on.

We divide this section based on the reporting policy.

5.2.1. HonestyWhen the relative performance parameter is small enough, we know from Proposition 1 that information will be

revealed truthfully. Hence, the analysis remains unaltered and we have

νM3 = ν0 + P1(γ) − L

2κ, (29)

νS3 = ν0 + ηP1(γ) + (1 − η)E1/R

2κ. (30)

5.2.2. Probabilistic liesRandomization makes the analysis more interesting. Remember that the insider will now be using a mixed strategy

where in the bad state he lies with probability

p(ν, γ) = 1 − ν(γ)

ν(γ)

ν

1 − ν. (31)

When choosing the project, he will need to take into consideration the effect of his choice on this probability.For this part of the parameter space, we can write the manager’s objective as

ν(L + C) + (1 − ν)[p(ν, γ)(L + C − C) + (1 − p(ν, γ))L] − κ(ν − ν0)2. (32)

The first part represents the fact that in case of good news investors attach probability ν(γ) to the good state andthe manager gets Pν(γ)(γ) = L + C.37 The second term is split into two: First, in the case of bad news and lies he getsPν(γ)(γ) − C, and in the case of honesty he gets his outside option. The first-order condition for this problem gives us

νM2 = ν0 + C

2κ. (33)

For an interior solution, we need κ > (C)/(2(1 − ν0)), and from (25) we have

κ >ηP1(γ) + (1 − η)E1/R

2(1 − ν0)>

C

2(1 − ν0). (34)

36 If we allow the reporting to be done by the shareholders, we need to change the communication game. Shareholders’ objectives are differentfrom the manager’s, as they may be able to hold on to their shares if not hit by the liquidity shock. This delivers a modified equilibrium. However,the main ideas continue to be valid. The statement and proof of a modified Proposition 1 for the case of liquidity shocks is available from the authorupon request.37 See the proof of Proposition 1.


If the shareholders choose the project and take the effect of this choice on the reporting policy into consideration,through (∂p(ν, γ))/(∂ν), their objective can be written as

η{ν(L + C) + (1 − ν)[p(ν, γ)(L + C) + (1 − p(ν, γ))0]} + (1 − η)νE1/R − κ(ν − ν0)2, (35)

where the first term indicates the fact that in the presence of a liquidity shock and the realization of the good statethey get Pν(γ)(γ). The second term represents their payoff in the presence of a liquidity shock in the bad state. In thisinstance if the manager lies the shareholders get Pν(γ)(γ) again, however, if he is honest the company has no marketvalue. The third term represents their payoff without a liquidity shock. The solution for this maximization problem hasthe following format:

νS2 = ν0 + η{[1 − p(ν, γ)] + (1 − νS2)∂p(νS, γ)/∂ν}(L + C) + (1 − η)E1/R

2κ. (36)

Solving for the optimal project, we have

νS2 = ν0 + η(L + C)/ν(γ) + (1 − η)E1R

2κ. (37)

It is clear that, once more, νS2 > νM2.In terms of restrictions on the parameters, we want to guarantee that νS2 ∈ (ν0, 1). Then we need to assume

κ >ηL + C/ν(γ) + (1 − η)E1/R

2(1 − ν0). (38)

But notice that ν(γ) ∈ (ν0, 1). So the maximum value the right-hand-side of this inequality can achieve is

ηL + C/ν0 + (1 − η)E1/R

2(1 − ν0), (39)

and this is the reason for the third term in the restriction (13).

5.2.3. LiesFinally, we come to the most interesting part of the analysis. In this section, we assume that the manager is always

lying in the case of bad news, independently of who is choosing the investment. We show later that the subset of theparameter space where this happens is non-empty.

We can write the manager’s objective as

νPν(γ) + (1 − ν)(Pν(γ) − C) − κ(ν − ν0)2, (40)

where the first term represents the fact that if the state is good he does not lie, but, given his strategy, agents do not believehim, so he collects the price that obtains with ex-ante beliefs, which is exactly his choice variable. The second term is sim-ilar, however taking into consideration the expected costs of lies. This problem has the following first-order condition38:

νM1 = ν0 + C

2κ+ 1

2κ

[E1

R− (σ2

H − σ2L + (1 − 2νM1)(E1)2)

ρm(1 − γ)

R(1 − γλ)

], (41)

so:

νM1 =[

1 − (E1)2ρm(1 − γ)

κR(1 − γλ)

]−1{ν0 + 1

2κ

[C + E1

R− ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]}. (42)

We need νM1 < 1, and this is guaranteed by

κ >C + E1/R − ρm(1 − γ)/R(1 − γλ)(σ2

H − σ2L − (E1)2)

2(1 − ν0), (43)

the second term in (13). We can now pin down the parameter γ1 in Proposition 1, PνM1 (γ1) = L + C.

38 SOC requires κ > ((E1)2ρm(1 − γ))/(R(1 − γλ)), which is guaranteed by (13).


We now proceed to the shareholders’ problem:

maxν

{ηPν(γ) + (1 − η)νE1

R− κ(ν − ν0)2}. (44)

The first term represents the fact that, if hit by a liquidity shock, he will get the same price for his share independentof the state of the world, and he is not punished, given that the manager is the reporting agent. The solution to thisproblem is given by39

νS1 =[

1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

]−1{ν0 + 1

2κ

[E1

R− η

ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]}. (45)

It is easy to show that if C > (1 − η)ρm(1 − γ)/R(1 − γλ)(σ2H − σ2

L + (E1)2) we have νS1 < νM1! Notice that

σ2H − σ2

L + (E1)2 = E[(VH )2] − (E1)2 − E[(V L)

2] + (E1)2 < 0. (46)

Clearly,

C > 0 > (1 − η)ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2), (47)

and we have νS1 < νM1.Notice that, in this case, νS1 < νM1 < 1 since we have shown νM1 < 1. However, we still need to guarantee νS1 ≥ ν0.

