Board bias, information, and investment efficiency

Abstract

We identify a novel force behind the benefit of misaligned preferences in corporate governance. Our model entails a CEO who encounters a project and, after gathering information, decides whether to seek the approval of a corporate board. The CEO may be able to choose the properties of the collected information—this may happen if the project is “novel,” i.e., explores a new technology, business concept, or market and directors are less knowledgeable about it. We find that only sufficiently conservative and expansion-cautious directors can discipline the CEO’s empire-building tendency and opportunistic information collection. Such directors, however, underinvest in projects that are not novel. The board that maximizes firm value is either conservative or neutral (has interests aligned with the shareholders) and always overinvests in innovations. Boards with greater expertise are more likely to be conservative, but their bias is less severe. The board’s commitment power and bias are substitutes.

1 Introduction

It is not uncommon for CEOs to harbor aspirations of creating “corporate empires” (Hope and Thomas 2008; Decaire and Sosyura 2021). Despite being outweighed by leadership skills, strategy, and vision—the reasons these CEOs were hired in the first place—such aspirations can still pose challenges for shareholders. Incentive contracting may not fully curb empire-building ambitions, or the costs of doing so may be prohibitively high (Shleifer and Vishny 1989; Gregor and Michaeli 2023). Consequently, to safeguard the shareholders’ interests, companies entrust their corporate boards with the approval of significant investment opportunities, such as cash acquisitions of other firms or major product launches (Useem 2006). Grasping the intricate tradeoffs in approval decisions necessitates an understanding that directors may have private benefits, costs, or innate biases influencing their investment preferences. A stream of prior literature (e.g., Adams and Ferreira 2007; Baldenius et al. 2014) indicates that aligning the preferences of boards and CEOs enhances the communication of exogenously given information. What if investment-relevant information (such as data about an acquisition target, customer demand, or technological feasibility) needs to be actively collected, with CEOs potentially guiding this process (for instance, by selecting the due diligence team or focus group)? This paper examines how CEOs’ ability to influence information gathering affects the optimal alignment of interests within the leadership team and the efficiency of corporate investments.

We build a model in which a CEO (“she”) finds an investment project and decides whether or not to present it for approval to a board of directors. The preferences of the shareholders, the CEO, and the board are such that each favors investments with a value exceeding a player-specific threshold. These thresholds determine the alignment of interests among the players and parsimoniously reflect pecuniary and non-pecuniary benefits, costs, as well as inherent characteristics and attitudes toward corporate investment and expansion. In line with classic agency theory (Jensen 1986; Shleifer and Vishny 1989; Stulz 1990; Jensen and Murphy 1990), we posit that the CEO is an empire-builder whose threshold is below that of the shareholders. The board’s threshold can assume any value: in comparison to the shareholders, the board can be classified as “expansionist” (when most directors favor company expansion due to entrepreneurial background, perks, or social connections with the CEOs, so their threshold is relatively low), “conservative” (when most directors are cautious about company expansion as a result of a desire to maintain the status quo and financial prudence, prioritize stability over growth, or focus on current operations, so their threshold is high), or “neutral” (a balanced mix of expansionists and conservatives, so the board overall is aligned with the shareholders). Throughout the paper, we refer to the player-specific thresholds as “types” or “biases.”

When bringing the project for approval to the board, the CEO gathers investment-relevant information. The nature of the project may render the CEO unable to control the properties of the information that she collects, especially when the project is similar to previous operations or the directors are knowledgeable about the industry. We dub such a project “routine,” and assume its value is (without loss of generality) fully revealed from the collected information. Conversely, when the project involves a new technology, business concept, or market, the CEO has more flexibility and thus can select the properties of the information. Such a project is referred to as “novel,” and the CEO’s decision-making process is modeled as a Bayesian persuasion problem. In our model, this simplifies to an ex ante choice of a reporting cutoff, whereby projects with a value above (below) the cutoff are reported as high (low). Selecting a greater cutoff increases the expected value of the project, conditional on either report. The CEO can “veto” a project in the sense that she can choose not to present it for approval.

We commence our analysis with a benchmark case in which the board commits to approving projects that meet a predetermined hurdle rate. For routine projects, where the value is perfectly revealed, we find that the board aligns the hurdle precisely with its threshold (type). For novel projects, the CEO can strategically report various projects as having high value; the critical consideration here is that the project’s expected value, given a high report, must be sufficiently high to meet the pre-committed hurdle rate. Evidently, if the board is less expansionist than the CEO and sets the hurdle for novel projects at its threshold (type), an empire-building CEO pools some low-value projects with high-value projects, securing approval for all. To counter this, the board optimally chooses a higher hurdle for novel projects. In the aftermath, because of the CEO’s veto power, only projects—whether routine or novel—that meet or surpass the higher of the CEO’s and the board’s types/biases are undertaken. Consequently, efficient investments in the benchmark case with commitment arise when the board’s preferences are perfectly aligned with those of the shareholders; that is, when the optimal board with commitment power is neutral.

We proceed with the analysis of our main model, where corporate boards cannot commit to approval policies. Suppose the CEO encountered a novel project. If the board’s preferences perfectly align with those of the CEO, meaning their types are the same, then the CEO can simply set the reporting cutoff at her threshold (type) and present only the novel projects reported as having high value, which the board will approve. This is true also when the preferences are only slightly misaligned or when the board has a strong pro-expansion bias. However, if the board is very conservative, such a strategy results in rejection. To avoid this outcome, the CEO optimally increases the reporting cutoff so that the expected value of novel projects reported as high is just enough for the board’s approval. Faced with an opportunistically set reporting cutoff, the board may underinvest or overinvest in novel projects from the shareholders’ perspective. Notably, a board with a significantly high conservative bias elicits a cutoff choice that results in efficient novel investments—this result comports with empirical evidence about the impact of activist pressure for the nomination of cost-cutting directors (Brav et al. 2018; Maffett et al. 2022) and director independence requirements (Rim and Sul 2020). As we explain below, this beneficial disciplining effect of misaligned preferences on persuasion has not been studied in prior literature. One could draw a parallel to our benchmark setting. There, disciplining the CEO is achieved via a commitment to a high hurdle rate. In the main model, this effect is achieved via a conservative bias. This observation implies that boards’ commitment power and bias are substitutes.

When the CEO encounters a routine project, she has no ability to persuade the board via a strategically constructed report—approval is granted if the revealed project value exceeds the board’s type. Thus, while a very conservative board invests efficiently in innovations, it can reject a routine project against the shareholders’ interest. So which board type maximizes firm value? In our setting, a board with pro-expansion bias can never be optimal because it distorts decisions about both sorts of projects. A board that is only mildly conservative is also suboptimal—it not only distorts decisions about routine investments but also fails to discipline the CEO’s opportunistic reporting about novel projects. We find that only one of two board types can be optimal—neutral or very conservative—but neither can fully undo the CEO’s empire-building. In equilibrium, the board approves some (but not all) empire-building projects: firms overinvest in innovations but may or may not underinvest in routines.

Which of the two board types is optimal depends on the relative magnitude of the expected gain from improved approvals of novel projects and the expected loss from distorted approvals of routine ones. When the CEO is more likely to find a routine opportunity, the expected loss is relatively large and outweighs the expected gain—thus, the optimal board is more likely neutral. However, when the CEO has a severe tendency to overinvest, a conservative board that disciplines the CEO is preferable from the shareholders’ perspective. This finding aligns with recent anecdotal evidence. For example, a Korn Ferry briefing on mergers and acquisitions notes that “some directors may find themselves trying to temper, if not fight back against, their CEOs’ urge to merge” and adds that it is “important ... for acquiring company directors to exercise fiscal prudence and strategic oversight.”¹ Overall, our results predict that in environments with heterogeneous projects the distribution of optimal boards is bimodal. Companies managed by mildly biased CEOs have neutral boards, and companies managed by extreme empire-builders have conservative boards.

We extend the analysis by allowing the board to acquire costly information about novel projects (information about routines is already perfect). After reviewing the CEO’s report, the board chooses a cutoff such that a good (bad) message is generated if the value is above (below) the cutoff. From the board’s perspective, the most useful message is when the cutoff coincides with its innate threshold/type. However, doing so is costly, and the incurred cost increases with the probability that the acquired information inspires the board to change the decision it would have taken solely based on the CEO’s report. Therefore, the board acquires only imperfect information. What can the CEO do in response? We find that she is indifferent between adjusting the reporting cutoff to provide the same information that the board would optimally acquire, sticking to the reporting cutoff she would have chosen in the main model, and choosing any cutoff in between.² While these options prompt different degrees of board’s learning at a varying cost, they all yield an identical ex post outcome. Though the CEO’s ability to get all her favored novel projects approved is restricted, our result that the optimal board is either neutral or very conservative remains valid. Our study reveals that when the cost of acquiring information is low (e.g., due to directors’ experience or qualifications), the optimal board is more likely to be biased (complementarity between bias and expertise). However, we also find that the optimal level of conservative bias decreases (substitution between bias and expertise). Consequently, we predict that conservative boards are more common in industries with professionals who have more experience and expertise. Nevertheless, the greater the directors’ expertise, the less conservative the board.

Related literature

Our paper contributes to several strands of literature. We predict that an intermediary (board of directors) having preferences that severely differ from those of an agent (CEO) elicits precise information that benefits a principal (shareholders). This is a significant departure from the predictions in a stream of literature that makes the case for aligned preference in settings where the agent is exogenously endowed with perfect information and can misrepresent it at no cost (e.g., Dessein 2002; Mitusch and Strausz 2005; Adams and Ferreira 2007; Harris and Raviv 2008; Baldenius et al. 2014; Chakraborty and Yilmaz 2017).³ The findings differ because in our model the sender may be able to choose the properties of the collected information. We believe this assumption is descriptive in many contexts, especially when directors are less knowledgeable about new concepts or markets, so management has more flexibility in guiding the information collection process.

