Managerial Compensation and Firm Value in the Presence of Socially Responsible Investors

Abstract

Shareholders with standard monetary preferences will give a manager incentives to increase firm profits, which can be achieved with equity grants. When shareholders are socially responsible, in the sense that they also value corporate social performance, it is not clear which incentives the manager should receive. Yet, in a standard principal–agent model, we show that the optimal contract is surprisingly simple: it consists in giving equity holdings to the manager. This is notably because the stock price will incorporate expected profits as well as the social performance of the firm, to the extent that it is valued by shareholders. Consequently, equity holdings give the manager incentives to jointly maximize the profits and the social performance of the firm according to shareholders’ preferences. To facilitate alignment of interests, more socially responsible firms will optimally hire more socially responsible managers. We conclude that neither the shareholder primacy model nor equity-based managerial compensation is necessarily inconsistent with the attainment of social objectives.

Appendix

Portfolio Choice

Denote by X(z, a) the argument of the utility function of any given shareholder as a function of the number of shares z purchased at t = 1 and the allocation a of firm resources. In this case, denote by $W(z)=\omega -zp$ the amount invested at t = 1 at the risk-free rate by the shareholder, which is a function of z, as determined optimally by the shareholder at t = 1. Using Eq. (1),

$$\begin{aligned} X(z,a) = z \left( \sqrt{1-a} + \tilde{\epsilon } + u_s \phi \sqrt{a} \right) + \omega -zp \end{aligned}$$

(15)

With CARA preferences with absolute risk aversion $\rho$ and a normally distributed risk $\tilde{\epsilon }$, we know (e.g., Grossman and Stiglitz 1980) that maximizing expected utility is equivalent to maximizing the following certainty equivalent with respect to z:

$$\begin{aligned} CE(z,a)\,=\,& {} {\mathbb {E}} \left[ X(z,a) \right] - \frac{\rho }{2} {\rm var} \left[ X(z,a) \right] \nonumber \\= & \, {} z \sqrt{1-a} + \omega -zp + z u_s \phi \sqrt{a} - \frac{\rho }{2} z^2 \sigma ^2. \end{aligned}$$

(16)

The solution to this optimization problem is given by the first-order condition, which after some rearranging yields

$$\begin{aligned} z = \frac{\sqrt{1-a} + u_s \phi \sqrt{a}-p }{\rho \sigma ^2 } \end{aligned}$$

(17)

Note that the optimal investment in firm stock by any given shareholder is independent from his wealth $\omega$, due to assumption of CARA utility. Given this set of optimal demands from n ex-ante identical shareholders, the stock price is given by the market clearing equation which equates the supply $1-z_m^{\text {LT}}$ of shares and the demand $n \times z$ of shares:

$$\begin{aligned} n \frac{\sqrt{1-a} + u_s \phi \sqrt{a}-p }{\rho \sigma ^2 } = 1-z_m^{\text {LT}} \end{aligned}$$

(18)

Solving this equation for p gives the t = 1 equilibrium stock price:

$$\begin{aligned} p = \sqrt{1-a} + u_s \phi \sqrt{a} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \end{aligned}$$

(19)

The stock price is simply equal to expected firm profits $\sqrt{1-a}$, plus the utility $u_s \phi \sqrt{a}$ of owning shares in a socially responsible firm, minus the risk premium due to the variability of firm profits ($\sigma ^2$) and shareholder risk aversion $\rho$. This risk premium is calculated based on $\frac{1-z_m^{\text {LT}}}{n}$, which is the fraction of the firm held by each shareholder in equilibrium. Indeed, substituting the stock price p from (19) in (17) gives

$$\begin{aligned} z = \frac{1-z_m^{\text {LT}}}{n} \end{aligned}$$

(20)

Proof of Proposition 1

Given that the stock price is concave in a, the resource allocation $a^{\star }$ that maximizes the stock price is the one that solves $\frac{{\text{d}}p}{{\text{d}}a}=0$. With the value of p derived in (4), this implies

$$\begin{aligned} \frac{\mathrm{d}p}{\mathrm{d}a} = -\frac{1}{2} \frac{1}{\sqrt{1-a^{\star }}} + u_s \phi \frac{1}{2} \frac{1}{\sqrt{a^{\star }}} = 0 \end{aligned}$$

(21)

Rearranging,

$$a^{\star }=\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }.$$

(22)

Comparing with (3) proves Proposition 1.

Proof of Proposition 2

With a contract consisting in a fixed wage w, $z_m^{\text {ST}}$ short-term equity holdings and $z_m^{\text {LT}}$ long-term equity holdings, and denoting $\xi =\{w,z_m^{\text {ST}},z_m^{\text {LT}}\}$, the argument in the utility function of a manager who exerts effort is

$$Y(\xi ,a)= w + z_m^{\text {ST}} p +z_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\epsilon } + u_s \phi \sqrt{a} \right) - C.$$

(23)

Substituting the stock price p from (4) gives

$$\begin{aligned} Y(\xi ,a)= \, & {} w + z_m^{\text {ST}}\left[ \sqrt{1-a} + u_s \phi \sqrt{a} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] \nonumber \\& +\,z_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\epsilon } + u_s \phi \sqrt{a} \right) - C \nonumber \\= \, & {} w + \left( \sqrt{1-a} + u_s \phi \sqrt{a} \right) \left[ z_m^{\text {ST}} + z_m^{\text {LT}}\right] - z_m^{\text {ST}}\frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 + z_m^{\text {LT}} \tilde{\epsilon } -C. \end{aligned}$$

(24)

The optimization problem of a manager who exerts effort is to choose a to maximize ${\mathbb {E}} [U(Y(\xi ,a))|e=1]$. Given that the problem is concave in a for $z_m^{\text {ST}} + z_m^{\text {LT}}>0$, the optimal value $a_{m}$ of a optimally chosen by the manager is described by the first-order condition of the manager’s expected utility with respect to a, which after some rearranging yields

$$\begin{aligned} a_m = \frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 } \end{aligned}$$

(25)

Note that $a_m$ is independent from the fixed wage w or from the equity holdings $z_m^{\text {ST}}$ and $z_m^{\text {LT}}$, as long as $z_m^{\text {ST}} + z_m^{\text {LT}}>0$. In addition, $a_m=a^{\star }=a^{\text{FB}}$: for any contract of the type $\xi =\left\{ w,z_m^{\text {ST}},z_m^{\text {LT}}\right\}$, the resource allocation optimally chosen by the manager maximizes the stock price and is the first-best resource allocation.

