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Strategic delegation under cost asymmetry revised
Operations Research Letters 47 (2019) 527–529
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Strategic delegation under cost asymmetry revised Stefano Colombo Largo A. Gemelli 1, I-20123, Università Cattolica del Sacro Cuore, Milano, Italy
article
info
Article history: Received 16 May 2019 Received in revised form 16 September 2019 Accepted 16 September 2019 Available online 21 September 2019 JEL classification: D43
a b s t r a c t We revisit the Cournot duopoly game with strategic delegation and asymmetric costs of Delbono et al. (2016). In particular, the authors claim that a Prisoner Dilemma always arises. However, we show that, by setting properly the admissible parameter set, if the firms are sufficiently different, the efficient firm is better off when both firms delegate production. Therefore, in contrast with the traditional view, we claim that a Prisoner Dilemma is not an inevitable outcome in a strategic delegation game. © 2019 Elsevier B.V. All rights reserved.
Keywords: Strategic delegation Cost asymmetry Cournot competition
1. Introduction The owners of firms commonly delegate production decisions to managers, whose objective function is different from that of the owners. Starting from the seminal contributions of [2,5] and [4], literature has shown that, in the case of Cournot duopolistic competition, each owner has the incentive to delegate production to managers who are more interested in revenue maximization than in profits maximization. In this way, each owner commits to be more aggressive in the production stage. However, the final outcome is detrimental for owners: as each owner wants to be more aggressive, the unique equilibrium entails both owners delegating to managers, but the resulting profits are lower than in the case of no delegation, because of the fiercer competition induced by delegation. Therefore, the delegation game is typically associated with a Prisoner Dilemma. Recently, [3] questioned this result, by showing that, in the case of asymmetric production costs, also asymmetric delegation equilibria are possible and Prisoner Dilemma can be avoided. However, [1] show that the conclusions in [3] heavily depend on the fact that the owners are not allowed to freely chose the delegation parameter. After removing the binary choice restriction in [3], [1] claim that the unique equilibrium is characterized by delegation by both owners. Moreover – and more importantly – [1] argue that a Prisoner Dilemma always arises, thus confirming the traditional view that strategic delegation ultimately reduces the profits of all firms. The aim of this note is to show that the conclusions in [1] are based on an undue restriction of the parameter set. In contrast, when the admissible parameter set is correctly defined, we E-mail address: [email protected]. https://doi.org/10.1016/j.orl.2019.09.008 0167-6377/© 2019 Elsevier B.V. All rights reserved.
show that a Prisoner Dilemma is not an inevitable outcome. In particular, we show that in almost 35% of the cases, the more efficient firm is better off in the case of symmetric delegation than in the case of symmetric no delegation. Thus, the traditional view of Prisoner Dilemma as an inevitable outcome in the case of strategic delegation is not supported in the case of asymmetric firms. The rest of the paper proceeds as follows. The set-up is in Section 2. In Section 3 we derive our main result, which contrasts with [1]. Section 4 concludes. 2. The model The model is identical to [1]. We briefly summarize the model. There are two owners (or firms, simply), A and B, that compete a la Cournot for a homogeneous product, whose demand is given by p = 1 − qA − qB , with qJ being the quantity produced by Firm J = A, B. The two firms have different (constant) marginal production costs. Let us indicate by cJ the marginal cost of Firm J = A, B. As in [1], it is assumed that 0 < cA < cB < 1. Therefore, Firm A (B) is the more (less) efficient firm. Each firm might delegate the production decision to a manager, whose objective function, MJ , is a mixture of revenues and profits, that is: MJ = αJ πJ + (1 −αJ )pJ qJ , where πJ indicates the profits and αJ is the delegation parameter set by Firm J. Note that, as in [2], we posit no restrictions on αJ . The more αJ is close to 0, the greater is the distortion from pure profit maximization. The game is two-stage. In the first-stage each firm chooses whether to delegate to a manager (m) or not (e). In the case of delegation, the owner chooses the delegation parameter. In the second stage, the owners or the managers (depending on the first period decision) set simultaneously the quantity.
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S. Colombo / Operations Research Letters 47 (2019) 527–529
Fig. 1. Relevant parameter sets.
3. Main result
Table 1 The pay-off matrix.
