jleo, v35 n3 information sharing and incentives in ......koeppl, jeff mcgill, matt mitchell, mikhail...

Information Sharing and Incentives in Organizations

Jean-Etienne de Bettignies

Queen’s University

Jan Zabojnik*

Queen’s University

We study an organization, consisting of a manager and a worker, whose suc-

cess depends on its ability to estimate a payoff-relevant but unknown param-

eter. If the manager has private information about this parameter, she has an

incentive to conceal it from the worker in order to motivate him to search for

additional information. Due to a time-inconsistency problem, the manager con-

ceals her information more often than if she could commit to an information

sharing policy, but even a manager with commitment power shares her infor-

mation less than would be efficient. We also show that managers who are more

likely to get informed are more willing to share their information and that unless

the manager’s information substantially improves the worker’s productivity,

managerial and worker abilities are substitutes in the firm’s profit function.

(JEL D21 D82, L23)

1. Introduction

In many situations, a decision-maker who needs to decide on the bestcourse of action relies on information collected by someone else, say asubordinate or a consultant. For example, when deciding what features oramenities to include when developing a new product, a product developermay rely on a report provided by a firm employee or management con-sultant who conducted a market feasibility study. A legislator who wantsto implement a reform may draft the legislation based on the input of alobbyist who collected information about the optimal feasible policy

An earlier version of this article was circulated under the title “On the Diffusion of

Information in Organizations.” We thank the Editor, two anonymous referees, and

Ricardo Alonso, Jim Brander, Chris Cotton, Tom Davidoff, Anne Duchene, April Franco,

Alberto Galasso, Denis Gromb, Thomas Hellmann, Ig Hortsmann, Navin Kartik, Thor

Koeppl, Jeff McGill, Matt Mitchell, Mikhail Nediak, Tom Ross, Nate Schiff, Kostas

Serfes, Peter Thompson, Mihkel Tombak, Ralph Winter, and seminar participants at

Drexel University, HEC Montreal, Instituto de Empresa, Queen’s University, University

of British Columbia, University of Toronto, and the EARIE Meetings 2013 (Evora) for

helpful comments. Jonathan Lee and Jing Liang provided excellent research assistance.

*Department of Economics, Queen’s University, 94 University Ave, Kingston, ON

K7L3N6, Canada. Email: [email protected]

The Journal of Law, Economics, and Organization, Vol. 35, No. 3doi:10.1093/jleo/ewz008Advance Access published June 20, 2019� The Author(s) 2019. Published by Oxford University Press on behalf of Yale University.All rights reserved. For permissions, please email: [email protected]

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reform. A partner at a law firm building a civil case against a defendant

may decide how to try the case based on the evidence collected by her team

of associates.In each of the above examples, the decision-maker may already have

some idea about the environment she faces even before she obtains a

report from the consultant, the lobbyist, or the team of associates. If the

decision-maker indeed has some prior knowledge about the problem,

should she share it with the party she has tasked with collecting additional

information? This is the question we address in this article.Much of the economics literature views information sharing in organ-

izations as desirable, and the research has focused on how incentives can

be structured to promote information sharing (e.g., Snyder and Levitt

1997) or on the consequences of imperfect information sharing (e.g.,

Dessein 2002). We show that in settings such as those described above,

information sharing has a downside because it can be detrimental to em-

ployees’ incentives to generate additional information.The above examples suggest that the question applies to a variety of

different settings. We capture the common features of these settings in a

model of an organization whose probability of success depends on its

ability to precisely estimate the true value of payoff-relevant but unknown

parameters. Our setup underscores the idea that organizational decisions

based on accurate information ought to lead to greater profits than deci-

sions based on imprecise information.1 Going back to the above examples,

a new product is more likely to succeed the closer it meets the market

needs, new legislation is more likely to pass the closer it is to the optimal

feasible policy, and a lawyer is more likely to win her case the closer her

arguments are lined up with available evidence.The manager in our model may or may not have observed a private

signal about a parameter of interest. In either case, she tasks a subordinate

(worker) with collecting additional information about the parameter. If

the worker discovers new information, the manager combines it with her

own signal and obtains a more precise estimate, thus increasing the or-

ganization’s success probability. In most of our analysis, monetary incen-

tives are not feasible, but the worker receives a private benefit if the firm is

successful, so he shares with the manager the goal of maximizing the or-

ganization’s success probability. The likelihood that the worker discovers

new information increases with the effort that he exerts in his search, but

effort is costly and difficult to measure, which leads to agency concerns.

1. This could be, for example, because a more accurate assessment of the environment it

faces allows the organization to better adapt to it. The idea that adaptive ability is an im-

portant factor affecting profits has a long tradition in economics. It goes back at least to

Barnard (1938) and has seen renewed attention in the recent literature on adaptive coordin-

ation, which includes Dessein and Santos (2006, 2016), Alonso et al. (2008, 2013), Rantakari

(2013), Calvo-Armengol et al. (2015), and Dessein et al. (2016).

620 The Journal of Law, Economics, & Organization, V35 N3D

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We show that these agency concerns are exacerbated when the managershares her information with the worker; in fact, even the very possibilitythat the manager is informed dampens the worker’s incentives, comparedto the situation where the manager never receives a signal. This is becausemanagerial information reduces the marginal impact of the worker’s efforton this distribution, and in turn on the worker’s expected payoff. Whenthe manager shares information with the worker, she reveals with cer-tainty that she is informed, and thus mutes the worker’s incentives tocollect additional information. We refer to this adverse effect of informa-tion sharing on incentives as the certainty effect because it is driven by thefact that when the manager discloses her information, the worker is nowcertain that the marginal productivity of his effort is low. Our first result isthat in this simple setup, the certainty effect always induces the informedmanager to withhold her information from the worker in order to avoidmuting his incentives.

But what happens when the manager can commit to an informationdisclosure strategy? In practice, such a commitment could take the form ofthe “open book” management approach adopted by some firms, whichconsists of disclosing to the firm’s employees detailed operating informa-tion, such as its financial records and the sources of its profits (Davis1997). By committing to more frequent information disclosure to theworker when she is informed, the manager generates a positive beliefseffect on the worker’s incentives: she increases the worker’s belief thatwhen she discloses no information she is indeed uninformed, a state ofthe world in which as mentioned above the worker’s marginal product ofeffort is higher. A higher likelihood of a high marginal product of effortstate in turn leads to greater worker incentives and effort. This positivebeliefs effect of information disclosure works against the negative cer-tainty effect; and in fact may dominate it. Indeed, our second resultshows that if commitment is feasible, the manager commits to alwaysdisclose her information to the worker, as long as the worker’s cost ofeffort function is not too convex.

The time-inconsistency problem characterizing the difference betweenthe commitment and no-commitment scenarios stems from an externalitythat the informed managerial type imposes upon the uninformed type: Ifthe worker knew for sure that the manager did not receive a signal, hewould have a strong incentive to collect more information. However, theworker understands that the manager tends to withhold information fromhim to avoid negatively affecting his incentives; he therefore holds back onhis effort even when the manager is in fact uninformed. An ex ante com-mitment to share her information with the worker allows the manager tointernalize this effect.

Importantly, we show that commitment strengthens the worker’s incen-tives when the manager is uninformed and the marginal benefit of add-itional information is large, while weakening the worker’s incentives whenthe manager is informed and the marginal benefit of additional

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information is small. Commitment thus increases the efficiency with whichthe worker allocates his effort, and represents a Pareto improvement overno commitment. Even under commitment, however, the manager tends toshare her information inefficiently too little because she ignores the posi-tive impact of information sharing on the worker’s expected utility.

After establishing these key results, we enrich the model by assumingthat if the worker learns the manager’s information, he is able to betterdirect his search for new information, so that for any given effort level, hediscovers new information with a higher probability. We show that thisproductivity effect can induce the manager to share her information evenin the absence of commitment power, but only if the resulting productivityimprovement exceeds a minimum threshold level.

We then use this richer framework to examine whether more able man-agers share their information with subordinates more frequently andwhether more able managers should be matched with more able workers.In our model, a natural way to measure the manager’s ability is throughthe probability that she will observe a private signal about the parameterof interest. In particular, we show that a better-informed manager is morelikely to share her information with the worker. Importantly, this holdsnot only because information sharing is feasible only when the manager isinformed but also because, once informed, the manager has a strongerincentive to share her information.

The worker’s ability can be measured in a similar way, as the probabilitythat any given level of his effort will yield new information. With quad-ratic costs of effort, managerial and worker abilities tend to be substitutesin the firm’s profit function, unless the manager’s information has a largepositive effect on the productivity of the worker’s effort. If this product-ivity effect is so large that the manager always wants to share her infor-mation, the two players’ abilities become complements.

Finally, we allow for monetary incentives, under the assumption thattransfers from the worker to the manager are not feasible. Even when themanager can offer a contract in which the worker’s pay is conditional onthe project’s outcome, our main results regarding how information shar-ing affects the worker’s incentives and how this feeds back to the man-ager’s willingness to share her information continue to hold.