Like us, several studies also call for misalignment, but their predictions are driven by different forces than ours—we contribute by studying a novel force behind the benefit of diverged preferences. In Dewatripont and Tirole (1999), multitasking leads to competition for information acquisition among multiple agents, and this is best utilized by the principal when the agents have different preferences. Che and Kartik (2009) study differences in prior beliefs (arising only under uncertainty), whereas we study differences in preferred policy (arising even under certainty). In their disclosure model, a greater difference between priors affects the cost-benefit ratio of information, which incentivizes the sender to choose higher precision. In our persuasion model, misalignment does not affect the cost-benefit ratio, as information acquisition is cost-free for the sender (CEO). Importantly, Che and Kartik (2009) demonstrate that disagreement over priors works differently than divergent preferences: if the sender and the receiver in their model were to have different preferences but a common prior, then the receiver would always prefer an unbiased sender, which is different than our findings.

In Baldenius et al. (2019), friendly directors receive more precise information from the CEO, whereas antagonistic ones search for information independently. Therefore, antagonistic boards can be optimal when the information from outsiders is more valuable. In our model, this channel is absent, as the CEO has access to perfectly precise information (i.e., the board has no informational advantage) and can prevent the board from learning by her choice of reporting properties. In Aghamola and Hashimoto (2020), an unfriendly intermediary is more likely to fire the CEO—a thread that incentivizes the CEO to boost productivity by reducing the bias in her report. In our model, this incentive is missing, as we abstract from CEO turnover. In Ball and Gao (2024), the benefit from misalignment arises due to the interplay between the agent’s bias and a restriction on the available policies. In our model, this channel is absent because the binary board’s action (approve or reject the project) cannot be restricted. Misalignment can also be beneficial when the agent interacts with third parties such as suppliers, business partners and competitors: in oligopolies, delegating the product decisions to an agent who competes aggressively serves as a pre-commitment device (Fershtman and Judd 1987) and can shape the managers’ disclosure choices (Bagnoli and Watts 2015). In static bargaining, appointing a less interested agent forces the bargaining partner to reduce her share of the surplus (Segendorff 1998).

Our paper also relates to the capital budgeting literature (e.g., Antle and Eppen 1985; Bernardo et al. 2001; Baldenius 2003; Dutta 2003; Baldenius et al. 2007; Dutta and Fan 2009; Baiman et al. 2013; Heinle et al. 2014; Bastian-Johnson et al. 2013, 2017) where a principal commits to an ex ante hurdle for investing. Typically, this literature recommends that the hurdle is set sufficiently high, which results in ex post underinvestment. Our analysis differs in several dimensions: First, we predict overinvestment in innovations and potential underinvestment in routines. Second, the CEO in our model does not have private information but rather strategically designs a system to generate public information. Third, in our main model, the board of directors has no commitment power, and its ex post approval decision depends on the board’s expansion bias. We compare this main setting with a benchmark case where the board has commitment power (similar to the capital budgeting literature) and show that commitment and conservative bias are substitutes. Notably, in our benchmark case, there is no ex post investment distortion (in contrast to capital budgeting studies)—this difference is due to the endogenous nature of information in our model.

Our observation that commitment and bias are substitutes is related to findings in Laux (2008) where, if the board is sequentially rational and cannot commit to not renegotiating, the shareholders prefer a board that is friendly to the CEO over an independent one. The main driving force in Laux (2008) is that it is cheaper for a friendly board to motivate the CEO’s effort and extract the CEO’s private information at the contract renegotiation stage. In contrast, information in our model is symmetric at every point in time and, therefore, cannot be extracted via a contract. However, a conservative (i.e., unfriendly to the CEO) board can discipline the CEO’s endogenous acquisition of public information.

Lastly, we contribute to the Bayesian persuasion literature.⁴ Unlike the extant work, our paper focuses on the optimal alignment of interests between senders and receivers. In an extension, we consider the board’s learning option, which relates to the literature incorporating the receiver’s information acquisition.⁵ The threat of the board’s learning disciplines the CEO (similar to other models with monitoring threat, e.g., Townsend 1979). This result relates to the findings of Matysková and Montes (2023); Caplin et al. (2019); Huang (2016), but unlike these studies, our focus is on the optimal misalignment between the interests of the sender and the receiver as well as the effect of learning costs on this misalignment.

2 Model

We consider a risk-neutral CEO (“she”) and a risk-neutral board of directors running a firm on behalf of a group of risk-neutral shareholders. The CEO finds a significant investment opportunity (“project”) and decides whether to present it ($d=1$) or not ($d=0$) to the board for consideration. The board approves ($a=1$) or rejects ($a=0$) the implementation.

Players’ preferences and biases

. The ex post utility of player $j\in \{S,C,B\}$ (shareholders, CEO, board) is

$$\begin{aligned} v_j(a,d, \theta )=a \cdot d \cdot (\theta -\theta _j), \end{aligned}$$

(1)

where $\theta \in [\theta _{min}, \theta _{max}]$ is the project value with $-\infty \le \theta _{min}<0<\theta _{max} \le + \infty$. The player-specific parameter $\theta _j\in [\theta _{min}, \theta _{max}]$ is common knowledge and captures misalignment of interests in a parsimonious way: each player prefers that projects with $\theta \ge \theta _j$ are undertaken (presented and approved) rather than not.⁶ This parameter reflects the players’ pecuniary and non-pecuniary benefits, costs or inherent investment attitudes. We refer to $\theta _j$ as player j’s “type” or “bias” and assume the following:

• (i) Without loss of generality, $\theta _S=0$ so that the shareholders naturally benefit from projects with a positive value and lose from projects with a negative value.

• (ii) For the CEO, $\theta _C<\theta _S$. That is, the CEO is an empire-builder (Baumol 1959; Marris 1964; Williamson 1964; Jensen 1986; Shleifer and Vishny 1989; Stulz 1990; Jensen and Murphy 1990; Hart 1995) and prefers that not only value-enhancing projects but also those with $\theta \in [\theta _C, \theta _S]$ are presented and approved.⁷

• (iii) The bias $\theta _B$ can assume any value so that the board can be “neutral” ($\theta _B=\theta _S$), “expansionist” ($\theta _B<\theta _S$), or “conservative” ($\theta _B>\theta _S$). Directors could have a pro-expansion bias if they are close with the CEO (e.g., same social circle), have an entrepreneurial background, or favor company expansion for other reasons (e.g., perks). They could have a conservative bias and be cautious about company expansion if they desire to maintain the status quo (Bertrand and Mullainathan 2003), prioritize stability over growth (Deloitte 2015), focus on current operations and financial prudence (Kroszner and Strahan 2001; Baldenius et al. 2019), or face pecuniary and non-pecuniary benefits and costs (Gregor and Michaeli 2023).⁸ The overall bias of the board depends on the bias of the majority of the members—neutral boards are a balanced mix.⁹

Project novelty and information structure

The project could involve a new technology, business concept, or market. In such cases, the project is observably classified as “novel.” Otherwise, it is labeled “routine.” A fraction $p\in (0,1)$ of the opportunities in the economy are routine, and the rest are novel. To capture any potential differences between the two categories of projects, we assume that the value of a routine project is drawn from a differentiable cumulative distribution G (with a corresponding probability density function g), and the value of a novel project is drawn from a differentiable cumulative distribution F (with probability density f). Finding a project means drawing a project from the pool of investment opportunities.

At the onset of the game, none of the players observes the underlying value $\theta$, but the CEO, after finding a project and before involving the board, can initiate the collection of information about it. For novel projects, the CEO can control the properties of the collected information due to the board’s lack of familiarity with the concept or the market.¹⁰ In particular, the CEO chooses a report structure, i.e., a distribution of report realizations and a distribution of the project value conditional on any given report realization. For instance, the CEO could select the due diligence team for an acquisition target, choose the focus group evaluating market demand, fix the experiment protocol for determining technological feasibility, or design the medical trial for drug effectiveness. This is essentially a persuasion problem, and, within the confines of our model, the CEO is at least weakly better off selecting binary reports with realizations that are supported by disjoint intervals of project values (see Online Appendix for proof). The CEO’s choice of report structure is then fully characterized by the choice of a reporting cutoff $\theta _R \in [\theta _{min}, \theta _{max}]$ such that a low report $r = l$ is generated if $\theta < \theta _R$ and a high report $r = h$ is generated otherwise.¹¹

To streamline the analysis, we define $H(\cdot )$ as the inverse function of the h-conditional expected value $\mathbb {E}[\theta |r = h, \theta _R]$ and $L(\cdot )$ as the inverse function of the l-conditional expected value $\mathbb {E}[\theta |r = l, \theta _R]$. Each of these functions yields a reporting cutoff $\theta _R$ that corresponds to a given conditional expected value. The report is verifiable and cannot be withheld if the project is presented (see discussion in Section 5.2.4). As for routine projects, the CEO lacks control over the information, potentially because directors possess knowledge about such projects, which renders the CEO unable to steer the collection of information. In this case, a public signal $s=\theta$ is generated.¹² Based on the observed information, the CEO decides whether or not to present the project to the board. There are no payoff consequences for not presenting.

Fig. 1

Timeline of events

Timeline

Figure 1 presents the timeline of the events. At date 1, the CEO finds a project from an observable category (novel or routine). At date 2, if the project is novel, the CEO chooses the reporting cutoff $\theta _R$, and an observable report $r\in \{l,h\}$ is generated. If the project is routine, an observable signal $s=\theta$ is generated. At date 3, the CEO decides whether or not to present the project to the board. At date 4, the board approves or rejects the project. At date 5, the payoffs are realized.