For this, it is sufficient to have[1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

]−1

ν0 ≥ ν0. (48)

To simplify this restriction, we first use the fact that(1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

)>

(1 − (E1)2ρm(1 − γ)

κR(1 − γλ)

)> 0, (49)

where the first inequality follows from η < 1 and the second from (20). Then we manipulate it to obtain the requirementthat

η(E1)2ρm(1 − γ)

κR(1 − γλ)> 0, (50)

which is trivially satisfied.

5.3. Summary

We summarize the results of this section in a proposition concerning solely the project choice. In the next section,we proceed to the main proposition, that puts the pieces together, and finish with a discussion.

Proposition 2. Assume that the manager is the reporting agent, and that the cost of choosing (and managing) aninvestment project that attaches probability ν to the good state of the world is given by

Q(ν) = κ(ν − ν0)2. (51)

Then we have that

(i) If the reporting policy is truthful:

νM3 = ν0 + P1(γ) − L

2κ, (52)

39 SOC requires κ > (η(E1)2ρm(1 − γ))/(R(1 − γλ)), which is guaranteed by (13).


νS3 = ν0 + ηP1(γ) + (1 − η)E1/R

2κ, (53)

where νM3 ≤ νS3.(ii) If a mixed strategy is being played as described in the semi-separating equilibrium:

νM2 = ν0 + C

2κ; (54)

νS2 = ν0 + η(L + C)/ν(γ) + (1 − η)E1/R

2κ, (55)

where νM2 ≤ νS2.(iii) Finally, if there are always lies in the bad state:

νM1 = ν0 + 1/2κ[C + E1/R − ρm(1 − γ)/R(1 − γλ)(σ2H − σ2

L + (E1)2)]

1 − (E1)2ρm(1 − γ)/κR(1 − γλ), (56)

νS1 = ν0 + 1/2κ[E1/R − ηρm(1 − γ)/R(1 − γλ)(σ2H − σ2

L + (E1)2)]

1 − η(E1)2ρm(1 − γ)/κR(1 − γλ), (57)

where νM1 > νS1.

Notice that the difference between νM1 and νS1 is driven in essence by C and η. The fact that the manager is theagent doing the reporting (and potentially being punished) and the different horizons of manager and shareholders arein the heart of our result. The relevance of these characteristics of our model is accentuated depending on the value ofthe relative performance parameter. Whenever γ is such that there are lies in equilibrium, managers always overinvestrelative to shareholders. As we show next, given the multiplicity of equilibria to be analyzed in more detail below, thissubset of the parameter space is larger than it appears.

The desire and temptation to later manipulate prices through the release of misleading information leads, ex-ante,to an added inefficiency: investment projects are chosen in order to facilitate later stock price inflation and not tomaximize shareholder well-being. Overinvestment is the direct result of the incentives to falsify information, and theseincentives come about due to the excessive weight put on stock price in the manager’s compensation package.

6. Equilibrium analysis and implications

Now we put all the results in a combined framework analyzing the equilibrium properties and discussing of theimplications of the model.

6.1. Equilibrium

In this part of the paper we compare the project choice that results when the manager is in charge with the oneunder shareholders’s choice, and compare both with the level that maximizes investors’ utility and firm value. Thenthe ensuing information manipulation problem is analyzed, where it is argued that more manipulation will occur undermanager’s choice, and thus the market will be less informed about the true state of the world. Last, but not least, thewelfare of the manager is analyzed. We compare how well he does under the discretionary reporting policy vis-a-visthe exogenous reporting policy. In other words, we analyze the effects that lack of credible commitment has on hisex-ante expected utility.

We first state the equilibrium of the whole game, basically putting all the parts analyzed so far together. Only theequilibrium where the shareholder chooses the project is presented.40 We have eight first-order conditions, and we needto guarantee that each one has an interior solution, i.e., (1/2) − ν0 < 0 ≤ νM,S − ν0 ≤ 1 − ν0. But, we have shownabove that these restrictions are all satisfied given our assumption (13).

40 For the case of the manager, the result is the same just putting νM• in place of νS•, with one exception that we point out below.


Proposition 3. Let γS1 , γS

2 , γS3 be such that PνS1 (γS

1 ) = L + C, PνS2 (γS2 ) = L + C and P1(γS

3 ) = L + C. Then,

(i) If γ ≥ γS1 , M always issues a g report for the company stock. The shareholder would choose νS1.

(ii) If γS3 < γ ≤ γS

2 , in the case of good news M always says g, and in the case of bad news he says g with probabilityp1, defined as

p1(γ) = νS2

1 − νS2

1 − ν(γ)

ν(γ), (58)

where ν(γ) is such that

Pν(γ)(γ) = L + C, ∀γ ∈ [γS3 , γS

2 ].

In this region, we have P(G|g) = ν(γ), and if there is a bad report then agents attach probability one to thebad state, P(G|b) = 0. The shareholder of the firm would choose νS2.

(iii) Finally, if γ ≤ γS3 , M always tells the truth and is fully believed by investors. The shareholder chooses νS3.

Since νS1 > νS2, we have that γS1 < γS

2 . Then, for any γ ∈ (γS1 , γS

2 ), we have multiple equilibria.41Either we havean equilibrium where the manager has semi-separating reporting (as described in (ii) above) with choice νS2 by theshareholders. Or, we have an equilibrium where the manager has pooling reporting (always issues a g report) withchoice νS1 by the shareholders.

