Asymmetric Performance Evaluation Under Quantity and Price Competition with Managerial Delegation

Jumpei Hamamura; Vinay Ramani

doi:10.1515/bejeap-2023-0135

Article

Asymmetric Performance Evaluation Under Quantity and Price Competition with Managerial Delegation

Jumpei Hamamura and Vinay Ramani

Published/Copyright: April 16, 2024

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 24 Issue 3

Abstract

In this paper, we consider asymmetric performance evaluation contracts under different product market configurations with managerial delegation and specify the optimal decision-making by the social and relative performance evaluation firms. We present a reversal result on the owner’s choice of the social performance and relative performance evaluation contract as the product market competition type changes from quantity to price competition. Surprisingly, results indicate that the consumer surplus increases as the degree of product substitution increases under quantity competition in a specific economic environment. A firm that considers social performance evaluation produces less, charges a higher price, and earns a lower profit than a firm that uses relative performance evaluation. We also endogenize the choice of performance evaluation systems. While relative performance emerges as the endogenous choice under both modes of product market competition, it leads to lower consumer surplus and social welfare in comparison to an asymmetric performance evaluation system.

Keywords: managerial delegation; price competition; quantity competition; relative performance; social performance

Corresponding author: Vinay Ramani, Department of Industrial & Management Engineering, Indian Institute of Technology, Kanpur, 208016, India, E-mail: vramani@iitk.ac.in

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP21K13409

Award Identifier / Grant number: The Japanese Association of Management Accounting

Acknowledgments

We would like to sincerely thank two anonymous referees and the editor, Prof. Till Requate for their helpful comments and suggestions on earlier versions of this paper. Any remaining errors are our responsibility.

Research funding: This work was partially supported by JSPS KAKENHI Grant Number JP21K13409, and The Japanese Association of Management Accounting (Study Group in 2022).

Appendix A: Additional Analysis: Firm 1 Emphasizes Social Welfare

To check the robustness of our results, in this appendix, we analyze the case in which the objective function of Firm 1’s (social performance firm) manager considers social welfare. The objective function of Firm 2 remains the same as in the main model, i.e. V ₂ = π ₂ + α ₂ π ₁. Several earlier social corporate responsibility or mixed oligopoly studies which consider socially concerned firms examine the effect of emphasizing social welfare on the decision-making and profits (e.g. Hino and Zennyo 2017; Liu, Wang, and Chen 2018; Matsumura 1998; Matsumura and Ogawa 2014; Nakamura 2018; Pal 1998). We aim to understand whether the findings reported in our main analysis with consumer surplus as the social performance measure continue to hold with social welfare applied as the social performance measure.

Quantity Competition

First, we consider the case of quantity competition wherein the objective function of Firm 1’s manager considers social welfare: i.e. V ₁ = π ₁ + α ₁ SW. By backward induction, we specify outcomes in this case and obtain the Outcome 3 in Appendix B.

In this analysis, the rival’s profit is included in the objective function of Firm 1. Therefore, our additional analysis yields different outcomes compared to Outcome 1 in Appendix B. However, comparing the effect of emphasizing the rival’s profit (reducing quantity) with the effect of emphasizing consumer surplus (increasing quantity), the latter effect is important. Therefore, the effects of θ on strategies have only a few differences between the main and additional models. Figure A.1 presents the effects of θ on α i Ccs and α i Csw where the superscript Csw denotes quantity (Cournot) competition with social welfare as the objective. From this figure, the inflection point remains largely unchanged.

$Figure A.1: Relation between α i Ccs ${\alpha }_{i}^{\mathit{Ccs}}$ and α i Csw ${\alpha }_{i}^{\mathit{Csw}}$ under a = 1.$

Figure A.1:

Relation between α i Ccs and α i Csw under a = 1.

In addition, based on Figure A.1, one can consider the relation between α i Ccs and α i Csw . We formalize the result in the following proposition.

Proposition A.1.

The weight assigned to social welfare by Firm 1 in the quantity competition case, α 1 Csw , always exceeds the weight placed on consumer surplus by Firm 1 in the quantity competition case, α 1 Ccs . In addition, for Firm 2, the weight it places on the rival firm’s profit is greater when Firm 1 considers CS as the objective compared to when it considers SW. Formally, α 1 Csw > α 1 Ccs and α 2 Csw < α 2 Ccs are always satisfied.

Proof.

The proof is presented in Appendix D.

Again, Figure A.1 presents the relation between α i Ccs and α i Csw in this case. From this figure, one can confirm that α 1 Csw > α 1 Ccs and α 2 Csw < α 2 Ccs is always satisfied. Therefore, π ₂, which is included in SW, has a negligible effect on the decision-making of the socially concerned firm under quantity competition. In this case, because social welfare includes the rival’s profit and consumer surplus, the weight placed on the rival’s profit is important in committing to an aggressive strategy in product market competition. Therefore, Firm 1 can increase quantity using a more aggressive (large) α ₁ to obtain a significant market share in the product market. This effect leads to a large α ₁ to threaten the rival.