Following [1] the equilibrium quantities in all the possible subgames are the following: q∗J (e, e) =
1 − 2cJ + c−J
q∗J (m, m) = q∗J (m, e) = qJ (e, m) = ∗
3 2(1 − 3cJ + 2c−J ) 5 1 − 2cJ + c−J 2 1 − 3cJ + 2c−J 4
(1) (2) 1−8c +2c
(3) (4)
By referring to these quantities, [1] impose the following restriction: 1+7c Admissible parameter set in [1]: cB ≤ cˆB ≡ 8 A ˆ [1] claim that condition cB < cB is needed to avoid that the 1−c efficient firm’s output exceeds the monopoly output (that is, 2 A ) when this firm is managerial and the other is not (see [1], p.444), that is in case (m, e). However, it is immediate to verify that Firm A’s output is never smaller than the monopoly output in case (m, e) under the model assumptions in [1]. Indeed, q∗A (m, e) > 1−cA is always verified when 0 < cA < cB < 1. This is not 2 surprising. Indeed, when only Firm A delegates, it is aggressive (as the manager cares less about the costs). For example, suppose Firm B does not produce, that is qB = 0. If Firm A does not 1−c delegate, it produces 2 A (that is, the monopolistic outcome).
On the other hand, if Firm A delegates (that is αA < 1), it 1−α c 1−c produces 2 A , which is greater than 2 A . Indeed, the condition cB ≤ cˆB is related to subgame (m, m) rather than subgame (m, e). In subgame (m, m), when cB ≥ cˆB , the output of the efficient firm is larger than the monopolistic output (that is, 1−c q∗A (m, m) ≥ 2 A ). Note that, since the efficient firm is managerial, it is not surprising that its equilibrium output is larger than the monopolistic one: indeed, the output is chosen by the manager, that does not maximize Firm A’s profits, but a weighted sum of profits and revenues. Still one might ask whether the efficient firm could profitably deviate in output or contracts. The answer is negative. Consider first deviation in output. Given the equilibrium
J −J delegation parameters (which are αJ∗ (m, m) = − ), the 5cJ output is chosen by managers, which maximize their own objective function by choosing (in equilibrium) q∗J (m, m). Consider now deviation in contracts. Given αB∗ (m, m), Firm A might deviate by choosing to be entrepreneurial (that is, setting αA = 1) and fixing the monopolistic output. In this case, the profits of Firm A are 19cA +16cB ) . It is easy to see that πAd < πA∗ (m, m) when πAd = (1−cA )(3−40
cB ̸ = cˆB and πAd = πA∗ (m, m) when cB = cˆB . Therefore, deviation is never profitable. It follows that condition cB < cˆB is undue. The admissible parameter set is determined by imposing that all quantities in (1)–(4) are non negative. That is: 1+2c Admissible parameter set revised: cB ≤ c B ≡ 3 A . As cˆB < c B , the assumption in [1] is too restrictive. Following [1], the equilibrium profits are summarized in Table 1. By comparing the equilibrium profits, it is immediate to show that there is a unique equilibrium where both firms delegate production to managers, (m, m), thus confirming the result in [1]. However, in contrast with [1], in what follows, we show that the Prisoner Dilemma is not an inevitable outcome. First, by comparing πB∗ (m, m) and πB∗ (e, e), we see that Firm B always gets higher profits in case (e, e). Therefore, the less efficient firm is always better off when no one delegates. Second, by comparing πA∗ (m, m) and πA∗ (e, e), we observe that Firm A might get higher profits in case (m, m). This happens when cB ≥ √ 15 2(1−cA ) ˜cB ≡ 58cA −11+47 . Therefore, the more efficient firm is better off when both firms delegate if the two firms are sufficiently different. Note that c B ≥ c˜B ≥ cˆB . This explains the error in [1]. By over-restricting the set of admissible parameters, [1] exclude the possibility of observing πA∗ (m, m) > πA∗ (e, e).
S. Colombo / Operations Research Letters 47 (2019) 527–529
529
The intuition of Proposition 1 is the following. Standard computations show that the best-reply of Firm J is qJ (q−J ) = 1−q−J −αJ cJ
. Therefore, moving from symmetric no-delegation to 2 symmetric delegation entails pushing the best-reply functions out. When cB is sufficiently high, the equilibrium changes from C to L (Fig. 2), that is, Firm A’s quantity increases, but Firm B’s quantity decreases. This is due to the fact that, because of the higher marginal costs, in equilibrium the Firm B’s manager’s objective function is less distorted towards revenues than the Firm A’s manager’s objective function, that is αA∗ (m, m) < αB∗ (m, m). This mitigates the negative effect of delegation on the equilibrium price. As Firm A’s quantity goes up thanks to delegation whereas the price reduces only slightly, the profits of Firm A are greater under symmetric delegation. 4. Conclusions
Fig. 2. Best-reply functions.
