Board of discord: Conflicting preferences and performance pay

Abstract

This study examines the interaction of conflicting preferences among directors, performance pay, and group effort. I model a corporate board in which directors voluntarily choose to research (or not research) an investment decision made by the board on behalf of the firm. Free-riding among directors creates a need for performance pay to motivate this costly research. The study shows that board diversity, modeled as heterogeneous personal preferences among directors over the chosen investment, can act as a substitute for costly performance pay and, in equilibrium, benefit the firm. This creates a direct financial incentive for firms to increase board diversity. The study then shows how the optimal level of diversity changes with board and firm characteristics and generates a set of testable empirical predictions.

Keywords:

• board of directors
• board diversity
• voluntary effort
• free-riding
• heterogeneous preferences

1. Introduction

Performance pay has become a widely used tool to increase the performance of corporate boards by aligning the incentives of directors and shareholders (Jensen & Murphy, 1990; Yermack, 2004). At the same time, numerous studies have sought to empirically document the effect of diversity on the performance of corporate boards. These studies, briefly summarized below, have not produced a consensus on the link between diversity and firm performance. In the policy realm, there have been multiple initiatives seeking to increase the diversity of corporate boards, often citing a business case for doing so, despite the lack of empirical consensus.¹ This study connects these topics by constructing a model that examines the interaction of board diversity and director performance pay. I find that diversity can act as a substitute for costly performance pay, making the firm’s shareholders better off, while maintaining the same level of utility for directors. This provides a direct causal link between board diversity and firm performance.

In the model, the board makes an investment decision for the firm. Directors receive utility in two ways. First, each director receives performance pay in the form of a share of firm profits. Second, each director has hidden heterogeneous preferences over the outcome of the investment decision. These preferences reflect private benefits that a director receives from the investment—possibilities ranging from financial incentives outside of the director’s official pay contract with the firm to non-monetary utility received from the project’s impact on personal, social or environmental causes. I model diversity as the variance of these hidden preferences. A more diverse board has directors with hidden preferences that are further (in expectation) from those of the other directors.

The investment decision is made in a framework of uncertainty—its success depends on an unobserved state of the world. However, directors can pay a utility cost to do research and receive a noisy signal about the state of the world. A director that does not do research is unable to affect the investment decision, however the research phase is not contractable. This introduces a free-riding problem—an individual director may want to avoid paying the research cost and rely on other directors to do the work of selecting a project. In a typical framework, this free-riding is eliminated through the use of performance pay (Grossman & Hart, 1983; Holmstrom, 1979). Here, a director’s hidden preferences provide an additional incentive to induce effort. By paying the research cost, a director is able to participate in the project selection process, both tipping the project in their preferred direction, and adding additional information to the board’s decision. As the diversity of the board increases, this incentive becomes more powerful and diversity becomes a valuable non-monetary tool that the firm can use to overcome free-riding among directors.

Numerous studies have sought to determine the empirical link between board diversity and firm performance. Much of this work has focused on gender as its measure of diversity. Ferreira (2015) summarizes this literature and concludes that “current research does not really support a business case for board gender quotas.” A brief and non-exhaustive selection of these studies follows. In the non-academic literature, several studies performed by the non-profit group Catalyst, including Joy et al. (2007) and Carter and Wagner (2011), show that boards in the top quartile of WBD (women board directors) are correlated with better financial results than boards in the bottom quartile of WBD. These non-peer-reviewed studies, while frequently cited in policy discussions, do not address causality and only consider correlation. In the academic literature, Carter et al. (2003), Erhardt et al. (2003), and Perryman et al. (2016) find that more diverse boards are associated with better financial performance. However, Adams and Ferreira (2009) include firm fixed effects and find a negative relationship. Ahern and Dittmar (2012) use a policy change in Norway as an exogenous shock to the number of women on boards to also show a negative relationship between diversity and performance. As discussed by Adams et al. (2010), joint endogeneity, omitted variables, and reverse causation concerns make the empirical link between the board and firm performance particularly difficult to study. This study differs from the empirical literature by modeling a direct avenue by which board diversity board can positively affect firm finances—by creating savings in costly incentive contracts for directors.