Price Competition

Next, we consider the case of price competition wherein the objective function of Firm 1’s manager considers social welfare: i.e. V ₁ = π ₁ + α ₁ SW. In this case, the objective function of Firm 2 stays the same. We are interested in confirming whether the results obtained under price competition remain the same when Firm 1 considers social welfare as the objective instead of consumer surplus. The equilibrium solutions of this model are reported as Outcome 4 in Appendix B. Here, the superscript Bsw denotes price (Bertrand) competition with social welfare as the objective. Furthermore, α 1 Bsw < 0 for all θ ∈ (0, 1] and α 2 Bsw > 0 for 0 < θ ≤ ( 17 − 1 ) / 4 .

Comparing the optimal solutions of Firm 1 and Firm 2, we can infer that p 1 Bsw > p 2 Bsw , q 1 Bsw < q 2 Bsw and π 1 Bsw < π 2 Bsw , which is identical to the case in which Firm 1 considers weight on consumer surplus. Next, consumer surplus and social welfare are obtained as follows.

C S Bsw = a 2 [ 16 − 4 A Bsw − θ ( 32 − 6 A Bsw ) − θ 2 ( 28 − 10 A Bsw ) + θ 3 ( 76 + 58 θ − θ 3 + A Bsw ) ] 64 θ 2 ( 1 + θ ) 2 , S W Bsw = a 2 [ − 16 + 4 A Bsw + 2 θ A Bsw + θ 2 ( 76 − 2 A Bsw ) + θ 3 ( 68 − 2 θ − θ 2 ( 8 − θ ) − A Bsw ) ] 64 θ 2 ( 1 + θ ) 2 .

Proposition A.2.

The weight placed on social welfare by Firm 1 in the price competition case, α 1 Bsw , is always lower than the weight placed on consumer surplus by Firm 1 in the price competition case. In addition, for Firm 2, the weight it places on Firm 1’s profit is lower when Firm 1 considers SW compared to the case in which Firm 1 considers CS. Formally, α 1 Bsw < α 1 Bcs and α 2 Bsw < α 2 Bcs are always satisfied.

Proof.

The proof is presented in Appendix D.

Comparison of the weights between the two models presents an interesting reversal result. Although Firm 1 attaches a higher weight to social welfare than consumer surplus under quantity competition, it assigns a lower weight to social welfare than consumer surplus under price competition. For Firm 2, the weights under social welfare are lower than that under consumer surplus. The intuition for the result is the following. Social welfare includes consumer surplus and the profits of both Firm 1 and Firm 2. Under price competition, Firm 1 adopts a less aggressive strategy under the social welfare objective than the consumer surplus objective. As a result, Firm 2 also chooses a lower α ₂ when Firm 1 assigns weight to social welfare.

Figure A.2 depicts the comparison pictorially. It is particularly interesting that α 2 BSw is non-monotonically increasing as θ increases. A higher θ implies that goods are close substitutes, intensifying the price competition. This increases the consumer surplus. As a result, Firm 1 assigns a higher weight in the case of consumer surplus, as opposed to social welfare. Consequently, Firm 2 also increases the weight on Firm 1’s profit in the case of consumer surplus.

$Figure A.2: Relation between α i Bcs ${\alpha }_{i}^{\mathit{Bcs}}$ and α i Bsw ${\alpha }_{i}^{\mathit{Bsw}}$ under a = 1 for i = 1, 2.$

Figure A.2:

Relation between α i Bcs and α i Bsw under a = 1 for i = 1, 2.

Finally, we can observe from the optimal solutions that ∂ 1 Bsw / ∂ θ < 0 for all θ ∈ (0, 1] and ∂ 2 Bsw / ∂ θ > 0 for 0 < θ < 0.524 and ∂ 2 Bsw / ∂ θ < 0 for 0.524 < θ < 1. In addition, consumer surplus is increasing in the degree of product substitution: ∂CS ^Bsw/∂θ > 0 for all θ ∈ (0, 1].

Appendix B: Outcomes

In Appendix B, we label the outcomes as Outcome to refer in our manuscript. All analysis are provided in Appendix D Additionally, A ^Ccs, B ^Ccs, A ^Bcs, A ^Csw, and A ^Bsw are defined in Appendix C.

Outcome 1.