Discussion of assumptions

In Section 5 and the Online Appendix, we discuss the robustness of our results to relaxing some of the model’s assumptions.

3 Benchmark with commitment to approval policy

We commence with a brief benchmark case where, at the game’s onset (date 0), the board can commit to approving a presented project if its expected value meets or exceeds a specified “hurdle” rate that depends on the project category, $\{\rho ^{novel},\rho ^{routine}\}$.

Lemma 1

Under the benchmark with commitment power:

• (i) If $\theta _B \le \theta _C$, the board chooses hurdle rates $\rho ^{routine} \in [\theta _{min},\theta _C]$ and $\rho ^{novel} \in [\theta _{min},H^{-1}(\theta _C)]$.

• (ii) If $\theta _B > \theta _C$, the board chooses hurdle rates $\rho ^{routine} = \theta _B$ and $\rho ^{novel} = H^{-1}(\theta _B) > \theta _B$.

The CEO sets reporting cutoff $\theta _R = \max \{\theta _C,\theta _B\}$. She presents novel projects with $r=h$ and routine ones with $s \ge \max \{\theta _C,\theta _B\}$. The firm undertakes all novel and routine projects with $\theta \ge \max \{\theta _C,\theta _B\}$.

First, consider a board that is more biased towards expansion than the CEO ($\theta _B \le \theta _C$). While such a board would like to undertake all routine projects with $\theta \ge \theta _B$, it cannot achieve this, as the CEO only presents, for approval, those with $s=\theta \ge \theta _C$. Thus, the board, in this case, is indifferent between any $\rho ^{routine} \le \theta _C$; with either of these hurdle rates, only routine projects with $\theta \ge \theta _C$ are undertaken. For novel projects, the CEO achieves her most desired outcome by simply setting $\theta _R = \theta _C$ and presenting only projects with $r=h$. The expansionist board in this case is indifferent between any $\rho ^{novel} \le H^{-1}(\theta _C)$.¹³ All of these hurdle rates yield that only novel projects with $\theta \ge \theta _C$ are undertaken. To summarize, when the board is strongly biased toward expansion, the firm implements routine and novel projects with a value of at least $\theta _C$.

Second, consider a board that is more conservative about expansion than the CEO ($\theta _B > \theta _C$). Now, there are no projects that are favored by the board but not by the CEO. The opposite, however, is not true. Thus, while the CEO’s veto power is no longer a threat, her opportunistic choice of reporting cutoff is. By committing to $\rho ^{routine}= \theta _B$ and $\rho ^{novel} = H^{-1}(\theta _B)$, the board can curb the CEO’s empire-building and ensure that routine and novel projects with $\theta \ge \theta _B$ are presented and approved; i.e., the board achieves its most desired outcome.¹⁴ It is now straightforward to see that under the benchmark with commitment power, the shareholders are best off if the board is neutral, $\theta _B = \theta _S > \theta _C$. By Lemma 1 (ii), such a board chooses $\left( \rho ^{routine}, \rho ^{novel} \right) = (\theta _S, H^{-1}(\theta _S))$, which ensures efficient investments, i.e., that all projects favored by the shareholders (those with value $\theta \ge \theta _S$) are undertaken.¹⁵

Lemma 2

From a shareholder’s perspective, the optimal board with commitment power is neutral with $\theta _B=\theta _S$ and invests efficiently in both routine and novel projects.

Our benchmark case implies that the optimal board with commitment power chooses $\rho ^{novel}>\rho ^{routine}$. This result provides a novel explanation of why hurdle rates for innovations in practice are higher and exceed what a standard application of the NPV rule would have suggested. The board’s commitment power to such hurdles fully mitigates the CEO’s empire-building tendency and ensures efficient investments.

4 Benefit from conservative boards

We now return to the main model (where the board cannot commit to hurdle rates) and illustrate the benefit of nominating conservative boards in such settings.

Board’s approval decision

A sequentially rational board (i.e., a board without commitment power) approves a presented routine project at date 4 if and only if the (revealed by the signal s) value $\theta$ is at least $\theta _B$. If the presented project is novel, the board grants an approval if its interim (expected at date 4) payoff from an undertaken project, $\mathbb {E}[\theta |r, \theta _R] - \theta _B$, at least weakly exceeds the zero-payoff from rejection. To characterize the report-specific decision, $a_r$, we note that at $\theta _R=L(\theta _B)$ and $\theta _R=H(\theta _B)$ the board is indifferent (between approving and rejecting) after low and high reports, respectively.¹⁶

Lemma 3

(Board’s approval of novel projects for a given reporting cutoff)

• (i) When $\theta _B \le \mathbb {E}[\theta ]$, the board rejects the presented novel project if and only if the report is low and the reporting cutoff is below $L(\theta _B)$; that is, $a_h = 1$ and $a_l= \mathbbm {1}_{\theta _R \ge L(\theta _B)}$.

• (ii) When $\theta _B > \mathbb {E}[\theta ]$, the board approves the novel project if and only if the report is high and the reporting cutoff is at least $H(\theta _B)$; that is, $a_l = 0$ and $a_h= \mathbbm {1}_{\theta _R \ge H(\theta _B)}$.

Figure 2 illustrates Lemma 3 and distinguishes between four problem regions. A board with relatively low type, $\theta _B \le \mathbb {E}[\theta ]$, is easily convinced to ratify the project. When the reporting cutoff is large (region $\mathcal {P}_1$), the expected value of the novel project, conditional on either report realization, is high and exceeds the board’s type—thus, the board always approves. However, when the reporting cutoff is small (region $\mathcal {P}_2$), the expected value of the project, conditional on $r=l$, is too low for approval even by a board with low type—as a result, the board ratifies the novel project only if the report is high. Furthermore, a board with a relatively high type, $\theta _B > \mathbb {E}[\theta ]$, is not easily convinced to approve. It ratifies novel projects with an expected value that is sufficiently high—this happens following $r=h$ with a sufficiently high $\theta _R$ (region $\mathcal {P}_3$) but not otherwise (region $\mathcal {P}_4$).

Fig. 2

Board’s report-specific approval of a presented novel project

CEO’s reporting and presentation decisions

The CEO presents an encountered routine project if and only if the (revealed by the signal) value is at least $\theta _C$; that is, $d_s=\mathbbm {1}_{s=\theta \ge \theta _C}$.¹⁷ The more interesting case is that of a novel project: then the problem of the CEO is to choose an observable reporting cutoff $\theta _R$ and report-specific presentation decisions $(d_l, d_h)$ that maximize her payoff. The best possible outcome from the CEO’s perspective is when all novel projects with value $\theta \ge \theta _C$ are presented and approved and the ones with value $\theta < \theta _C$ are either not presented or rejected. This can be implemented by setting $\theta _R=\theta _C$ and making sure that (i) $a_l\cdot d_l=0$ and (ii) $a_h\cdot d_h=1$. Ensuring (i) is straightforward—all the CEO needs to do is not present the novel project if the report is low, i.e., $d_l=0$.¹⁸ Ensuring (ii) is more challenging. The first necessary condition is that the CEO presents after observing a high report, $d_h=1$. The second necessary condition is that the board approves after a high report, $a_h=1$. While this is the case for problem regions $\mathcal {P}_1$–$\mathcal {P}_3$ in Figure 2, it is not for $\mathcal {P}_4$. In this last region, making sure that the board approves a presented novel project with a high report requires increasing the reporting cutoff to the board’s indifference point, $\theta _R=H(\theta _B)$.¹⁹

Lemma 4

(Reporting, presentation and approval of novel projects) For novel projects, the CEO chooses a reporting cutoff $\theta _R^* =R(\theta _B)\equiv \max \{\theta _C, H(\theta _B)\}$ at date 2 and presents at date 3 if and only if the report is high. At date 4, the board approves the presented project.

A board with $\theta _B <H^{-1}(\theta _C)$ faces a reporting cutoff of $\theta ^*_R =\theta _C<\theta _S$ and approves all novel projects favored by the CEO; such a board overinvests from the shareholders’ perspective (Fig. 3). As $\theta _B$ increases beyond $H^{-1}(\theta _C)$, the reporting cutoff also increases: the board now approves fewer novel projects and is less likely to overinvest. When the board has a sufficiently conservative bias, $\theta _B=\theta _H \equiv H^{-1}(\theta _S) > \theta _S$, it faces cutoff $\theta _R^*=\theta _S$ and invests efficiently in innovations. Any board with $\theta _B>\theta _H$ underinvests, as then $\theta _R^*>\theta _S$.

Fig. 3

Optimal reporting cutoff of a novel project (in bold line)

Corollary 1

The CEO’s optimal reporting cutoff about novel projects is (weakly) increasing in $\theta _B$ and $\theta _C$ and is independent of $\theta _S$. There exists a unique value $\theta _H=H^{-1}(\theta _S)>\theta _S$, such that a board with bias $\theta _B<\theta _H$ approves some shareholder-value-destroying novel projects and a board with bias $\theta _B>\theta _H$ rejects some shareholder-value-enhancing ones.

Before we analyze the board bias that maximizes firm value, we briefly summarize the outcomes for routine and novel projects. Given that both players have veto power over the project, the firm undertakes all routine projects with value $\theta \ge \max \{\theta _C,\theta _B\}$ and all novel projects with value $\theta \ge \theta _R^*=R(\theta _B)=\max \{\theta _C,H(\theta _B)\}$. Efficient investments for routine projects are thus achieved when the board is neutral ($\theta _{B} = \theta _S>\theta _C$), and efficient investments for novel projects are achieved when the board is strongly conservative ($\theta _{B} = \theta _H> \theta _S>\theta _C$). One can draw a parallel to the benchmark setting, where the CEO’s empire-building and opportunistic reporting are curbed via a commitment to a sufficiently high hurdle rate. Here, in contrast, the same disciplining effect is achieved by a conservative bias. This observation implies that commitment and bias serve as substitutes in influencing CEO behavior.