Given that $a_m=a^{\text{FB}}$ for $z_m^{\text {ST}} + z_m^{\text {LT}}>0$, we now derive the optimal values of w, $z_m^{\text {ST}}$, and $z_m^{\text {LT}}$ such that the manager accepts the contract and exerts effort. At the time of contracting (t = 0), there are n shareholders, each of whom owns the same fraction of the firm. Any shareholder will accordingly bear a fraction 1 / n of the cost of managerial compensation, which consists in the fixed wage w and the liquidation value $z_m^{\text {ST}} p$ of short-term equity holdings at t = 1. In addition, each shareholder will own the same fraction $\frac{1-z_m^{\text {LT}}}{n}$ of the firm from t = 1 to t = 2 (see (20)), given that the manager will own a fraction $z_m^{\text {LT}}$ of the shares. Using the certainty equivalent approach, the optimization problem of any shareholder at t = 0 is⁷

$$\max _{w,z_m^{\text {ST}},z_m^{\text {LT}}} \frac{1-z_m^{\text {LT}}}{n} \left( \sqrt{1-a} + u_s \phi \sqrt{a} \right) - \frac{1}{n}\left( w+z_m^{\text {ST}}p \right) - \frac{\rho }{2} \frac{\left( 1-z_m^{\text {LT}}\right) ^2}{n^2} \sigma ^2$$

(26)

given $a=a^{\text{FB}}$, subject to the following constraints:

$$\begin{aligned}{\mathbb {E}}\left[ U\left( Y\left( \xi ,a^{\text{FB}}\right) \right) |e=1\right]\ge & {} {\mathbb {E}}[U(Y(\xi ,a))|e=0]\end{aligned}$$

(27)

$$\begin{aligned}{\mathbb {E}}\left[ U\left( Y\left( \xi ,a^{\text{FB}}\right) \right) |e=1\right]\ge & {} \bar{U}, \end{aligned}$$

(28)

where $Y(\xi ,a)$ conditional on e = 1 is given in (24), and $Y(\xi ,a)$ conditional on e = 0 is $w + z_m^{\text {ST}}\left[ - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] +z_m^{\text {LT}} \tilde{\epsilon }$. The incentive constraint (27) ensures that the expected utility of a manager who exerts effort is larger than the expected utility of a manager who does not exert effort, thus ensuring that the contract elicits effort. There is a continuum of contracts which achieve incentive compatibility and have the same implications for resource allocation, cost of compensation, and managerial expected utility (indeed, as will be further explained below notably in footnote 7, $z_m^{\text {ST}}$ can be set above the level which satisfies (27) as an equality, and the fixed wage correspondingly lowered to leave expected pay unchanged); as in Edmans et al. (2009), we choose the maximum between the level of $z_m^{\text {ST}}$ which satisfies (27) as an equality and zero (so that $z_m^{\text {ST}} \ge 0$). The participation constraint (28) ensures that the expected utility of a manager who accepts the contract is in equilibrium (conditional on e = 1 and $a=a_m$) is larger than his reservation level of utility $\bar{U}$.

Using the certainty equivalent approach, the incentive constraint (27) may be rewritten as

$$\begin{aligned}&w + z_m^{\text {ST}}\left[ \sqrt{1-a^{\text{FB}}} + u_s \phi \sqrt{a^{\text{FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] \nonumber \\&\quad +z_m^{\text {LT}} \left( \sqrt{1-a^{\text{FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) -C -\frac{\rho }{2} {z_m^{\text {LT}}}^2 \sigma ^2 \nonumber \\&\quad \ge w + z_m^{\text {ST}}\left[ - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] -\frac{\rho }{2} {z_m^{\text {LT}}}^2 \sigma ^2. \end{aligned}$$

(29)

Removing offsetting terms and rearranging yields

$$z_m^{\text {ST}}+z_m^{\text {LT}} \ge \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} }.$$

(30)

Denoting by $\bar{W}$ the reservation wage which is implicitly defined by $U(\bar{W}) \equiv \bar{U}$, and given that the equity holdings are such that the manager exerts effort, the participation constraint (28) may be rewritten with the certainty equivalent approach as

$$\begin{aligned}&w + z_m^{\text {ST}}\left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] +z_m^{\text {LT}} \left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) \nonumber \\&\quad -C -\frac{\rho }{2} {z_m^{\text {LT}}}^2 \sigma ^2 \ge \bar{W}. \end{aligned}$$

(31)

We denote by $\mu$ and $\lambda$ the Lagrange multipliers associated with the constraints (30) and (31), respectively.

The first-order conditions of the optimization problem in (26)–(28) with respect to w, $z_m^{\text {ST}}$, and $z_m^{\text {LT}}$ are then, respectively,

$$-1+\lambda =0$$

(32)

$$\begin{aligned}&-\left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] \nonumber \\&+ \lambda \left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] + \mu = 0 \end{aligned}$$

(33)

$$\begin{aligned}&-\left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) - z_m^{\text {ST}}\frac{\rho \sigma ^2}{n} + \rho \frac{1-z_m^{\text {LT}} }{n} \sigma ^2\nonumber \\&+ \lambda \left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} + z_m^{\text {ST}}\frac{\rho \sigma ^2}{n} -\rho z_m^{\text {LT}} \sigma ^2 \right) + \mu = 0 \end{aligned}$$

(34)

Equation (32) gives $\lambda =1$, which used in (33) implies $\mu =0$.⁸ With $\lambda =1$ and $\mu =0$, the first-order condition (34) can be rewritten as

$$\begin{aligned} z_m^{\text {LT}} = \frac{1}{1+n} \end{aligned}$$

(35)

Plugging this value of $z_m^{\text {LT}}$ in (30), equating both sides, and using $z_m^{\text {ST}}\ge 0$ gives

$$\begin{aligned} z_m^{\text {ST}} = \max \left\{ \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} } - \frac{1}{1+n} , 0\right\} \end{aligned}$$

(36)

Finally, substituting for $z_m^{\text {ST}}$ and $z_m^{\text {LT}}$ in (31) gives the fixed wage w, which satisfies the participation constraint as an equality ($\lambda =1$ implies that the participation constraint is binding due to the complementary slackness condition):

$$\begin{aligned} w= & {} \bar{W} - \max \left\{ \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} } - \frac{1}{1+n} , 0\right\} \nonumber \\&\quad \left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-\frac{1}{1+n}}{n}\rho \sigma ^2 \right] \nonumber \\&-\frac{1}{1+n} \left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) +C +\frac{\rho }{2} \frac{1}{(1+n)^2} \sigma ^2 \end{aligned}$$

(37)