Another area of research explores the interaction of personal preferences and incentives to participate in decision-making. Che and Kartik (2009) consider an environment where an advisor with personal opinions about a decision strategically discloses information to a decision-maker. There, some difference of opinion can benefit the decision-maker. Tan and Wen (2020) explores the selection of heterogeneous experts voting between two alternatives. They find that the optimal level of bias depends on whether the selection mechanism is sequential or simultaneous. They also explore the welfare effects of simple majority versus supermajority mechanisms. Most closely related is Cai (2009), which explores the effect of heterogeneous preferences on committee members’ willingness to gather information and ultimately committee size and the principal’s expected payoff. This study’s model is an extension of Cai (2009) that adds performance pay as an additional tool for the principal to overcome free-riding, and embeds the model in the context of a corporate board.

Section 2 presents the model and discusses the assumptions and modelling choices. Section 3 solves the model, discusses the results, and presents empirical predictions. Section 4 concludes.

2. Model

The model is an extension of Cai (2009) with the addition of performance pay. Where possible, I use the same notation. The firm has a board with $N$ directors and needs to make an investment decision that depends on an unknown state of the world. Directors have many duties on the board, often grouped into two categories: monitoring and advising (Adams & Ferreira, 2007). In the monitoring role, directors oversee firm management and supervise agency conflicts between executives and shareholders. In the advising role, directors provide advice, information, and other resources for the firm. In the advising role, directors make large investment and business decisions, and this study’s model focuses on the role of diversity in the advising context.

The timing of the model is as follows:

1. The firm chooses a share of profits to pay the directors.

2. Directors choose whether or not to gather information.

3. Participating directors receive a private and noisy signal about the state of the world. They also privately learn their personal preferences over the investment.

4. Participating directors report their findings to the board. These reports need not be “truthfull.”

5. The project is selected, the state of the world revealed, and payoffs received.

Each part of the model is presented in detail below.

2.1. Firm

The firm must make an investment decision $x\in \mathbb{R}$. The success of the project depends on the true (but unknown ex-ante) state of the world $\theta \in \mathbb{R}$. The project success is given by $1-(x-\theta {)}^2$ and the optimal project for the firm is $x=\theta$.

The firm has a board consisting of $N$ directors and one of the board’s tasks is to choose $x$. Director pay contracts typically include a flat base salary as well as performance pay such as stock options and stock grants that depend on firm performance (Yermack, 2004). Because this paper focuses only on the project selection aspect of the role of the board, I set aside consideration of base pay and focus on performance pay. In a full model of the board base pay may play an important role in the willingness of a director to work for a firm and to address any risk aversion among directors. Here, the firm pays each director a share $\alpha \in [0,1/N]$ of firm profits. Directors are protected by limited liability. A simple linear contract is chosen for simplicity and is supported by the literature on team incentives, for example Dai and Toikka (2022).

The final project payoff to the firm is then given by

(1) $\pi =(1-N\alpha )A\left(1-{(x-\theta )}^2)\right).$(1)

where $A$ represents the size of the project and becomes important when discussing director utility below.

2.2. Directors

Directors benefit from the project selection process in two ways. First, each director has personal intrinsic preferences over which project is selected, and second, the director is paid a share of firm profits. The personal preferences are different among directors and unknown ex-ante to the firm, other directors, and the director herself. This assumption allows for the possibility that the general preferences of a director are known ex-ante, however the way in which particular projects meet a director’s private goals is unknown until the project is researched. The process by which a director discovers their preferences for a project are discussed in detail in a later section. The personal preference component of director $i$’s utility given decision $x$ and state of the world $\theta$ is given by

(2) $1-(x-\theta -t_i{)}^2$(2)

where $t_i$ represents director $i$’s personal preference, or bias, in project selection. When only considering a director’s private preferences, the optimal project for director $i$ is $x=\theta +t_i$. The director’s preferences are not perfectly aligned with the firm because of the bias term. To address this, the firm can set $\alpha >0$ in order to partially align the directors preferences with the firm. The director’s utility with performance pay is given by

(3) $u_i(x,\theta ,t_i)=1-(x-\theta -t_i{)}^2+A\alpha \left(1-{(x-\theta )}^2\right).$(3)

Now $A$ has an important interpretation as the relative impact of the project on firm profits, and therefore a directors’ pay contract, versus the private benefit of the project to a director. A project with a larger $A$ could be interpreted as one with less scope for private benefit to directors or as a project that has a large impact on firm profits. For example, a decision about the firm’s stance on a social issue would be represented by a low $A$, while a capital structure decision would be a higher $A$.