Optimal board bias

Taking into account the reporting, presentation, and approval decisions, the optimal board bias from the shareholders’ perspective, which we denote by $\widetilde{\theta }_B^*$, maximizes the shareholders’ welfare $W (\theta _B)\equiv p W^{routine}(\theta _B) + (1 - p) W^{novel}(\theta _B)$, i.e., a convex combination of the firm value in case of routine projects, $W^{routine}(\theta _B) \equiv \int _{\max \{\theta _C, \theta _B\}}^{\theta _{max}} (\theta - \theta _S) g(\theta ) d \theta ,$ and the firm value in case of novel projects, $W^{novel}(\theta _B) \equiv \int _{\theta _R^*}^{\theta _{max}} (\theta - \theta _S) f(\theta ) d \theta .$ It is easy to see from our analysis in the preceding section that the ex ante optimal board has a bias between $\theta _S$ and $\theta _H$. Any other bias is associated with prohibitively large investment inefficiencies: an expansionist board with $\theta _B<\theta _S$ overinvests and a conservative board with $\theta _B>\theta _H$ underinvests in both routine and novel projects. This observation allows us to focus on the interval $[\theta _S, \theta _H]$ (we refer to it as “the relevant” interval) and streamline the analysis.

In the relevant interval, $W^{routine}(\theta _B)$ is a decreasing function: an increase in the board’s type beyond $\theta _S$ leads to underinvestment in routine projects and, thereby, a decrease in firm value. To analyze the shape of $W^{novel}(\theta _B)$, it is instructive to classify boards into two subsets depending on the intensity of their conflict of interest with the CEO.

Definition 1

(Conflict of interest) The conflict of interest between a board with bias $\theta _B$ and a CEO with bias $\theta _C$ is weak if $\theta _B < H^{-1}(\theta _C)$ and strong otherwise.

An increase of $\theta _B$ in the region of weak conflict has no effect on the reporting cutoff and the novel project decision: the CEO continues to set $\theta _R=\theta _C$, and the board continues to approve all novel projects with value above this cutoff. As a result, in the region of weak conflict, $W^{novel}(\theta _B)$ remains constant. In contrast, an increase of $\theta _B$ in the region of strong conflict increases the reporting cutoff, reduces overinvestment, and raises $W^{novel}(\theta _B)$.²⁰

The preceding discussion implies that an increase of $\theta _B$ in the region with weak conflict between the board and CEO over novel projects is associated with a decrease in welfare $W (\theta _B)$ because the shareholders only incur a “cost” from deterioration in decisions about routine operations. However, in the region with strong conflict, the shape of $W (\theta _B)$ depends on the relative magnitude of deterioration in routine projects and a “benefit” from improved decisions about novel projects. As a result, the shareholders’ welfare is not necessarily single-peaked. This complicates the identification of the optimal board. We proceed in two steps. First, in Lemma 5, we show that only two types of boards can be optimal: neutral and conservative. Second, in Proposition 1, we describe the necessary conditions on primitives for either of these types to be optimal.

Lemma 5

(Candidates for optimal board) The optimal board is (i) conservative and in a strong conflict with the CEO, i.e., $\widetilde{\theta }_B^* \in (\theta _S, \theta _H)$ with $\widetilde{\theta }_B^*>H^{-1}(\theta _C)$; or (ii) neutral and in a weak conflict with the CEO, i.e., $\widetilde{\theta }_B^* = \theta _S$ with $\widetilde{\theta }_B^*<H^{-1}(\theta _C)$.

The candidate board in Lemma 5 part (i) has an interior bias between $\theta _S$ (the bias that ensures efficient routine investments) and $\theta _H$ (the bias that ensures efficient novel investments). This optimum arises if the shareholders’ welfare—a convex combination of the firm value in case of routine projects and that in case of novel projects—is single-peaked which happens when the conflict between the CEO and a neutral board, $\theta _B = \theta _S$, is strong.

What happens when the shareholders’ welfare is not single-peaked? This is the scenario in part (ii) of Lemma 5, which establishes that a neutral board ($\theta _B=\theta _S$) may be optimal yet a strongly conservative board ($\theta _B=\theta _H$) may not. The intuition behind this is as follows: A change from a strongly conservative board mitigates underinvestments, whereas a change from neutral board does not mitigate overinvestments if the conflict between the CEO and neutral board is weak. This property makes neutral boards locally optimal.

Three implications of Lemma 5 stand out. First, optimal neutral boards are associated with weak conflicts, and optimal biased boards are associated with strong conflicts. Second, the shareholders always face inappropriate approvals of novel projects (because $\widetilde{\theta }_B^*<\theta _H$) and may face inappropriate rejections of routine projects (because $\widetilde{\theta }_B^*\ge \theta _S$). Put differently, in equilibrium, investments in routine projects are either efficient or insufficient, but investments in novel projects are always excessive. Third, we predict that the distribution of optimal boards is bimodal in the exogenous characteristics of the investment opportunities, directors, and CEOs, as companies can be classified into those with optimally neutral boards and those with optimally conservative ones.

The conditions for optimality depend on the parameters, especially on the probability p and the distributions F and G. Without imposing additional restrictions, we can only formulate necessary conditions on primitives for the optimality of each of the two candidate board types:²¹

Proposition 1

(Conditions for optimality of the board candidates)

• (i) A necessary condition for the optimal board to be neutral is that the latter is in a weak conflict with the CEO, i.e., $\theta _S<H^{-1}(\theta _C)$.

• (ii) A necessary condition for the optimal board to be biased against project approval is that $(1 - p) \, [ W^{novel} (\theta _H) - W^{novel}(\theta _S) ] \ge p \, [ W^{routine}(\theta _S) - W^{routine}(\max \{\theta _S,H^{-1}(\theta _C)\})].$

The intuition behind (i) is straightforward: if this condition is violated, the reporting cutoff set by the CEO is $\theta _R=H(\theta _S)>\theta _C$. In this case, the neutral board cannot be optimal, as making it more conservative (by increasing the bias beyond $\theta _S$) alleviates overinvestment in novel projects (by pushing the reporting cutoff closer to the one most favored by the shareholders). To understand condition (ii), note that the right-hand side of the inequality represents the expected minimal loss from distortions in routine projects committed by a conservative board—accordingly, we refer to it as “the minimal cost of establishing a strong conflict with the CEO.” The left-hand side of the inequality represents the ex ante gain from alleviating distortions in novel projects thanks to a strongly biased board that can undo the CEO’s empire-building ($\theta _B=\theta _H$), compared with a neutral board. Because this is the maximum gain that can be achieved by a conservative board, we refer to it as “the maximal benefit from establishing a strong conflict with the CEO.” For a biased board to ever be optimal, the maximal benefit that shareholders can gain has to exceed the minimal cost. An alternative way to describe this condition is to say that the cost of switching from a weak to a strong conflict, measured by the loss from distorted decisions on routine operations, should not be prohibitively large.

Fig. 4

Evaluation of the necessary condition (ii) in Proposition 1 for biased board

To gain further intuition about Proposition 1, consider a neutral board that is in a weak conflict with the CEO. A small increase in $\theta _B$ does not affect approved novel projects (as the CEO continues to set the reporting cutoff at $\theta _C$), but it distorts approved routine projects (as some projects with value $\theta >\theta _S$ are rejected). As long as $\theta _B <H^{-1}(\theta _C)$, the shareholders incur costs from distorted approvals of routine projects without gaining the benefits related to novel projects. However, an increase in $\theta _B$ beyond $H^{-1}(\theta _C)$—resulting in a switch from weak to strong conflict—raises the quality of approved novel projects. For the optimal board to be biased, it is necessary that the cumulative cost of achieving a strong conflict (minimal cost) be smaller than the benefit in eliminating all distortions in novel projects (maximal benefit). Figure 4 panel (a) illustrates a case where the minimal cost is prohibitively large so that the inequality in condition (ii) is violated. In this case, the shareholders’ welfare (in bold) for a board with $\theta _B\ne \theta _S$ is lower than the welfare with a neutral board—thus, the optimal board is neutral. In contrast, panel (b) presents a case where the maximal benefit outweighs the minimal cost—the inequality in condition (ii) is satisfied. Because this is only a necessary condition, the fact that it is satisfied still does not imply that the optimal board is biased. For this to happen, the total effect on shareholders’ welfare (in bold) has to exceed the welfare with a neutral board for some $\theta _B$. In the scenario of panel (b), this is true so that the optimal board is sufficiently (but not extremely) conservative.²²

Our next result identifies a condition on the frequency of routine projects, p, that determines whether the necessary condition for the optimal board to be biased is satisfied.

Corollary 2

There exists a unique value $p^*\in (0,1]$ such that condition (ii) in Proposition 1 is violated when $p>p^*$ and satisfied otherwise.

Intuitively, when routine projects occur with a sufficiently high frequency, the expected cost from distorted decisions about routine operations is large and outweighs any expected benefit from improved decisions about novel projects. The opposite holds when routine projects are less likely—then, the expected benefit exceeds the expected cost.

Condition (ii) of Proposition 1 depends also on $\theta _C$. While we can say that the condition is more likely met when the CEO has a stronger empire-building tendency, little beyond that can be said for general distributions.²³

Before concluding the main analysis, we emphasize that the difference between a model where the board can commit to hurdle rates (benchmark case) and a model where it cannot (main model) is that the desired outcome is achieved with a different instrument. In the benchmark, a neutral board achieves the outcome through a hurdle rate (sets the investment policy directly), whereas in the main model the shareholders achieve the outcome through board bias (i.e., they implement the investment policy indirectly).²⁴ Our analysis thus illustrates that the board’s commitment power and the board’s bias are substitutes.