Consider the following numerical example. Set cA = 0.2 and cB = 0.4. Note that the non-negative constraint of the quantities is satisfied in any possible subgame. Furthermore, the relevant profits are πA∗ (m, m) = 0.115, πA∗ (e, e) = 0.111, πB∗ (m, m) = 0.003 and πB∗ (e, e) = 0.017. Therefore, it is immediate to note that the Prisoner Dilemma does not emerge. The following proposition summarizes the discussion above: Proposition 1. The less efficient firm is always better off when no firm delegates; the more efficient firm is better off when both firms delegate if it is sufficiently more efficient than the rival. In Fig. 1, the admissible parameter space is represented by triangle GFX, whereas the parameter space where the Prisoner Dilemma does not occur is represented by triangle DFX. As Fig. 1 shows, Prisoner Dilemma does not occur in many of the relevant cases (approximately in 35% of the relevant cases, calculated as the area of triangle DFX over the area of triangle GFX).
Strategic delegation games have been typically considered yielding a Prisoner Dilemma outcome, that is, a situation where the unique equilibrium is characterized by both firms choosing delegation even if both firms would be better off in the case of no delegation. This traditional view has been extended by [1] to the case of asymmetric firms. In this note, we have shown that the argument in [1] is based on an undue restriction of the admissible parameter set. When the admissible parameter set is correctly defined, we have shown that in almost 35% of cases (corresponding to those situations where the firms are sufficiently different) the more efficient firm is better off when both firms delegate. Therefore, in contrast with the traditional view, the Prisoner Dilemma is not an inevitable outcome of a strategic delegation game when the firms are asymmetric. References [1] F. Delbono, L. Lambertini, L. Marattin, Strategic delegation under cost asymmetry, Oper. Res. Lett. 44 (2016) 443–445. [2] C. Fershtman, K.L. Judd, Equilibrium incentives in oligopoly, Amer. Econ. Rev. 77 (1987) 927–940. [3] D. Sen, G. Stamatopoulos, When an inefficient competitor makes higher profits than its efficient rival, Oper. Res. Lett. 43 (2015) 148–150. [4] S. Sklivas, The strategic choice of managerial incentives, RAND J. Econ. 18 (1987) 452–458. [5] J. Vickers, Delegation and the theory of the firm, Econ. J. 95 (1985) 138–147.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Strategic delegation under cost asymmetry revised Stefano Colombo Largo A. Gemelli 1, I-20123, Università Cattolica del Sacro Cuore, Milano, Italy
article
info
Article history: Received 16 May 2019 Received in revised form 16 September 2019 Accepted 16 September 2019 Available online 21 September 2019 JEL classification: D43
a b s t r a c t We revisit the Cournot duopoly game with strategic delegation and asymmetric costs of Delbono et al. (2016). In particular, the authors claim that a Prisoner Dilemma always arises. However, we show that, by setting properly the admissible parameter set, if the firms are sufficiently different, the efficient firm is better off when both firms delegate production. Therefore, in contrast with the traditional view, we claim that a Prisoner Dilemma is not an inevitable outcome in a strategic delegation game. © 2019 Elsevier B.V. All rights reserved.
Keywords: Strategic delegation Cost asymmetry Cournot competition
1. Introduction The owners of firms commonly delegate production decisions to managers, whose objective function is different from that of the owners. Starting from the seminal contributions of [2,5] and [4], literature has shown that, in the case of Cournot duopolistic competition, each owner has the incentive to delegate production to managers who are more interested in revenue maximization than in profits maximization. In this way, each owner commits to be more aggressive in the production stage. However, the final outcome is detrimental for owners: as each owner wants to be more aggressive, the unique equilibrium entails both owners delegating to managers, but the resulting profits are lower than in the case of no delegation, because of the fiercer competition induced by delegation. Therefore, the delegation game is typically associated with a Prisoner Dilemma. Recently, [3] questioned this result, by showing that, in the case of asymmetric production costs, also asymmetric delegation equilibria are possible and Prisoner Dilemma can be avoided. However, [1] show that the conclusions in [3] heavily depend on the fact that the owners are not allowed to freely chose the delegation parameter. After removing the binary choice restriction in [3], [1] claim that the unique equilibrium is characterized by delegation by both owners. Moreover – and more importantly – [1] argue that a Prisoner Dilemma always arises, thus confirming the traditional view that strategic delegation ultimately reduces the profits of all firms. The aim of this note is to show that the conclusions in [1] are based on an undue restriction of the parameter set. In contrast, when the admissible parameter set is correctly defined, we E-mail address: [email protected]. https://doi.org/10.1016/j.orl.2019.09.008 0167-6377/© 2019 Elsevier B.V. All rights reserved.