1.1 Related Literature

This article is closely related to Prendergast’s (1993) theory of “yes men”:In both papers, a principal enlists the help of an agent to improve anestimate of some parameter of interest, and organizational profitsdepend on the precision of this estimate. However, in Prendergast’smodel, the worker is rewarded when his report is close enough to themanager’s estimate, and the main focus of the analysis is on the worker’sincentive to bias his report toward the manager’s estimate. In our model,the worker’s payoff is conditional on the success of the firm’s project, and


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the main question we study is whether the manager will find it optimal toconceal her information from the worker.

Two other papers in which agents exert effort to collect information areChe and Kartik (2009) and Angelucci (2017). Che and Kartik (2009) showthat in order to strengthen the agent’s incentives, the principal may delib-erately choose an agent with a prior belief that differs from her own prior.The cost is that this difference in beliefs leads the agent to withhold someof his information from the principal. In our model, it is the principal whowithholds information from the agent. Angelucci (2017) examines a set-ting with multiple agents and shows that from the incentive point of view,it is better to organize the agents in a team, so that they are jointly re-sponsible for generating a vector of signals, rather than tasking each agentwith producing his own signal.

The theme of information gathering by an agent makes our article alsorelated to the growing literature on delegated expertise, which includesLambert (1986), Demski and Sappington (1987), Core and Qian (2002),Gromb and Martimort (2007), and Malcomson (2009), among others.These models focus on environments in which agents need to be motivatedto both collect information about available projects and to choose theproject to be undertaken. In contrast, our agent does not have formaldecision-making authority (about the adaptive action to be selected).Furthermore, the literature on delegated expertise typically assumes thatit is prohibitively costly for the agent to convey to the principal his infor-mation about the available projects, whereas communication between theprincipal and the agent is at the heart of our analysis.

Ours is not the first article to underline the positive effects of “ignor-ance” on incentives. Delaying information release (a milder form of in-formation concealment) has been shown to improve effort incentives indelegated expertise models (Demski and Sappington 1986) and in dynamicteam production settings (Campbell et al. 2013), and to favor cooperationin partnership games (Abreu et al. 1991). In Zabojnik (2002), a principal’sinformation negatively affects an agent’s incentives if it conflicts with theagent’s own signal. In repeated principal-agent settings where the princi-pal privately observes the agent’s output, providing no feedback on agentperformance may improve effort incentives when the agent receives noindependent signal about his performance (Fuchs 2007) or when thissignal is poorly correlated with the principal’s information (Maestri2012). Our model contributes to this line of research by introducing andexploring the trade-off between the incentive effects and the productivityeffect of information sharing and in highlighting the importance of com-mitment in resolving this trade-off.

Finally, there exists a large literature on the impact of commitment onagency relationships. Within this literature, the most closely related to ourwork are the papers in which the principal’s ability to commit to an ex postcourse of action has positive effects on the agent’s ex ante effort. Thesepositive effects may come from the principal’s ability to commit (a) not to


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renegotiate to provide full insurance to a risk-averse agent (Fudenberg

and Tirole 1990); (b) not to terminate projects devised by the agent

(Rotemberg and Saloner 1993, 2000; Levitt and Snyder 1997; Van den

Steen 2005); (c) not to interfere with the agent’s project choice (Aghion

and Tirole, 1997; Burkart et al. 1997); (d) not to change corporate strategy

after strategy-specific investments have been made by the agent (Ferreira

and Rezende 2007); or (e) to monitor the agent’s effort directly once

exerted (Jost 1996). In contrast, in our model, the impact of commitment

on incentives works through the beliefs effect discussed above: by commit-

ting to more frequent information sharing when she is informed, the prin-

cipal can increase the agent’s belief that the marginal product of his effort

is high when no information is shared, thus augmenting his incentives to

exert effort.Other papers within this literature do not necessarily focus on agents’

effort, but a number of them are related in that a principal’s ability to

commit has beneficial incentive effects. These papers include Dessein

(2002), Alonso et al. (2008), Krishna and Morgan (2008), Rantakari

(2008), Bouvard et al. (2015), and Shadmehr and Bernhardt (2015). In

the last two papers, the impact of commitment has a similar flavor as

our beliefs effect, albeit in different contexts. Specifically, in Shadmehr

and Bernhardt’s (2015) political economy model, a ruler commits to

reduce censorship in order to decrease the citizens’ belief that no news is

bad news, which decreases their incentives to revolt. In Bouvard et al.

(2015), a bank regulator commits to disclosing negative information

about individual banks to increase investors’ belief that the state is good

when no information is disclosed, which helps prevent bank runs. Finally,

the first four papers above are cheap talk models of organizational design

in which a principal uses delegation of authority to commit not to interfere

with an agent’s preferred project. As in our model, commitment in these

papers helps the organization to make better-informed project choices,

although the mechanism is different.2

The article proceeds as follows. In Section 2, we present the model. In

Section 3, we consider the case without commitment. We analyze how the

manager’s information affects the worker’s incentives and characterize the

manager’s optimal strategy for sharing her information with the worker.

In Section 4, we allow the manager to commit to an information sharing

policy. We highlight the manager’s time-inconsistency problem and exam-

ine the efficiency properties of the equilibrium. Section 5 introduces direct

productivity benefits of information sharing, provides comparative statics

results with respect to the manager’s ability, and examines whether

2. Commitment also plays a role in the context of adverse selection. Most notably per-

haps, Laffont and Tirole (1988) show that the principal’s inability to commit to long-term

contracts may render communication from the informed agent more difficult—the so-called

ratchet effect—due to the agent anticipating that information revelation may be used oppor-

tunistically by the principal in the future.


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managerial and worker abilities are complements or substitutes. Section 6explores a setting in which monetary incentives are feasible. Section 7concludes. Formal proofs are in the Appendix.

2. The Model

We consider an organization composed of a manager (she) and a worker/subordinate (he), both of them risk neutral. The manager’s goal is toestimate as precisely as possible the true value of a parameter of interestZ, for example a random shock to which the organization will have toadapt. She has at her disposal up to three pieces of information (describedbelow in more detail)—a prior belief p0 about the distribution of Z; and,possibly, imperfect signals of Z observed by herself and by the worker—which she can use to form a posterior belief p about the distribution of Z.

We follow Prendergast (1993) in assuming that expected profits areproportional to the negative of the variance s2 of the posterior distribu-tion p. Specifically, the organization generates a successful project withprobability � ¼ maxf0; 1� Ks2g ¼ maxf0; 1� K=hg, where h is the pre-cision of p and K is a positive constant.3 This is a reduced-form way ofcapturing the idea that managerial decisions based on accurate informa-tion lead to higher profits than decisions based on imprecise information.

The organization’s profit from the project, ~P, is ~P ¼ P > 0 if theproject is successful and ~P ¼ 0 if the project fails (which happens withprobability 1� �). To ensure an interior equilibrium (� > 0), it will beassumed that K < h0, where h0 is the precision of the players’ priorbelief p0 ¼ NðZ0; 1=h0Þ about the distribution of Z.

2.1 Priors and Signals

At the beginning of the game, both the manager and the worker have thesame prior p0. After the worker is hired, with probability � < 1 the man-ager observes an imperfect signal of Z, Zm ¼ Z+ em, with em � Nð0; 1=hmÞ.With probability 1� �, the manager does not observe any signal. Both thesignal and whether or not she has received it are the manager’s privateinformation, but the worker knows the probability � and the distributionof em.

The worker can search for additional information about �. If he exertssearch effort e, then with probability e, he obtains a signal Zw ¼ Z+ ew,where ew � Nð0; 1=hwÞ. With probability 1� e, he does not obtain anysignal. We assume that Z, em, and ew are independently distributed.The worker’s personal cost of effort function C : ½0; 1�!R is twicecontinuously differentiable, with Cð0Þ ¼ C0ð0Þ ¼ 0; C0ðeÞ > 0 fore 2 ð0; 1�; C0 > 0, and C0ð1Þ > Kb 1

h0� 1

h0+hw

� �. These conditions, together

3. For expositional convenience, we will work with precisions rather than variances,

where given a variance s2, the corresponding precision is defined by h ¼ 1=s2.


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with the restriction that K < h0, ensure an interior solution e 2 ð0; 1Þ forthe worker’s effort.

2.2 Information Sharing and Updating

If informed, the manager decides whether to share her information withthe worker before the worker exerts his effort. The manager’s informationis not verifiable, which means that she can choose to misrepresent it, hide apart or all of it, or even pretend that she is informed when she is not.However, as will become apparent in the course of the analysis, in ourmodel neither the manager nor the worker will have an incentive to mis-represent their information, divulge only a part of it, or pretend to beinformed when they are not. The only misrepresentation we need toworry about is that an informed manager may have an incentive to pre-tend she has not received any signal. We will discuss the worker’s and themanager’s incentives to communicate truthfully in greater detail later; fornow, we will proceed under the assumption that the worker always revealshis signal fully and truthfully, and that the manager either pretends to beuninformed or reveals her signal fully and truthfully. We will say that themanager “conceals” her information in the former case, and “divulges” or“shares” it in the latter case.