5 Model variations and discussions

5.1 Extension with board’s information acquisition

In this section, we extend our results by allowing the board to acquire costly information (e.g., hire a consultant or exert an effort to learn) about the value of a presented novel project after reviewing the CEO’s report on date 3.²⁵ Similar to the CEO, the board acquires a binary message. Formally, for any CEO’s report r that induces presentation ($d_r=1$) and implies that $\theta \in [\underline{\theta },\overline{\theta }]$, the board chooses a cutoff, $\theta _M \in [\underline{\theta },\overline{\theta }]$, such that a message $m=b$ is generated when $\theta <\theta _M$ and a message $m=g$ is generated otherwise. We say that the board learns if $\theta _M$ is in the interior of this interval. Otherwise, if $\theta _M$ is in a corner, the board does not learn. We assume that the board incurs an information acquisition cost $C(\theta _M, \kappa ) = \kappa \cdot \Pr ( a_m \ne a_r \mid r, d_r = 1)$, where $\kappa > 0$ is a cost parameter while $a_r \in \{0,1\}$ and $a_m \in \{0,1\}$ are the optimal pre- and post-message decisions of the board, respectively. This specification means that the cost increases in the probability of reversing pre-message decisions, i.e., the likelihood of the acquired information being useful. In the Online Appendix, we employ a different modeling approach whereby we consider entropy cost and the full board’s signal space.

Lemma 6

For any report r that induces presentation ($d_r=1$) and implies that $\theta \in [\underline{\theta },\overline{\theta }]$:

• (i) The board sets ${\theta }_M = \max \{ \theta _B - \kappa , \underline{\theta } \}$ and rejects the projects with $\theta \in [\underline{\theta },\theta _M)$ if its pre-message decision is to accept the presented novel project, $a_r = 1$.

• (ii) The board sets $\theta _M = \min \{ \theta _B + \kappa , \overline{\theta } \}$ and approves projects with $\theta \in [\theta _M, \overline{\theta }]$ if its pre-message decision is to reject the presented novel project, $a_r = 0$.

Consider the case of part (i). Without further information, the board accepts the project, $a_r=1$, which implies that $\theta _B< \overline{\theta }$ because $\mathbb {E}[\theta |r]\ge \theta _B$. If $\theta _B \le \underline{\theta }$, the board favors all presented projects and has no incentives to learn at a cost; consequently, it optimally sets the message cutoff at $\theta _M=\underline{\theta }$. However, if $\theta _B \in (\underline{\theta },\overline{\theta }]$, learning can be beneficial. Setting $\theta _M=\theta _B$ would yield the most useful information, as it would eliminate all inappropriate approvals. This, however, is costly, so the board optimally reduces the cutoff to $\theta _M= \theta _B - \kappa$ and learns imperfectly unless the cost parameter is prohibitively high, $\kappa >\theta _B-\underline{\theta }$, in which case the board sets $\theta _M=\underline{\theta }$ and does not learn.²⁶

Now consider the case of part (ii), where the pre-message decision of the board is to reject the project, $a_r=0$. For the board to be willing to reject the project, it needs to be the case that $\theta _B > \mathbb {E}[\theta |r]$, which in turn implies $\theta _B > \underline{\theta }$. If $\theta _B \ge \overline{\theta }$, it is never optimal to set an interior cutoff $\theta _M$ and learn, because no project with $\theta <\overline{\theta }$ is valuable enough to be pursued from the board’s point of view. It remains to consider $\theta _B\in (\underline{\theta }, \overline{\theta })$. Setting $\theta _M=\theta _B$ eliminates all inappropriate rejections, but this is costly. So the board sets $\theta _M= \min \{ \theta _B + \kappa , \overline{\theta } \}$ and learns imperfectly unless the cost parameter is prohibitively high, $\kappa >\overline{\theta }-\theta _B$.²⁷

Next, we consider the optimal binary reporting and presentation strategy of the CEO in anticipation of the board’s information acquisition.

Lemma 7

At date 2, the CEO sets $\theta _R \in [R(\theta _B), \widehat{R}(\theta _B, \kappa )]$, where $R(\theta _B)= \max \{\theta _C, H(\theta _B)\}$ as defined in Lemma 4 and $\widehat{R}(\theta _B, \kappa ) \equiv \max \{ \theta _C, H(\theta _B), \theta _B - \kappa \}$. At date 3, the CEO presents the novel project with $r=h$, and the board sets $\theta _M = \widehat{R}(\theta _B, \kappa )$. At date 4, the board approves projects with $\theta \ge \theta _M$ and rejects the rest.

On the one hand, if the CEO sets the reporting cutoff at the level that is optimal without board’s learning, i.e., $\theta _R=R(\theta _B)$ as defined in Lemma 4, the board may acquire further information (Lemma 6). On the other hand, by setting $\theta _R= \widehat{R}(\theta _B, \kappa )$, the CEO provides exactly the same information that the board may choose to acquire at a cost—in such a case, it would be irrational for the board to learn.²⁸ So what is the CEO’s choice? She is indifferent between setting any $\theta _R \in [R(\theta _B), \widehat{R}(\theta _B, \kappa )]$. Either of these cutoffs yields an identical outcome for the CEO, with the only difference being the cost incurred by the board.²⁹

The natural follow-up question is, What type of board invests efficiently in novel projects? From the shareholders’ perspective, the most desired outcome for novel projects is when the board has a type $\widehat{\theta }_H = \widehat{R}^{-1}(\theta _S; \kappa ) = \min \{ H^{-1}(\theta _S), \theta _S + \kappa \}$. It is easy to see that this is weakly below $\theta _H= H^{-1}(\theta _S)$; that is, the board with a learning option that invests efficiently in novel projects is less conservative than the one without a learning option. Notably, when $\kappa$ is lower, $\widehat{\theta }_H$ is lower. This implies that the CEO’s opportunistic reporting is mitigated either by appointing a more conservative board or by improving the board’s access to information—in our model, the two are substitutes. Does the introduction of a learning option always mitigate investment inefficiencies? Our next result shows that this, surprisingly, is not always the case, at least for a given board type.

Lemma 8

For a given $\theta _B$, the introduction of a learning option mitigates investment distortions if $\theta _B \in [\theta _{min}, \widehat{\theta }_H]$ and amplifies them if $\theta _B \in [\theta _H, \theta _{max}]$. If $\theta _B \in (\widehat{\theta }_H, \theta _H)$, investment distortions for $\theta < \theta _S$ are eliminated but new distortions for some $\theta > \theta _S$ are introduced.

Before concluding, we comment on how learning affects the optimal board type. Now the interval of relevant board biases is narrower, $[\theta _S, \widehat{\theta }_H] \subseteq [\theta _S, \theta _H]$. In addition, for any relevant board bias, the firm value in the case of a novel project increases from $W^{novel}(\theta _B)= \int _{{R}(\theta _B)}^{\theta _{max}} (\theta -\theta _S) f(\theta ) d \theta$ to $\widehat{W}^{novel}(\theta _B) \equiv \int _{\widehat{R}(\theta _B, \kappa )}^{\theta _{max}} (\theta -\theta _S) f(\theta ) d \theta$. These changes do not affect qualitatively our findings that there are only two candidates for the optimal board.

Proposition 2

High learning cost, $\kappa$, increases the likelihood that the optimal board is neutral.

High $\kappa$ increases the minimal cost of establishing a strong conflict between the CEO and the board while keeping the maximal benefit of a biased board constant. Thus, the necessary condition for a conservative board to be optimal is more likely violated; the board is more likely neutral. The effect of $\kappa$ on $\widehat{\theta }_B^*$ when the optimal board is already conservative is ambiguous. Nevertheless, the asymptotic effects are clear: as $\kappa \rightarrow \infty$, the optimal board type converges to the level without learning, $\lim _{\kappa \rightarrow \infty } \widehat{\theta }_B^*=\widetilde{\theta }_B^* \ge \theta _S$, whereas as $\kappa \rightarrow 0^+$, the board sets $\theta _M=\theta _B$ and so it is best for it to be neutral, $\lim _{\kappa \rightarrow 0^+} \widehat{\theta }_B^* = \theta _S$. Overall, as the learning cost decreases, (i) it becomes more likely that a conservative bias is optimal, but (ii) the bias level decreases asymptotically. Together, observations (i) and (ii) imply that the optimal board type might be nonmonotonic.³⁰ Taking into account that low $\kappa$ can be interpreted as higher board expertise, observation (i) reflects complementarity between bias and expertise, whereas observation (ii) reflects substitution of board bias and expertise.

5.2 Discussion of other extensions

5.2.1 Board bias that depends on the project category

In the main model, we assume that there is only one board in charge of all investment opportunities, and its bias does not depend on the project category (routine or novel). This is consistent with the board bias reflecting benefits, costs, or inherent attitudes toward company expansion that do not depend on the nature of the project. If the board’s preferences were to depend on the project category—i.e., bias $\theta _B^{routine}$ for routine projects and bias $\theta _B^{novel}$ for innovations—then, the discussion following Corollary 1 implies that the optimal board would be strongly conservative about novel projects (with $\theta _B^{novel}=\theta _H$) and aligned with the shareholders about routine ones (with $\theta _B^{routine}=\theta _S$).³¹

5.2.2 Delegation of approval decisions

Consistent with empirical evidence that management is tasked with searching and boards are tasked with ratification of significant opportunities (Useem 2006), we take the delegation of approval to the board as given in the main model. This also aligns with recent empirical evidence that more emphasis is given to the board’s monitoring role (Faleye et al. 2011). Our analysis in Section 4 implies that it is clearly suboptimal to delegate decisions about all projects to the CEO or to delegate decisions about novel projects to the CEO and those about routine projects to the board. The only alternative division of approval powers that could be considered then is to delegate decisions about routine projects to the CEO and have the board specialize in novel projects. Under this scenario, the forces at play in our model are reinforced, and the optimal board is strongly conservative.