Proof of Claim 2

Using the value of $z_m^{\text {ST}}$ in (10) with $a=a^{\text {FB}}$ (cf. Claim 1 and Proposition 2), we have

$$\begin{aligned} \frac{\mathrm{d}z_m^{\text {ST}}}{\mathrm{d}u_s} = \frac{\mathrm{d}}{\mathrm{d} u_s} \left\{ \frac{C}{\sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} } - \frac{1}{1+n} \right\} \end{aligned}$$

$$\begin{aligned} = -C \frac{-\frac{1}{2} \frac{2u_s \phi ^2}{(1 + u_s^2 \phi ^2)^2} \left( \frac{1}{1 + u_s^2 \phi ^2 } \right) ^{-0.5} + \frac{2u_s \phi ^2 \sqrt{1 + u_s^2 \phi ^2} -u_s^3 \phi ^4 (1 + u_s^2 \phi ^2)^{-0.5} }{1 + u_s^2 \phi ^2} }{ \left( \sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} \right) ^2 } \end{aligned}$$

$$\begin{aligned} = -C \frac{u_s \phi ^2}{(1 + u_s^2 \phi ^2)^{3/2}} \frac{-1 + 2(1 + u_s^2 \phi ^2) -u_s^2 \phi ^2 }{ \left( \sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} \right) ^2 } < 0 \end{aligned}$$

(38)

Unobservable Actions

This section revisits the model under the assumption that the resource allocation a is unobservable by shareholders. We now assume that profits at t = 2 are equal to $e\sqrt{1-a} + \tilde{\theta }_{\pi } + \tilde{\epsilon }$, and CSP is equal to $e \phi \sqrt{a} + \tilde{\theta }_{\phi }$, where as in the baseline model $e \in \{0,1\}$ and $a \in [0,1]$ are optimally chosen by the manager at t = 0. The random variables $\tilde{\theta }_{\pi }$ and $\tilde{\theta }_{\phi }$ are normally distributed with mean zero and respective variance $\sigma _{\pi }^2$ and $\sigma _{\phi }^2$. Both $\tilde{\theta }_{\pi }$ and $\tilde{\theta }_{\phi }$ are realized at t = 1. We assume that $\tilde{\theta }_{\phi }$ is independent from other random variables. Since both $\tilde{\epsilon }$ and $\tilde{\theta }_{\pi }$ affect firm profits, we assume that they are correlated, with ${\rm cov}(\tilde{\epsilon },\tilde{\theta }_{\pi }) \equiv \varrho$, and for simplicity we also assume that the variance of $\tilde{\epsilon }$ does not depend on the realization of $\tilde{\theta }_{\pi }$ at t = 1, i.e., ${\rm var}(\tilde{\epsilon }|\theta _{\pi })={\rm var}(\tilde{\epsilon })== \sigma ^2$.

At t = 1, before making portfolio choices, shareholders observe two signals, namely $s_{\pi }=e\sqrt{1-a} + \theta _{\pi }$ and $s_{\phi }=e \phi \sqrt{a} + \theta _{\phi }$, where $\theta _{\pi }$ and $\theta _{\phi }$ denote the realizations of $\tilde{\theta }_{\pi }$ and $\tilde{\theta }_{\phi }$, respectively. As in the baseline model, $\tilde{\epsilon }$ is realized at t == 2.

These assumptions capture the notions that the effect of investments in business operations and in CSP have effects on profits and the provision of social goods which are uncertain, and that managerial actions are unobservable. For example, shareholders only observe the polluting emissions of the firm, but pollution-reducing efforts may fail. More generally, a number of factors beyond the manager’s control may affect the profits and CSP of a firm.

We now establish that the main results of the paper remain unchanged under these assumptions that shareholders observe signals which are imperfectly informative about the manager’s effort e and resource allocation a if $\varrho =0$, and we characterized the manager’s contract when $\varrho \ne 0$.

As in the Portfolio choice section, denoting by X(z, a) the argument of the utility function of any given shareholder as a function of the number of shares z purchased at t == 1 and the allocation a of firm resources, in equilibrium we have

$$\begin{aligned} X(z,a) = z \left( \sqrt{1-a} + \theta _{\pi } + \tilde{\epsilon } + u_s (\phi \sqrt{a} + \theta _{\phi }) \right) + \omega -zp \end{aligned}$$

(39)

The solution z of this optimization problem is

$$\begin{aligned} z = \frac{\sqrt{1-a} + \theta _{\pi } + + u_s (\phi \sqrt{a} + \theta _{\phi })-p }{\rho \sigma ^2 } \end{aligned}$$

(40)

The t = 1 equilibrium stock price is

$$\begin{aligned} p = \sqrt{1-a} + \theta _{\pi } + u_s \left( \phi \sqrt{a} + \theta _{\phi } \right) - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \end{aligned}$$

(41)

Substituting the stock price p from (19) in (17) gives the optimal portfolio allocation:

$$\begin{aligned} z = \frac{1-z_m^{\text {LT}}}{n} \end{aligned}$$

(42)

It immediately follows from (41) that the stock price maximizing resource allocation $a^{\star }$, as defined in (22), is unchanged. Moreover, the first-best optimal resource allocation maximizes

$${\mathbb {E}} \left[ U \left( \sqrt{1-a}+\tilde{\theta }_{\pi }+\tilde{\epsilon } + u_s (\phi \sqrt{a}+\tilde{\theta }_{\phi })-C \right) \right].$$

(43)

That is, it maximizes the certainty equivalent $\sqrt{1-a}+ u_s \phi \sqrt{a} - \frac{\rho }{2} \left[ \sigma ^2 + \sigma _{\pi }^2+\sigma _{\phi }^2 + 2 \varrho \right] -C$, which yields the same value of $a^{\text {FB}}$ as in Claim (1). With $a^{\star }$ and $a^{\text {FB}}$ both as in the baseline model, it immediately follows that Proposition 1 still holds in this setting.

A manager who exerts effort now chooses the resource allocation a to maximize

$${\mathbb {E}}[U(Y(\xi ,a))|e=1] \equiv {\mathbb {E}} \left[ U \left( w + z_m^{\text {ST}} p +z_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\theta }_{\pi } + \tilde{\epsilon } + u_s \left( \phi \sqrt{a}+ \tilde{\theta }_{\phi } \right) \right) - C \right) \right].$$

(44)

For $z_m^{\text {ST}}+z_m^{\text {LT}} >0$, the action $a_m$ that maximizes this expression is the same as in the baseline model (defined in (25)), and we have $a_m=a^{\text {FB}}$.