If the manager shares her information, or tells the worker she is unin-formed, the worker uses the information to update his prior and thensearches for additional information. If he is successful, he reports hissignal to the manager, who combines the worker’s report with her ownsignal (if any) to form a posterior belief, p ¼ NðZ; 1=hÞ. Given the proper-ties of Bayesian updating under normal distributions, this posterior beliefrepresents a normal distribution, with

Z ¼h0Z0+dhwZw+mhmZm

h0+dhw+mhmand h ¼ h0 + dhw + mhm;

where m¼ 1 if the manager is informed, m¼ 0 if she is uninformed, d ¼ 1 ifthe worker has generated an additional signal, and d¼ 0 if he has not.

2.3 Contracting and the Worker’s Payoff

The worker’s payoff consists of a private benefit ~b ¼ b > 0 from a suc-cessful project and ~b ¼ 0 from an unsuccessful project, capturing the ideathat the worker cares about the success of the organization.

In order to better isolate the effects of information sharing, we abstractfrom monetary transfers throughout the majority of the article. Thisimplies both that contingent contracts are not feasible and that the man-ager cannot extract from the worker his private benefits through an up-front payment. The noncontractibility assumption is relaxed in Section 6,where we allow the worker’s pay to depend on realized profits. We showthere that if the worker is protected by limited liability (so that agency


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concerns are not assumed away), the main results of our model continue tohold.

2.4 Timing of the Game

The game evolves in five stages:

At date 0, the worker is hired and Nature draws parameterZ to be estimated.At date 1, with probability �, the manager receives a private

signal, Zm. She then decides whether or not to divulge herinformation to the worker.At date 2, the worker searches for an additional signal

about Z and, with probability e determined by his searcheffort, observes Zw.At date 3, if the worker obtained an additional signal, he

reports it to the manager, who uses it to form her posteriorbelief p ¼ NðZ; 1=hÞ.At date 4, profits (and managerial payoff) P or 0 and

private benefits b or 0 are realized.

3. Analysis of the Case Without Commitment

We start by analyzing the case in which the manager is unable to committo a particular information sharing strategy. In the next section, we willconsider the case in which the manager is able to commit, at the time ofcontracting, to a particular probability with which she will share her in-formation with the worker if she is informed.

3.1 The Effects of Managerial Information Sharing on Worker Effort

Proceeding by backward induction, at date 3 the firm’s expected profit,conditional on the posterior distribution p, is given by

E½ ~Pjp� ¼ maxf0; 1� K=hgP ¼ ð1� K=hÞP; ð1Þ

where the last equality follows from K < h04h. Similarly, the worker’sexpected benefit conditional on p is

E½ ~bjp� ¼ ð1� K=hÞb: ð2Þ

As can be seen from equations (1) and (2), once the worker’s effort issunk, he shares with the principal the goal of maximizing the probabilityof success. Consequently, the worker always has an incentive to report hisposterior belief to the manager truthfully, as doing so maximizes the pay-offs. The two expressions also reveal that both of the expected payofffunctions are strictly increasing and strictly concave in h. This makes in-tuitive sense: a more precise posterior distribution leads to a better


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estimation of parameter Z, and in turn to a greater expected probability

that the project will be successful. But due to decreasing marginal returns,

as the posterior distribution becomes more precise, the impact of further

increases in this precision on the expected probability of success becomes

smaller.Suppose the manager has observed a signal at date 1 and has shared it

with the worker. The worker’s expected private benefit is then

BIðeÞ � eb 1�K

h0+hw+hm

� �+ð1� eÞb 1�

K

h0+hm

� �; ð3Þ

where the subscript I indicates that the manager is informed. Denote by e�Sthe worker’s optimal effort at date 2 given that the manager shared with

him her signal. This effort maximizes the worker’s expected payoff

BIðeÞ � CðeÞ, and is therefore given by the first-order condition

Kb1

h0+hm�

1

h0+hw+hm

� �¼ C0ðe�SÞ: ð4Þ

The left-hand side (LHS) of equation (4), which captures the worker’s

marginal benefit of effort, shows that an increase in the worker’s effort

raises the expected precision of the manager’s posterior distribution. As

already discussed, this in turn leads to a better estimation of parameter Z,which increases the probability of the project’s success.

Now suppose the manager has not communicated any information to

the worker at date 1. This could be either because she has not observed any

signal or because she has concealed her information from the worker. In

this case, the worker makes a conjecture g about the manager’s strategy g,where the latter is the probability that the manager will divulge her infor-

mation to the worker if she is informed. The worker uses this conjecture to

form a posterior belief � about the probability that the manager is in-

formed. Using Bayes’ rule, we can express this belief as

� ¼�ð1� gÞ

�ð1� gÞ+1� �: ð5Þ

Conditional on whether the manager is informed or not, the worker’s

expected private benefit is either BIðeÞ given in expression (3) or

BUðeÞ � eb 1�K

h0+hw

� �+ð1� eÞb 1�

K

h0

� �;

where the subscript U indicates that the manager is uninformed. The

worker then chooses her effort so as to maximize

�BIðeÞ+ð1� �ÞBUðeÞ � CðeÞ: ð6Þ


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To indicate that no information is shared, we denote the worker’s op-timal effort in this case as e�N. This effort solves the first-order condition

�Kb1

h0+hm�

1

h0+hw+hm

� �+ð1� �ÞKb

1

h0�

1

h0+hw

� �¼ C0ðe�NÞ: ð7Þ

Lemma 1. For all � < 1, the worker exerts greater effort if he does notreceive information from the manager than if he does: e�N > e�S. Moreover,contingent on not receiving information from the manager, the worker’soptimal effort strictly decreases in his belief, �, that the manager is in-formed:

qe�Nq�< 0.

Intuitively, the worker anticipates that for any given level of effort e,the expected precision of the manager’s posterior distribution will begreater if the manager received a signal than if she did not:h0+ehw+hm > h0+ehw. But as discussed above, the worker’s expectedpayoff is strictly concave in the manager’s posterior precision h, andhence in the worker’s effort, implying that the marginal impact of anincrease in effort on E½ ~bjp� is lower when the manager is informed thanwhen she is not. Thus, contingent on not receiving information from themanager, the greater is the worker’s belief � that the manager is informedbut concealed her information, the lower is the worker’s expected mar-ginal impact of his effort and, hence, also his equilibrium effort.

In contrast to e�N, the worker’s optimal effort conditional on receivinginformation from the manager, e�S, does not depend on �, as is apparentfrom equation (4). This differing effect of the worker’s belief about themanager’s strategy on his effort will play an important role in determiningthe manager’s equilibrium information sharing strategy and the worker’sequilibrium effort.

3.2 Equilibrium Sharing of Managerial Information

In this section, we characterize the manager’s equilibrium informationsharing strategy, g, under no commitment.4 Note that given the effectsof information sharing on the worker’s effort described in the previoussection, an uninformed manager has no incentive to pretend she is in-formed, or to misrepresent her information. Telling the worker that sheis informed would only mute his incentives, and this would happen even ifshe were uninformed, as long as the worker believed her.

4. In principle, the manager could condition her information sharing decision on the

realization of her signal Zm. However, given that the expected payoff functions do not

depend upon Zm and given that due to the nature of normal updating, the actual realization

of Zm has no impact on the worker’s incentives, it should be clear that the manager cannot

benefit from such a conditional disclosure strategy.

Nevertheless, if Z were not distributed normally, the precision of the worker’s posterior,

and therefore also his incentives, could depend upon the realization of Zm in which case the

manager might benefit from conditioning her strategy on Zm.


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Also note that the disincentive effect of information sharing only de-pends on the precision of the manager’s signal, which is known to theworker. Therefore, once the manager decides to reveal to the worker thatshe has received a signal, she cannot benefit from hiding a part of it. It istherefore optimal for the manager either to hide her signal or to share itwith the worker fully and truthfully, as we have assumed.

Thus, suppose the manager has received a signal. Using equation (1),her expected profit is

EpS ¼ e�SP 1�K

h0+hw+hm

� �+ð1� e�SÞP 1�

K

h0+hm

� �ð8Þ

if she shares her information with the worker, and

EpP ¼ e�NP 1�K

h0+hw+hm

� �+ð1� e�NÞP 1�

K

h0+hm

� �ð9Þ

if she keeps her signal private.Let g� denote the equilibrium probability with which an informed man-

ager shares her information with the worker.

Proposition 1. If the manager is unable to commit to sharing her informa-tion with the worker, the unique equilibrium entails full concealment.That is, the manager never shares her information with the worker,g� ¼ 0, and the worker’s effort, e�N, is given by equation (7) evaluated at� ¼ �.

Proposition 1 tells us that without the ability to commit to a particularinformation sharing strategy, the manager always ends up concealing herinformation from the worker. The reason is that sharing her informationbrings about certainty in the worker’s mind that she is in fact informed. Butas we have seen in the previous subsection, it is doubt as to whether themanager is informed which motivates the worker to exert more effort. Thus,the manager realizes that by eliminating this doubt, sharing information hasa negative “certainty” effect on the worker’s effort and in turn on her ex-pected profit. As a result, the manager prefers to conceal her information.