5.2.3 Project search

In this paper, we assume that the CEO considers the first investment opportunity she encounters. The Online Appendix considers a scenario that allows the CEO to draw from the pool of opportunities until she finds a project from the category (routine or novel) that she prefers. This does not change our results qualitatively.

5.2.4 Report verifiability

The assumption maintained in this paper that the report is verifiable fits our setting. A proposal for an investment project of significant importance has to be supported by convincing evidence of the project feasibility—e.g., the results of a due diligence process, market test, experiment, or drug trial—before involving the board (Useem 2006). Once collected, this evidence is available within the company, and the board can request it. Thus it is hard for the CEO to conceal or misrepresent it, which may also be associated with harsh legal consequences. For example, the former CEOs of Kmart and Kentucky aluminum company faced significant legal charges for providing misleading information to their boards (Peterson 2003; Associated Press 2020). To this end, the assumption that the report is verifiable arises naturally. However, this assumption is not critical:

• (i) Allowing the CEO to withhold the report yields an identical outcome to that in the main analysis as the information contained in the report unravels (Gentzkow and Kamenica 2017b; Kamenica 2019).

• (ii) The Online Appendix allows the CEO to receive additional (private) information and send non-verifiable messages at no cost (cheap talk) after the optimally constructed public report is generated. This does not alter our results.

• (iii) The Online Appendix also lets the CEO privately observe r and misrepresent it at a cost. Our main results continue to hold qualitatively.

5.2.5 Contracting

This paper focuses on preference misalignment for given compensation. Concurrent research finds that it is often not optimal to eliminate agents’ innate characteristics contractually. For example, Gregor and Michaeli (2023) find that (i) it may not be optimal to fully eliminate a CEO’s empire-building tendency via contracting, and (ii) it may be optimal to strengthen a board’s conservative tendency via incentive contracting. As information in our setting is symmetric at every point of time, a screening contract cannot be written. The possibility that information design may be contracted (or, more broadly, determined by the board) is captured in our model by the presence of routine projects.

6 Concluding remarks

We study the optimal bias of corporate boards tasked with approving investment opportunities proposed by empire-building CEOs. In line with the empirical evidence (Maffett et al. 2022), we find that reducing the alignment of interests between CEOs and boards (that lack the ability to commit to approval policy) may improve investment efficiency due to the novel force explored in this paper. Accounting for the heterogeneity of project nature, we also predict a bimodal distribution of boards in the economy: a peak where boards are neutral (aligned with shareholders), and a peak where they are strongly conservative (expansion-cautious). We expect optimal boards in firms with a large share of novel projects and managed by CEOs with strong aspirations toward empire-building to be less likely to be neutral. Our model also predicts that firms overinvest in novel projects and sometimes underinvest in routine operations. The effect of directors’ expertise is nontrivial: we anticipate that boards with higher expertise are more likely to be conservative (but do not expect them to be strongly conservative). Our results provide testable predictions about the link between board composition and investment efficiency.

Our paper also illuminates the ongoing debate about board independence. Prior analytical research suggests that CEOs are less forthcoming about exogenously acquired non-verifiable information when faced with directors whose preferences are not aligned with theirs—a finding that may raise concerns about unintended consequences of regulations mandating a minimum number of independent directors or enabling activists to intervene, such as by electing directors with a cost-cutting agenda (e.g., the SEC’s 2021 Universal Proxy Rules for Director Elections). Our supplemental analysis in the Online Appendix shows that CEOs prefer to commit to gathering and communicating verifiable rather than non-verifiable information. Because, in this case, nominating a neutral or conservative board is optimal for shareholders, our study highlights the positive effect of such requirements.

Proofs

Proof of Lemma 1

The solution concept is backward induction. On date 4, the board approves ($a=1$) routine projects if $\mathbb {E}[\theta |s ] \ge \rho ^{routine}$ and novel projects if $\mathbb {E}[\theta |r, \theta _R ] \ge \rho ^{novel}$. Since s is perfectly informative, we can restate the former condition as $\theta \ge \rho ^{routine}$. To streamline the analysis, we can focus on $r=h$ and restate the latter condition as $\theta \ge H(\rho ^{novel})$, where $H(\cdot )$ is the inverse function of $\mathbb {E}[\theta |r = h, \theta _R]$.³² On date 3, exercising her veto power, the CEO presents ($d=1$) only routine projects with value $\theta \ge \max \{ \rho ^{routine}, \theta _C \}$.³³ For innovations, if $\theta _C \ge H(\rho ^{novel})$, the CEO can set (on date 2) $\theta _R = \theta _C$ and present (on date 3) if $r=h$. Then, because in this case $\theta _R=\theta _C \ge H(\rho ^{novel})$, the expected value of the presented project (at least weakly) exceeds the hurdle rate. This secures subsequent project approval of all favored by the CEO projects (those with value $\theta \ge \theta _C$). However, if $\theta _C < H(\rho ^{novel})$, setting $\theta _R = \theta _C$ will lead to subsequent project rejection, as its expected value is below the hurdle. To secure approval of projects with $r=h$, the CEO must increase $\theta _R$.³⁴ The question is: By how much? It turns out that it is optimal to increase $\theta _R$ only up to the point where the expected project value conditional on high report equals the hurdle rate, $\mathbb {E}[\theta |r=h, \theta _R] = H^{-1}(\theta _R) = \rho ^{novel}$, or equivalently, $\theta _R = H(\rho ^{novel}) > \theta _C$. Any increase beyond that point sacrifices favored-by-the-CEO projects that the board is willing to approve. The remainder of the proof follows from the discussion in the main text and is omitted.$\square$

Proof of Lemma 2

Follows from the discussion in the main text and is omitted.$\square$

Proof of Lemma 3

To begin, let $f_r(\theta )$ be the probability density function after $r \in \{ l, h \}$ is observed. This probability is obtained by truncation of the prior distribution F at the reporting cutoff $\theta _R$, because after $r=h$ the board realizes that the project value is in $[\theta _R, \theta _{max}]$, and after $r=l$ the board realizes that the project value is in $[\theta _{min}, \theta _R)$:

$$ f_h(\theta ) = {\left\{ \begin{array}{ll} 0, & \text{ if } \, \theta \in [\theta _{min}, \theta _R), \\ \frac{f(\theta )}{1 - F(\theta _R)}, & \text{ if } \, \theta \in [\theta _R, \theta _{max}]. \end{array}\right. } $$

$$ f_l(\theta ) = {\left\{ \begin{array}{ll} \frac{f(\theta )}{F(\theta _R)}, & \text{ if } \, \theta \in [\theta _{min}, \theta _R) \\ 0, & \text{ if } \, \theta \in [\theta _R, \theta _{max}]; \end{array}\right. } $$

Now note that the interim (expected at date 4, i.e., after observing a report r with cutoff $\theta _R$) project value is as follows:

$$ \mathbb {E}[\theta |r, \theta _R] = \int _{\theta _{min}}^{\theta _{max}} \theta f_r(\theta ) d \theta . $$

The board’s interim (expected at date 4) payoff from approval is then,

$$\begin{aligned} \mathbb {E}[v_B(a=1, \theta )|r, \theta _R] = \int _{\theta _{min}}^{\theta _{max}} \theta f_r(\theta ) d \theta - \theta _B. \end{aligned}$$

(2)

When deciding whether to approve the project, the board compares the interim payoff from approval in (2) with the zero-payoff from project rejection. Thus, the approval decision of the board is report-specific and is given by

$$\begin{aligned} a_r= \mathbbm {1}_{ \int _{\theta _{min}}^{\theta _{max}} \theta f_r(\theta ) d \theta - \theta _B \ge 0}, \ r \in \{l,h\} \end{aligned}$$

where the indicator function equals 1 if the board’s interim payoff from approval is at least zero and equals zero otherwise. Because the board’s interim payoff from approval is increasing in $\theta _R$ and decreasing in $\theta _B$ (for any report r), a board facing a higher (given) reporting cutoff $\theta _R$ (i.e., a higher expected project value conditional on the report) and/or a board with lower bias $\theta _B$ is more likely to approve the project proposed by the CEO.

For a given $\theta _R$, the range of the interim project value when $r = l$ is $[\theta _{min}, \mathbb {E}[\theta ]]$ (see $\mathbb {E}[ \theta |r=l, \theta _R]\le \mathbb {E}[\theta ]$), whereas the range of the interim project value when $r = h$ is $[\mathbb {E}[\theta ], \theta _{max}]$ (see $\mathbb {E}[\theta |r=h, \theta _R]\ge \mathbb {E}[\theta ]$). The two intervals overlap only in the expected value of the project, $\mathbb {E}[\theta ]$. This implies that $\theta _B \lesseqgtr \mathbb {E}[\theta ]$ determines the range of the indicator functions.