Using the certainty equivalent approach, the optimization problem of shareholders at t = 0 is

$$\begin{aligned}&\max _{w,z_m^{\text {ST}},z_m^{\text {LT}}} {\mathbb {E}} \left[ U \left( \frac{1-z_m^{\text {LT}}}{n} \left( \sqrt{1-a} +\tilde{\theta }_{\pi }+\tilde{\epsilon } + u_s \left( \phi \sqrt{a} + \tilde{\theta }_{\phi }\right) \right) \right. \right. \nonumber \\&\quad \left. \left. - \frac{1}{n} \left( w+z_m^{\text {ST}}p \right) \right) \right] \nonumber \\&\quad \Leftrightarrow \; \max _{w,z_m^{\text {ST}},z_m^{\text {LT}}} \frac{1-z_m^{\text {LT}}}{n} \left( \sqrt{1-a} + u_s \phi \sqrt{a} \right) - \frac{1}{n} \left( w+z_m^{\text {ST}}{\mathbb {E}}[p] \right) \nonumber \\&\quad - \frac{\rho }{2} \left[ \frac{\left( 1-z_m^{\text {LT}}\right) ^2}{n^2} \sigma ^2 + \frac{\left( 1-z_m^{\text {LT}}-z_m^{\text {ST}}\right) ^2}{n^2} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) \right. \nonumber \\&\quad \left. + \, 2\frac{\left( 1-z_m^{\text {LT}}\right) \left( 1-z_m^{\text {LT}}-z_m^{\text {ST}}\right) }{n^2} \varrho \right] \end{aligned}$$

(45)

given $a=a^{\text {FB}}$, subject to the following constraints at t = 0 (note that the stock price p, which is realized at t = 1, is a random variable at t = 0):

$$\begin{aligned} {\mathbb {E}}\left[ U\left( Y\left( \xi ,a^{\text {FB}}\right) \right) |e=1\right]\ge & {} {\mathbb {E}}[U(Y(\xi ,a))|e=0] \end{aligned}$$

(46)

$$\begin{aligned} {\mathbb {E}}\left[ U\left( Y\left( \xi ,a^{\text {FB}}\right) \right) |e=1\right]\ge & {} \bar{U}\end{aligned}.$$

(47)

Using the certainty equivalent approach, the incentive constraint (46) may be rewritten as

$$\begin{aligned}&w + z_m^{\text {ST}}\left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] +z_m^{\text {LT}} \left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) -C \nonumber \\&\quad -\frac{\rho }{2} \left[ {z_m^{\text {LT}}}^2 \sigma ^2 + \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) ^2 \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + 2 z_m^{\text {LT}} \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \varrho \right] \nonumber \\&\quad \ge w + z_m^{\text {ST}}\left[ - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] -\frac{\rho }{2} \left[ {z_m^{\text {LT}}}^2 \sigma ^2 + \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) ^2 \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) \right. \nonumber \\&\quad \left. + \, 2 z_m^{\text {LT}} \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \varrho \right]. \end{aligned}$$

(48)

Removing offsetting terms and rearranging yields

$$z_m^{\text {ST}}+z_m^{\text {LT}} \ge \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} },$$

(49)

which is the same inequality as in the baseline model (cf. (30)). Denoting by $\bar{W}$ the reservation wage which is implicitly defined by $U(\bar{W}) \equiv \bar{U}$, and given that the equity holdings are such that the manager exerts effort, the participation constraint (47) may be rewritten with the certainty equivalent approach as

$$w + z_m^{\text {ST}}\left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] +z_m^{\text {LT}} \left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) -C$$

$$-\frac{\rho }{2} \left[ {z_m^{\text {LT}}}^2 \sigma ^2 + \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) ^2 \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + 2 z_m^{\text {LT}} \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \varrho \right] \ge \bar{W}.$$

(50)

We denote by $\mu$ and $\lambda$ the Lagrange multipliers associated with the constraints (49) and (50), respectively.

The first-order conditions of the optimization problem in (45)–(47) with respect to w, $z_m^{\text {ST}}$, and $z_m^{\text {LT}}$ are then, respectively,

$$-1+\lambda =0$$

(51)

$$\begin{aligned}&-\left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2 \right] \nonumber \\&+ \rho \left[ \frac{1-z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) +\frac{\left( 1-z_m^{\text {LT}}\right) }{n} \varrho \right] \nonumber \\&+ \lambda \left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} - \frac{1-z_m^{\text {LT}}}{n}\rho \sigma ^2\nonumber \right. \nonumber \\&\left. - \rho \left( \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + z_m^{\text {LT}} \varrho \right) \right] + \mu = 0\end{aligned}$$

(52)

$$\begin{aligned}&-\left( \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} \right) - z_m^{\text {ST}}\frac{\rho \sigma ^2}{n} + \rho \left[ \frac{1-z_m^{\text {LT}} }{n} \sigma ^2\right. \nonumber \\&\left. + \frac{1-z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + \frac{2-2z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \varrho \right] \nonumber \\&+ \lambda \left[ \sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} + z_m^{\text {ST}}\frac{\rho \sigma ^2}{n} -\rho \left( z_m^{\text {LT}} \sigma ^2 + \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) \right. \right. \nonumber \\&\left. \left. + \left( 2 z_m^{\text {LT}}+z_m^{\text {ST}} \right) \varrho \right) \right] + \mu = 0 \end{aligned}$$

(53)

Equation (51) gives $\lambda =1$, which allows to rewrite (52) as

$$\begin{aligned}&\rho \left[ \frac{1-z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) +\frac{\left( 1-z_m^{\text {LT}}\right) }{n} \varrho \right] \nonumber \\&\quad - \rho \left( \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \left( \sigma _{\pi }^2 + \sigma _{\phi }^2 \right) + z_m^{\text {LT}} \varrho \right) + \mu = 0 \nonumber \\&\quad \Leftrightarrow \qquad \rho \left[ \frac{1-(n+1)\left( z_m^{\text {LT}}-z_m^{\text {ST}}\right) }{n} \left( \sigma _{\pi }^2 +u_s^2 \sigma _{\phi }^2 \right) + \frac{\left( 1-(n+1)z_m^{\text {LT}}\right) }{n} \varrho \right] + \mu = 0 \end{aligned}$$

(54)

Likewise, (53) can be rewritten as follows:

$$\begin{aligned}&\rho \left[ \frac{1-z_m^{\text {LT}} }{n} \sigma ^2 + \frac{1-z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + \frac{2-2z_m^{\text {LT}}-z_m^{\text {ST}}}{n} \varrho \right] \\&\quad -\rho \left[ z_m^{\text {LT}} \sigma ^2 + \left( z_m^{\text {LT}}+z_m^{\text {ST}} \right) \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) + \left( 2 z_m^{\text {LT}}+z_m^{\text {ST}} \right) \varrho \right] + \mu = 0\\&\quad \Leftrightarrow \; \rho \left[ \frac{1-(n+1)z_m^{\text {LT}} }{n} \sigma ^2 + \frac{1-(n+1)(z_m^{\text {LT}}-z_m^{\text {ST}})}{n} \left( \sigma _{\pi }^2 + u_s^2 \sigma _{\phi }^2 \right) \right. \\&\quad \left. + \frac{2-(n+1)(2z_m^{\text {LT}}+z_m^{\text {ST}})}{n} \varrho \right] + \mu = 0 \end{aligned}$$

Plugging $\mu$ from (54), this gives

$$\begin{aligned}&\rho \left[ \frac{1-(n+1)z_m^{\text {LT}}}{n} \sigma ^2 + \frac{1-(n+1)(z_m^{\text {LT}}+z_m^{\text {ST}})}{n} \varrho \right] = 0 \nonumber \\&\quad \Leftrightarrow \; \; z_m^{\text {LT}} = \frac{\frac{1}{n+1} \sigma ^2+ \left[ \frac{1}{n+1} -z_m^{\text {ST}} \right] \varrho }{\sigma ^2 + \varrho } \end{aligned}$$

(55)

Note that, if either $z_m^{\text {ST}}=0$ or $\varrho =0$, then $z_m^{\text {LT}} = \frac{1}{n+1}$.