4. Analysis of the Case with Commitment

4.1 Optimal Information Sharing Policy

Given that information sharing has detrimental effects on the worker’sincentives, the full concealment equilibrium described in Proposition 1may seem natural and unsurprising. However, in this section, we showthat this stark result is a consequence of the principal’s inability to committo sharing her information. When commitment is possible, there is a rangeof parameter values such that the principal finds it optimal to commit tofull information disclosure, which improves both her expected payoff andefficiency. This is demonstrated in the next proposition.


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Proposition 2. Suppose the manager can commit to a probability g withwhich she will share her signal with the worker and let g�� denote theprobability to which she optimally commits. Then there exists a constantC� > 0 such that if C0ðeÞ

C0ðeÞ e < C� for all e, the manager commits to alwaysshare her information, i.e., g�� ¼ 1.

Proposition 2 tells us that if the worker’s cost function is not too convex

as measured by the relative Arrow–Pratt coefficient C00ðeÞC0ðeÞ e, a manager who

can publicly commit to the frequency with which she will share her infor-mation with her subordinate would choose full information disclosure.This result can be understood by considering the two opposing effectsthat an increase in the conditional probability with which the managershares her information has on her expected profit.

On the one hand, sharing information brings about certainty in themind of the worker that the manager is in fact informed. As shown in theno-commitment scenario (see Lemma 1 and Proposition 1), this reducesthe worker’s effort from e�N to e�S, which in turn reduces the manager’sexpected profit. Increasing the disclosure probability g increases the like-lihood of the low effort and, hence, of the low expected profit. This is thecertainty effect of information disclosure discussed in the no-commitmentscenario.

On the other hand, by committing to more frequent information shar-ing, the manager reduces the worker’s belief � ¼ �ð1�gÞ

1��+�ð1�gÞ that she is ac-tually informed when no information is shared. Thus, when the workerreceives no information from the manager, he now views it more likelythat this is because the manager is uninformed (rather than informed andconcealing). This in turn increases the worker’s expected marginal benefitfrom effort, and indeed his effort choice, whenever he receives no infor-mation from the manager. This “beliefs” effect of information disclosurebenefits the manager both when she is not informed, and when she isinformed but conceals it from the worker.

Which effect dominates? This is where the relative Arrow–Pratt coeffi-cient comes into play. When the worker’s cost of effort function is not tooconvex (C

00ðeÞC0ðeÞ e is sufficiently small), an increase in information disclosure g,

via its effect on the worker’s beliefs, will have a large positive effect on theworker’s effort level. In that case, the beliefs effect is sufficiently strong todominate the certainty effect and to ensure that increasing the disclosureprobability has a positive impact on the manager’s expected profit. In thatcase, the manager’s optimal strategy is to always reveal her signal.

More generally, commitment eliminates the manager’s time-inconsist-ency problem. Forcing the informed managerial type to share her infor-mation imposes a positive externality on the uninformed type by reducingdoubt in the worker’s mind about whether a manager who does not shareinformation is really uninformed. In the absence of commitment, the in-formed type does not take this externality into account and thereforeconceals her information too often.


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Note that the low convexity requirement of Proposition 2 holds for anumber of simple and standard cost functions including, for example, thequadratic function.

Proposition 3. Suppose CðeÞ ¼ e2=2. Then if the manager has commitmentpower, she commits to always share her information, that is, g�� ¼ 1.

To sum up, the analysis of this section shows that optimal organiza-tional design often includes policies that allow the organization to committo information sharing between managers and their subordinates, a pre-diction that fits well with the “open book” management approach that hasbeen adopted by a number of firms, including Southwest Airlines, HomeDepot, and Whole Foods. The essence of the open book managementstrategy is that the company explains to its employees its goals and com-mits to share with them all of its operating information (Dixon et al.2004), similar to what happens in our model.

4.2 Commitment versus No Commitment

The above analyses reveal that the difference between the no-commitmentscenario and the commitment scenario is driven by how a deviation fromthe manager’s strategy affects the worker’s effort. Under commitment,any changes in the manager’s strategy g are fully reflected in the worker’supdating process. In contrast, in the absence of commitment, the workercannot directly observe the principal’s choice of g. Instead, he forms aconjecture about it, which means that a deviation by the principal from gcannot change the worker’s belief about the likelihood the principal isinformed and, therefore, does not affect his effort. Thus, while in theno-commitment case information disclosure only brings about the nega-tive certainty effect on the manager’s payoff and hence never takes place,in the commitment case it also gives rise to the positive beliefs effect which,when strong enough, induces the manager to commit to informationsharing.

How does the ability to commit to an information sharing probabilityaffect welfare? We address this in the following proposition:

Proposition 4. If the commitment case yields an equilibrium informationsharing probability g�� 2 ð0; 1�, the commitment outcome of Proposition 2represents a Pareto improvement relative to the no-commitment outcomeof Proposition 1.

To understand the intuition behind Proposition 4, consider the date 0expected welfare, EW, as a function of an equilibrium information dis-closure probability g:

EWðgÞ ¼ �½gEWS+ð1� gÞEWPðgÞ�+ð1� �ÞEWUðgÞ; ð10Þ


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where

EWS ¼ e�SðP+bÞ 1�K

h0+hw+hm

� �+ð1� e�SÞðP+bÞ 1�

K

h0+hm

� �� Cðe�SÞ

is the welfare if the manager is informed and shares her information with

the worker,

EWPðgÞ ¼ e�NðgÞðP+bÞ 1�K

h0+hw+hm

� �+�1� e�NðgÞ

�ðP+bÞ 1�

K

h0+hm

� �

� C�e�NðgÞ

�

is the welfare if the manager is informed but keeps her signal private, and

EWUðgÞ ¼ e�NðgÞðP+bÞ 1�K

h0+hw

� �+�1� e�NðgÞ

�ðP+bÞ 1�

K

h0

� �� C

�e�NðgÞ

�

is the welfare if the manager is uninformed. Recall that e�S and e�N are given

by equations (4) and (7), respectively, and note that e�N is a function of the

worker’s belief �, which itself is a function of the manager’s information

disclosure probability g. Hence, for expositional clarity, and to emphasize

this relationship, we write e�N as e�NðgÞ.Using equation (10), we can write the equilibrium expected welfare

under the no-commitment outcome of Proposition 1 (when the manager

never discloses) and under the commitment outcome of Proposition 2

(when the manager always discloses) as EWð0Þ ¼ �EWPð0Þ+ð1� �ÞEWU

ð0Þ and EWð1Þ ¼ �EWS+ð1� �ÞEWUð1Þ, respectively. The welfare

impact of commitment can then be expressed as

EWð1Þ � EWð0Þ ¼ �½EWS � EWPð0Þ�+ ð1� �Þ½EWUð1Þ � EWUð0Þ�:

ð11Þ

The logic behind Proposition 4 follows directly from expression (11).

On the one hand, commitment to an information disclosure probability

reduces worker effort when the manager is informed, due to the certainty

effect discussed above. This is captured by the first factor on the RHS of

equation (11).On the other hand, commitment increases worker effort when the man-

ager is uninformed, by enabling the worker to infer that the manager truly is

not informed when not sharing information. This is the beliefs effect dis-

cussed above and captured by the second term on the RHS of equation (11).Importantly, however, the marginal impact of worker effort is greater

when the manager is not informed than when she is informed. Thus, the


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second effect dominates the first, explaining why the ability to commityields a welfare improvement.5

4.3 Further Efficiency Properties of the Equilibrium

Is the manager’s information strategy in the commitment equilibrium ofProposition 2 efficient? Observe that the first-best benchmark in which asocial planner who is equally well informed as the manager controls boththe probability with which the manager’s information is disclosed and theeffort exerted by the worker is not useful here because in this case, there isno need for the worker to learn the manager’s information. We thereforefocus on a second-best benchmark in which the planner is equally wellinformed as the manager and can (publicly) choose the probability, g, withwhich the informed manager will share her information with the worker,but is unable to observe and control the worker’s effort. Rather, the work-er’s effort is determined by first-order conditions (4) and (7), respectively.Under this second-best benchmark, the date 0 expected welfare as a func-tion of g is again given by expression (10) presented above, except that g isreplaced by g.

Let gSB be the information sharing strategy that maximizes the second-best welfare expression (10). In the next result, this efficient sharing level iscompared with the manager’s optimal information sharing strategy undercommitment, g��.

Proposition 5. Compared to the second-best information sharing level, themanager with commitment power shares her information too infre-quently, that is, g��4gSB. Moreover, if g�� 2 ð0; 1Þ, she shares her infor-mation strictly less than would be efficient, that is, g�� < gSB.

Proposition 5 says that even when the manager is able to commit toinformation sharing, she tends to conceal her information inefficiently toooften, and strictly so whenever she commits to share her information witha strictly positive probability less than 1. The logic behind this resultcomes from the fact that the worker’s expected utility is strictly increasingin g, the probability the manager’s signal is disclosed when she is informed.Whereas in the second-best scenario the planner considers the marginalimpact of g on both players’ payoffs, in the commitment scenario themanager ignores the positive marginal impact of a higher g on the work-er’s expected utility. It is this externality that leads to too little disclosurebeing selected in the commitment scenario.