Exploiting this observation, we characterize the board’s indifference over the project approval in the two-dimensional space of cutoffs, $(\theta _B, \theta _R) \in [\theta _{min},\theta _{max}]^2$. First, consider the low report realization. The function $L: [\theta _{min}, \mathbb {E}[\theta ]] \rightarrow [\theta _{min},\theta _{max}]$ that yields cutoff $\theta _R$ such that the board is indifferent between project approval and rejection for a low report, is implicitly characterized by $\int _{\theta _{min}}^{\theta _{max}} \theta f_l(\theta ) d \theta = \theta _B$ such that $\theta _R = L(\theta _B)$. Equivalently, $L^{-1}(\theta _R) = \int _{\theta _{min}}^{\theta _{max}} \theta f_l(\theta ) d \theta$. Similarly, consider the high report realization case and specify a function $H:[\mathbb {E}[\theta ], \theta _{max}] \rightarrow [\theta _{min},\theta _{max}]$ that yields a cutoff $\theta _R$ such that the board is indifferent between project approval and rejection for a high report. The function is implicitly characterized by $\int _{\theta _{min}}^{\theta _{max}} \theta f_h(\theta ) d \theta = \theta _B$ such that $\theta _R = H(\theta _B)$. Equivalently, $H^{-1}(\theta _R) = \int _{\theta _{min}}^{\theta _{max}} \theta f_r(\theta ) d \theta$.

To characterize $a_r$, we exploit that the board’s interim payoff from approval is increasing in $\theta _R$. For $\theta _B \le \mathbb {E}[\theta ]$, consider a low report realization. The board is indifferent at $\theta _R = L(\theta _B) \in [\theta _{min}, \theta _{max}]$. We have $a_l = 0$ if $\theta _R < L(\theta _B)$ and $a_l= 1$ if $\theta _R \ge L(\theta _B)$. For a high report realization, observe that $a_h = 1$ because the board’s interim payoff from approval is

$$ H^{-1}(\theta _R) - \theta _B \ge H^{-1}(\theta _R) - \mathbb {E}[\theta ] \ge H^{-1}(\theta _{min}) - \mathbb {E}[\theta ] = 0. $$

For $\theta _B > \mathbb {E}[\theta ]$, consider a high report realization. The board is indifferent at $\theta _R = H(\theta _B) \in [\theta _{min}, \theta _{max}]$. We have $a_h= 0$ if $\theta _R \le H(\theta _B)$ and $a_h = 1$ if $\theta _R \ge H(\theta _B)$. For a low report realization, observe that $a_l = 0$ because the board’s interim payoff from approval is

$$ L^{-1}(\theta _R) - \theta _B < L^{-1}(\theta _R) - \mathbb {E}[\theta ] \le L^{-1}(\theta _{max}) - \mathbb {E}[\theta ] = 0. $$

$\square$

Proof of Lemma 4

The proof follows from the discussion in the text.$\square$

Proof of Corollary 1

The properties of the CEO’s optimal reporting cutoff $\theta _R^*$ are given by the properties of R function in Lemma 4. Uniqueness of $\theta _H$ follows from the fact that H function (and also its inverse $H^{-1}$ function) is increasing. The observation that any $\theta _B \ne \theta _H$ distorts novel investments follows from the fact that a conflict between shareholders and CEO, $\theta _S > \theta _C$, implies that R function is in its increasing part, $R(\theta _H) = H(\theta _H) > \theta _C$.$\square$

Proof of Lemma 5

Three separate claims eliminate all other candidates for the optimal board type.$\square$

Claim 1

If the optimal board induces a weak conflict over novel project, it must be neutral: $R(\widetilde{\theta }_B^*) = \theta _C \Rightarrow \widetilde{\theta }_B^* = \theta _S$

Suppose not: $\theta _S < \widetilde{\theta }_B^*$. Then, from the shape of $R(\theta _B)$, a weak conflict is induced for any $\theta _B \in [\theta _S, \widetilde{\theta }_B^*]$. (By Definition 1 and Lemma 4, the weak conflict exists for relevant board types, $\theta _B \in [\theta _S, \theta _H]$, on an interval $\theta _B \in [\theta _S, H^{-1}(\theta _C)]$.) On this interval, however, $W^{routine}$ is decreasing in $\theta _B$ whereas $W^{novel}$ is constant in $\theta _B$, and therefore the maximizer is the minimal board type $\theta _S$, which contradicts $\theta _S < \widetilde{\theta }_B^*$.

Claim 2

If the optimal board induces a strong conflict over a novel project, it cannot be a severely biased board: $R(\widetilde{\theta }_B^*) > \theta _C \Rightarrow \widetilde{\theta }_B^* < \theta _H$

To begin, note that a severely biased board induces a strong conflict always, $R(\theta _H) = \theta _S > \theta _C$; by continuity of $R(\theta _B)$ and $\theta _S - \theta _C > 0$, it is inducing a strong conflict also on a left neighborhood of $\theta _H$. We will now analyze $W(\theta _B)$ on this neighborhood. We know that the marginal benefit for novel projects decreases to zero when $\widetilde{\theta }_B^*$ approaches $\theta _H$, $\lim _{\theta _B \rightarrow \theta _H^{-}} \frac{d \, W^{novel}(\theta _B)}{d \theta _B} = 0.$ At the same time, the marginal cost for routine projects is negative, $\lim _{\theta _B \rightarrow \theta _H^{-}} \frac{d \, W^{routine}(\theta _B)}{d \theta _B} = - \theta _H g(\theta _H) < 0.$ To combine, the shareholders’ welfare is decreasing when all value-destroying novel projects are eliminated, $\lim _{\theta _B \rightarrow \theta _H^{-}} \frac{d \, W(\theta _B)}{d \theta _B} = - p \theta _H g(\theta _H) < 0,$ and therefore the optimal board is $\widetilde{\theta }_B^* < \theta _H$.

Claim 3

If the optimal board induces a strong conflict over a novel project, it cannot be neutral: $R(\widetilde{\theta }_B^*)> \theta _C \Rightarrow \widetilde{\theta }_B^* > \theta _S$

Suppose not: $\widetilde{\theta }_B^* = \theta _S$ and $R(\theta _S) > \theta _C$. Then, by the shape of $R(\theta _B)$, a strong conflict is induced for any relevant board type, $\theta _B \in [\theta _S, \theta _H]$. At $\theta _B = \theta _S$, the marginal cost for routine projects is zero, $\frac{d \, W^{routine}(\theta _B)}{d \theta _B}\Big \vert _{\theta _B = \theta _S} = - \theta _S g(\theta _S) = 0.$ In contrast, the marginal benefit for novel projects is positive for the strong conflict (where $R(\theta _S) = H(\theta _S) > \theta _C$ when the conflict is strong), $\frac{d \, W^{novel}(\theta _B)}{d \theta _B}\Big \vert _{\theta _B = \theta _S} = 1 - F(H(\theta _S)) > 0.$ By combining the two effects, the ex ante shareholders’ payoff is increasing at $\theta _B = \theta _S$, $\frac{d \, W(\theta _B)}{d \theta _B}\Big \vert _{\theta _B = \theta _S} = (1 - p)[1 - F(H(\theta _S))] > 0.$ As a result, the optimal board has $\widetilde{\theta }_B^* > \theta _S$, which is a contradiction.

Proof of Proposition 1

For each condition, we construct a separate claim.$\square$

Claim 4

If $\widetilde{\theta }_B^* = \theta _S$, then $R(\theta _S) = \theta _C$.

Suppose not: $\widetilde{\theta }_B^* = \theta _S$ and $R(\theta _S) > \theta _C$. Then, by Claim 3 from Proof of Lemma 5, $\widetilde{\theta }_B^* > \theta _S$, which is a contradiction.

Claim 5

If $\widetilde{\theta }_B^* > \theta _S$, then it holds that $(1 - p) [ W^{novel} (\theta _H) - W^{novel}(\theta _S) ] \ge p[ W^{routine}(\theta _S) - W^{routine}(\max \{\theta _S,H^{-1}(\theta _C)\})]$

Recall that the left-hand side of the inequality (LHS) is the maximal benefit of a strong conflict (novel projects side), and the right-hand side of the inequality (RHS) is the minimal cost of a strong conflict (routine projects side). We prove by contradiction: suppose the optimal board is biased but the maximal benefit is below the minimal cost (LHS below RHS). Three cases exist:

• $H^{-1}(\theta _C) \le \theta _S$. Then, a strong conflict is induced for any relevant board type $\theta _B \ge \theta _S$. The minimal cost is zero but the maximal benefit is positive, $W^{novel}(\theta _H) - W^{novel}(\theta _S) > 0$. This contradicts that the maximal benefit is below the minimal cost.

• $H^{-1}(\theta _C) > \theta _S$ and $\widetilde{\theta }_B^* \in [\theta _S, H^{-1}(\theta _C)]$ (weak conflict induced). By Claim 1 in Proof of Lemma 5, the optimal board is neutral if the conflict is weak; this is a contradiction.

• $H^{-1}(\theta _C) > \theta _S$ and $\widetilde{\theta }_B^* \in [H^{-1}(\theta _C), \theta _H]$ (strong conflict induced). On this interval, the shareholders’ payoff is bounded by $\overline{W}$ from above,

  $$\begin{aligned} W(\theta _B)= & p \, W^{routine}(\theta _B) + (1 - p) \, W^{novel}(\theta _B)\\< & p \, W^{routine}(H^{-1}(\theta _C)) + (1 - p) \, W^{novel}(\theta _H) \equiv \overline{W}, \end{aligned}$$

  because on this interval, $W^{routine}$ is maximized at $\theta _B = H^{-1}(\theta _C)$ and $W^{novel}$ is maximized at $\theta _H$. Now, we use that the condition that the maximal benefit exceeds the minimal cost can be rearranged into $W(\theta _S) \le \overline{W}$. Therefore, since we suppose the maximal benefit is below the minimal cost, we have $W(\theta _S) > \overline{W}$. This contradicts that $\overline{W}$ is an upper bound of $W(\theta _B)$.

Proof of Corollary 2

Like in Proof of Proposition 1, LHS denotes the left-hand side of the inequality in condition (ii) (the maximal benefit of a strong conflict), and RHS denotes the right-hand side of the inequality (the minimal cost of a strong conflict). The proof follows from the following three observations:

• LHS is decreasing in p, whereas RHS is increasing in p; therefore, a unique cutoff $p^* \in [0,1]$ exists.