Proof. See Appendix B. �Intuitively, the multiplicity problem arises because for γS

1 < γ < γS2 there are enough incentives to always lie as long

as the project choice is appropriate (i.e., νS1). However, if the choice is smaller (νS2), γ is not high enough to supportconstant lies, and hence we have the semi-separating equilibrium. It should be made clear that the proposition wouldentail different thresholds in the γ-parameter space if it was the manager making the investment decision because thethresholds are determined by the project choice in each region and νM• = νS•. Of course γ3 is an exception, where wewould have γM

3 = γS3 (:= γ3), since this parameter is independent of the individual project choice.

Now, we are ready to state Proposition 4, that deals with the precise comparison of investment policies.

Proposition 4. Under the conditions o f Proposition 3, we have

νM1 > νM2 ≥ νM3;

νM1 > νS1 > ν∗ > νS2 ≥ νS3 > νi;

νM2 ≥ νM3 ≥ νi;

vSk ≥ vMk, for k ∈ {2, 3}

(59)

where νi is the project that maximizes investors’ discounted ex-ante utility and ν∗ is the one maximizing discountedexpected firm value. Furthermore, if L ≥ ηC, νS3 ≥ νM2 so that

νM1 > νS1 > ν∗ > νS2 ≥ νS3 ≥ νM2 ≥ νM3 > νi. (60)

Proof. See Appendix C. �The result that there exists an area of the parameter space such that managers would like to work harder and choose

an investment project that attaches too high a probability to the good state is striking, given that on every other instancethe shareholders would like a “higher” investment level. The interesting point is that this reversal happens when themarket conditions grant it to be extremely profitable for the manager to issue untruthful reports. Intuitively, in this

41 Note that γS1 > γS

2 ⇔ νS1 < νS2.


region his payoff is extremely sensitive to his choice. Not only he affects the probability of the good state (when he isnot punished), but he directly affects the price, through beliefs generated by his lies. In the case of low γ , he affectsthe probability of the good state, and only through this indirect channel he can affect his payoff because he is alwaystruthful. For intermediate γ , increasing the likelihood of the good state is the only way to alter his payoff as well. Hestill cannot directly affect prices, which should satisfy his indifference condition for any project he chooses, beingequal to L + C, since it has to be optimal to randomize in this region.

Another point worth mentioning is the fact that νM1 > νS1 > ν∗ > νi. This helps us better understand the prominenceof the result. It is not the case that the manager’s choice brings the company closer to the ideal for investors, or for theeconomy (maximization of firm value). On the contrary, in this region the manager will distort the choice even furtherthan the shareholders would. The difference between objectives of the shareholders and investors is that they may needto sell the shares if they are hit by a liquidity shock, while the investors buy their shares in the market and hold on tothem until the final value is realized. Therefore, independently of whose point of view one adopts, the manager willbe overinvesting.

In the current framework, the sensitivity of prices to project choice leads to an overshooting effect. It is true thatthis sensitivity increases incentives to invest in a good project, but it does it so much as to lead to an investment policythat is not optimal-overinvestment. This result is due to the fact that the price informativeness is, to some degree, underthe manager’s control, via his reporting policy. Under some policies, company stock prices may be too sensitive to theproject choice, as compared to the sensitivity of the company’s final value.

Now let us address the issue of the multiplicity of equilibria and how this actually may be seen as enhancing theimportance of our result of overinvestment. It is straightforward to see that the parameters defined in our propositionsatisfy 0 < γ3 < γM

1 < γS1 < γS

2 < γM2 < γi

1 < 1. So, for γ < γ3, independently of who is investing, the unique equi-librium has truthful reporting, and hence the comparison νS3 > νM3 is appropriate. For γ3 < γ < γM

1 , the comparisonνS2 > νM2 applies. And, for γM

2 < γ , the comparison νS1 < νM1 is apposite. For all the other regions, multiplicity isa problem. We analyze these separately.

• γ ∈ (γM1 , γS

1 ): If the manager is investing, he can choose νM2 with a semi-separating equilibrium in the communicationstage. Or he can choose νM1 with a pooling equilibrium. If the shareholders are choosing, then the only equilibriumhas them choosing νS2 and the manager playing the semi-separating equilibrium in the communication game. Wehave shown before that νM1 > νS2 > νM2. Hence, the possibility for overinvestment is also present here.

• γ ∈ (γS1 , γS

2 ): Now, on top of the cases above, we can also have an equilibrium where the shareholders choose νS1

with the manager playing the pooling equilibrium in the communication game. The possibilities satisfy νM1 > νS1 >

νS2 > νM2. Again, overinvestment is a possibility.• γ ∈ (γS

2 , γM2 ): In this case, we can have all the equilibria mentioned so far except the one with νS2, so the comparison

becomes νM1 > νS1 > νM2.

This analysis clarifies that, as long as γ ∈ (γM1 , 1), we have the potential for overinvestment. Furthermore, overin-

vestment becomes the unique possibility if γ ∈ (γM2 , 1) ⊂ (γM

1 , 1). Thus, our earlier assumption that the manager andthe shareholders play the same equilibrium only weakens our results. The fact that the reporting policy is determinedendogenously has an even higher effect on the type of distortion generated by the manager’s control over project choice.

It is worth noting that our novel result, where managers overinvest relative to shareholders’ optimal, emanates fromthree main assumptions: different horizons for the agents, the fact that the manager is the one reporting to the marketand, most importantly, the assumption that the reporting policy is discretionary being under management’s control.Without this last assumption, we are back to the traditional results where managers want to shirk/under-invest relativeto shareholders.