But why is the worker’s expected utility increasing in g? The reason isthat conditional on the manager or planner being informed, the workeralways prefers to be told about it because then he exerts the optimalamount of effort given the information available. If the fact that the man-ager is informed is concealed from him, he exerts too much effort, which iscostly.

5. We would like to thank the Editor for pointing out this intuition.


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To sum up, commitment does not always lead to efficient informationsharing, although it tends to induce the principal to share her informationmore efficiently than in the absence of commitment.

5 Direct Benefits of Information Sharing

In this section, we further explore the no-commitment model, but we addrealism by allowing for the possibility that sharing information with theworker directly improves the productivity of the worker’s subsequentsearch effort. Within this framework, we first derive the manager’s opti-mal information sharing strategy. We then examine how this informationsharing strategy depends on her ability and how the managerial andworker abilities interact in the expected profit function. Finally, we dem-onstrate that our insights carry over to a setting in which contingent con-tracts are feasible.

5.1 Baseline Results

In this subsection, we formalize the idea that by sharing her informationwith the worker, the manager can point him in the right direction and,thus, save him the effort of exploring avenues that are unlikely to yieldadditional information about the firm’s project. We refer to this kind ofinformation sharing as “directional advice.”

Thus, assume from now on that if the manager shares her informationwith the worker, the worker’s subsequent search yields a signal with prob-ability qe, where q> 1. If the manager does not share her information withthe worker or if she communicates to him information that is not truthfulor complete, the worker’s search effort is productive with probability e, asin the previous analysis.6 Clearly, this modeling change does not affect theworker’s optimization problem when no information is shared, which istherefore still characterized by the first-order condition (7). However, theworker’s problem when information is shared changes to

maxe

eqb 1�K

h0+hw+hm

� �+ð1� eqÞb 1�

K

h0+hm

� �� CðeÞ;

which, assuming an interior solution, yields the worker’s optimal effort inthis case, e�SðqÞ, as the solution to the first-order condition

Kbq1

h0+hm�

1

h0+hw+hm

� �¼ C0

�e�SðqÞ

�: ð12Þ

Thus, as expected, q> 1 implies that the worker provides more effortunder information sharing than in the baseline case in which information

6. One could alternatively assume that q is higher the more information the manager

discloses, but this would only complicate the model without adding any insights. The

reason is that such partial information sharing would never happen in equilibrium.


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sharing had no effect on the productivity of his effort (which corresponds

to q¼ 1). Moreover, e�SðqÞ increases in q.

Proposition 6. Holding all other parameters fixed, there exist finite q1 and

q2, 1 < q1 < q2, such that:

1. if q � q2, the manager always shares her information with the worker,

that is, g�ðqÞ ¼ 1;2. if q4q1, the manager always conceals her information, that is,

g�ðqÞ ¼ 0;3. if q 2 ðq1; q2Þ, the manager shares her information with the worker

with conditional probability g�ðqÞ 2 ð0; 1Þ. Moreover, g�ðqÞ strictly in-

creases in q.

The result of Proposition 6 hinges on a trade-off between the certainty

effect of information sharing familiar from the previous sections and the

direct benefit q of information sharing introduced in this section. When

the manager shares her information with the worker, she brings certainty

in the worker’s mind that she is in fact informed, and as before this has a

negative impact on the worker’s incentives. But sharing information also

directly increases the worker’s marginal productivity, by increasing the

probability that his search effort will yield an informative signal. When

q4q1, this direct productivity benefit is relatively small and therefore

dominated by the negative certainty effect. As in the basic no-commitment

scenario, the manager therefore always conceals her information from the

worker. As q increases, the direct productivity benefit of information

sharing becomes more prominent. For q > q1, it gets large enough to

induce the manager to disclose her information at least with some prob-

ability, and for q � q2 with certainty.Despite the fact that for sufficiently large q the manager finds it optimal

to share her information at least sometimes even without commitment, she

still shares her information too infrequently from an ex ante point of view.

Proposition 7. Let g��ðqÞ be the information sharing probability to

which the manager would commit if such commitment were feasible.

Then, g�ðqÞ4g��ðqÞ for all q and g�ðqÞ < g��ðqÞ for q 2 ðq1; q2Þ, where q1and q2 are as in Proposition 6.

Proposition 7 extends the insights of Sections 3 and 4 to the case where

information sharing has direct productivity benefits. In particular, it

shows that in the absence of commitment, the manager again suffers

from a time-inconsistency problem, due to which she shares her informa-

tion less than she would if commitment were feasible. Moreover, it is

straightforward that even in the presence of productivity benefits of in-

formation sharing, the logic behind Proposition 5 continues to hold in the

present setting. Thus, the manager shares her information inefficiently too

little, both when she can commit and when she can not: g�4g��4gSB.


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5.2 Managerial Ability

We now turn our attention to how the optimal information sharing ar-rangement depends on the manager’s ability. A natural way of capturingmanagerial ability in our model is through the probability � with whichthe manager observes a signal: a more able manager is better at gleaninginformation. Should we expect better informed managers to share theirinformation with subordinates more frequently? Our next result answersthis question in the affirmative.

Proposition 8. Let q1 and q2 be as in Proposition 6. An increase in � leavesq2 unchanged, decreases q1, and leads to an increase in g�ðqÞ for eachq 2 ðq1; q2Þ. That is, a manager who is more likely to be informed is alsomore likely to share her information with the worker, conditional on beinginformed.

Intuitively, if �, the manager’s probability of being informed, in-creases, so does the worker’s belief �. When no information is shared, ahigher � leads the worker to believe it is more likely that the manager is infact informed and has concealed her information. Accordingly, the workerreduces his effort e�N. In contrast, when information is shared with theworker, an increase in � has no impact on his effort e�S. This differentialeffect of � on worker effort increases the appeal of information sharingover concealing.

The results of Proposition 8 can be used to formulate potentially test-able predictions. A possible interpretation of our setup is that sharinginformation with a subordinate takes the form of advising or mentoringhim. Our results then suggest that managers who are more likely to beinformed about a particular project (say, because they are more experi-enced or because they oversee fewer projects) should be more likely toadvise and mentor their subordinates. Moreover, this prediction does nothold just because managers who are more likely to be informed are auto-matically more often in a position to be able to offer useful advice, but alsobecause they are more inclined to advise the worker conditional on beinginformed.

Finally, we note that when the manager has commitment power, as inSection 4, the belief effect of commitment that we described there adds acomplicated feedback through which � affects the manager’s optimalstrategy. As a result, the effect of � on the frequency of informationsharing is in general ambiguous under commitment. Thus, we wouldexpect more able managers to more frequently share their informationwith subordinates in organizations that cannot easily commit to informa-tion sharing policies, but not (systematically) in organizations in whichcommitment is possible.

5.3 Interaction Between Managerial and Worker Abilities

The result of Proposition 8, according to which more able managers aremore willing to share their information with subordinates, of course, does


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not mean that the firm would always want to hire a manager of the highestpossible ability. Presumably, managerial ability is valued by the labormarket and more able managers command higher wages. A more inter-esting question, therefore, is whether a high ability manager is more or lessvaluable when paired with a high ability worker.

To introduce a measure of worker ability, suppose the worker’s costfunction can be expressed as CðeÞ ¼ e2=2y, where y is a positive constant.Thus, a larger y decreases the worker’s marginal cost, or, equivalently,increases his effort and therefore the probability that he receives a newsignal. The parameter y can therefore be viewed as capturing the worker’sability, analogous to how � captures the manager’s ability. Assuming

again an interior solution, which obtains if Kby 1h0� 1

h0+hw

� �< 1 and

q < q � ðh0+hmÞðh0+hw+hmÞyKbhw

, we get the following result.

Proposition 9. Suppose CðeÞ ¼ e2=2y, and let Epð�; yÞ be the firm’s ex-pected profit when the manager is of ability � and the worker of abilityy. In any interior equilibrium, there exists a q3 2 ðq2; qÞ such that:

(i) for q < q3, managerial and worker abilities are substitutes in the

firm’s profit function, that is, q2Epð �;y Þq�qy < 0;

(ii) for q 2 ðq3; qÞ, managerial and worker abilities are complements in

the firm’s profit function, that is, q2Epð�;yÞq�qy > 0.

Proposition 9 has implications for organizational design. In particular,suppose that more able workers command higher wages in the labormarket. The proposition then implies that unless the manager’s directionaladvice is extremely valuable, firms with more able managers find it optimalto hire less able workers. By the same token, a firm with a less skilledworkforce would want to hire a more able manager, who, by Proposition8, would then also be more willing to share her information with theworkers.