• At the limit, as $p\rightarrow 0$, LHS becomes $W^{novel} (\theta _H) - W^{novel}(\theta _S) > 0$, whereas RHS becomes 0 so that the condition is satisfied. Therefore, the cutoff is positive, $p^* > 0$.

• At the limit, as $p\rightarrow 1$, LHS becomes 0, whereas RHS becomes

  $$\begin{aligned} W^{routine}(\theta _S) - W^{routine}(\max \{ \theta _S, H^{-1}(\theta _C) \} ) \ge 0. \end{aligned}$$

  If $\theta _S < H^{-1}(\theta _C)$, RHS is positive, the condition is violated and the cutoff is $p^* < 1$. Otherwise, both LHS and RHS are zero, the condition is satisfied, and $p^* = 1$.

$\square$

Proof of Lemma 6

The board updates its prior belief F into a posterior belief $F^r$, where $F^r(\theta ) = \frac{F(\theta ) - F(\underline{\theta })}{ F(\overline{\theta }) - F(\underline{\theta }) }$ for $\theta \in [\underline{\theta }, \overline{\theta }]$.

Part (i): Suppose the board’s preferred decision after r is acceptance, $a_r = 1$. If $\underline{\theta } \ge \theta _B$, the board will approve any project under full information. Thus, the board’s value of information is zero and, consequently, the board does not learn, i.e., sets $\theta _M = \underline{\theta }$. If $\underline{\theta } < \theta _B$, the board wants to revise the acceptance decision for low-value projects by setting a cutoff $\theta _M \in (\underline{\theta }, \theta _B]$. With this cutoff, the information acquisition cost is $C(\theta _M, \kappa ) = \kappa F^{r}(\theta _M)$. By increasing $\theta _M$, the marginal board’s information acquisition cost is $\frac{d C(\theta _M, \kappa )}{d \theta _M} = \kappa f^r(\theta _M)$, and the marginal board’s benefit from a higher $\theta _M$ is $(\theta _B - \theta _M) f^r(\theta _M)$. The interior optimum is at $\theta _B - \kappa < \theta _B$; to account for the lower corner, $\theta _M \ge \underline{\theta }$, the optimum is $\theta _M = \max \{ \theta _B - \kappa , \underline{\theta } \}$.

Part (ii): Suppose the board’s preferred decision after r is rejection, $a_r = 0$. If $\overline{\theta } \le \theta _B$, the board will reject any project under full information. Thus the board’s value of information is zero and, consequently, the board does not learn, i.e., $\theta _M = \overline{\theta }$. If $\overline{\theta } > \theta _B$, the board wants to revise the rejection decision for high-value projects by setting a cutoff $\theta _M \in [\theta _B, \overline{\theta })$. With this cutoff, the information acquisition cost is $C(\theta _M, \kappa ) = \kappa [1 - F^{r}(\theta _M)]$. By decreasing $\theta _M$, the marginal board’s information acquisition cost is $\frac{d C(\theta _M, \kappa )}{d - \theta _M} = \kappa f^r(\theta _M)$, and the marginal board’s benefit from a lower $\theta _M$ is $(\theta _M - \theta _B) f^r(\theta _M)$. The interior optimum is at $\theta _B + \kappa > \theta _B$; to account for the upper corner, $\theta _M \le \underline{\theta }$, the optimum is $\theta _M = \min \{ \theta _B + \kappa , \overline{\theta } \}$.$\square$

Proof of Lemma 7

For $\theta _B \le \theta _C$, the optimal reporting cutoff is clearly $\theta _R = \theta _C$, as the board then approves the project without learning. For $\theta _B > \theta _C$, we construct the board’s optimal report (which is also the board’s decision cutoff) as a function of $\theta _R$: (i) For $\theta _R < H(\theta _B)$, the board’s preferred decision is rejection, and $\theta _M = \min \{ \theta _{max}, \theta _B + \kappa \}$. (Notice that this interval is empty when $H(\theta _B) < \theta _C$.) (ii) For $\max \{ \theta _C, H(\theta _B) \} \le \theta _R \le \theta _B$, the board’s preferred decision is acceptance, and $\theta _M = \max \{ \theta _R, \theta _B - \kappa \}$. (iii) For $\theta _R > \theta _B$, the board accepts and does not learn, $\theta _M = \theta _R$.

Note that $\theta _M \ge \theta _C$ and therefore the CEO seeks to minimize $\theta _M$. Clearly, this is in the interval (ii), where $\theta _M = \max \{ \theta _R, \theta _B - \kappa \}$. Any report from the interval (i) is clearly worse for the CEO than the report (and decision cutoff) $\theta _R = \theta _B$, because $\theta _M = \min \{ \theta _{max}, \theta _B + \kappa \} > \theta _B$. Also, any report from the interval (iii) is clearly worse for the CEO than the report (and decision cutoff) $\theta _R = \theta _B$, because $\theta _M = \theta _R > \theta _B$.

The interval (ii) is characterized by the reports $\theta _R \in [\max \{ \theta _C, H(\theta _B) \}, \theta _B]$ and the board’s decision cutoff $\theta _M = \max \{ \theta _R, \theta _B - \kappa \}$. On this interval, the value of $\theta _M$ is minimized (closest to the value $\theta _C$) when $\theta _R \ge \max \{ \theta _C, H(\theta _B) \}$ and $\theta _R \le \theta _B - \kappa$. Using $R(\theta _B) = \max \{ \theta _C, H(\theta _B) \}$ and introducing $\widehat{R}(\theta _B) = \max \{ \theta _C, H(\theta _B), \theta _B - \kappa \}$, this is equivalent for $\theta _R \in [R(\theta _B), \widehat{R}(\theta _B)]$. The board’s decisions at dates 3 and 4 follow from Lemma 6.$\square$

Proof of Lemma 8

Introducing a learning option affects only decisions for novel projects. Without a learning option, the projects $\theta \ge R(\theta _B)$ are approved. With a learning option, the projects $\theta \ge \widehat{R}(\theta _B) \ge R(\theta _B)$ are approved. The approval is less likely as projects $\theta \in [R(\theta _B), \widehat{R}(\theta _B)]$ are now rejected. There are three cases: (i) If $\theta _B \le \widehat{\theta }_H$, we have $R(\theta _B) \le \widehat{R}(\theta _B) \le \theta _S$; board learning mitigates investment distortions. (ii) If $\theta _B \ge \theta _H$, we have $\theta _S \le R(\theta _B) \le \widehat{R}(\theta _B)$. The board’s learning amplifies investment distortions. (iii) In the intermediate case, the board’s learning mitigates investment distortions for $\theta \in [R(\theta _B),\theta _S)$ but introduces new distortions for $\theta \in [\theta _S, \widehat{R}(\theta _B))$.$\square$

Proof of Proposition 2

First, we analyze the effect of a higher learning cost on shareholders’ welfare. The shareholders’ optimal board problem is to maximize $W(\theta _B)$ subject to the constraint $\theta _R = \widehat{R}(\theta _B)$. The relevant board types are $\theta _B \in [\theta _S, \widehat{\theta }_H)]$; for these types, $\theta _R \in [\theta _C, \theta _S]$. Intuitively, in the parameter space of $(\theta _B, \theta _R)$, the shareholders want as low a $\theta _B$ as possible (to not distort routine projects) and also as high a $\theta _R$ as possible (to not distort novel projects). A higher $\kappa$ reduces $\theta _B - \kappa$ function and consequently (weakly) reduces the reporting cutoff function, $\widehat{R}(\theta _B) = \max \{ \theta _C, H(\theta _B), \theta _B - \kappa \}$. A higher $\kappa$ thus makes the shareholders’ constraint, $\theta _R = \widehat{R}(\theta _B)$, tighter, which implies that the maximized shareholders’ welfare is (weakly) reduced.

Second, we analyze the effect of a higher learning cost on the type of board (neutral or biased). A higher $\kappa$ (weakly) reduces $\widehat{R}(\theta _B)$. We know that the shareholders’ optimal board is chosen from two conditionally optimal boards: (i) for low $\theta _B$ such that $\widehat{R}(\theta _B) = \theta _C$ (weak conflicts), the conditionally optimal board is neutral, whereas for (ii) high $\theta _B$ such that $\widehat{R}(\theta _B) > \theta _C$ (strong conflicts), the conditionally optimal board is biased. The shareholders’ welfare associated with the neutral board in a weak conflict is constant in the cost $\kappa$, since the pair $(\theta _B, \theta _R) = (\theta _S,\theta _C)$ is invariant in the cost, whereas the welfare associated with the biased board in a strong conflict is (weakly) decreasing in the cost $\kappa$, as $\widehat{R}(\theta _B)$ (weakly) decreases in $\kappa$. This implies that the shareholders may switch from the biased board to the neutral board when the learning cost increases, but not vice versa.

To complete the argument, we also have to analyze the effect of a higher learning cost on the existence of a weak conflict for a neutral board, which is a necessary condition for a neutral board. This effect is positive; a decrease in $\widehat{R}(\theta _B)$ implies that it is more likely that a weak conflict begins to exist for a neutral board, i.e., it is more likely that we switch from a case with $\widehat{R}(\theta _S) > \theta _C$ to a case with $\widehat{R}(\theta _S) < \theta _C$. To sum up, with a higher learning cost, it is more likely that a weak conflict begins to exist for a neutral board, and also more likely that the board switches from the biased board to the neutral board (conditional on the existence of weak conflict for the neutral board); the two effects thus jointly imply that with a higher learning cost, the board is more likely neutral.$\square$