There are then two cases to consider. First, if

$$\frac{1}{n+1} \ge \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} },$$

(56)

then the incentive constraint in (49) is nonbinding with $z_m^{\text {ST}}=0$ and the implied value of $z_m^{\text {LT}}$ in (55). Standard cost minimization arguments then show that the optimal contract is such that $z_m^{\text {ST}}=0$ and $z_m^{\text {LT}}=\frac{1}{n+1}$, as in Proposition 2. Second, if (56) does not hold, then the incentive constraint in (49) is binding, so using (49) and the value of $z_m^{\text {LT}}$ in (55), we have

$$z_m^{\text {ST}} = \left( \frac{C}{\sqrt{1-a^{\text {FB}}} + u_s \phi \sqrt{a^{\text {FB}}} } - \frac{1}{n+1} \right) \bigg / \left( 1-\frac{\varrho }{\sigma ^2+\varrho } \right) > 0.$$

(57)

That is, with $\varrho =0$ or when (56) holds, the optimal short-term and long-term equity holdings, $z_m^{\text {ST}}$ and $z_m^{\text {LT}}$, are still as in Proposition 2.

With $\varrho \ne 0$ and when (56) does not hold, the optimal values of $z_m^{\text {LT}}$ and $z_m^{\text {ST}}$, in equations (55) and (57), respectively, are not exactly as in Proposition 2. Intuitively, optimal risk sharing is different when the two shocks on firm profits realized at t = 1 and t = 2, $\tilde{\theta }_{\pi }$ and $\tilde{\epsilon }$, are correlated. Indeed, the manager is exposed to the t = 1 shock $\tilde{\theta }_{\pi }$ because of his short-term and long-term equity holdings $\left( z_m^{\text {ST}}\,{\text {and}}\,z_m^{\text {LT}}\right)$, whereas he is only exposed to the t = 2 shock $\tilde{\epsilon }$ because of his long-term equity holdings $\left( z_m^{\text {LT}}\right)$. In the absence of correlation between these two shocks ($\varrho =0$), the optimal risk-sharing rule is simply for the manager to bear a fraction $\frac{1}{n+1}$ of each shock, which is achieved with $z_m^{\text {LT}}=\frac{1}{n+1}$ and $z_m^{\text {ST}}=0$; if these equity holdings are insufficient to elicit effort, then short-term equity holdings will increase, because they do not increase the manager’s risk exposure as much as long-term equity holdings, but they provide just as much effort incentives. However, with $\varrho > 0$ (respectively $\varrho < 0$), this increase in short-term equity holdings will expose the manager to the risk common to $\tilde{\theta }_{\pi }$ and $\tilde{\epsilon }$ over and above (resp. below) the optimal risk-sharing rule. To decrease (resp. increase) the manager’s risk exposure to this “common risk” while maintaining adequate effort incentives, the less risky short-term equity holdings will be increased (resp. decreased) relative to the case with $\varrho =0$, while the more risky long-term equity holdings will be decreased (resp. increased).

Proof of Proposition 3

First, substituting the value of $a=a^{\text {FB}}$ (cf. Claim 1 and Proposition 2) in the stock price p from (4), firm value at t = 1 is

$$p = \sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2.$$

So

$$\begin{aligned} \frac{{\text{d}}p}{{\text{d}} \phi }= & {} \sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} \nonumber \\= & {} -\frac{1}{2} \frac{2u_s^2 \phi }{(1+u_s^2 \phi ^2)^2} \left( 1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 } \right) ^{-0.5} \nonumber \\&\quad + u_s \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + \frac{1}{2} u_s \phi \frac{2u_s^2 \phi }{(1+u_s^2 \phi ^2)^2} \left( \frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 } \right) ^{-0.5} \nonumber \\= & {} -\frac{1}{2} \frac{2u_s^2 \phi }{(1+u_s^2 \phi ^2)^2} \frac{1 }{\left( 1 + u_s^2 \phi ^2\right) ^{-0.5} } \nonumber \\&\quad + u_s \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + \frac{1}{2} \frac{2u_s^2 \phi }{(1+u_s^2 \phi ^2)^2} \frac{1 }{\left( 1 + u_s^2 \phi ^2 \right) ^{-0.5}} > 0. \end{aligned}$$

(58)

The first part of the Proposition is proven.

Second, denoting by ${\mathbb {E}}[\cdot ]$ the mathematical expectation operator, in equilibrium (with $a=a^{\text {FB}}$), the expected stock return ${\mathbb {E}}[r(a,p)]$ from t = 1 to t = 2 is

$${\mathbb {E}}[r(a^{\text {FB}},p)] \equiv {\mathbb {E}} \left[ \frac{\sqrt{1-a^{\text {FB}}} + \tilde{\epsilon } -p}{p} \right]$$

$$\begin{aligned}&= \frac{\sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} - \left( \sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \right) }{\sqrt{1-\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 }\nonumber \\&= \frac{ - u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }}+ \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 }{p}. \end{aligned}$$

(59)

It follows that

$$\begin{aligned}&\frac{{\text{d}}{\mathbb {E}}[r(a^{\text {FB}},p)]}{{\text{d}} \phi } = \frac{{\text{d}}}{{\text{d}} \phi } \left\{ \frac{ - u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }}+ \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 }{p} \right\} \\&\quad =\frac{ - \left( u_s \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }} + \frac{1}{2} u_s \phi \frac{2 u_s^2 \phi }{(1 + u_s^2 \phi ^2 )^2} \left( \frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 } \right) ^{0.5} \right) p - \left( - u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }}+ \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \right) \frac{{\text{d}}p}{{\text{d}} \phi } }{p^2} < 0 \end{aligned}$$

where the inequality follows from the assumption that the stock price p and the expected stock return are positive, i.e., $- u_s \phi \sqrt{\frac{u_s^2 \phi ^2 }{1 + u_s^2 \phi ^2 }}+ \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \ge 0$ (using (59)), and the fact that $\frac{{\text{d}}p}{\text{d} \phi } > 0$ (cf. (58)). The second part of the Proposition is proven.

Stock Returns over Time

In this section, we extend the model to let the firm produce and its stock price be established over multiple periods. To this end, we make a number of simplifying assumptions. The purpose is to establish the robustness of the results stated in Proposition 3.