The logic behind the above result is easiest to see when the value ofthe manager’s advice is intermediate, q 2 ðq1; q2Þ. We know fromProposition 6 that for these parameter values the manager’s informationsharing strategy g equalizes the expected profits from sharing and notsharing information: EpS ¼ EpP. The firm’s ex ante expected profit is

therefore Epð�; yÞ ¼ �EpS+ð1� �ÞEpU, where EpU is the profit if themanager is uninformed. The marginal benefit from an increase in the

manager’s ability � is therefore qEpð�;yÞq� ¼ EpS � EpU. Now suppose the

worker’s ability y increases. This decreases the worker’s cost of effortand, therefore, leads to more effort both when the manager has sharedher information and when she has not, but the impact of the higher efforton the probability of the project’s success is smaller when the manager is

informed than when she is not. Thus, qEpð�;yÞq� decreases with y.


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The reason why managerial and worker abilities become complementswhen the managerial advice is very valuable (q > q3) is that under infor-mation sharing, an increase in the worker’s ability has a larger impact onthe marginal productivity of his effort when the manager’s directionaladvice is more valuable—that is, y and q are complements in the worker’sobjective function.7 For sufficiently large q, this complementarity effect isboth large and comes into play frequently because the manager alwaysshares her information. This effect therefore swamps the considerationsdescribed in the previous paragraph. The implication is that if directionaladvice is very valuable, more able managers should be matched with moreable workers.

6. Monetary Incentives

In this section, we demonstrate that our core arguments continue to holdwhen monetary incentives are feasible, as long as contracting is imperfect.Specifically, assume that it is possible to write contracts in which theworker’s pay, ~w, is contingent on whether the project was successful.Assume also that the worker is protected by limited liability, and haszero initial wealth, zero reservation utility, and no private benefits,b¼ 0. Let ~w ¼ w > 0 be the bonus the worker gets if the project is suc-cessful and generates P, and note that, as is standard in limited liabilitymodels, he will optimally get ~w ¼ 0 if the project fails. Realistically, ~wcannot be contingent on whether the manager is informed and/or onwhether the manager shares her information with the worker.8

In this setting, the worker’s efforts eS and eN are given by first-orderconditions identical to equations (12) and (7), except that the privatebenefit b is replaced by the bonus w. Similarly, the manager’s conditionalexpected profits EpUðwÞ; EpSðwÞ, and EpPðwÞ are identical to equations(A2), (8), and (9), except that P is replaced with ðP� wÞ. At date 0, themanager thus chooses w, eS, eN, and g to solve the following optimizationprogram:

maxw;eS;eN;g

�½gEpSðwÞ+ð1� gÞEpPðwÞ�+ð1� �ÞEpUðwÞ;

subject to

Kwq1

h0+hm�

1

h0+hw+hm

� �¼ C0

�e�SðqÞ

�; ð13Þ

7. This can be seen from the first-order condition for the worker’s effort,

e�S ¼ yqKb 1h0+hm� 1

h0+hw+hm

� �.

8. If contracts conditional on whether the manager has shared her information were

feasible, then it should also be feasible to write contracts that would reward the worker

directly for communicating to the manager new information, instead of rewarding him in-

directly for the success of the project. Under such a direct contract, the worker’s effort would

no longer depend on his belief regarding the likelihood that the manager is informed. Thus,

the manager would always share her information.


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�Kw1

h0+hm�

1

h0+hw+hm

� �+ð1� �ÞKb

1

h0�

1

h0+hw

� �¼ C0ðe�NÞ; ð14Þ

g 2 argmaxg 0

g0

EpSðwÞ+ð1� g0

ÞEpPðwÞ; and ð15Þ

w � 0: ð16Þ

Here, equations (13) and (14) are the worker’s incentive compatibilityconstraints for efforts eS and eN, respectively, and condition (15) is themanager’s incentive compatibility constraint that ensures that if informed,she is willing to divulge her signal to the worker with probability g. Theworker’s participation constraint is implied by conditions (13), (14), and(16), and we omit it in the statement of the problem.

As in our model with direct benefit of information sharing presented inSection 5.1, the trade-off due to information sharing is reflected in thedifference between the worker’s two IC constraints (13) and (14): by shar-ing her information, the manager increases the productivity of the work-er’s effort, but also decreases the worker’s perception of the marginalbenefit of discovering an additional signal—the certainty effect of infor-mation sharing discussed earlier. What is different here is that unlike theprivate benefit b in the baseline model, bonus w is chosen by the managerand will in general depend on the parameters of the environment, such asthe signals’ precisions and the quality q of the directional advice.Nevertheless, Proposition 10 shows that this difference is immaterial forthe qualitative properties of the manager’s optimal information sharingstrategy.

Proposition 10. When monetary incentives are feasible, the manager’s op-timal information sharing strategy continues to be characterized by twocutoff levels q1 and q2 as described in Proposition 6. Moreover, if w�

denotes the bonus in the optimal contract, then the cutoff levels q1ðw�Þ

and q2ðw�Þ are identical to the cutoff levels q1ðbÞ and q2ðbÞ that would be

implied by Proposition 6 for the case of a private benefit b ¼ w�.Proposition 10 shows that availability of monetary incentives does not

alter our previous qualitative conclusions about the effects informationsharing has on incentives and about the manager’s optimal informationsharing strategy. The manager’s optimization problem is somewhat morecomplex here because she chooses an optimal bonus for the worker, butthis additional consideration does not affect the fundamental trade-offsthat we identified and studied. The reason is that at the time the managerdecides whether to share her information with the worker, she treats thebonus w as given. Her goal at this point is therefore to simply maximize theprobability of success, which is the same problem she faced in the baselinemodel with private benefits.


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7. Concluding Remarks

In this article, we examine optimal information flows between a managerand a worker when the manager must estimate as precisely as possible thetrue value of an unknown but payoff-relevant parameter. The manager maypossess information about this parameter of interest, and must decidewhether to share it with the worker. We have shown that if the manager’sinformation is shared with the worker, the worker’s incentives to collectadditional information are dampened, which decreases the effort theworker is willing to provide. Thus, when effort is difficult to measure andcontrol, it is optimal for the informed manager to conceal her information.

However, we demonstrate that the lack of information sharing is aconsequence of a time-inconsistency problem that the manager facesdue to not being able to commit to an information sharing policy. If shehad the commitment power, the manager would often commit to alwaysdisclose her information to the worker. We also show that managers whoare less likely to be informed have a stronger incentive to withhold theirinformation from the worker and that unless the manager’s information isextremely helpful in guiding the worker toward additional information,more able managers should be matched with less able workers.

In order to provide clear and intuitive results, we have abstracted fromsome considerations that may be important in practice. For example, whilewe believe that the assumption that the manager’s and the agent’s informa-tion are substitutes is a natural one to make, we can also imagine situationsin which the two pieces of information might be complementary. To fit suchsituations, the model would have to be somewhat modified. However, weexpect that the analysis would have the same flavor and the insights would beanalogous to the current ones, albeit with different signs. In particular,rather than pretending to be uninformed, the principal would now havean incentive to pretend that she is informed—which might be credible, forinstance, if communicating information requires time and effort, or is un-desirable for strategic reasons, such as trade secrets protection.

We have also assumed that both the precision of the manager’s signaland the probability that she receives a signal are exogenous. In reality,managers may be able to influence both of these parameters by expendingcostly effort. Our framework could readily incorporate such an extension,and could then be used to address additional questions of interest, such aswhether the manager might have an incentive to publicly commit to limitthe amount of effort she expends on information gathering, so as not todilute the worker’s incentives.

A related consideration that is absent from our model but potentiallyimportant in reality is that the worker may not know the precision hm ofthe manager’s signal. In that case, a “cheap talk” scenario would emerge inwhich the manager would have an incentive to convince the worker thather signal is noisy. As is standard in cheap talk games, there would exist a“babbling” equilibrium in which the manager would send an uninforma-tive message about her signal’s precision and the worker would rely on his


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prior belief about hm. In this equilibrium, all our results should followthrough. Even in an informative equilibrium, the tensions regarding man-agerial information disclosure and worker effort should still be present,yielding qualitatively similar results. A careful examination of such anextension would make an interesting avenue for future research.

Funding

Both authors thank the Social Sciences and Humanities Research Councilof Canada for financial support.

Conflict of interest statement. None declared.

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Appendix A

To simplify the proofs, define H1 �1h0� 1

h0+hwand H2 �

1h0+hm� 1

h0+hw+hm,

and notice that H1 > H2 > 0.

Proof of Lemma 1. BecauseH2 decreases in hm, it follows from a compari-son of conditions (4) and (7) that e�N > e�S whenever � < 1, and fromcondition (7) that

qe�Nq�< 0.«

Proof of Proposition 1. We have qEpiqe�j¼ PKH2 > 0; i ¼ S;P; j ¼ S;N.

