Managerial ownership with rent-seeking employees

Abstract

In some cases, the incentives of the manager will affect the behavior of the firm’s employees. A manager with low-powered incentives will discourage employees from engaging in destructive rent-seeking activities. Union members will need to cooperate with this poorly compensated manager if the firm will have any chance to succeed. The elimination of rent-seeking costs can increase the value of owners’ stakes in the firm. Thus, value can be maximized by giving control to a CEO with an ownership stake strictly less than 100 percent.

Appendix

1.1 Proof of Lemma 1

The Lemma says that a satisfied $I\!R_{M}$ constraint implies that the $I\!R_{O}$ constraint is also satisfied. The right hand side (RHS) of both constraints are identical; therefore, the left hand side (LHS) of $I\!R_{O}$ must be greater than or equal to the (LHS) of $I\!R_{M}$. If we subtract the two constraints, $I\!R_{O}$ – $\textit{IR}_{M}\ge 0$.

To prove that $\textit{IR}_{O}$ – $I\!R_{M}\ge 0$ is the case, it is sufficient to show that the complement $I\!R_{O}$ – $I\!R_{M} < 0$ is impossible. The following is a proof by contradiction:

$$\begin{aligned} I\!R_O -I\!R_M&= E\{R\}-w-c(q)-\alpha (E\{R\}-w-u)-u+c(q)<U_M -U_M\nonumber \\ I\!R_O -I\!R_M&= (1-\alpha )(E\{R\}-w)-(1-\alpha )u<0 \end{aligned}$$

(13)

The LHS above will be smallest when $u$, the manager’s base wage is the largest. The maximum $u=E\{R\}$ – $w$ from Sect. 2. If we substitute this into (13) above, we get the contradiction, $0 < 0$. This is what we wanted to show.$\square$

1.2 Deriving the upper and lower bounds of $\underline{\alpha }$ in Eq. (12)

Rearranging Eq. (11) we get the following expression for the optimal level of share ownership under the min union strategy.

$$\begin{aligned} \underline{\alpha }=\frac{U_M -\underline{u}}{{\overline{e}}R-{\overline{c}}-\underline{u}} \end{aligned}$$

(14)

If we rearrange Eq. (1), we know that ${\overline{e}}R-{\overline{c}}>U_M$. If $U_{M}$ approaches but does not reach its upper bound, ${\overline{e}}R-{\overline{c}},$then $\underline{\alpha }$ approaches 1. This allows us to conclude that the upper bound of $\underline{\alpha }$ is less than unity. That is, $\mathop {\lim }\nolimits _{U_M \rightarrow ({\overline{e}}R-{\overline{c}})^{-}} \underline{\alpha }\rightarrow \left( {\frac{({\overline{e}}R-{\overline{c}})^{-}-\underline{u}}{{\overline{e}}R-{\overline{c}}-\underline{u}}} \right) =1^{-}<1.$

We know that $\underline{u}$ in Eq. (10) is non-negative and is strictly less than $U_{M}$. If either $U_{M}$ approaches zero, or $\underline{u}$ approaches, but does not reach $U_{M}$, then $\underline{\alpha }$ approaches but does not reach zero. That is, either $\mathop {\lim }\nolimits _{U_M =0^{+}} \underline{\alpha }\rightarrow 0^{+}>0,$ or $\mathop {\lim }\nolimits _{\underline{u}\rightarrow (U_M )^{-}} \underline{\alpha }\rightarrow \left( {\frac{0^{+}}{{\overline{e}}R-{\overline{c}}-\underline{u}}} \right) =0^{+}>0.$ This allows us to conclude that $\underline{\alpha }$ is greater than 0.

This is the justification for the upper and lower bounds in Eq. (12) being strictly less than unity and strictly greater than zero, respectively. $\square$

1.3 Proof of Theorem 4

The compensation of the manager consists of expected share compensation and wages. Suppose there are two identical firms where each manager faces the same total monitoring costs of $\underline{C}$. In the firm denoted by the superscript “A”, where the $I\!C_{M}$ binds and $I\!R_{M}$ is slack, expected share compensation is denoted $S^{A }$ and the CEO’s fixed wages are $u^{A}$. (The superscript “A” denotes a firm in which controlling managerial agency costs are the most important concern.) That is,

$$\begin{aligned} I\!C_M^A :&S^{A}-\underline{C}=0\end{aligned}$$

(15)

$$\begin{aligned} I\!R_M^A :&S^{A}-\underline{C}>U_M -u^{A} \end{aligned}$$

(16)

Total compensation for the manager is defined as

$$\begin{aligned} T^{A}\equiv S^{A}+u^{A}. \end{aligned}$$

(17)

Further, substituting the right hand side (RHS) of Eq. (15) into the left hand side (LHS) of Eq. (16) and rearranging, it becomes clear that the manager’s wage must exceed her opportunity cost. That is,

$$\begin{aligned} u^{A}>U_M. \end{aligned}$$

(18)

In contrast, in the firm where $I\!R_{M}$ binds and $I\!C_{M}$ is slack, we will denote the share and fixed wage components of compensation by the superscript “$D$.” This is consistent with the “delegation” solution given by Eq. (12). The managerial constraints are the following:

$$\begin{aligned} I\!C_M^D :&S^{D}-\underline{C}>0\end{aligned}$$

(19)

$$\begin{aligned} I\!R_M^D :&S^{D}-\underline{C}=U_M -u^{D} \end{aligned}$$

(20)

Total “delegation” compensation for the manager is defined as

$$\begin{aligned} T^{D}\equiv S^{D}+u^{D}. \end{aligned}$$

(21)

Further, substituting the right hand side (RHS) of Eq. (20) into the left hand side (LHS) of Eq. (19) and rearranging, it must be the case that

$$\begin{aligned} U_M >u^{D}. \end{aligned}$$

(22)

Subtracting the binding constraint in Eq. (20) from the binding constraint in Eq. (15), we get the following:

$$\begin{aligned} S^{A}-S^{D}=u^{D}-U_M \end{aligned}$$

Rearranging this relationship, we derive the following relationship:

$$\begin{aligned} S^{A}+U_M =S^{D}+u^{D} \end{aligned}$$

(23)

Equation (23) allows us to compare the total compensation when the $I\!C_{M}$ binds given in Eq. (17) and the total compensation when the $I\!R_{M}$ binds given by Eq. (21). Since $u^{A} > U^{M}$ according to Eq. (22), then the LHS of (23) is less than total compensation in Eq. (17). In short,

$$\begin{aligned} T^{A}\equiv S^{A}+u^{A}>S^{A}+U_M =S^{D}+u^{D}\equiv T^{D}. \end{aligned}$$

(24)

This is what we wanted to show. For identical monitoring costs and identical opportunity costs for each manager, the total compensation, $T^{A}$, in a firm where the manager is compensated with a binding $I\!C_{M}$ constraint and a slack $I\!R_{M}$ constraint exceeds the total compensation, $T^{D}$, of a manager whose $I\!R_{M}$ constraint binds and the $I\!C_{M}$ constraint is slack. In short, Eq. (24) shows that $T^{A} > T^{D}$. $\square$