Under the quantity competition, the optimal quantities chosen by the two firms, prices, and profits are, respectively

q 1 Ccs = a 12 − 4 θ − 5 θ 2 − 2 θ 3 8 − A Ccs − 4 θ − 8 θ 2 − 2 θ 3 − θ 4 8 1 − θ 1 + θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 , q 2 Ccs = a 8 A Ccs − 4 + 4 3 A Ccs + 8 θ − 2 7 A Ccs − 16 θ 2 − 5 A Ccs + 56 θ 3 + 22 θ 4 + 4 θ 5 − 3 θ 6 8 θ 1 − θ 1 + θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 , p 1 Ccs = a ( 2 − θ ) 4 , p 2 Ccs = a 8 4 − A Ccs + 4 A Ccs + 24 θ − 2 A Ccs + 32 θ 2 + 5 A Ccs − 24 θ 3 + 2 A Ccs + 13 θ 4 + 24 θ 5 + 9 θ 6 + 2 θ 7 8 θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 , π 1 Ccs = a 2 ( 2 − θ ) 12 − 4 θ − 5 θ 2 − 2 θ 3 8 − A Ccs − 4 θ − 8 θ 2 − 2 θ 3 − θ 4 32 1 − θ 1 + θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 , π 2 Ccs = a 2 B Ccs 8 A Ccs − 4 + 4 3 A Ccs + 8 θ − 2 7 A Ccs − 16 θ 2 − 5 A Ccs + 56 θ 3 + 22 θ 4 + 4 θ 5 − 3 θ 6 64 θ 2 1 − θ 1 + θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 2 .

Outcome 2.

Under the price competition, the optimal prices charged by the two firms, the optimal quantities, and profits are, respectively,

p 1 Bcs = a ( θ 3 + A Bcs ) 8 ( 1 + θ ) , p 2 Bcs = a ( A Bcs − 4 + 2 θ + 4 θ 2 − θ 3 ) 8 θ ( 1 + θ ) , q 1 Bcs = a ( 2 + θ ) 4 ( 1 + θ ) , q 2 Bcs = a ( 4 + 6 θ − θ 3 − A Bcs ) 8 θ ( 1 + θ ) , π 1 Bcs = a 2 ( 2 + θ ) ( θ 3 + A Bcs ) 32 ( 1 + θ ) 2 , π 2 Bcs = a 2 ( 2 − θ ) A Bcs − 8 + 10 θ 2 − θ 4 16 θ 2 ( 1 + θ ) .

Outcome 3.

When the socially concerned firm emphasizes social welfare, under the quantity competition, the optimal weights chosen by the two firms, quantity, prices, and profits are, respectively

α 1 Csw = A Csw − 4 + 2 θ + 4 θ 2 + − 3 θ 3 2 2 − θ − 2 θ 2 + θ 3 , α 2 Csw = − A Csw − 4 + 2 θ + 2 θ 2 ( 2 − θ ) θ 2 , q 1 Csw = a A Csw − θ 3 8 1 − θ 2 , q 2 Csw = a A Csw − 4 − 2 θ + 4 θ 2 + θ 3 8 θ θ 2 − 1 , p 1 Csw = a ( 2 − θ ) 4 , p 2 Csw = a A Csw − 4 + 6 θ − θ 3 8 θ , π 1 Csw = a 2 ( 2 − θ ) A Csw − θ 3 32 1 − θ 2 , π 2 Csw = a 2 A Csw − 4 + 6 θ − θ 3 A Csw − 4 − 2 θ + 4 θ 2 + θ 3 64 θ 2 θ 2 − 1 .

Outcome 4.

When the socially concerned firm emphasizes social welfare, under the price competition, the optimal weights chosen by the two firms, the equilibrium prices, quantities, and profit of the two firms are, respectively

α 1 Bsw = − 4 − θ 2 + θ ( 2 − 3 θ ) − A Bsw 2 ( 2 − θ ) ( 1 − θ ) ( 1 + θ ) , α 2 Bsw = 8 − ( 2 + θ ) A Bsw − 2 θ 2 5 − 2 θ + θ 2 ( 2 − θ ) θ 2 12 + 4 θ − θ 2 ( 5 − 2 θ ) , p 1 Bsw = a 8 − A Bsw − θ 2 ( 6 − θ ) 8 ( 1 + θ ) , p 2 Bsw = a 4 − A Bsw + θ 2 − θ ( 2 + θ ) 8 θ ( 1 + θ ) , q 1 Bsw = a ( 2 + θ ) 4 ( 1 + θ ) , q 2 Bsw = a A Bsw + 6 θ + θ 2 ( 6 − θ ) − 4 8 θ ( 1 + θ ) , π 1 Bsw = a 2 ( 2 + θ ) 8 − A Bsw − θ 2 ( 6 − θ ) 64 θ 2 ( 1 + θ ) 2 , π 2 Bsw = a 2 A Bsw + 6 θ + θ 2 ( 6 − θ ) − 4 4 + 2 θ − θ 2 ( 2 + θ ) − A Bsw 64 θ 2 ( 1 + θ ) 2 .

Appendix C: Definitions

Appendix C shows definitions of the replaced variables in our analysis. In Section 5, the following variables are used.