We now turn to the issue of information manipulation. We can describe how much information is being revealedto the market place depending on who is in charge of investment. Chart 1 presents the results in a schematic form.For example, we see that if γM

1 < γ < γS1 and the manager is in charge, then we observe either full lies (pooling

equilibrium) or probabilistic lies (semi-separating equilibrium). On the other hand, if the shareholders are investing,there is no possibility for the existence of a pooling equilibrium. In that sense, we can assert that giving the projectchoice to the shareholders weakly increases the information revealed to the market. In general, we can conclude thatwhenever there is a possibility for a pooling equilibrium under the shareholders’ choice, this possibility is also presentif the manager is in control. Furthermore, there are instances where the manager’s choice could generate a pooling


Chart 1. Project Choice and its Effect on Information Manipulation.

equilibrium but this would not happen if the shareholders were to choose. Accordingly, one could argue that givingthe choice to the manager weakly increases information manipulation decreasing the amount of information revealedto investors, and hence making prices less informative.

Finally, we discuss the manager’s ex-ante expected utility in Proposition 5, showing that his inability to commit totell the truth affects him negatively and he would rather have a third party reveal the information concerning the qualityof the company. The proof is in Appendix D.

Proposition 5. Ex-ante the manager is made worse off by having the reporting policy under his control. His ex-ante expected utility is higher under the exogenous reporting policy than under the discretionary reporting policyequilibrium. This result is independent of which equilibrium he is playing in the multiple equilibria region.

6.2. Empirical implications

We can now discuss the empirical implications of the model. First, it is important to notice that fraud and over-investment are at the heart of our paper, so one should think carefully about how to measure such quantities. Onepath that can be taken is to use earnings restatement as a measure of lies/fraud (since this usually happens after SECintervention) and use investment relative to the firm’s industry as a rough measure of overinvestment. Albeit far fromideal, these measures should capture some of the ideas behind our definitions of lies and overinvestment. Second, wesplit the implications into cross-sectional and time-series, although they are closely related.

1. Cross-sectional implication: If companies have different γ ′s then companies with higher γ are more prone to fraudand have an increased probability of overinvestment.

As γ increases so does demand so, one way to test this implication is to associate γ with a sort ofexcess/captive demand. And, a way to identify such captive demand is to look at 401(k) holdings of companystock. Employees tend to hold own company stock in their 401(k) plan and considerable allocations could beinterpreted as a form of captive demand.42 Hence, one would expect that firms whose employees’ 401(k) funds

42 In Cohen (2004), an empirical study shows that loyalty explains apparent overinvestment in own company stock in 401(k) funds. So, one cansee that this demand is less affected by fundamentals then by external factors (such as loyalty), hence it is, to some extent, captive.


are highly concentrated on own company stock to experience more lies and have a higher likelihood of over-investing.

2. Time-series: Similarly to the cross-sectional implication, times when the market as a whole has a high average γ

are times where fraud and overinvestment are more likely.A way to test this implication is to notice that our model is very similar to a model where institutional investors

perceive the variance of the firm to be (1 − γ)σ2ν and the other investors perceive it to be just σ2

ν . So, our model issimilar to one with heterogenous beliefs or overconfidence. Using measures of heterogeneity of beliefs one shouldfind a positive relation between these and fraud and overinvestment. Such measures are discussed (and used) inScheinkman and Xiong (2003); Bolton et al. (2004) and Mei et al. (2005).

3. General implications: Both cross-sectionally and in the time domain we should observe a positive correlationbetween overinvestment and fraud, with overinvestment leading fraud. This is so because, as shown in the model,it is the desire to later manipulate prices by releasing misleading information that leads the manager to overinvest.Also, fraud should be positively related to the sensitivity of the manager’s compensation package to stock prices.

In support of our general implications, Kedia and Philippon (2007a) presents evidence that during periods of suspi-cious accounting, firms invest excessively, while managers exercise options. Peng and Roell (2004) and Bergstresserand Philippon (2006) provide evidence that earnings manipulation is more pronounced at firms where the CEO’scompensation is more closely tied to the value of stock and option holdings. This provides support to the con-clusion of our model that it is the desire to profit from inflated prices that leads the manager to release falseinformation.

We could derive similar implications regarding the cost of fraud, C. For instance, in times of strict monitoring onthe part of the SEC one would assume that this cost is higher, hence on average less manipulation and overinvestmentwould take place.

Testing such implications is left for future research.

7. Conclusion

In this paper, we present a full blown equilibrium analysis of the choices of a manager. We put together four mainelements: the fact that managers (and/or shareholders) have the choice to engage in value increasing activity, differinghorizons of different agents, the important issue of endogenous information revelation and the fact that prices are notexogenous, and hence, respond to the previous choice variables.

The main contribution of this work is to analyze the choice of investment and the reporting policy together. Mostpapers so far would only consider each one in isolation. We show that putting them together may generate importantconsequences, due to the feedback effect they have on each other. Investment policy depends on how information isrevealed, and vice-versa.

Interestingly enough, we show that, contrary to the norm, our equilibrium model may produce cases where man-agers and shareholders would both be overinvesting (or working too hard) as compared to the value maximizinglevel. This occurs because both insiders have objectives that are at least partially dependent on stock price, and theprice has the tendency to be too sensitive to investment project choice. Hence, with these objectives and with thefreedom of choosing the way to report the state of world to the public, they choose the “wrong” project. More impor-tantly, we observe managers overinvesting even more than shareholders would. Therefore, the extra sensitivity ofmanagers’ payoffs to the project choice leads them to overinvest above and beyond the optimal for either investors orshareholders.

We also show that the results are not weakened even in the presence of multiple equilibria. Much on the contrary,multiplicity of equilibria allows us to draw stronger conclusions concerning the result that managers may overinvest.The fact that the reporting policy is determined endogenously has an even higher effect on the type of distortiongenerated by the manager’s control over project choice.