Suppose that a firm with capacity for CSP $\phi$ lives for T periods, with $T \ge 2$. For simplicity, in every period the firm produces the same expected profits $\sqrt{1-a}$ and has the same CSP $\phi \sqrt{a}$. We consider a standard model of portfolio choice with overlapping generations of investors. Every period, a new generation of n shareholders is born who lives for two periods. As in the baseline model, each generation of investors values CSP at rate $u_s$. At the beginning of the first period of their lives, say period t, they invest at the risk-free rate and in the firm stock at price $p_t$. CSP is realized in every period, while profits are realized at the end of each period and fully paid off to investors at the end of each period. At the beginning of the second period of their lives, period $t+1$, this generation of investors sells firm stocks at price $p_{t+1}$, and the new generation of investors invests at this price. As in the baseline model, in any period t, the price $p_t$ adjusts so that there is adequate demand by “young” investors (“old” investors are forced sellers). Portfolio choices and price formation are thus as in the baseline model, except that, in every period apart from the last, investors now receive an additional payoff in the form of the resale value of their stocks, as valued at the next period stock price.

At the beginning of the last period (T), the stock price is as in the baseline model:

$$p_{T} = \sqrt{1-a} + u_s \phi \sqrt{a} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2.$$

(60)

In addition, for $t \le T-1$, we have

$$p_{t} = \sqrt{1-a} + u_s \phi \sqrt{a} + p_{t+1} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2.$$

(61)

Setting $t=T-1$ in (61) and substituting from (60),

$$p_{T-1} = \sqrt{1-a} + u_s \phi \sqrt{a} + p_{T} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 = 2 p_{T}.$$

(62)

Iterating, for any integer $\tau \le T-1$, we have $p_{T - \tau } = (\tau +1) p_{T}$. At any point in time, the stock price (or firm value) is strictly increasing in $\phi$, as in Proposition 3.

As in the proof of Proposition 3, we calculate the expected return in period $T-\tau$:

$${\mathbb {E}}[r(a,p_{T-\tau })] = \frac{\sqrt{1-a}+ p_{T-\tau +1} -p_{T-\tau }}{p_{T-\tau }} = \frac{ -u_s \phi \sqrt{a} + \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2}{(\tau +1) \left[ \sqrt{1-a} + u_s \phi \sqrt{a} - \frac{1-z_m^{\text {LT}}}{n} \rho \sigma ^2 \right] }.$$

(63)

Note that the expression for the (expected) return is standard: it is based on the (expected) profit paid off to investors as dividends during the period, and on the stock prices at times t and $t+1$ (or in this case $T-\tau$ and $T-\tau +1$). Assuming that stock prices are positive, the expected return in (63) is strictly decreasing in $\phi$, as in Proposition 3.

Portfolio Choice with Heterogeneous Preferences

This section follows the same lines as the “Portfolio choice” section, and is therefore abbreviated. For a shareholder with preferences $u_s^i$ for CSP, maximizing ${\mathbb {E}}[U(X(\bar{z}_i,a))]$ is equivalent to maximizing the following certainty equivalent with respect to $\bar{z}_i$:

$$CE(\bar{z}_i,a)\,=\,\bar{z}_i \sqrt{1-a} + \omega -\bar{z}_i \bar{p} + \bar{z}_i u_s^i \phi \sqrt{a} - \frac{\rho }{2} \bar{z}_i^2 \sigma ^2.$$

(64)

The solution to this optimization problem is given by the first-order condition, which after some rearranging yields

$$\bar{z}_i = \frac{\sqrt{1-a} + u_s^i \phi \sqrt{a}-\bar{p} }{\rho \sigma ^2 }.$$

(65)

Given this set of optimal demands from n ex-ante identical shareholders, the stock price is given by the market clearing equation which equates the supply $1-\bar{z}_m^{\text {LT}}$ of shares and the demand $\sum _{i=1}^n \bar{z}_i$ of shares:

$$\sum _{i=1}^n \frac{\sqrt{1-a} + u_s^i \phi \sqrt{a}-\bar{p} }{\rho \sigma ^2 } = 1-\bar{z}_m^{\text {LT}}.$$

(66)

Solving this equation for $\bar{p}$ gives the t = 1 equilibrium stock price:

$$\bar{p} = \sqrt{1-a} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{a} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2.$$

(67)

Substituting the stock price $\bar{p}$ from (67) in (65) gives the fraction of the firm held by shareholder i in equilibrium:

$$\bar{z}_i = \frac{\left( u_s^i -\frac{\sum _{h=1}^n u_s^h}{n}\right) \phi \sqrt{a}}{\rho \sigma ^2 } + \frac{1-\bar{z}_m^{\text {LT}}}{n}.$$

(68)

With heterogeneous preferences, a shareholder who values CSP more will hold a larger fraction of a socially responsible firm in equilibrium.

First-Best with Heterogeneous Preferences

We redefine the first-best, i.e., the outcome in the absence of agency problems, in the setting with heterogeneous shareholder preferences.

Denote by $z_i^{\text {FB}}$ the fraction of shares held by shareholder i at t = 1 at the first-best. It is the outcome of the optimal portfolio allocation when shareholders directly manage the firm, and is therefore given by (68) with $\bar{z}_m^{\text {LT}}=0$, where the value of a is taken as given at this stage:

$$z_i^{\text {FB}} = \frac{\left( u_s^i -\frac{\sum _{h=1}^n u_s^h}{n}\right) \phi \sqrt{a}}{\rho \sigma ^2 } + \frac{1}{n}.$$

(69)

With $\bar{z}_m^{\text {ST}}=\bar{z}_m^{\text {LT}}=0$, the stake of shareholder i at t = 0 is the same.

In accordance with the criterion proposed by Grossman and Hart (1979), the first-best resource allocation in which shareholders directly manage the firm, $\bar{a}^{\text {FB}}$, is the value of a that maximizes

$$\sum _{i=1}^n \left\{ z_i^{\text {FB}} \left( \sqrt{1-a} + u_s^i \phi \sqrt{a} \right) - \frac{\rho }{2} {z_i^{\text {FB}}}^2 \sigma ^2 \right\} -C.$$

(70)

Denoting $\sigma _{u_s}^2 \equiv \sum _{i=1}^n \frac{1}{n} \left( u_s^i -\frac{\sum _{h=1}^n u_s^h}{n} \right) ^2$, the first-order condition with respect to a is

$$-\frac{1}{2} \frac{1}{\sqrt{1-a}} + \frac{n \sigma _{u_s}^2 \phi ^2}{\rho \sigma ^2} + \frac{1}{2} \frac{\sum _{i=1}^n u_s^i}{n} \phi \frac{1}{\sqrt{a}} - \frac{n \sigma _{u_s}^2 \phi ^2}{2\rho \sigma ^2} = 0.$$

(71)

As the expression in (70) is concave in a, the optimum $\bar{a}^{\text {FB}}$ is given by the first-order condition in (71), which after some rearranging gives

$$\bar{a}^{\text {FB}} = \frac{\left( \frac{\sum _{i=1}^n u_s^i}{n} \right) ^2 \phi ^2}{1+\left( \frac{\sum _{i=1}^n u_s^i}{n} \right) ^2 \phi ^2}.$$

(72)

Likewise, the stock price $\bar{p}$ in (67) is concave in a, so that the value of a that maximizes $\bar{p}$ is given by the first-order condition. Simple calculations show that the value of a that maximizes the stock price is equal to $\bar{a}^{\text {FB}}$.