A Ccs ≡ θ 6 + 4 θ 5 + 16 θ 4 − 24 θ 3 − 28 θ 2 + 16 θ + 16 , B Ccs ≡ 2 θ 7 + 9 θ 6 + 24 θ 5 + 2 A Ccs + 13 θ 4 + 5 A Ccs − 24 θ 3 − 2 A Ccs + 32 θ 2 + 4 A Ccs + 24 θ − 8 A Ccs − 4 , C Ccs ≡ − 4 θ 14 + 20 θ 13 + 85 θ 12 − 4 ( 159 A Ccs + 2888 ) θ 6 − 8 ( 19 A Ccs − 80 ) θ 5 + 16 ( 137 A Ccs + 720 ) θ 4 − 32 ( A Ccs − 32 ) θ 3 − 64 ( 25 A Ccs + 32 ) θ 2 + 128 ( A Ccs − 8 ) θ + 256 ( A Ccs − 4 ) + 4 ( A Ccs + 22 ) θ 11 + 6 ( 2 A Ccs + 7 ) θ 10 + ( 37 A Ccs + 908 ) θ 9 + ( 30 A Ccs + 2212 ) θ 8 − 14 ( 17 A Ccs + 64 ) θ 7 , D Ccs ≡ − − 4 θ 14 + 12 θ 13 + 75 θ 12 − 20 ( 29 A Ccs + 1416 ) θ 6 − 8 ( 649 A Ccs + 512 ) θ 5 + 48 ( 85 A Ccs + 864 ) θ 4 + 32 ( 133 A Ccs − 176 ) θ 3 − 64 ( 39 A Ccs + 272 ) θ 2 − 128 ( 5 A Ccs − 24 ) θ − 256 ( A Ccs − 4 ) − 4 ( A Ccs − 164 ) θ 11 + ( 20 A Ccs + 718 ) θ 10 + ( 59 A Ccs − 92 ) θ 9 + 6 ( 55 A Ccs − 142 ) θ 8 + ( 422 A Ccs + 9376 ) θ 7 .

Under 0.45 < θ < 1, A ^Ccs < 4 is satisfied.

In Section 6, the following variable is used.

A Bcs ≡ 16 + 16 θ − 12 θ 2 − 16 θ 3 − 4 θ 4 + θ 6 .

In Section A, the following variable is used.

A Csw ≡ θ 6 − 4 θ 4 + 16 θ 3 − 12 θ 2 − 16 θ + 16 .

In Section A, the following variables are used.

A Bsw ≡ 16 ( 1 − θ ) − 28 θ 2 + θ 3 24 + 16 θ − θ 2 ( 4 − θ ) .

Appendix D: Proofs

Analysis for Outcome 1

By backward induction, we can specify the optimal strategies and profits under quantity competition as Outcome 1. First, we consider the stage 2 problem for managers. In the quantity competition case, managers decide q ₁ and q ₂ to maximize V ₁ = π ₁ + α ₁ CS and V ₂ = π ₂ + α ₂ π ₁. From the first-order conditions, we obtain the following Best-response functions of the stage-2 problem.

(D.1) q 1 ( q 2 ) = a − ( 1 − α 1 ) θ q 2 2 − α 1 ,

(D.2) q 2 ( q 1 ) = a − ( 1 + α 2 ) θ q 1 2 .

From Eqs. (D.1) and (D.2), an increase in the quantity of the rival firm j, causes firm i to decrease its quantity, indicating that quantities are strategic substitutes even when the objective of one firm moves away from considering relative profits. Solving Eqs. (D.1) and (D.2), we obtain stage 2 solutions.

(D.3) q 1 ( α 1 , α 2 ) = a ( 2 − ( 1 − α 1 ) θ ) 4 − α 1 2 − ( 1 + α 2 ) θ 2 − ( 1 + α 2 ) θ 2 ,

(D.4) q 2 ( α 1 , α 2 ) = a ( 2 − α 1 − α 2 θ − θ ) 4 − α 1 2 − ( 1 + α 2 ) θ 2 − ( 1 + α 2 ) θ 2 .

Based on this outcome, we obtain the optimal solutions of weights. In stage 1, the owner of the firm 1 decides the weight placed on consumer surplus. The owner of the firm 2 decides the weight assigned to the rival firm’s profit. Using Outcome 1, we obtain Proposition 1 straightforwardly. □

Proof of Proposition 2

Considering ∂ α 1 Ccs / ∂ θ and ∂ α 2 Ccs / ∂ θ , we obtain

(D.5) ∂ α 1 Ccs ∂ θ = ( 2 − 2 θ 2 + θ − θ 3 ) 2 − 4 θ − 9 θ 2 − 3 θ 5 + 10 θ 4 + 32 θ 3 − 36 θ 2 − 28 θ + 8 A Ccs − ( 1 − 4 θ ) ( 4 − A Ccs + 2 θ − 2 θ 2 − 3 θ 3 ) 2 ( 2 + θ − 2 θ 2 − θ 3 ) 2 ,