Maximizing director utility in Equation 3(3) $u_i(x,\theta ,t_i)=1-(x-\theta -t_i{)}^2+A\alpha \left(1-{(x-\theta )}^2\right).$(3) with respect to $x$ reveals that the optimal project for a director with bias $t_i$ and contract $\alpha$ is given by

(4) $x=\theta +\frac{t_i}{1+\alpha A}.$(4)

As seen in Equation 4(4) $x=\theta +\frac{t_i}{1+\alpha A}.$(4) , as the firm increases profit sharing $\alpha$, the director’s optimal project gets closer to the firm’s optimal project. As expected, as the firm increases profit sharing with directors, directors’ incentives are more aligned with the firm. However, as the director’s bias $t_i$ moves away from zero, the director prefers a project that is less profitable to the firm.

2.3. Research and project selection

Each director chooses whether or not to voluntarily research the project. If director $i$ participates, they pay a utility cost $c$ and receive two pieces of information. First, they receive a noisy signal ${\theta }_i$ about the true state of the world, where ${\theta }_i=\theta +{\text{isin}}_i$ and the noise terms are i.i.d. draws ${\text{isin}}_i\sim N(0,{\sigma }_{\text{isin}}^2)$.

In the process of researching the project, each director also receives information about their personal preferences over the project. For an inside director, this could be information about how the project benefits their career path, the teams that they preside over, or any pay bonuses outside of the model. For an outside director, the project might benefit their other investments or firms that the director has a stake in. Additionally the project might affect causes they personally care about such as the environment, social issues, or community impact. Importantly, directors do not find out how a project affects their personal preferences unless they take the time and effort to investigate the investment decision. A director’s personal bias in project selection is represented by an i.i.d. draw $t_i\sim N(0,{\sigma }_t^2)$.

The variance of the bias draw ${\sigma }_t^2$ is an important variable in the model and is interpreted as the diversity of the board. It is known ex-ante to the firm and the directors themselves. Before the research phase, the actual realizations of director personal preferences are unknown, but the expected variance of the preferences is known. A board that is more diverse in terms of industry background, education, demographics, and life experiences will have a greater expected variance in the bias draws, leading to conflicting personal preferences.

If a director researches the project, they are able to participate in the project selection process by making a report $s_i({\theta }_i,t_i)\in \mathbb{R}$. Crucially, the director only reports one number and not separate numbers for each of their signals. In this way, a director presents information about their research, but is able to distort the information to reflect their personal bias. For example, in a decision about the desired scale of a new factory, a director concerned about the environment might report a smaller than optimal scale that could be powered by renewable energy. An inside director that will preside over the factory may prefer a larger than optimal project that will increase their visibility in the firm. From the single reported number, the other directors are not able to separate out “truthful” information from ${\theta }_i$ and personal bias from $t_i$.

If a director does not pay the cost $c$ to investigate the project, they are not able to participate in the decision-making process, as it is evident to the rest of the board that the director has not done the requisite work. In this sense, effort is observable, but the payoff to the director does not depend directly on this observability. Despite not participating in this one particular decision, the director still receives their pay contract, including the $\alpha$ share of profits.

Even though directors still receive pay if they do not participate in the project selection, they might voluntarily choose to participate for two reasons. First, their signal ${\theta }_i$ adds additional information about the true state of the world, and this information can increase the project’s success and therefore firm profits, of which the director receives a share. Second, by participating, the director is able to learn their personal preference and steer the project in that direction.

I assume that the firm commits to a mean decision rule where the project selected is the mean of the individual director reports: $x=1/N{\sum }_i{s}_i({\theta }_i,t_i)$.²

3. Equilibrium

3.1. Director reporting

Following Cai (2009), there is a simple reporting equilibrium where directors report a combination of their signal about $\theta$ and their personal preference $t_i$. Here, however, profit sharing reduces the weight put on the personal bias term.