Finally, we demonstrate that ex-ante the manager is made worse off by having the reporting policy under his control.So, the lack of commitment affects him negatively, and he would rather have a third party reveal the informationconcerning the quality of the company.

We also discuss the empirical implications of the model.


Appendix A. Proof of Proposition 1

Fix the initial beliefs at a given ν, i.e., take the project choice as given. Notice that we have assumed

1

R[ν0E1 − mρσ2

0 ] < L + C <1

Rν0E1. (62)

This can be translated into the following condition concerning the prices in the pooling equilibrium: Pν(0) <

L + C < limγ→1

Pν(γ). We also have P1(0) < L + C < limγ→1

Pν(γ) ≤ limγ→1

P1(γ). So, for small γ , it is not worth lying.

M is better off quitting and telling the truth. However, for γ close to 1, it is optimal to lie. In order to show theexistence of a unique pair (γ1, γ2) with the desired properties, we first need to show that Pν(γ) is strictly increas-ing in its argument. It is easy to see that Pν(γ) is an increasing, continuous and convex function of γ . Now wedefine the parameters Pν(γ1) = L + C; P1(γ2) = L + C. Given the above properties of the price function and theassumptions we made, we know that these parameters exist and are unique. And, since P1(γ) > Pν(γ)∀γ , we haveγ1 > γ2.

Assume that γ ≥ γ1. The optimal strategy is to lie in the case of bad news and sell the shares if and only ifPν(γ) − C > L, which is verified given our definitions. In the case of good news, the insider also sells the shares sincePν(γ) > L + C > L. So, quitting is not optimal.

Now let γ ≤ γ2. In this case, we have P1(γ) − C < L by definition of γ2. Assume that the investors believe thereports. In the case of bad news then, if the informed agent is truthful, he can only get L. If he deviates and lies, heexpects to get P1(γ) − C < L. So, he sticks with the truth and quits the firm (before, he issues a bad report). In thecase of good news, if he follows the equilibrium, he can get P1(γ) by selling the shares or L by quitting. So, he sellsas long as P1(γ) ≥ L, otherwise he quits and gets L. If he deviates he can only get L, so there are no incentives todeviate.

Finally, assume γ2 < γ < γ1. We first need to show the existence of the postulated ν(γ) function.We know that Pν(γ) = L + C and P1(γ2) = L + C. Hence, we must have ν(γ1) = ν and ν(γ2) = 1. We also proved

that (∂Pν(γ))/(∂γ) > 0 and (∂Pν(γ))/(∂ν) > 0. Notice that Pν(γ) is a continuous function of both its arguments.43 So,it is trivial to see that for every γ2 < γ < γ1 there exists a ν, potentially dependent on γ , such that if beliefs are equalto ν we have Pν(γ)(γ) = L + C.44

Let us turn now to the strategy of the informed agent and the beliefs he generates. First, conjecture that M adoptsthe following strategy: if the state is good, he says g with probability p(γ) and b with complementary probability (andquits the firm). If the state is bad, he says g with probability p1(γ). If this is the strategy followed by the insider, theninvestors will have beliefs given by

P(G|g) = νp(γ)

νp(γ) + (1 − ν)p1(γ). (63)

In order to have P(G|g) = ν(γ), we need

p1(γ)

p(γ)= ν

1 − ν

1 − ν(γ)

ν(γ). (64)

It is clear that the proposed probabilities, p(γ) = 1 and p1(γ) = ν/1 − ν1 − ν(γ)/ν(γ), work fine.

43 This is true as long as γ ∈ [0, 1).44 We know that

dν

dγ= − ∂Pν(γ)/∂γ

∂Pν(γ)/∂ν.

Consequently, one could think of constructing the function using the following approximations:

ν(γ1 + dγ) = ν0 − ∂Pν(γ)/∂γ

∂Pν(γ)/∂νdγ,

where dγ < 0.


Now, notice that the ex-ante expected cost of punishment for lies, for any strategy, is equal to

{ν(1 − p(γ)) + (1 − ν0)p1(γ)}C. (65)

Since we have ν > (1/2), it is trivial to see that ν > 1 − ν. Hence, we want to minimize (1 − p(γ)) and this deliversthe desired result: p(γ) = 1, ∀γ .

We finally need to check that there is no other strategy that dominates this one, that is, we need to check that thereare no incentives to deviate.

Suppose we are in the bad state. Consider deviating to p′1(γ). We need to show that

p1(γ)Pν(γ)(γ) + [1 − p1(γ)]L − p1(γ)C ≥ p′1(γ)Pν(γ)(γ) + [1 − p′

1(γ)]L − p′1(γ)C. (66)

By rearranging, we know that this is true if and only if

[p1(γ) − p′1(γ)][Pν(γ)(γ) − C − L] ≥ 0. (67)

This requires that for p1(γ) > p′1(γ), Pν(γ)(γ) ≥ C + L, and that for p1(γ) < p′

1(γ), Pν(γ)(γ) ≤ C + L. Hence, forthis to hold for any deviation, we need that Pν(γ)(γ) = C + L. This is exactly what the definition of ν(γ) guarantees.So, there are no incentives to deviate in the case of bad news.

Let us turn now to the case of good news. In this case, the only possible deviation is to a p′(γ) < p(γ) = 1. Weneed to show that45

Pν(γ)(γ) ≥ p′(γ)Pν(γ)(γ) + [1 − p′(γ)]L ⇔ [1 − p′(γ)][Pν(γ)(γ) − L] ≥ 0. (68)

The last inequality holds, actually in its strict form, since Pν(γ)(γ) = C + L > L. This completes the proof.