(D.6) ∂ α 2 Ccs ∂ θ = − 2 A App θ 3 ( 2 θ 3 + 5 θ 2 + 4 θ − 12 ) 2 A Ccs < 0 .

where A ^App ≡ − θ ¹⁰ − 2θ ⁹ − 10θ ⁸ + (A ^Ccs + 138)θ ⁷ + 2(8A ^Ccs + 75)θ ⁶ + 4(3A ^Ccs − 106)θ ⁵ + 2(A ^Ccs − 228)θ ⁴ + 4(A ^Ccs + 48)θ ³ − 80(A ^Ccs − 10)θ ² − 48A ^Ccs θ + 96(A ^Ccs − 4). From the numerator of Eq. (D.5), under θ ̲ Ccs ≃ 0.796 < θ < 1 , ∂ α 1 Ccs / ∂ θ < 0 holds. □

Proof of Proposition 4

After considering q 1 Ccs − q 2 Ccs , we obtain the following outcome:

(D.7) q 1 Ccs − q 2 Ccs = − a B App 8 ( 1 − θ ) θ 2 A Ccs + 8 − A Ccs − 4 θ − 16 θ 2 − θ 3 − θ 4 < 0 ,

where B ^App ≡ 8(A ^Ccs − 4) + 4(A ^Ccs − 8)θ − 6(A ^Ccs − 8)θ ² − 3(A ^Ccs − 16)θ ³ − 2(A ^Ccs − 1)θ ⁴ − 26θ ⁵ − 7θ ⁶ − 2θ ⁷. □

Proof of Proposition 5

We consider ∂CS ^Ccs/∂θ and get the following outcome:

(D.8) ∂ C S Ccs ∂ θ = D App − E App − F App − G App ,

where C ^App ≡ 6θ ⁵ + 20θ ⁴ + 64θ ³ − 72θ ² − 56θ + 16, D App ≡ ( a 2 ( 56 θ 13 + 260 θ 12 + 2 C App θ 11 / A Ccs + 1020 θ 11 + 44 ( A Ccs + 22 ) θ 10 + 6 C App θ 10 / A Ccs + 60 ( 2 A Ccs + 7 ) θ 9 + 37 C App θ 9 / 2 A Ccs + 9 ( 37 A Ccs + 908 ) θ 8 + 15 C App θ 8 / A Ccs + 8 ( 30 A Ccs + 2212 ) θ 7 − 119 C App θ 7 / A Ccs − 98 ( 17 A Ccs + 64 ) θ 6 − 318 C App θ 6 / A Ccs − 24 ( 159 A Ccs + 2888 ) θ 5 − 76 C App θ 5 / A Ccs − 40 ( 19 A Ccs − 80 ) θ 4 + 1096 C App θ 4 / A Ccs + 64 ( 137 A Ccs + 720 ) θ 3 − 16 C App θ 3 / A Ccs − 96 ( A Ccs − 32 ) θ 2 − 800 C App θ 2 / A Ccs − 128 ( 25 A Ccs + 32 ) θ + 64 C App θ / A Ccs + 128 ( A Ccs − 8 ) + 128 C App / A Ccs ) ) / 64 θ 2 ( θ 2 − 1 ) ( θ 4 + θ 3 + 16 θ 2 + ( A Ccs − 4 ) θ − 2 ( A Ccs + 8 ) ) 2 , E App ≡ ( a 2 ( 4 θ 14 + 20 θ 13 + 85 θ 12 + 4 ( A Ccs + 22 ) θ 11 + 6 ( 2 A Ccs + 7 ) θ 10 + ( 37 A Ccs + 908 ) θ 9 + ( 30 A Ccs + 2212 ) θ 8 − 14 ( 17 A Ccs + 64 ) θ 7 − 4 ( 159 A Ccs + 2888 ) θ 6 − 8 ( 19 A Ccs − 80 ) θ 5 + 16 ( 137 A Ccs + 720 ) θ 4 − 32 ( A Ccs − 32 ) θ 3 − 64 ( 25 A Ccs + 32 ) θ 2 + 128 ( A Ccs − 8 ) θ + 256 ( A Ccs − 4 ) ) ) / ( 32 θ 3 ( θ 2 − 1 ) ( θ 4 + θ 3 + 16 θ 2 + ( A Ccs − 4 ) θ − 2 ( A Ccs + 8 ) ) 2 ) , and F App ≡ ( a 2 ( 4 θ 14 + 20 θ 13 + 85 θ 12 + 4 ( A Ccs + 22 ) θ 11 + 6 ( 2 A Ccs + 7 ) θ 10 + ( 37 A Ccs + 908 ) θ 9 + ( 30 A Ccs + 2212 ) θ 8 − 14 ( 17 A Ccs + 64 ) θ 7 − 4 ( 159 A Ccs + 2888 ) θ 6 − 8 ( 19 A Ccs − 80 ) θ 5 + 16 ( 137 A Ccs + 720 ) θ 4 − 32 ( A Ccs − 32 ) θ 3 − 64 ( 25 A Ccs + 32 ) θ 2 + 128 ( A Ccs − 8 ) θ + 256 ( A Ccs − 4 ) ) ) / ( 32 θ ( θ 2 − 1 ) 2 ( θ 4 + θ 3 + 16 θ 2 + ( A Ccs − 4 ) θ − 2 ( A Ccs + 8 ) ) 2 ) , and G App ≡ − ( a 2 ( 4 θ 3 + 3 θ 2 + C App θ / 2 A Ccs + 32 θ + A Ccs − C App / A Ccs − 4 ) ( 4 θ 14 + 20 θ 13 + 85 θ 12 + 4 ( A Ccs + 22 ) θ 11 + 6 ( 2 A Ccs + 7 ) θ 10 + ( 37 A Ccs + 908 ) θ 9 + ( 30 A Ccs + 2212 ) θ 8 − 14 ( 17 A Ccs + 64 ) θ 7 − 4 ( 159 A Ccs + 2888 ) θ 6 − 8 ( 19 A Ccs − 80 ) θ 5 + 16 ( 137 A Ccs + 720 ) θ 4 − 32 ( A Ccs − 32 ) θ 3 − 64 ( 25 A Ccs + 32 ) θ 2 + 128 ( A Ccs − 8 ) θ + 256 ( A Ccs − 4 ) ) ) / ( 32 θ 2 ( θ 2 − 1 ) ( θ 4 + θ 3 + 16 θ 2 + ( A Ccs − 4 ) θ − 2 ( A Ccs + 8 ) ) 3 ) . Under 0 < θ < 0.966 ≃ θ ̄ , ∂CS ^Ccs/∂θ > 0 holds.

