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.
Notes
For a comprehensive survey of managerial delegation, please refer to Lambertini (2017).
When the firm’s marginal costs are constant, a linear combination of profits and revenues can be arranged as profits and outputs.
Murphy (1999) provides a comprehensive survey of executive compensation.
Although in a slightly different context from non-cooperative delegation, Pal (2010) shows that if managers must choose R&D in addition to output levels, owners cannot achieve the fully collusive outcome, even under cooperative managerial delegation. In contrast, Hamada (2020) shows that if owners cooperatively offer managers an incentive scheme consisting of a linear combination of profit, revenue, and cost, they can recover the fully collusive outcome.
When \(\alpha _{i}=1\), \(O_{i}=\pi _{i}\) corresponds to the \(\widehat{O}_{i}\) evaluated by the limit at which \(\beta _{i}\) approaches positive infinity.
Because market share \(q_{i}/Q\) is not a quadratic function of the output, we consider a second-order Taylor approximation of market share around equilibrium outputs \((q_{1}^{*},q_{2}^{*})\). We define \(f_{i}=q_{i}/Q\equiv g(q_{1},q_{2})\) and the first- and second-order partial derivatives by \(g_{i}\equiv \partial g/\!\partial q_{i}\) and \(g_{ij}\equiv \partial ^{2} g/\!\partial q_{i}\partial q_{j}\), respectively. The second-order Taylor approximation of market share around \((q_{1}^{*},q_{2}^{*})\) is \(g(q_{1},q_{2}) =g(q_{1}^{*},q_{2}^{*}) +g_{1}(q_{1}-q_{1}^{*}) +g_{2}(q_{2}-q_{2}^{*}) +\frac{1}{2} \bigl [ g_{11}(q_{1}-q_{1}^{*})^{2} +2g_{12}(q_{1}-q_{1}^{*})(q_{2}-q_{2}^{*}) +g_{22}(q_{2}-q_{2}^{*})^{2} \bigr ]\), where we evaluate the partial derivatives at \((q_{1}^{*},q_{2}^{*})\). By arranging this equation, we obtain \(g(q_{1},q_{2}) =\frac{1}{2}g_{11}q_{1}^{2} +\frac{1}{2}g_{22}q_{2}^{2} +g_{12}q_{1}q_{2} +H_{1}q_{1} +H_{2}q_{2} +X(q_{1}^{*},q_{2}^{*})\), where \(H_{i}\equiv g_{i}-g_{ii}q_{i}^{*}-g_{ij}q_{j}^{*}\), and \(X(q_{1}^{*},q_{2}^{*})\) is a constant remainder term.
The case of \(A=C=D=0\) corresponds to the no delegation case, that is, the usual profit maximization case.
Note that (6) holds even when \(A=C=D=0\).
Vickers (1985) and Fershtman and Judd (1987) focus on strategic delegation when the manager’s objective is a linear combination of profit and output or of profit and revenue, respectively. Indeed, since Lambertini and Trombetta (2002), it is well-known that the two approaches are formally equivalent and have the same equilibrium result.
Alternatively, following Fershtman and Judd’s (1987) original method, we can directly derive the equilibrium profit when managerial rewards are revenue-oriented. Because manager i’s objective is \(\widehat{O}_{i}=\beta _{i}\pi _{i}+pq_{i}\), we obtain the equilibrium output from the first-order condition for the maximization as follows: \(q_{i}(\beta _{i},\beta _{j}) = (\beta _{i}\beta _{j}+2\beta _{i}-\beta _{j})(1-c)\!/3(\beta _{i}+1)(\beta _{j}+1)\). Substituting \(q_{i}(\beta _{i},\beta _{j})\) into \(\pi _{i}\) and maximizing \(\pi _{i}\) with respect to \(\beta _{i}\), we obtain \(\beta _{i} =-6\). From \(q_{i}(\beta _{i},\beta _{j})\), we immediately obtain \(q_{i}=2(1-c)\!/5\) and \(\pi ^{*}_{R}=2(1-c)^{2}\!/25\).
From the perspective of empirical research, whether CSR and profitability are positively correlated is unresolved. Orlitzky et al. (2003) find a positive correlation between CSR and profitability in their meta-analysis of existing empirical studies. In contrast, Servaes and Tamayo (2013) show that CSR activities can add value to the firm, but only under certain conditions. CSR and firm value are positively related for firms with high customer awareness, whereas the relation is either negative or insignificant for firms with low customer awareness.
However, we should note that in strategic managerial delegation, consumer-oriented managerial incentives function as a commitment device to mitigate market competition. Thus, consumer surplus decreases, contrary to managers’ intentions. We are grateful to an anonymous reviewer for pointing out this result.
In (23), we define \(H\equiv 21\beta ^{2}-14(4A-C)\beta +36A^{2}-16AC+C^{2}\) and \(I\equiv 3\beta ^{3}-10(2A+C)\beta ^{2}+(40A^{2}+16AC-5C^{2})\beta -4A(2A+C)(3A-C)\).
Throughout our study, we check the condition only for local maximization, that is, the second-order condition in the equilibrium value \(\beta ^{*}\), but not the condition for global maximization. Thus, \(\pi (\beta )\) might not satisfy strict concavity when \(\beta\) is sufficiently far from \(\beta ^{*}\). Nevertheless, because the manager’s objective function \(\widehat{O}_{i}\) is at most quadratic on \(q_{i}\), we do not need to consider any cases for which \(\beta ^{*}\) is locally, but not globally, optimal.
We can confirm that \(HD-I(1-c)>0\) is satisfied in all the following classified cases, although we omit the detailed derivation because it requires a long calculation process. In all cases classified in Table 2, the second-order condition of the owners is satisfied.
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This work was supported by JSPS KAKENHI [grant numbers 16H03612, 16K03615, 20K01629].
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I am very grateful to the Editor and two anonymous reviewers for their helpful suggestions. This work was supported by JSPS KAKENHI [Grant Nos. 16H03612, 16K03615, 20K01629]. The author is solely responsible for any errors.
Appendix
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:
The second-order partial derivatives are
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:
The second-order condition for profit maximization is
Under \(\beta _{i}=\beta _{j}\equiv \beta\), we arrange (22) as follows:Footnote 13
The second-order condition is satisfied if \(\bigl [ 3D+(2A+C)(1-c) \bigr ] \bigl [ HD-I(1-c) \bigr ] >0\).Footnote 14 Because \(HD-I(1-c)>0\) is satisfied, the sufficient condition for the second-order condition is \(3D+(2A+C)(1-c)>0\).Footnote 15
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Hamada, K. Generalization of strategic delegation. JER 74, 199–214 (2023). https://doi.org/10.1007/s42973-020-00070-8
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DOI: https://doi.org/10.1007/s42973-020-00070-8