Appendix B. Proof of Proposition 3

The proof is basically identical to the proof of Proposition 1, but now it is as if we had three propositions insideone—one for each project choice. This happens due to the fact that the project choice determines ex-ante beliefs, andthese beliefs change the structure of the proposition.

First recall that we had defined two parameters in Proposition 1: Pν(γ1) = L + C; P1(γ2) = L + C. It shouldbe clear then that γ2 = γM

3 = γS3 . The idea of the proof is to fix the project choice, derive the equilibrium of the

communication stage and then see where this project choice fits.46 Suppose, for example, that we fix ν = νS1. For thisex-ante probability of the good state, we can derive an equilibrium as in Proposition 1. Then, we see that this choiceis the relevant one for γ > γS

1 since, given the definition of the parameter, for these values the manager will be lyingand hence choosing νS1optimally. Now if we fix ν = νS2, we can do the same. However, we have to keep in mind thatmultiple equilibria may exist. For ν = νS3, we are in safe ground again because γS

3 is independent of beliefs.We know that 0 < γS

3 < γS1 < γS

2 < 1. So, for γ < γS3 , the unique equilibrium has truthful reporting with choice

νS3. For γS3 < γ < γS

1 , the unique equilibrium has semi-separating reporting with choice νS2. For γS1 < γ < γS

2 , wehave multiple equilibria. Either we have an equilibrium where the manager has semi-separating reporting with projectchoice νS2 by the shareholders. Or, we have an equilibrium where the manager has pooling reporting with choice νS1 bythe shareholders. Finally, for γ > γS

2 , we again have a unique equilibrium where the manager has pooling reporting withchoice νS1 by the shareholders. The multiplicity problem arises because for γS

1 < γ < γS2 there are enough incentives

to always lie, as long as the project choice is appropriate (i.e., νS1). However, if the project chosen is smaller (νS2), γ

is not high enough to support constant lies; hence, the semi-separating equilibrium obtains.

45 One should notice that we are assuming that quitting is an uttermost right of the manager. That is, if lies lead to quitting, then these lies willultimately not be punished. So, in the good news case, if the manager lies and issues a bad report, beliefs will be such that the asset is worthless torisk-averse buyers. Hence, he quits and no punishment is inflicted on him. This is the reason why there are no punishment costs associated with thedeviation. It holds true that even if there were costs associated with this deviation, this strategy would still be optimal.46 The details are omitted since they are identical to those of Proposition 1, we just sketch the argument.


Appendix C. Proof of Proposition 4

Notice that, given our restrictions,

νM1 ≥ ν0 + 1

2κ

[C + E1

R− ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]> ν0 + 1

2κC = νM2. (69)

We obtain νM2 ≥ νM3 since full honesty only happens in the region where P1(γ) − L ≤ C. This also implies νS2 ≥ νS3

because

νS2 − νS3 = η[(L + C)/ν(γ) − P1(γ)

]2κ

≥ η[L + C − P1(γ)/ν(γ)

]2κ ≥ 0, (70)

where the second-to-last inequality follows from ν(γ) < 1. We can also calculate

where the first inequality follows from

[1η(E1)2ρm(1 − γ)

κR(1 − γλ)

]< 1, (74)

and the second follows from the fact that, given our assumptions on the moments of the asset payoff distribution,

−ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2) > 0. (75)

Finally, the third inequality follows since ν(γ) > ν0 and the fourth from our assumption on the manager’s outsideoption value.

Now, notice that η < 1 and [1 − (E1)2ρm(1 − γ)/κR(1 − γλ)] ∈ (0, 1] together entail

[1 − (E1)2ρm(1 − γ)

κR(1 − γλ)

]−1

>

[1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

]−1

. (76)


And, C > 0, η < 1 and (σ2H − σ2

L + (E1)2) < 0 imply[1 − (E1)2ρm(1 − γ)

κR(1 − γλ)

]−1{ν0 + 1

2κ

[C + E1

R− ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]}

>

[1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

]−1{ν0 + 1

2κ

[E1

R− η

ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]}.

Therefore, we have νM1 > νS1. By looking at the right hand side of the above inequality, one can easily see that itis increasing in η. Therefore,[

1 − η(E1)2ρm(1 − γ)

κR(1 − γλ)

]−1{ν0 + 1

2κ

[E1

R− η

ρm(1 − γ)

R(1 − γλ)(σ2

H − σ2L + (E1)2)

]}> ν0 + 1

2κ

E1

R, (77)

or, νS1 > ν∗.Recall that

νS2 = ν0 + η(L + C)/ν(γ) + (1 − η)E1/R

2κ, (78)

and notice that (L + C)/ν(γ) < (E1)/(R), since ν(γ)(E1)/(R) > Pν(γ)(γ) = (L + C), for any γ in the region whereνS2 applies. Then, ν∗ > νS2.

Finally, one can easily see that, if L > ηC,

νS3 = ν0 + ηP1(γ) + (1 − η)E1/R

2κ> ν0 + ηL + (1 − η)(L + C)

2κ= ν0+C + (L − ηC)

2κ> ν0+ C

2κ= νM2,

(79)

in view of the fact that P1(0) ≥ L and E1/R > L + C.

Appendix D. Proof of Proposition 5

First, recall that 0 < γ3 < γM1 < γM

2 < 1, so we have to consider four different regions of the parameter space. Weshow that, for each of these regions, the manager is weakly worse off under the proposed equilibrium (discretionaryreporting) than under truthful/exogenous reporting. Clearly, for 0 < γ < γ3, this statement is trivial to prove, since inthis region the equilibrium involves truth-telling, meaning the expected utility is the same under either reporting policy.