Hence, EpP > EpS whenever e�N > e�S. Moreover, as shown in Lemma 1,e�N > e�S for all � < 1. Furthermore, it must be � < 1 given that the trueprobability with which the manager is informed is � < 1. Hence,EpP > EpS, which means that concealing her signal from the worker is


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the manager’s strictly dominant strategy, that is, g� ¼ 0. The worker’s

belief, therefore, has to be � ¼ �. «

Proof of Proposition 2. If the manager commits to share her signal with

probability g, her date 0 expected payoff is

EpðgÞ ¼ �½gEpS+ð1� gÞEpP�+ð1� �ÞEpU; ðA1Þ

where EpS and EpP are given by equations (8) and (9) and EpU is her

expected profit if she is uninformed:

EpU ¼ e�NP 1�K

h0+hw

� �+ð1� e�NÞP 1�

K

h0

� �: ðA2Þ

The efforts e�S and e�N are given by equations (4) and (7), respectively.

Note that e�N is a function of the worker’s belief �, which in turn is a

function of the manager’s information disclosure probability g (because

under commitment g ¼ g). Applying the Implicit Function Theorem to

the first-order condition (7) yields

qe�Nqg¼

Kb

C00ðe�NÞðH1 �H2Þ

�ð1� �Þ

ð1� � gÞ2> 0: ðA3Þ

Also, using the expressions for EpP; EpS, and EpU, we can write

EpP � EpS ¼ ðe�N � e�SÞPKH2;qEpS

qe�N¼ 0;

qEpP

qe�N¼ PKH2;

qEpU

qe�N¼ PKH1:

Thus, differentiating equation (A1) with respect to g and rearranging yields

qEpðgÞqg

¼ �ð1� gÞqEpP

qe�N+ð1� �Þ

qEpU

qe�N

� �qe�Nqg� �ðEpP � EpSÞ

¼ PK½�ð1� gÞH2+ð1� �ÞH1�qe�Nqg� �ðe�N � e�SÞPKH2;

so that qEpðgÞqg > 0 iff

ð1� �gÞ½�H2+ð1� �ÞH1�qe�Nqg

> �ðe�N � e�SÞH2:


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Using the first-order condition (7), we can replace the content of the

square brackets on the LHS by C0ðe�NÞ=bK . Further, replacingqe�

N

qg by its

expression from equation (A3), we get

ð1� � gÞC0ðe�NÞ

bK

bK

C00ðe�NÞðH1 �H2Þ

�ð1� �Þ

ð1� � gÞ2

� �> �ðe�N � e�SÞH2:

Divide both sides byC0ðe�NÞ

C00ðe�NÞand by H2. After simplifying, we get that for

� > 0; qEpðgÞqg > 0 iff

ð1� �Þ

ð1� � gÞH1 �H2

H2

� �>

C00ðe�NÞ

C0ðe�NÞðe�N � e�SÞ: ðA4Þ

Given thatC00ðe�NÞ

C0ðe�NÞ> 0, a sufficient condition for this inequality to

be satisfied is that it holds (weakly) for e�S ¼ 0 and g ¼ 0, that is,

C00ðe�NÞ

C0ðe�NÞe�N4ð1� �Þ

H1 �H2

H2

� �� C�:«

Proof of Proposition 3. Using � ¼ �ð1�gÞ1� g � ; CðeÞ ¼ e2=2 implies

e�N ¼ �KbH2+ð1� �ÞKbH1; e�S ¼ KbH2;

e�N � e�S ¼ ð1� �ÞKbðH1 �H2Þ; andqe�Nqg¼ KbðH1 �H2Þ

�ð1� �Þ

ð1� � gÞ2:

Plugging all these expressions into condition (A4) and using

1� � ¼ 1��1� g �, we know that for � > 0; qEpðgÞ

qg > 0 iff

ð1� �Þ

ð1� � gÞH1 �H2

H2

� �>

1

e�N

1� �

1� g �

� �KbðH1 �H2Þ:

Simplifying and rearranging, this holds iff e�N > H2Kb, or, using

e�N ¼ �KbH2+ð1� �ÞKbH1, iff

�H2+ð1� �ÞH1 > H2:

Given that H1 > H2, this holds for all � 2 ð0; 1Þ.«


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Proof of Proposition 4. Clearly, the manager is at least as well off under g��

as she would be under g� because g� was a feasible commitment. Thus, to

show Pareto improvement, we only need to prove that the commitment

outcome makes the worker better off.

The worker’s ex ante (before the manager’s decision whether to share

her information) utility as a function of his belief g is

EU ðgÞ ¼ �g½BIðe�SÞ � Cðe�SÞ�+�ð1� gÞ½BIðe

�NÞ � Cðe�NÞ�+ð1� �Þ½BUðe

�NÞ

� Cðe�NÞ�:

ðA5Þ

The Envelope Theorem then yields9

EU0

ðgÞ ¼ �½BIðe�SÞ � Cðe�SÞ� � �½BIðe

�NÞ � Cðe�NÞ�;

so that EU0

ðgÞ > 0 iff BIðe�SÞ � Cðe�SÞ > BIðe

�NÞ � Cðe�NÞ. But this has to

hold because the function BIðeÞ � CðeÞ is uniquely maximized at e�S.In the game without commitment, the worker’s equilibrium belief is

g ¼ g� ¼ 0, and his ex ante expected utility is EU ðg� ¼ 0Þ. In the game

with commitment, the worker knows with certainty the manager’s equi-

librium strategy, and his equilibrium belief is g ¼ g��. Since g�� > g� ¼ 0

and EU0

ðgÞ > 0, we get EU ðg��Þ > EU ðg�Þ.«

Proof of Proposition 5. If the planner commits to disclose the manager’s

signal with probability g, the worker’s belief is g ¼ g. The date 0 expected

welfare as a function of g can then be expressed as the sum of the man-

ager’s expected profit and the worker’s expected utility,

EW ðgÞ ¼ EpðgÞ+EU ðgÞ, with EpðgÞ and EU ðgÞ defined in expressions

(A1) and (A5).

If g�� ¼ 0, then gSB 2 ½0; 1� immediately implies g��4gSB. Next, sup-

pose g�� 2 ð0; 1Þ. Since this is a unique interior solution, Ep0ðg��Þ ¼ 0

holds. But as shown in the proof of Proposition 4, EU0ðgÞ > 0

for all g 2 ½0; 1�. It must therefore be that

EW0ðg��Þ ¼ Ep0ðg��Þ+EU0ðg��Þ ¼ EU0ðg��Þ > 0, which yields gSB 2 ðg��; 1�.Finally, if g�� ¼ 1, it must be Ep0ðg�� ¼ 1Þ � 0, and since EU0ðgÞ > 0 for

all g 2 ½0; 1�, we must have EW0ðg�� ¼ 1Þ > 0. This implies g�� ¼ gSB ¼ 1.

«

Proof of Proposition 6. Let �ðq; gÞ be the informed manager’s net payoff

from sharing her information with the worker as compared to not sharing,

9. To see this, note that using expression (5) we can re-write �ð1� gÞ½BIðe�NÞ � Cðe�NÞ�+ð1

��Þ½BUðe�NÞ � Cðe�NÞ� as ð1� �gÞ½�

�BIðe

�NÞ � Cðe�NÞ

�+ð1� �Þ

�BUðe

�NÞ � Cðe�NÞ

��. The square

bracket is simply the worker’s program when no information has been disclosed to him, and

corresponds to expression (6), which is maximized at e�N. Hence, using the Envelope Theorem,

we know that g does not affect EU ðgÞ through e�N.


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when the worker’s belief about the manager’s strategy is g. Thus,�ðq; gÞ � EpS � EpP, where EpP is given by equation (9) and

EpS ¼ qe�SP 1�K

h0+hw+hm

� �+ð1� qe�SÞP 1�

K

h0+hm

� �:

Substituting the expressions for EpS and EpP into the definition of �ðq; gÞ yields

�ðq; gÞ ¼ PK½qe�S � e�NðgÞ�H2;

so that EpS > EpP if qe�S > e�NðgÞ and EpS < EpP if qe�S < e�NðgÞ.

1. As can be readily verified from equation (7), e�NðgÞ strictly increases in g,with e�Nð0Þ > 0 and (by assumption on K) e�Nð1Þ <1. On the other

hand, e�SðqÞ is independent of g but strictly increases in q, with qe�SðqÞ!

1 as q!1. Moreover, as shown in Subsection 3.1, it must be e�Sðq ¼ 1Þ

< e�NðgÞ for any g. Hence, by continuity, there exists a q2 > 1 such that

q2e�Sðq2Þ ¼ e�Nðg ¼ 1Þ and qe�SðqÞ > e�Nðg ¼ 1Þ for any q > q2. That is, for

these values of q, the manager finds it optimal to always share her in-

formation. Hence, in this case g� ¼ 1 and, in equilibrium, g ¼ g�.2. Because (a) e�Sðq ¼ 1Þ < e�NðgÞ for any g, (b) qe

�SðqÞ strictly increases in q,

and (c) e�NðgÞ strictly increases in g, there exists a q1 2 ð1; q2Þ such that

q1e�Sðq1Þ ¼ e�Nðg ¼ 0Þ and qe�SðqÞ < e�Nðg ¼ 0Þ for all q < q1. In this case,

the manager finds it optimal to conceal her information. That is, g� ¼ 0

and, in equilibrium, g ¼ 0 also.3. For q 2 ðq1; q2Þ, we have qe�SðqÞ > e�Nðg ¼ 0Þ and qe�SðqÞ < e�Nðg ¼ 1Þ.