show that a Prisoner Dilemma is not an inevitable outcome. In particular, we show that in almost 35% of the cases, the more efficient firm is better off in the case of symmetric delegation than in the case of symmetric no delegation. Thus, the traditional view of Prisoner Dilemma as an inevitable outcome in the case of strategic delegation is not supported in the case of asymmetric firms. The rest of the paper proceeds as follows. The set-up is in Section 2. In Section 3 we derive our main result, which contrasts with [1]. Section 4 concludes. 2. The model The model is identical to [1]. We briefly summarize the model. There are two owners (or firms, simply), A and B, that compete a la Cournot for a homogeneous product, whose demand is given by p = 1 − qA − qB , with qJ being the quantity produced by Firm J = A, B. The two firms have different (constant) marginal production costs. Let us indicate by cJ the marginal cost of Firm J = A, B. As in [1], it is assumed that 0 < cA < cB < 1. Therefore, Firm A (B) is the more (less) efficient firm. Each firm might delegate the production decision to a manager, whose objective function, MJ , is a mixture of revenues and profits, that is: MJ = αJ πJ + (1 −αJ )pJ qJ , where πJ indicates the profits and αJ is the delegation parameter set by Firm J. Note that, as in [2], we posit no restrictions on αJ . The more αJ is close to 0, the greater is the distortion from pure profit maximization. The game is two-stage. In the first-stage each firm chooses whether to delegate to a manager (m) or not (e). In the case of delegation, the owner chooses the delegation parameter. In the second stage, the owners or the managers (depending on the first period decision) set simultaneously the quantity.
528
S. Colombo / Operations Research Letters 47 (2019) 527–529
Fig. 1. Relevant parameter sets.
3. Main result
Table 1 The pay-off matrix.
Following [1] the equilibrium quantities in all the possible subgames are the following: q∗J (e, e) =
1 − 2cJ + c−J
q∗J (m, m) = q∗J (m, e) = qJ (e, m) = ∗
3 2(1 − 3cJ + 2c−J ) 5 1 − 2cJ + c−J 2 1 − 3cJ + 2c−J 4
(1) (2) 1−8c +2c
(3) (4)
By referring to these quantities, [1] impose the following restriction: 1+7c Admissible parameter set in [1]: cB ≤ cˆB ≡ 8 A ˆ [1] claim that condition cB < cB is needed to avoid that the 1−c efficient firm’s output exceeds the monopoly output (that is, 2 A ) when this firm is managerial and the other is not (see [1], p.444), that is in case (m, e). However, it is immediate to verify that Firm A’s output is never smaller than the monopoly output in case (m, e) under the model assumptions in [1]. Indeed, q∗A (m, e) > 1−cA is always verified when 0 < cA < cB < 1. This is not 2 surprising. Indeed, when only Firm A delegates, it is aggressive (as the manager cares less about the costs). For example, suppose Firm B does not produce, that is qB = 0. If Firm A does not 1−c delegate, it produces 2 A (that is, the monopolistic outcome).
On the other hand, if Firm A delegates (that is αA < 1), it 1−α c 1−c produces 2 A , which is greater than 2 A . Indeed, the condition cB ≤ cˆB is related to subgame (m, m) rather than subgame (m, e). In subgame (m, m), when cB ≥ cˆB , the output of the efficient firm is larger than the monopolistic output (that is, 1−c q∗A (m, m) ≥ 2 A ). Note that, since the efficient firm is managerial, it is not surprising that its equilibrium output is larger than the monopolistic one: indeed, the output is chosen by the manager, that does not maximize Firm A’s profits, but a weighted sum of profits and revenues. Still one might ask whether the efficient firm could profitably deviate in output or contracts. The answer is negative. Consider first deviation in output. Given the equilibrium
J −J delegation parameters (which are αJ∗ (m, m) = − ), the 5cJ output is chosen by managers, which maximize their own objective function by choosing (in equilibrium) q∗J (m, m). Consider now deviation in contracts. Given αB∗ (m, m), Firm A might deviate by choosing to be entrepreneurial (that is, setting αA = 1) and fixing the monopolistic output. In this case, the profits of Firm A are 19cA +16cB ) . It is easy to see that πAd < πA∗ (m, m) when πAd = (1−cA )(3−40
cB ̸ = cˆB and πAd = πA∗ (m, m) when cB = cˆB . Therefore, deviation is never profitable. It follows that condition cB < cˆB is undue. The admissible parameter set is determined by imposing that all quantities in (1)–(4) are non negative. That is: 1+2c Admissible parameter set revised: cB ≤ c B ≡ 3 A . As cˆB < c B , the assumption in [1] is too restrictive. Following [1], the equilibrium profits are summarized in Table 1. By comparing the equilibrium profits, it is immediate to show that there is a unique equilibrium where both firms delegate production to managers, (m, m), thus confirming the result in [1]. However, in contrast with [1], in what follows, we show that the Prisoner Dilemma is not an inevitable outcome. First, by comparing πB∗ (m, m) and πB∗ (e, e), we see that Firm B always gets higher profits in case (e, e). Therefore, the less efficient firm is always better off when no one delegates. Second, by comparing πA∗ (m, m) and πA∗ (e, e), we observe that Firm A might get higher profits in case (m, m). This happens when cB ≥ √ 15 2(1−cA ) ˜cB ≡ 58cA −11+47 . Therefore, the more efficient firm is better off when both firms delegate if the two firms are sufficiently different. Note that c B ≥ c˜B ≥ cˆB . This explains the error in [1]. By over-restricting the set of admissible parameters, [1] exclude the possibility of observing πA∗ (m, m) > πA∗ (e, e).