Proposition 1

It is an equilibrium for each director $i$ to report $s_i^{\ast }({\theta }_i,t_i)={\theta }_i+N\frac{t_i}{1+A\alpha }$.

Proof.

I show that the director’s problem is the same as in Cai (2009), but with $t_i$ replaced by $t_i/(1+A\alpha )$. Here, the director wishes to make a report that maximizes

$E\left[1-{(x-\theta -t_i)}^2+A\alpha \left(1-{(x-\theta )}^2)\right)\right]$

$=E\left[1-\left({(x-\theta )}^2+t_i^2-2(x-\theta )t_i\right)+A\alpha -A\alpha {(x-\theta )}^2\right]$

$=E\left[(1+A\alpha )-\left({(x-\theta )}^2+t_i^2-2(x-\theta )t_i+A\alpha {(x-\theta )}^2\right)\right]$

$=E\left[{(1}_A\alpha )-\left((1+A\alpha )(x-\theta {)}^2+t_i^2-2(x-\theta )t_i\right)\right]$

$=(1-A\alpha )E\left[1-{(x-\theta )}^2-\frac{t_i^2}{1+A\alpha }+\frac{2(x-\theta )t_i}{1+A\alpha }+\frac{t_i^2}{{(1+A\alpha )}^2}-\frac{t_i^2}{{(1+A\alpha )}^2}\right]$

$=(1-A\alpha )E\left[1-\left({(x-\theta )}^2-\frac{2(x-\theta )t_i}{1+A\alpha }+\frac{t_i^2}{{(1+A\alpha )}^2}\right)+\frac{t_i^2}{{(1+A\alpha )}^2}-\frac{t_i^2}{1+A\alpha }\right]$

$=(1-A\alpha )E\left[1\underset{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{o}\mathrm{n}x}{\underbrace{-{\left(x-\theta -\frac{t_i}{(1+A\alpha )}\right)}^2}}+\frac{t_i^2}{{(1+A\alpha )}^2}-\frac{t_i^2}{1+A\alpha }\right]$

where only the underbraced term depends on $x$. This is the same as in Cai but with $t_i$ replaced by $\frac{t_i}{1+A\alpha }$. The proof then follows from the proof of Proposition 4 in Cai.

Recall from before that director $i$’s preferred project is given by $\theta +t_i/(1+A\alpha )$. Their equilibrium report reflects this, but with the bias term exaggerated $N$ times, as the mean decision rule divides each directors’ report by $N$.

3.2. Director participation

In this equilibrium, the expected payoff to the firm (assuming all $N$ directors participate) is given by

(5) $E[\pi ]=(1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right].$(5)

First, the firm pays each director $\alpha$ share of the profits, leaving the firm with the remaining $1-N\alpha$ share. Focusing on the terms in the square brackets, the ideal investment pays out 1. Remaining uncertainty about the state of the world after receiving the directors’ reports subtracts ${\sigma }_{\text{isin}}^2/N$ from this. Noise introduced by director bias subtracts another $N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}$ from the ideal investment. The investment success is then multiplied by firm size $A$.

For the firm, there are trade-offs to hiring additional directors. The firm benefits from the director’s additional information about $\theta$. However, each additional director gets paid $\alpha$ share of the profits. Additionally, each director introduces some noise to the decision process because of their personal biases. The two variance terms clearly affect firm profits in a negative way. Variance in signals about $\theta$ make directors’ signals less precise, while an increase in board diversity increases the noise that directors introduce in an effort to steer the project to their personal ideal.

This expression for firm profit assumes that all directors participate, which is not guaranteed ex-ante. In equilibrium, each participating director has expected utility (assuming all other directors participate) given by

(6) $E[u_i{]}_P=(1+A\alpha )\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-(N-1)\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]-\frac{A\alpha }{1+A\alpha }{\sigma }_t^2-c.$(6)

If a director does not participate, but the other $N-1$ directors do, then the non-participating director’s expected utility is given by

(7) $E[u_i{]}_{NP}=(1+A\alpha )\left[1-\frac{{\sigma }_{\text{isin}}^2}{N-1}-(N-1)\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]-{\sigma }_t^2.$(7)

Therefore the marginal utility to a director of participating is given by

(8) $MB=E[u_i{]}_P-E[u_i{]}_{NP}=(1+A\alpha )\left[\frac{{\sigma }_{\text{isin}}^2}{N-1}-\frac{{\sigma }_{\text{isin}}^2}{N}\right]+\frac{{\sigma }_t^2}{1+A\alpha }-c$(8)

A director benefits in two ways by participating. First, they benefit from a more precise signal about $\theta$. Second, they benefit from pulling the project toward their bias $t_i$. A director will participate if these benefits outweigh the cost $c$ of participation.

