Generalization of strategic delegation

Abstract

This study revisits the strategic delegation game in a duopoly setting by generalizing the managerial incentives. Prior studies considered only the case when firms’ managers are incentivized by a linear combination of profits and a specific objective such as revenue or market share. By extending managerial incentives to a linear combination of profits and a quadratic function of firm outputs, we aim to determine the type of managerial incentives that can achieve the highest profit. We show the following results. First, in any managerial incentive structure, the equilibrium profit is equal to or greater than that in the revenue-oriented case. Second, when the coefficient of the squared term is equal to the coefficient of the product of both firms’ outputs, the equilibrium profit is equal to that in the revenue-oriented case. Third, the firm achieves the highest profit with a managerial incentive consisting of a linear combination of profits and functions that increase with the product of both firms’ outputs and decrease with its output but that do not depend on the squared term of its output. Fourth, the equilibrium profit is strictly less than that in the no delegation case.

Appendix

1.1 The first- and second-order partial derivatives of the equilibrium output

From (2) and (3), we obtain the first-order partial derivatives of the equilibrium output with respect to $(\beta _{i},\beta _{j})$ as follows:

$$\begin{aligned} \frac{\partial q_{i}}{\partial \beta _{i}}&= -\frac{2[3D+(2A+C)(1-c)](\beta _{j}-A)(\beta _{j}-2A+C)}{[4(\beta _{1}-A)(\beta _{2}-A)-(\beta _{1}-C)(\beta _{2}-C)]^{2}},\end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial q_{j}}{\partial \beta _{i}}&= \frac{[3D+(2A+C)(1-c)](\beta _{j}-C)(\beta _{j}-2A+C)}{[4(\beta _{1}-A)(\beta _{2}-A)-(\beta _{1}-C)(\beta _{2}-C)]^{2}}. \end{aligned}$$

(18)

The second-order partial derivatives are

$$\begin{aligned} \frac{\partial ^{2}q_{i}}{\partial \beta _{i}^{2}}&= \frac{4[3D+(2A+C)(1-c)](\beta _{j}-A)(\beta _{j}-2A+C)(3\beta _{j}-4A+C)}{[4(\beta _{1}-A)(\beta _{2}-A)-(\beta _{1}-C)(\beta _{2}-C)]^{3}}, \end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial ^{2} q_{j}}{\partial \beta _{i}^{2}}&=-\frac{2[3D+(2A+C)(1-c)](\beta _{j}-C)(\beta _{j}-2A+C)(3\beta _{j}-4A+C)}{[4(\beta _{1}-A)(\beta _{2}-A)-(\beta _{1}-C)(\beta _{2}-C)]^{3}}. \end{aligned}$$

(20)

1.2 The first- and second-order conditions for owner i’s profit maximization

Arranging (4), which is the first-order condition for owner i, we obtain the optimal managerial incentive weight $\beta _{i}$ as follows:

$$\begin{aligned} \beta _{i} = \textstyle { \frac{(2A-C) [ D(6A-C)+2A(2A+C)(1-c) ] - [ D(16A-7C)+(12A^{2}+2AC-3C^{2})(1-c) ] \beta _{j} + [ 4D+(4A+C)(1-c) ] \beta _{j}^{2}}{(\beta _{j}-C) [ (1-c)\beta _{j}-2(1-c)A-D ] }. } \end{aligned}$$

(21)

The second-order condition for profit maximization is

$$\begin{aligned} \frac{\partial ^{2}\pi _{i}}{\partial \beta _{i}^{2}}&= (1-2q_{i}-q_{j}-c)\frac{\partial ^{2}q_{i}}{\partial \beta _{i}^{2}} -q_{i}\frac{\partial ^{2}q_{j}}{\partial \beta _{i}^{2}} -2\left( \frac{\partial q_{i}}{\partial \beta _{i}} +\frac{\partial q_{j}}{\partial \beta _{i}} \right)\frac{\partial q_{i}}{\partial \beta _{i}}<0. \end{aligned}$$

(22)

Under $\beta _{i}=\beta _{j}\equiv \beta$, we arrange (22) as follows:¹³

$$\begin{aligned} -\frac{2(\beta -2A+C)^{2}[3D+(2A+C)(1-c)][HD-I(1-c)]}{[4(\beta -A)^{2}-(\beta -C)^{2}]^{4}}<0. \end{aligned}$$

(23)

The second-order condition is satisfied if $\bigl [ 3D+(2A+C)(1-c) \bigr ] \bigl [ HD-I(1-c) \bigr ] >0$.¹⁴ Because $HD-I(1-c)>0$ is satisfied, the sufficient condition for the second-order condition is $3D+(2A+C)(1-c)>0$.¹⁵