Proof of Proposition 4

This proof follows the same lines as the proof of Proposition 2 and is therefore abbreviated. With a contract $\xi =\left\{ \bar{w},\bar{z}_m^{\text {ST}},\bar{z}_m^{\text {LT}}\right\}$, the argument in the utility function of a manager who exerts effort is

$$Y(\xi ,a)= \bar{w} + \bar{z}_m^{\text {ST}} \bar{p} +\bar{z}_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\epsilon } + u_s^j \phi \sqrt{a} \right) - C.$$

(73)

Substituting the stock price $\bar{p}$ from (13) gives

$$\begin{aligned} Y(\xi ,a)= & {} \bar{w} + \bar{z}_m^{\text {ST}}\left[ \sqrt{1-a} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{a} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&\quad +\bar{z}_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\epsilon } + u_s^j \phi \sqrt{a} \right) - C. \end{aligned}$$

(74)

The optimization problem of a manager who exerts effort is to choose a to maximize ${\mathbb {E}}[U(Y(\xi ,a))|e=1]$. Given that the problem is concave in a for $\bar{z}_m^{\text {ST}} \ge 0$ and $\bar{z}_m^{\text {LT}}>0$, the optimal value $\bar{a}_{m}$ of a optimally chosen by the manager is described by the first-order condition of the manager’s expected utility with respect to a, which after some rearranging yields $\bar{a}_m = \bar{a}^{\text {FB}}$ (here it is crucial that $u_s^j =\frac{1}{n}\sum _{i=1}^n u_s^i$).

We now derive the optimal values of $\bar{w}$, $\bar{z}_m^{\text {ST}}$, and $\bar{z}_m^{\text {LT}}$ such that the manager accepts the contract and exerts effort. Using the certainty equivalent approach and the Grossman and Hart (1979) criterion, the optimization problem of shareholders at t = 0 is⁹

$$\begin{aligned}&\max _{\bar{w},\bar{z}_m^{\text {ST}},\bar{z}_m^{\text {LT}}} \sum _{i=1}^n \left[ \bar{z}_i \left( \sqrt{1-a} + u_s^i \phi \sqrt{a} \right) -\left( \bar{z}_i + \frac{\bar{z}_m^{\text {LT}}}{n} \right) \left( \bar{w}+\bar{z}_m^{\text {ST}} \bar{p}\right) - \frac{\rho }{2} \bar{z}_i^2 \sigma ^2 \right] \end{aligned}$$

(75)

$$\begin{aligned}&\quad = \left( 1-\bar{z}_m^{\text {LT}}\right) \left( \sqrt{1-a} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{a} \right) +\frac{n\sigma _{u_s}^2 \phi ^2 a}{\rho \sigma ^2} - \left( \bar{w}+\bar{z}_m^{\text {ST}} \bar{p} \right) \nonumber \\&\qquad - \frac{\rho }{2} \left( \frac{n\sigma _{u_s}^2 \phi ^2 a}{\rho ^2 \sigma ^4} +\frac{\left( 1-\bar{z}_m^{\text {LT}}\right) ^2}{n} \right) \sigma ^2 \end{aligned}$$

(76)

given $a=\bar{a}^{\text {FB}}$, where we denoted $\sigma _{u_s}^2 \equiv \sum _{i=1}^n \frac{1}{n} \left( u_s^i -\frac{\sum _{h=1}^n u_s^h}{n} \right) ^2$, and we used (68) and

$\sum _{i=1}^n \left( u_s^i -\frac{\sum _{h=1}^n u_s^h}{n} \right) =0$. The objective function in (76) is maximized subject to the following constraints:

$$\begin{aligned}{\mathbb {E}}[U(Y(\xi ,\bar{a}^{\text {FB}}))|e=1]\ge & {} {\mathbb {E}}[U(Y(\xi ,a))|e=0] \end{aligned}$$

(77)

$$\begin{aligned}{\mathbb {E}}[U(Y(\xi ,\bar{a}^{\text {FB}}))|e=1]\ge & {} \bar{U}, \end{aligned}$$

(78)

where $Y(\xi ,a)$ conditional on e = 0 is given in (74), and $Y(\xi ,a)$ conditional on e = 0 is $\bar{w}+ \bar{z}_m^{\text {ST}}\left[ - \frac{1-\bar{z}_m^{\text {LT}}}{n}\rho \sigma ^2 \right] +\bar{z}_m^{\text {LT}} \tilde{\epsilon }$. As before, there is a continuum of contracts that achieve incentive compatibility; as in Edmans et al. (2009), we choose the maximum between the level of $\bar{z}_m^{\text {ST}}$ which satisfies (77) as an equality and zero (so that $\bar{z}_m^{\text {ST}} \ge 0$).

Using the certainty equivalent approach, the incentive constraint (77) may be rewritten as

$$\begin{aligned}&\bar{w} + \bar{z}_m^{\text {ST}}\left[ \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&\quad +\bar{z}_m^{\text {LT}} \left( \sqrt{1-\bar{a}^{\text {FB}}} + u_s^j \phi \sqrt{\bar{a}^{\text {FB}}} \right) -C -\frac{\rho }{2} \left( {\bar{z}_m}^{\text {LT}}\right) ^2 \sigma ^2 \end{aligned}$$

$$\ge \bar{w} + \bar{z}_m^{\text {ST}}\left[ - \frac{1-\bar{z}_m^{\text {LT}}}{n}\rho \sigma ^2 \right] -\frac{\rho }{2} \left( {\bar{z}_m^{\text {LT}}}\right) ^2 \sigma ^2.$$

(79)

Removing offsetting terms, using $u_s^j = \frac{\sum _{i=1}^n u_s^i}{n}$, and rearranging yields

$$\left( \bar{z}_m^{\text {ST}} +\bar{z}_m^{\text {LT}} \right) \left( \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} \right) \ge C.$$

(80)

Denoting by $\bar{W}$ the reservation wage which is implicitly defined by $U(\bar{W}) \equiv \bar{U}$, and given that the equity holdings are such that the manager exerts effort, the participation constraint (78) may be rewritten with the certainty equivalent approach as

$$\begin{aligned}&\bar{w} + \bar{z}_m^{\text {ST}}\left[ \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&\quad +\bar{z}_m^{\text {LT}} \left( \sqrt{1-\bar{a}^{\text {FB}}} + u_s^j \phi \sqrt{\bar{a}^{\text {FB}}} \right) -C -\frac{\rho }{2} (\bar{z}_m^{\text {LT}})^2 \sigma ^2 \ge \bar{W}. \end{aligned}$$

(81)

We denote by $\mu$ and $\lambda$ the Lagrange multipliers associated with the constraints (80) and (81), respectively.