From the firm’s perspective, performance pay affects a director’s willingness to participate in two ways. Increasing $\alpha$ aligns the director’s preferences with the firm and increases the importance of good project selection, and therefore the value of an addition signal about $\theta$. However, increasing $\alpha$ also decreases the importance for a director of pulling the project toward their bias. Equation 8(8) $MB=E[u_i{]}_P-E[u_i{]}_{NP}=(1+A\alpha )\left[\frac{{\sigma }_{\text{isin}}^2}{N-1}-\frac{{\sigma }_{\text{isin}}^2}{N}\right]+\frac{{\sigma }_t^2}{1+A\alpha }-c$(8) is then the participation constraint that must be greater than zero for a director to be willing to participate in the decision-making process.

3.3. Optimal contract and diversity

To begin, I only take $\alpha$ as endogenous. The firm has the ability to choose the amount of profit sharing with directors and takes the other parameters as given. The firm wishes to maximize expected profit subject to the participation constraint of the directors.

(9) $\underset{\alpha }{max}(1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]$(9)

$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}(1+A\alpha )\left[\frac{{\sigma }_{\text{isin}}^2}{N-1}-\frac{{\sigma }_{\text{isin}}^2}{N}\right]+\frac{{\sigma }_t^2}{1+A\alpha }\ge c$

If the participation constraint is not binding with $\alpha =0$, then an increase in board diversity ${\sigma }_t^2$ hurts the firm. This is readily seen in Equation 9(9) $\underset{\alpha }{max}(1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]$(9) as ${\sigma }^2$ only enters profit negatively and increases the benefit of participation for directors that are already willing to participate without any profit sharing.

If the participation constraint is costly and binding, then there is potential for board diversity to increase firm value.

Proposition 2

For a sufficiently costly participation constraint, an increase in board diversity increases firm value.

Proof.

Use the envelope theorem on the Lagrangian of the firm’s maximization problem in Equation 9.(9) $\underset{\alpha }{max}(1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]$(9)

(10) $\mathcal{L}\equiv (1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]+\lambda \left[(1+A\alpha )\left[\frac{{\sigma }_{\text{isin}}^2}{N-1}-\frac{{\sigma }_{\text{isin}}^2}{N}\right]+\frac{{\sigma }_t^2}{1+A\alpha }-c\right]$(10)

(11) $\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\sigma }_t^2}=\frac{\lambda (1+A\alpha )-AN(1-N\alpha )}{{(1+A\alpha )}^2}$(11)

This is positive when

(12) $\lambda >\frac{AN(1-N\alpha )}{1+A\alpha }$(12)

For a sufficiently high shadow price of the participation constraint $\lambda$, an increase in diversity benefits the firm. The question then becomes whether or not $\lambda$ ever crosses that threshold. Examining this inequality further, as the equilibrium $\alpha \to 0$, implying that the constraint is not costly and the firm does not need to provide incentive pay then the right hand side approaches $AN$; a shadow price of participation inconsistent with no incentive problem. In this situation, more diversity hurts the firm (as explained above).

However, as $\alpha \to 1/N$, i.e. the firm needing to pay out all profits to incentivize directors, then the right hand side approaches 0 and it is known that $\lambda \ge 0$ from the Kuhn-Tucker conditions. At some intermediate point between these two extremes, Equation 12(12) $\lambda >\frac{AN(1-N\alpha )}{1+A\alpha }$(12) holds and more diversity benefits the firm.

The next proposition details how the optimal level of profit sharing changes with exogenous parameters.

Proposition 3 Comparative statics on $\alpha$ when the participation constrain is binding.

1. $\alpha$ is decreasing in ${\sigma }_t^2$.

2. $\alpha$ is increasing in $N$.

3. $\alpha$ is decreasing in ${\sigma }_{\text{isin}}^2$.