No equilibrium in pure strategies therefore exists: If the worker expects

the manager to never share her information (g ¼ 0), he discovers newinformation with smaller probability if the manager does not reveal anyinformation to him than if the manager divulged her signal. Hence, themanager has an incentive to deviate by divulging her information. If the

worker expects the manager to always share her information (g ¼ 1), hediscovers new information with smaller probability when the managerreveals her information than he would if she concealed it. The managertherefore has an incentive to deviate by concealing her signal. Thus, forthese values of q, the equilibrium requires that the worker believes that

the manager plays a mixed strategy gðqÞ such that qe�SðqÞ ¼ e�NðgÞ. Sucha gðqÞ exists by continuity of e�NðgÞ in g. As always, equilibrium then

requires that g�ðqÞ ¼ gðqÞ.

Finally, the result that g�ðqÞ strictly increases in q for q 2 ðq1; q2Þ isobtained from qe�SðqÞ ¼ e�Nðg

�Þ and from the facts that e�NðgÞ strictly in-creases in g and qe�SðqÞ strictly increases in q.«

Proof of Proposition 7. From the proof of Proposition 2, the marginalprofit due to an increase in g under commitment is


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qEpðgÞqg

¼ �ð1� gÞqEpP

qe�N+ð1� �Þ

qEpU

qe�N

� �qe�Nqg

+�ðEpS � EpPÞ: ðA6Þ

Now, the proof of Proposition 6 shows that for q 2 ðq1; q2Þ; g�ðqÞ issuch that EpS ¼ EpP. Since the first term in equation (A6) is strictly posi-

tive for all g, evaluating equation (A6) at g�ðqÞ yields qEpðgÞqg jg¼g� > 0.

Moreover, EpS is independent of g, while EpP increases in g (through

eN). Thus, EpS � EpP > 0 for all g < g�, which means that qEpðgÞqg > 0

also for all g < g�. It must therefore be that g��ðqÞ > g�ðqÞ when

q 2 ðq1; q2Þ.When q � q2, so that g�ðqÞ ¼ 1, we have EpS � EpPðg ¼ 1Þ, which

implies qEpðgÞqg jg¼1 � 0. Thus, g��ðqÞ ¼ g�ðqÞ ¼ 1 for these values of q.

Finally, for q4q1, we have g�ðqÞ ¼ 0, so that g��ðqÞ � g�ðqÞ holdstrivially.«

Proof of Proposition 8. The proof of Proposition 6 implies that q1 and q2are determined by

q1e�Sðq1Þ ¼ e�Nðg ¼ 0Þ and ðA7Þ

q2e�Sðq2Þ ¼ e�Nðg ¼ 1Þ; ðA8Þ

respectively, and g�ðqÞ for q 2 ðq1; q2Þ is determined by

qe�SðqÞ ¼ e�Nðg�Þ: ðA9Þ

When g ¼ 1, we have � ¼ 0 for any � < 1. It is therefore immediate

from equation (7) that e�Nðg ¼ 1Þ is independent of � and from equation

(A8) that q2 is independent of �.On the other hand, for any g < 1, we have q�

q� > 0 so that from equation

(7) we getqe�Nq� < 0. Since qe�SðqÞ strictly increases in q, equation (A7) then

implies that q1 strictly decreases in �.Finally, for any given q 2 ðq1; q2Þ and for g� 2 ð0; 1Þ, equation (12)

shows that qe�SðqÞ is independent of both � and g�, while equation (7)

implies thatqe�Nq� < 0 and

qe�Nqg� > 0. It therefore follows from equation (A9)

that qg�ðqÞq� > 0.«

Proof of Proposition 9. The firm’s date 0 expected payoff is

Epð�; yÞ ¼ �½g�EpS+ð1� g�ÞEpP�+ð1� �ÞEpU; ðA10Þ


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where EpS is given by equation (8), EpP by equation (9), and EpU by

equation (A2).Suppose first q4q1 so that g� ¼ 0 and � ¼ �. Plugging CðeÞ ¼ e2=2y

into the first-order condition (7) yields e�N ¼ yKb½�H2+ð1� �ÞH1�. Note

that e�N < 1 becauseH2<H1 and, by assumption, yKbH1 < 1. We thus get

that Epð�;yÞq� is given by

Epð�; yÞq�

¼ EpP � EpU+�qEpP

q�

¼ PK½e�NH2+ð1� e�NÞH1�+yPK2�bH2ðH2 �H1Þ;

from which

q2Epð�; yÞq�qy

¼ PKqe�Nqy

+K�bH2

� �ðH2 �H1Þ:

Given thatqe�Nqy > 0, this implies q2Epð�;yÞ

q�qy < 0 for all q4q1.Next let q 2 ðq1; q2Þ. The proof of Proposition 6 shows that in this case

EpS ¼ EpP, so the firm’s profit equation (A10) reduces to

Epð�; yÞ ¼ �EpS+ð1� �ÞEpU. The same expression applies when q > q2in which case g� ¼ 1. Thus, for all q > q1, we get

qEpð�; yÞq�

¼ EpS � EpU ¼ PK½qe�SH2+ð1� e�NÞH1�;

from which


¼ PK qqe�Sqy

H2 �qe�Nqy

H1

� �: ðA11Þ

Now, as shown in the proof of Proposition 6, for q 2 ðq1; q2Þ, it must be

qe�SðqÞ ¼ e�Nðg�Þ.10 Since under CðeÞ ¼ e2=2y both e�S and e�N are linearly

increasing in y, it must also be that qqe�

S

qy ¼qe�Nqy for these values of q, which

yields q2Epð�;yÞq�qy < 0 for all q 2 ðq1; q2Þ.

Finally, when q > q2, we have g� ¼ 1 and � ¼ 0, which implies e�N ¼ yKbH1 and e�S ¼ minfyKbqH2; 1g. Thus, an interior solution for e�S obtains

for all q < q � 1yKbH2

, which has been imposed in this section. Notice that

since q2 is defined by q2e�Sðq2Þ ¼ e�Nðg

� ¼ 1Þ and q2 > 1, it must be

e�Sðq2Þ < e�Nðg� ¼ 1Þ < 1. Moreover, because e�S increases in q and q is

defined so that e�SðqÞ ¼ 1, we have that q > q2 and that there exists a q32 ðq2; qÞ such that e�S < e�N for q24q < q3 and e�S > e�N for q3 < q < q.

10. Note that q> 1 and e�N < 1 then implies e�SðqÞ < 1 for this range of q values.


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Now, using e�N ¼ yKbH1 and e�S ¼ yKbqH2, equation (A11) becomes


¼P

y2bðe�2S � e�2N Þ:

Hence, it must be that q2Epð�;yÞq�qy < 0 for q24q < q3 and q2Epð�;yÞ

q�qy > 0 for

q3 < q < q.«

Proof of Proposition 10. For any given contract w, the informed manager’s

choice of g is again driven by a comparison of EpS and EpP, where, asbefore, EpS denotes the informed manager’s expected profits if she shares

her information with the worker and EpP her expected profits if she does

not. These expected profits are given by expressions analogous to equa-

tions (8) and (9), the only difference being that P in these expressions is

replaced with P� w. Thus, as before, if �ðqÞ � EpS � EpU > 0 the man-

ager sets g¼ 1, if �ðqÞ < 0 she sets g¼ 0, and if �ðqÞ ¼ 0 she employs a

mixed strategy, with g 2 ð0; 1Þ.

Clearly, for any fixed w, this comparison is the same as before because

if we label P� w as P0

and w as b0

, the problem is identical to the one

whose equilibrium is characterized in Proposition 6. In particular, for any

fixed w, the equilibrium must be characterized by two cutoff levels q1 and

q2, as described in Proposition 6. Thus, the only new consideration here is

that unlike the private benefit b, which was exogenous, the bonus w is

endogenously chosen by the firm so as to maximize its expected profit.Now, let w� be the bonus in the optimal contract and note that it cannot

be optimal to set w� � P. Inspection of the expression

�ðqÞ ¼ ðP� w�ÞKðqe�S � e�NÞH2

then reveals that w� can affect the sign of �ðqÞ only through the worker’s

efforts eS and eN. Let e��S ¼ e��S ðw

�Þ and e��N ¼ e��N ðw�Þ be the solutions to

the first-order conditions (13) and (14), respectively. A comparison of

equation (13) with equation (12) and of equation (14) with equation (7)

reveals that b ¼ w� implies e��S ðw�Þ ¼ e�SðbÞ and e��N ðw

�Þ ¼ e�NðbÞ. Thus, the

cutoff levels q1ðw�Þ and q2ðw

�Þ must be the same as the cutoff levels q1ðbÞ

and q2ðbÞ that would be observed in the absence of monetary incentives

but with a private benefit b ¼ w�. «


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jleo, v35 n3 information sharing and incentives in ......koeppl, jeff mcgill, matt mitchell, mikhail...

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