The first-order conditions of the optimization problem in (76)–(78) with respect to $\bar{w}$, $\bar{z}_m^{\text {ST}}$, and $\bar{z}_m^{\text {LT}}$ are then, respectively,

$$-1+\lambda =0$$

(82)

$$\begin{aligned}&-\left[ \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&+ \lambda \left[ \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] + \mu = 0 \end{aligned}$$

(83)

$$\begin{aligned}&-\left( \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} \right) - \bar{z}_m^{\text {ST}}\frac{\rho \sigma ^2}{n} + \rho \frac{1-\bar{z}_m^{\text {LT}} }{n} \sigma ^2\nonumber \\&+ \lambda \left( \sqrt{1-\bar{a}^{\text {FB}}} + u_s^j \phi \sqrt{\bar{a}^{\text {FB}}} + \bar{z}_m^{\text {ST}}\frac{\rho \sigma ^2}{n} -\rho \bar{z}_m^{\text {LT}} \sigma ^2 \right) + \mu = 0. \end{aligned}$$

(84)

Equation (82) gives $\lambda =1$, which used in (83) implies $\mu =0$. With $\lambda =1$ and $\mu =0$, the first-order condition (84) can be rewritten as

$$\bar{z}_m^{\text {LT}} = \frac{1}{1+n}.$$

(85)

Plugging in (80) and equating both sides, and using $\bar{z}_m^{\text {ST}}\ge 0$ gives

$$\bar{z}_m^{\text {ST}} = \max \left\{ \frac{C}{\sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}}} - \frac{1}{1+n} , 0\right\}.$$

(86)

Finally, substituting for $\bar{z}_m^{\text {ST}}$ and $\bar{z}_m^{\text {LT}}$ in (81) gives the fixed wage, which satisfies the participation constraint as an equality ($\lambda =1$ implies that the participation constraint is binding due to the complementary slackness condition):

$$\begin{aligned} \bar{w}= & {} \bar{W} - \max \left\{ \frac{C}{\sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} } - \frac{1}{1+n} , 0\right\} \nonumber \\&\qquad \left[ \sqrt{1-\bar{a}^{\text {FB}}} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{\bar{a}^{\text {FB}}} - \frac{1-\frac{1}{1+n}}{n}\rho \sigma ^2 \right] \nonumber \\&\quad -\frac{1}{1+n} \left( \sqrt{1-\bar{a}^{\text {FB}}} + u_s^j \phi \sqrt{\bar{a}^{\text {FB}}} \right) +C +\frac{\rho }{2} \frac{1}{(1+n)^2} \sigma ^2 \end{aligned}.$$

(87)

Proof of Proposition 5

Let the manager have $u_s^j$ such that $u_s^j \ne \frac{1}{n}\sum _{i=1}^n u_s^i$. We will show that this results in inefficiencies relative to the case studied in Proposition 4.

First, suppose that $\bar{z}_m^{\text {LT}}=0$. Then risk sharing is not socially optimal, an inefficiency.

Second, suppose that $\bar{z}_m^{\text {LT}}>0$. Consider the case $u_s^j > \frac{1}{n}\sum _{i=1}^n u_s^i$ $\left( {\text {respectively}}\; u_s^j < \frac{1}{n}\sum _{i=1}^n u_s^i \right)$. Then the first-best optimal resource allocation coincides with the allocation that maximizes the stock price, but a manager with long-term equity holdings $\left( \bar{z}_m^{\text {LT}}>0\right)$ will optimally choose a resource allocation strictly higher (resp. lower) than at the first-best, given that the argument in the utility function of a manager who exerts effort is

$$\begin{aligned} Y(\xi ,a)= & {} \bar{w} + \bar{z}_m^{\text {ST}} \left[ \sqrt{1-a} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{a} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&\quad +\bar{z}_m^{\text {LT}} \left( \sqrt{1-a} + \tilde{\epsilon } + u_s^j \phi \sqrt{a} \right) - C \end{aligned}.$$

(88)

The certainty equivalent of ${\mathbb {E}}[U(Y(\xi ,a))|e=1]$ is therefore

$$\begin{aligned} {\text {CE}}(\xi ,a)= & {} \bar{w} + \bar{z}_m^{\text {ST}} \left[ \sqrt{1-a} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \sqrt{a} - \frac{1-\bar{z}_m^{\text {LT}}}{n} \rho \sigma ^2 \right] \nonumber \\&\quad +\bar{z}_m^{\text {LT}} \left( \sqrt{1-a} + u_s^j \phi \sqrt{a} \right) - C - \frac{\rho }{2} \bar{z}{_m^{\text {LT}}}^2 \sigma ^2 \end{aligned}.$$

(89)

The certainty equivalent is concave in a, so that the value of a optimally chosen by the manager is given by the following first-order condition:

$$\bar{z}_m^{\text {ST}} \left[ -\frac{1}{2} (1-a)^{-1/2} + \frac{\sum _{i=1}^n u_s^i}{n} \phi \frac{1}{2} a^{-1/2} \right] +\bar{z}_m^{\text {LT}} \left( -\frac{1}{2} (1-a)^{-1/2} + u_s^j \phi \frac{1}{2} a^{-1/2} \right) = 0$$

(90)

$$\Leftrightarrow \frac{(1-a)^{-1/2} }{a^{-1/2} } = \phi \frac{ \bar{z}_m^{\text {ST}} \frac{\sum _{i=1}^n u_s^i}{n} +\bar{z}_m^{\text {LT}} u_s^j }{ \bar{z}_m^{\text {ST}} + \bar{z}_m^{\text {LT}} }$$

(91)

$$\begin{aligned} \Leftrightarrow a = \frac{ \phi ^2\left( \bar{z}_m^{\text {ST}} \frac{\sum _{i=1}^n u_s^i}{n} +\bar{z}_m^{\text {LT}} u_s^j \right) ^2 }{(\bar{z}_m^{\text {ST}} + \bar{z}_m^{\text {LT}})^2 + \phi ^2\left( \bar{z}_m^{\text {ST}} \frac{\sum _{i=1}^n u_s^i}{n} +\bar{z}_m^{\text {LT}} u_s^j \right) ^2 } \end{aligned}.$$

(92)

With $\bar{z}_m^{\text {LT}}>0$ and $u_s^j \ne \frac{1}{n}\sum _{i=1}^n u_s^i$, the value of a as derived in (92) is different from the first-best resource allocation $\bar{a}^{\text {FB}}$ derived in (72), an inefficiency.¹⁰