4. $\alpha$ is decreasing in $c$.

5. $\alpha$ is decreasing in $A$.

Proof.

The analytical solution for $\alpha$ with a binding constraint can be solved for directly from the constraint holding with equality.

(13) $\alpha =\frac{cN(N-1)-2{\sigma }_{\text{isin}}^2+\sqrt{N(N-1)}\sqrt{c^{2}N(N-1)-4{\sigma }_{\text{isin}}^2{\sigma }_t^2}}{2A{\sigma }_{\text{isin}}^2}$(13)

1. Follows from ${\sigma }_t^2$ only entering the expression for $\alpha$ in a negative term.

2. Follows from $N$ only entering the expression for $\alpha$ in positive terms.

3. Follows from ${\sigma }_{\text{isin}}^2$ only entering the expression for $\alpha$ in negative terms in the numerator and in the denominator.

4. Follows from $c$ only entering the expression for $\alpha$ in positive terms.

5. Follows from $A$ only entering in the denominator of the expression for $\alpha$. In particular, $A\alpha$ is constant in $A$. An increase in $A$ leads to a proportionate decrease in $\alpha$.

First, $\alpha$ decreases in ${\sigma }_t^2$. That is, boards with more diversity need to pay less in profit sharing. When directors have larger biases, there is more intrinsic motivation to participate in the research phase and less incentive pay is needed to overcome the free-riding problem. In this environment diversity acts as a substitute to performance pay. The decrease in profit sharing does not mean however that directors are worse off, as the participation constraint still binds.

Second, $\alpha$ is increasing in the size of the board. On one hand this is costly – $\alpha$ is paid to every director, and an increase in $N$ increasing the marginal cost of increasing $\alpha$. However, the effect of increasing $N$ on the participation constraint outweighs this cost and drives the result. As $N$ increases, the marginal benefit of an additional signal about $\theta$ decreases and this exacerbates the free-riding problem, resulting in a need for more performance pay.

Third, $\alpha$ is decreasing in ${\sigma }_{\text{isin}}^2$. As the variance of signals about $\theta$ increase, the participation constraint is easier to satisfy and the firm needs less performance pay to overcome free-riding. Fourth, $\alpha$ is increasing in the cost of participation. Intuitively, as research is more costly for directors, the free-rider problem is worse and more performance pay is needed. Finally, $\alpha$ is decreasing in $A$. As $A$ increases, $\alpha$ decreases a proportionate amount so that the amount $A\alpha$ paid to directors remains constant.

Next, I consider the case where the firm is able to choose both the amount of profit sharing and the diversity of the board.

3.4. Endogenous diversity

The model now extends to the case where the firm chooses both $\alpha$ and ${\sigma }_t^2$ in order to examine how the optimal level of diversity changes with the remaining exogenous variables. The firm’s problem remains the same as in Equation 9(9) $\underset{\alpha }{max}(1-N\alpha )A\left[1-\frac{{\sigma }_{\text{isin}}^2}{N}-N\frac{{\sigma }_t^2}{{(1+A\alpha )}^2}\right]$(9) , with the addition of ${\sigma }_t^2$ as a choice variable in the maximization. As suggested in the previous section, the firm is potentially better off choosing a positive level of diversity. The following proposition examines how the optimal level of diversity changes with exogenous parameters when the participation constraint is binding, i.e. when free-riding is a concern to the firm.

Proposition 4

Comparative statics on ${\sigma }_t^2$ with a binding participation constraint.

1. ${\sigma }_t$ is decreasing in ${\sigma }_{\text{isin}}^2$.

2. ${\sigma }_t$ is increasing in $A$.

3. ${\sigma }_t$ is increasing in $c$.

Proof.

Solving the maximization problem with a binding participation constraint leads to the following analytical solution, where $\lambda$ is the Lagrange multiplier on the participation constraint.

(14) ${\sigma }_t^2=\frac{\sqrt{A^{2}cN(N-1)(A+N)\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}-A(A+N){\sigma }_{\text{isin}}^2}{A\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}$(14)

(15) $\alpha =\frac{cN(N-1)(A+N)}{\sqrt{A^{2}cN(N-1)(A+N)\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}}-\frac{1}{A}$(15)

(16) $\lambda =\frac{AN(N-1)\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}{\sqrt{A^{2}cN(N-1)(A+N)\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}}-N^2$(16)

1.