Additionally, we demonstrate the existence of the inflection point. It is larger than θ = 0.966 using numerical examples. Under a = 1, when θ = 0.966, ∂CS/∂θ = 0.000641386 > 0 is obtained. However, θ = 0.967, ∂CS/∂θ = −0.000427304 < 0 is obtained. Therefore, the inflection point exists in 0.966 < θ < 0.967. Because the existence of positive ∂CS/∂θ in this proposition is important, we propose our important results obtained using numerical examples. From the discussion presented above, under a = 1, when θ = 0.966, ∂CS/∂θ = 0.000641386 > 0 is obtained. In addition, when θ = 0.01, ∂CS/∂θ = 0.00124063 is obtained. From the discussion presented above, there exists θ ∈ ( 0 , θ ̄ ) such that ∂CS/∂θ > 0.

Figure D.1 presents the effect of θ on ∂CS ^Ccs/∂θ under a = 1. From this figure, one can confirm that there exists a case of ∂CS ^Ccs/∂θ > 0.□

Figure D.1:

Effect of θ on ∂CS ^Ccs/∂θ under a = 1.

Proof of Proposition 6

Solving Eqs. (D.21) and (D.22), we get the optimal contract weights as α 1 Bcs = − ( A Bcs − 4 − 2 θ + 4 θ 2 + 3 θ 3 ) / 2 ( 2 + θ − 2 θ 2 − θ 3 ) and α 2 Bcs = ( 4 + 2 θ − 2 θ 2 − A Bcs ) / θ 2 ( 2 + θ ) .

To demonstrate that α 1 Bcs < 0 , it suffices to prove that the numerator expressions A ^Bcs − 4 − 2θ + 4θ ² + 3θ ³ > 0. To illustrate this point, it is noteworthy that, on simplification, we obtain

(D.9) A Bcs − 4 − 2 θ + 4 θ 2 + 3 θ 3 = 8 ( 2 − θ 2 − θ 4 ) + 24 θ ( 1 − θ 2 ) > 0 .

Similarly, to prove that α 2 Bcs > 0 , it suffices to prove that the numerator expression 4 + 2θ − 2θ ² − A ^Bcs > 0. Regarding this point on simplification, we obtain

(D.10) 4 + 2 θ − 2 θ 2 − A Bcs = θ 3 ( 8 − θ 3 ) + 8 θ 4 > 0 .

This completes the proof. □

Proof of Proposition 7

From α 1 Bcs − α 1 B B , we get

(D.11) α 1 Bcs − α 1 B B = 2 ( 4 − A Ccs ) + 4 θ + A Ccs − 4 θ 2 − 6 θ 3 − 2 θ 4 + θ 5 2 ( θ − 1 ) ( θ + 1 ) ( θ + 2 ) θ 2 − 2 > 0 .