S. Colombo / Operations Research Letters 47 (2019) 527–529
529
The intuition of Proposition 1 is the following. Standard computations show that the best-reply of Firm J is qJ (q−J ) = 1−q−J −αJ cJ
. Therefore, moving from symmetric no-delegation to 2 symmetric delegation entails pushing the best-reply functions out. When cB is sufficiently high, the equilibrium changes from C to L (Fig. 2), that is, Firm A’s quantity increases, but Firm B’s quantity decreases. This is due to the fact that, because of the higher marginal costs, in equilibrium the Firm B’s manager’s objective function is less distorted towards revenues than the Firm A’s manager’s objective function, that is αA∗ (m, m) < αB∗ (m, m). This mitigates the negative effect of delegation on the equilibrium price. As Firm A’s quantity goes up thanks to delegation whereas the price reduces only slightly, the profits of Firm A are greater under symmetric delegation. 4. Conclusions
Fig. 2. Best-reply functions.
Consider the following numerical example. Set cA = 0.2 and cB = 0.4. Note that the non-negative constraint of the quantities is satisfied in any possible subgame. Furthermore, the relevant profits are πA∗ (m, m) = 0.115, πA∗ (e, e) = 0.111, πB∗ (m, m) = 0.003 and πB∗ (e, e) = 0.017. Therefore, it is immediate to note that the Prisoner Dilemma does not emerge. The following proposition summarizes the discussion above: Proposition 1. The less efficient firm is always better off when no firm delegates; the more efficient firm is better off when both firms delegate if it is sufficiently more efficient than the rival. In Fig. 1, the admissible parameter space is represented by triangle GFX, whereas the parameter space where the Prisoner Dilemma does not occur is represented by triangle DFX. As Fig. 1 shows, Prisoner Dilemma does not occur in many of the relevant cases (approximately in 35% of the relevant cases, calculated as the area of triangle DFX over the area of triangle GFX).
Strategic delegation games have been typically considered yielding a Prisoner Dilemma outcome, that is, a situation where the unique equilibrium is characterized by both firms choosing delegation even if both firms would be better off in the case of no delegation. This traditional view has been extended by [1] to the case of asymmetric firms. In this note, we have shown that the argument in [1] is based on an undue restriction of the admissible parameter set. When the admissible parameter set is correctly defined, we have shown that in almost 35% of cases (corresponding to those situations where the firms are sufficiently different) the more efficient firm is better off when both firms delegate. Therefore, in contrast with the traditional view, the Prisoner Dilemma is not an inevitable outcome of a strategic delegation game when the firms are asymmetric. References [1] F. Delbono, L. Lambertini, L. Marattin, Strategic delegation under cost asymmetry, Oper. Res. Lett. 44 (2016) 443–445. [2] C. Fershtman, K.L. Judd, Equilibrium incentives in oligopoly, Amer. Econ. Rev. 77 (1987) 927–940. [3] D. Sen, G. Stamatopoulos, When an inefficient competitor makes higher profits than its efficient rival, Oper. Res. Lett. 43 (2015) 148–150. [4] S. Sklivas, The strategic choice of managerial incentives, RAND J. Econ. 18 (1987) 452–458. [5] J. Vickers, Delegation and the theory of the firm, Econ. J. 95 (1985) 138–147.