$\frac{\mathrm{\partial }{\sigma }_t^2}{\mathrm{\partial }{\sigma }_{\text{isin}}^2}=-\frac{2AcN(N-1)(A+N)+c\sqrt{A^{2}cN(N-1)(A+N)\left[N(N-1)+{\sigma }_{\text{isin}}^2\right]}}{2A{(N(N-1)+{\sigma }_{\text{isin}}^2)}^2}$

(17) $<0asalltermswithinthefractionarepositive.$(17)

2.

(18) $\frac{\mathrm{\partial }{\sigma }_t^2}{\mathrm{\partial }A}=\frac{A{c}^{2}N\left[N^3-N^2+N+(N-1){\sigma }_{\text{isin}}^2\right]-2c{\sigma }_{\text{isin}}^2\sqrt{A^{2}c(A+N)(N^2-N)(N^2-N+{\sigma }_{\text{isin}}^2)}}{2\left(N^2-N+{\sigma }_{\text{isin}}^2\right)\sqrt{A^{2}c\left(N^2-N\right)(A+N)\left(N^2-N+{\sigma }_{\text{isin}}^2\right)}}$(18)

which is positive when

(19) ${\sigma }_{\text{isin}}^2<\frac{1}{8}\left(\sqrt{\frac{c{(N-1)}^2{N}^2(16A+c+16N)}{{(A+N)}^2}}+\frac{c(N-1)N}{A+N}\right)$(19)

Numerical investigation shows that this inequality holds over wide ranges of parameters.

3. Direct inspection shows that $c$ only enters the expression for ${\sigma }_t^2$ in a positive term.

First, the optimal level of board diversity is decreasing in the noisiness of the directors’ signals about $\theta$. When these signals are very noisy, the value of an additional signal goes up and the participation constraint is easier to satisfy. Second, the optimal level of diversity is increasing in the size of the project, $A$. The intuition for this result is less clear, as $A$ enters the firm’s objective and the participation constraint in several places. A larger project increases the importance of an additional signal about $\theta$, but this can be reduced by the firm endogenously choosing a lower level of profit sharing. Also as $A$ increases, the amount that a director wishes to distort the decision decreases, but this also is affected by the firm’s endogenous choice of $\alpha$. Last, the optimal level of diversity is increasing in $c$. This is straightforward, as diversity’s benefit in the model is in satisfying the participation constraint, and as research becomes more costly, diversity can be used to overcome that cost.

Missing from these comparative statics are the effects of a change in board size $N$. The analytical expression for how ${\sigma }_t^2$ changes with $N$ is impenetrable, however numerical analysis suggests that the optimal level of board diversity is increasing in the board size. A larger board decreases the marginal benefit of another signal about $\theta$, and increased diversity can offset this in the participation constraint. However, a larger board also increases the amount of noise entering the investment decision when director bias is high. Over all numerically investigated parameterizations, the first effect is larger and diversity increases with board size.

Figure 1 presents an example parameterization that is representative of the model over a broad range of parameter choices. The example sets the board size $N=10$, variance of signals about the true state of the world ${\sigma }_{\text{isin}}^2=1$, project size $A=15$, and the cost of participation $c=0.08$. In each set of figures, one exogenous parameter is varied over a specified range. In most cases, board diversity and profit sharing increase and decrease together, in accord with the interpretation that they are both tools that the firm can use to eliminate free-riding. However, when the project size $A$ gets large enough, the amount of profit sharing $\alpha$ decreases (as in the previous section with exogenous diversity) while the optimal level of diversity increases.

Figure 1. An example showing how ${\sigma }_t^2$ and $\alpha$ vary. Exogenous variables not varied in the plots are set at ${\sigma }_{\text{isin}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$, $\mathbf{N}\mathbf{=}\mathbf{10}$, $\mathbf{A}\mathbf{=}\mathbf{15}$, and $\mathbf{c}\mathbf{=}\mathbf{0.08}$.