This result implies α 1 Bcs > α 1 B B . □

Proof of Proposition 8

From the optimal solutions, we have

(D.12) ∂ α 1 Bcs ∂ θ = − θ 16 − 9 θ 4 − θ 5 − θ 6 − 4 θ 2 − 22 θ 3 + 24 θ − ( θ 2 − 3 θ + θ 3 ) A Bcs ( 1 − θ ) 2 ( 1 + θ ) 2 ( 2 + θ ) 2 A Bcs < 0 ,

(D.13) ∂ α 2 Bcs ∂ θ = 2 32 − 8 A Bcs + 2 θ 2 ( 4 − A Bcs ) + 8 θ ( 6 − A Bcs ) − θ 3 ( 20 − A Bcs ) − 12 θ 4 − 2 θ 5 − θ 6 θ 3 ( 2 + θ ) 2 A Bcs > 0 .

This completes the proof. □

Analysis for Outcome 2

We begin with the stage-2 problem for managers whereby the manager of Firm 1 chooses p ₁ to maximize V ₁ = π ₁ + α ₁ CS. The manager of Firm 2 chooses p ₂ to maximize V ₂ = π ₂ + α ₂ π ₁. Differentiating the objective functions of the managers, we have

(D.14) ∂ V 1 ∂ p 1 = ( 1 − α 1 ) a ( 1 − θ ) + θ p 2 − ( 2 − α 1 ) p 1 1 − θ 2 = 0 ,

(D.15) ∂ V 2 ∂ p 2 = a ( 1 − θ ) + θ ( 1 + α 2 ) p 1 1 − θ 2 = 0 ,

yielding the reaction functions

(D.16) p 1 ( p 2 ) = ( 1 − α 1 ) ( 1 − θ ) a + θ p 2 2 − α 1 , p 2 ( p 1 ) = 1 2 ( 1 − θ ) a + θ ( 1 + α 2 ) p 1 .

The second-order conditions are satisfied as ∂ V 1 2 / ∂ p 1 2 = − ( 2 − α 1 ) / ( 1 − θ 2 ) < 0 and ∂ V 2 2 / ∂ p 2 2 = − 2 / ( 1 − θ 2 ) < 0 . The Best-response functions of the stage-two problem are, respectively,

(D.17) p 1 ( p 2 ) = ( 1 − α 1 ) ( 1 − θ ) a + θ p 2 2 − α 1 ,

(D.18) p 2 ( p 1 ) = 1 2 ( 1 − θ ) a + θ ( 1 + α 2 ) p 1 .

From Eqs. (D.17) and (D.18), we can infer that an increase in the price of the rival firm j causes firm i to increase its price, indicating that prices are strategic complements even when the objective of one firm moves away from considering relative profits. Solving Eqs. (D.17) and (D.18), we obtain the stage 2 solutions as

(D.19) p 1 ( α 1 , α 2 ) = a ( 1 − α 1 ) ( 1 − θ ) ( 2 + θ ) 4 − 2 α 1 − ( 1 − α 1 ) ( 1 + α 2 ) θ 2 ,

(D.20) p 2 ( α 1 , α 2 ) = a ( 1 − θ ) 2 − α 1 + ( 1 − α 1 ) ( 1 + α 2 ) θ 4 − 2 α 1 − ( 1 − α 1 ) ( 1 + α 2 ) θ 2 .

We solve the stage 1 problem for the owners. Notably, each owner chooses the contract weight α _i to maximize profits. Substituting Eqs. (D.19) and (D.20) into the profits, and taking the derivative with respect to the weights, we have the stage 1 best-response functions as

(D.21) α 1 ( α 2 ) = − ( 1 + α 2 ) θ 2 2 − ( 1 + α 2 ) θ 2 ,

(D.22) α 2 ( α 1 ) = θ 2 + θ − ( 1 + θ ) α 1 4 + ( 2 − θ ) θ − ( 1 + θ ) α 1 .

Solving Eqs. (D.21) and (D.22), we obtain the optimal solutions for the weights. Proposition 6 formalizes the result. Substituting the reaction functions given in Eq. (D.16), we get the stage 1 profit functions of the owners as π ₁(α ₁, α ₂) and π ₂(α ₁, α ₂). Differentiating the objective functions of the two firms with respect to α ₁ and α ₂, we have the following reaction functions:

(D.23) α 1 ( α 2 ) = − ( 1 + α 2 ) θ 2 2 − ( 1 + α 2 ) θ 2 , α 2 ( α 1 ) = θ 2 + θ − ( 1 + θ ) α 1 4 + ( 2 − θ ) θ − ( 1 + θ ) α 1 .

From Eq. (D.23), we get the optimal solutions α 1 Bcs and α 2 Bcs as given in Proposition 6. Substituting the optimal solutions in Eq. (D.16), the demand functions as well as the profit functions, we get the optimal solutions as given by Outcome 2. □

Proof of Proposition 9

Comparing the optimal prices charged by the two firms, we have the following on simplifying the expressions:

(D.24) p 1 Bcs − p 2 Bcs = 4 a θ 4 ( 1 − θ ) ( 1 − θ 2 ) 8 θ ( 1 + θ ) > 0 .