Let us calculate the expected utility under exogenous reporting. By our previous analysis, this is trivially seen toequal

EUexo := ν3P1(γ) + (1 − ν3)L − κ(ν3 − ν0), (80)

where we omit the superscript M on the project levels for sake of notation.Under discretionary reporting we have three additional regions:

1. γ > γM2 :

EU1 := ν1Pν1 (γ) + (1 − ν1)(Pν1 (γ) − C) − κ(ν1 − ν0) = Pν1 (γ) − (1 − ν1)C − κ(ν1 − ν0); (81)

2. γM1 < γ < γM

2 : EU1 if he plays the pooling equilibrium, or

EU2 := ν2(L + C) + (1 − ν2)(L) − κ(ν2 − ν0); (82)

if he plays the semi-separating equilibrium.3. γ3 < γ < γM

1 : EU2.


Now, we can calculate the difference in expected utility, �EUi := EUexo − EUi:

�EU1 = ν3P1(γ) + (1 − ν3)L − κ(ν3 − ν0) − Pν1 (γ) + (1 − ν1)C + κ(ν1 − ν0)

= ν3P1(γ) − Pν1 (γ) + (1 − ν3)L + (1 − ν1)C − κ(ν3 − ν1)

= ν3 1

R

[E1 − mρσ2

H

(1 − γ)

1 − λγ

]− 1

R

[ν1E1 − mρσ2

ν1

(1 − γ)

1 − λγ

]+ (1 − ν3)L + (1 − ν1)C − κ(ν3 − ν1)

= (ν3 − ν1)

[E1

R− κ

]+ ν3mρ

(1 − γ)

1 − λγ

(σ2

ν1

ν3 − σ2H

)+ (1 − ν3)L + (1 − ν1)C. (83)

Notice that the second term in the last line is positive, since (σ2ν1 )/(ν3) > σ2

ν1 > σ2H by our distributional assumptions.

So, for this expression to be positive, it is sufficient that

(ν3 − ν1)

[E1

R− κ

]+ (1 − ν3)L + (1 − ν1)C > 0, (84)

but this is equivalent to

κ >E1

R+ 1 − ν3

ν3 − ν1 L + 1 − ν1

ν3 − ν1 C,

where we used the fact that ν3 < ν1. Then, it is easy to show that κ satisfies this condition because, from our secondrestriction in (13), we have

κ >(E1)/(R)

2(1 − ν0)>E1

R>E1

R+ 1 − ν3

ν3 − ν1 L + 1 − ν1

ν3 − ν1 C,

where the last inequality follows from ν3 < ν1. So, �EU1 > 0.In the next case,

�EU2 = ν3P1(γ) − ν2(L + C) − κ(ν3 − ν2) − (ν3 − ν2)L > ν3P1(γ) − ν2P1(γ)−κ(ν3 − ν2) − (ν3 − ν2)L,

(85)

where the inequality follows from P1(γ) > L + C. Then we can rewrite the last terms as

(ν3 − ν2)[P1(γ) − κ − L] > 0, (86)

where the inequality follows from ν3 < ν2 and the fact that, given our assumptions,

κ >P1(γ) − L

2(1 − ν0)> P1(γ) − L. (87)

To put it more precisely, (ν3 − ν2) < 0 and [P1(γ) − κ − L] < 0, and obviously the product of two negative termsis positive. Therefore, �EU2 > 0. This completes the proof. Independently of the parameter value for γ , and no matterwhich equilibrium is being played, when we have multiplicity, we can state that the manager would be weakly betteroff under an exogenous reporting policy than under a discretionary policy controlled by him!

Appendix E. Arbitrageurs

In this section, we generalize the model by allowing for the presence of so-called arbitrageurs. More precisely, weassume that there is a group of investors that are fully informed about the state of the world. Clearly, if this is the caseand the supply is fixed as we assumed in the main text, when we allow the remaining investors to condition on prices,the equilibrium unravels and we can only have a separating equilibrium where the state of the world is fully revealed.To maintain the interesting aspects of the model, we must then assume either that investors cannot/do not condition onprices or that they are naive and cannot back-engineer the true state of the world (they are adamant of their beliefs).


We could also allow m to be random, then prices would only partially reveal information and the equilibrium wouldnot unravel completely. We take the former path for ease of exposition. Prices are then determined by

λ1[E − PR]

ρσ2 + λ2[A − PR]

ρB2 + (1 − λ1 − λ2)1

1 − γ

[E − PR]

ρσ2 = m, (88)

where A and B represent the beliefs of the arbitrageurs and we assume that they constitute a measure λ2 of the market.Clearly, B2 = σ2

H or σ2L and A = 0 or E1, depending on the state of the world. So, price in the good state, when the

rest of the investors hold beliefs ν, is

PGν (γ) = 1

R

σ2Hσ2

ν (1 − γ)

λ2σ2ν (1 − γ) + (1 − γλ1 − λ2)σ2

H

(E1

σ2H

λ2 + νE1

σ2ν

(1 − γλ1 − λ2)

(1 − γ)− mρ

), (89)

and price in the bad state is

PBν (γ) = 1

R

σ2Lσ2

ν (1 − γ)

λ2σ2ν (1 − γ) + (1 − γλ1 − λ2)σ2

L

(νE1(1 − γλ1 − λ2)

σ2ν (1 − γ)

− mρ

). (90)

At this point we also assume that the cost parameter κ satisfies

κ > max

{(E1)2ρm(1 − γ)

R(1 − γλ),C + E1/R − ρm(1 − γ)/R(1 − γλ)(σ2

H − σ2L − (E1)2)

2(1 − ν0),

η[1 − ν0/ν0(L + C) − mρσ2H/R] + E1/R

2(1 − ν0)

}. (91)

This guarantees that solutions to the project choice problems are interior, i.e., the probability of the good state is setabove ν0 and below 1.