3.5. Empirical predictions

In moving to an empirical setting, it is necessary to further examine the types of diversity that apply to the model. A more diverse board is one in which the directors are more likely to have personal preferences over decisions made by the firm and one in which the directors have an ability to move those decisions in their preferred direction. While an appropriate measure of diversity will depend on the specific context, possibilities include the following:

• Industry background. Directors from different industries may want to select investments that directly benefit those industries or projects that would benefit from their industry-specific knowledge, thus making the director more relevant in the future operation of the firm. For example, Burak Güner et al. (2008) find that boards with more financial experts make different financing decisions.

• Demographics. Director demographics such as age, race, nationality, and gender identity could be correlated with personal preferences over social issues affected by the firm’s decisions.

• Educational background. Directors with diverse educational backgrounds may approach decisions with different viewpoints and biases.

Additionally, the this study focuses on the role of directors in assisting management in decision-making—the resource dependence role of the board. The board also plays instrumental roles in monitoring management and making hire/fire decisions about the CEO. Those activities lie outside of my model and could introduce different predictions.

Empirical predictions of the model follow from the discussion of the equilibrium. First, is that more diverse boards are associated with a lower level of performance pay. Because diversity helps to overcome the free-riding problem, less incentive pay in the form of stock grant, stock options, and bonuses based on relevant financial measures are needed to address the issue. This is in contrast to Adams and Ferreira (2009) who find that boards with more women are associated with more monitoring and governance, including a larger share of equity-based pay. The model does not make a prediction about the relationship between diversity and total compensation. Similarly, several studies, including Cook et al. (2019), Tee (2021), Sarhan et al. (2019), and Newman and Mozes (1999) examine the link between board diversity and CEO and top-level management compensation. The model in this study does not address this link, as it is focused on the role of diversity as a substitute for performance pay within the same group.

Second are a set of predictions about how the level of diversity in a board is correlated with firm characteristics. First, diversity is increasing in firm size and board size. This is consistent with Carter et al. (2003) who find that the proportion of women and minorities on the board increases with both board size and firm size. Second, board diversity is predicted to be decreasing in the variance of signals about the state of the world. This could be interpreted as less diverse boards in companies that operate in very uncertain or volatile markets. Last, board diversity is increasing in the cost of research.

4. Conclusion

This study examines the interaction between diversity, modeled as conflicting preferences among directors, and board performance pay in a corporate board setting. It finds that diversity can act as an additional motivating factor in overcoming free-riding among directors, thereby reducing the need for performance pay. However, the corporate board is hardly the only setting in which free-riding presents a problem. Additional research could reveal other settings, such as non-profits, university administration, and civic obligations, in which diversity could replace or complement other costly solutions to free-riding. Managers choosing groups in these settings can use conflicting personal preferences as an additional motivating factor, and thereby reduce the need for financial incentives used to induce effort.

In the corporate context, this model contributes to the literature seeking a financial justification for increased board diversity. As discussed in the introduction, the link between board characteristics and firm performance is both not established and empirically difficult to reason about. In contrast, this study offers a direct link between board diversity and firm finances. The model reveals that heterogenous preferences among directors can help offset costly performance pay that others have found to be excessive (Dah & Frye, 2017). Rather than trying to trace the effects of diversity through the decision-making machinery of a firm, it links directly to the costly performance pay given to directors. While financial performance is only one factor in the policy discussion on board diversity, this paper offers a financial justification for efforts to increase diversity on boards. A drawback of the current study is that the link between observable characteristics and conflicting preferences is unspecified. Future empirical work could help to determine which observable dimensions of diversity are most strongly linked to heterogeneous preferences over corporate investment decisions.

There is further work to be done in this area. First, is a more comprehensive examination of the relationship between board (or group) diversity and the heterogeneity of preferences in the group. Observable characteristics may not always be the best proxy for ex-ante disagreement among directors. Second, this study does not conduct an empirical examination of the relationship between diversity and the pay structure of directors, nor the does it test the empirical predictions generated by the model about the relationship between firm characteristics, board diversity, and the components of director pay. All are promising avenues for further research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The Better Boards Initiative, the European Union’s “Women on Boards” law, California’s board diversity mandate (California State Corporations Code Chapter 954, ss 301.3 and 2115.5) For example, Joy et al. (2007)

2. Rosar (2015) discusses mean versus median decision rules.