We could even have allowed the institutional investors to be informed—to be the arbitrageurs. Everything wouldfollow in the same lines, with the exception that prices would be given by

λ[E − PR]

ρσ2 + (1 − λ)1

1 − γ

[A − PR]

ρB2 = m, (92)

and we would need minor adjustments on our assumptions. All we need for the model to work is that demand increaseswith γ and beliefs, ν.

E.1. Information manipulation in the presence of arbitrageurs

We can then establish Proposition 6, which replaces Proposition 1. We omit the proof as it mimics the proof of thelatter.

Proposition 6 (or 1A). For any project choice ν ∈ (ν0, 1), the second stage (communication) has a Perfect BayesianEquilibrium determined by two thresholds that we call γA

1 < 1 and γA2 ≥ 0, with PB

ν (γA1 ) = L + C and PB

1 (γA2 ) =

L + C

(i) If γ ≥ γA1 , M always issues a g report for the company stock. Beliefs are equal to ν.

(ii) If γA2 < γ < γA

1 , there are no incentives either to always lie or to be always truthful. Hence, the manager uses amixed strategy. More precisely, in the case of good news he always says g, and in the case of bad news he says gwith probability pA

1 , defined as pA1 (γ) = ν/1 − ν1 − ν(γ)/ν(γ), where ν(γ) is such that

PBν(γ)(γ) = L + C, ∀γ ∈ [γA

2 , γA1 ]. (93)

In this region, we have P(G|g) = ν(γ) and P(G|b) = 0.(iii) Finally, if γ ≤ γA

2 , the manager always tells the truth and is fully believed by investors.


Given this result, one then proceeds as before to the investment stage. Even though we do not fully solve the modelagain, we show that qualitative results are the same, as the main mechanisms of the model are not changed. In essence,the only thing that changed is that prices are a little bit higher in the good state (since some agents know the true stateof the world), and a little bit lower in the bad state. It is still true that under exogenous reporting or low γ the managerwants to under-invest. However, due to algebraic complications in the model, we resort to proving a weaker result thanbefore for the case of full lies, as we explain below.47

E.2. Investment policy with arbitrageurs

We sketch the changes in the model presenting only the results that are different.

E.2.1. Probabilistic liesWe can write the manager’s objective as νPG

ν(γ) + (1 − ν)L − κ(ν − ν0)2, so νM2 = ν0 + PGν(γ) − L/2κ. If the

shareholder chooses the projects, he maximizes

η{νPGν(γ) + (1 − ν)p(ν, γ)(L + C)} + (1 − η)ν

E1

R− κ(ν − ν0)2, (94)

and νS2 = ν0 + η((L + C)/(ν(γ)) + PGν(γ) − L − C) + (1 − η)E1/R2κ. It is clear that, once more, νS2 > νM2.

E.2.2. LiesIn this case, we can write the manager’s objective as

νPGν (γ) + (1 − ν)(PB

ν (γ) − C) − κ(ν − ν0)2, (95)

with FOC given by

PGν (γ) − PB

ν (γ) + ν∂PG

ν (γ)

∂ν+ (1 − ν)

∂PBν (γ)

∂ν+ C − 2κ(ν − ν0) = 0. (96)

The shareholder’s problem is to maximize

η[νPGν (γ) + (1 − ν)PB

ν (γ)] + (1 − η)νE1

R− κ(ν − ν0)2, (97)

with FOC

η

[PG

ν (γ) − PBν (γ) + ν

∂PGν (γ)

∂ν+ (1 − ν)

∂PBν (γ)

∂ν

]+ (1 − η)

E1

R− 2κ(ν − ν0) = 0. (98)

As it turns out, solving these FOCs is not as easy as before. Hence, that is where our results get weaker in thepresence of arbitrageurs. We can no longer guarantee that the shareholders’ optimal is below the manager’s for anyparameter value, even though we conjecture that to be true. Instead, we establish a weaker result, as follows:

Proposition 7. There exists a γ < 1 such that for every γ ≥ γ we have νS1 < νM1.

Proof. We know that at the shareholders’ optimal

η(F − C) + (1 − η)E1

R= 2κ(νS1 − ν0), (99)

where

F = C + PGνS1 (γ) − PB

νS1 (γ) + νS1 ∂PGν (γ)

∂ν+ (1 − νS1)

∂PBν (γ)

∂ν. (100)

47 Given the above reasoning, it is my conjecture that a result as strong as the one in the main text may be proven. Unfortunately, as of the lastrevision of this paper, I have not been able to do so.


Now we can evaluate the manager’s FOC at the shareholders’ optimal to obtain

F −{

η(F − C) + (1 − η)E1

R

}= (1 − η)

(F − E1

R

)+ ηC. (101)

Furthermore, it is easy to show that limγ→1

F = E1/R + C, and hence, the FOC of the manager at the shareholders’

optimal goes to C as γ → 1. So, for γ high enough, we have that the FOC of the manager when evaluated at theshareholders’ optimal is positive, implying νS1 < νM1. �

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Further reading

Benabou, R., Laroque, G., 1992. Using privileged information to manipulate markets: insiders, gurus and credibility. Quarterly Journal of Economics107 (3), 921–958.

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12 (4), 653–686.Morgan, J., Stocken, P.C., 2003. An analysis of stock recommendations. Rand Journal of Economics 34 (1), 183–283.Nagar, V., Nanda, D., Wysocki, P.D., 2003. Compensation policy and discretionary disclosure. Journal of Accounting and Economics 34 (1),

283–309.

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