Comparing the profit of the two firms, we have the following on simplifying the expressions: π 1 Bcs − π 2 Bcs = ( − A Bcs ( 2 ( 1 + θ ) ( 2 − θ ) − θ 2 ( 2 + θ ) ) + 2 ( 1 + θ ) ( 8 − 10 θ 2 + θ 4 ) + θ 5 ( 2 + θ ) ) / 32 θ 2 ( 1 + θ ) 2 < 0 for all θ ∈ (0, 1]. Therefore, π 1 Bcs < π 2 Bcs . □

Analysis for Outcome 3

By backward induction, we specify the optimal strategies in equilibrium. First, we consider stage-2. In this stage, managers decide supplied quantities to maximize their objective functions. Therefore, both firms face following maximization problems.

(D.25) max q 1 V 1 = max q 1 π 1 + α 1 S W ,

(D.26) max q 2 V 2 = max q 2 π 2 + α 2 π 2 .

From these maximization problems, we obtain the following Best-response functions.

(D.27) q 1 ( q 2 ) = ( α 1 + 1 ) ( a − θ q 2 ) α 1 + 2 ,

(D.28) q 2 ( q 1 ) = a − ( α 2 + 1 ) θ q 1 2 .

From Eqs. (D.27) and (D.28), we get following strategies in stage-2.

(D.29) q 1 = a ( α 1 + 1 ) ( 2 − θ ) 4 − α 1 2 − ( α 2 + 1 ) θ 2 − ( α 2 + 1 ) θ 2 ,

(D.30) q 2 = a ( 2 − α 1 ( α 2 θ + θ − 1 ) − ( α 2 + 1 ) θ ) 4 − α 1 2 − ( α 2 + 1 ) θ 2 − ( α 2 + 1 ) θ 2 .

Next, we analyze stage-1. In this stage, owners choose α _i to maximize their own profits. Using outcomes found earlier, the owners face the following maximization problems.

(D.31) max α i π i ( α 1 , α 2 ) .

Solving the first-order conditions of both firms, we obtain α i Csw . In addition, using α i Csw , we get outcomes in Outcome 3. □

Proof of Proposition A.1

To demonstrate Proposition A.1, we consider α i Ccs − α i Csw .

(D.32) α 1 Ccs − α 1 Csw = − 1 2 A Ccs − 4 − 2 θ + 2 θ 2 + 3 θ 3 2 + θ − 2 θ 2 − θ 3 + A Csw − 4 + 2 θ + 4 θ 2 − 3 θ 3 2 − θ − 2 θ 2 + θ 3 < 0 ,

(D.33) α 2 Ccs − α 2 Csw = 1 θ 2 8 − A Ccs ( 2 − θ ) − 10 θ 2 − 4 θ 3 + 4 θ 4 + 2 θ 5 12 − 4 θ − 5 θ 2 − 2 θ 3 + A Csw − 4 + 2 θ + 2 θ 2 2 − θ > 0 .

□

Analysis for Outcome 4

Differentiating the objective functions of the managers, we have

(D.34) ∂ V 1 ∂ p 1 = a ( 1 − θ ) − ( 2 + α 1 ) p 1 − θ ( 1 + α 1 ) p 2 1 − θ 2 = 0 ,

(D.35) ∂ V 2 ∂ p 2 = a ( 1 − θ ) + θ ( 1 + α 2 ) p 1 − 2 p 2 1 − θ 2 = 0 ,

yielding the reaction functions

(D.36) p 1 ( p 2 ) = a ( 1 − θ ) + θ ( 1 + α 1 ) p 2 2 + α 1 , p 2 ( p 1 ) = 1 2 a ( 1 − θ ) + θ ( 1 + α 2 ) p 1 .

It is noteworthy that the second-order conditions are satisfied as ∂ V 1 2 / ∂ p 1 2 = − ( 2 + α 1 ) / ( 1 − θ 2 ) < 0 and ∂ V 2 2 / ∂ p 2 2 = − 2 / ( 1 − θ 2 ) < 0 . Substituting the reaction functions given in Eq. (D.36), we get the stage 1 profit functions of the owners as π ₁(α ₁, α ₂) and π ₂(α ₁, α ₂). Differentiating the objective functions of the two firms with respect to α ₁ and α ₂, we have the following reaction functions

(D.37) α 1 ( α 2 ) = − ( 1 + α 2 ) θ 2 ( 2 + θ ) ( 2 − θ ) 2 − ( 1 + α 2 ) θ 2 , α 2 ( α 1 ) = θ ( 1 + α 1 ) ( 2 + θ + α 1 ) ( 2 + α 1 ) 2 + θ ( 1 + α 1 ) − θ 2 ( 1 + α 1 ) .

From Eq. (D.37), we get the optimal solutions α 1 Bsw and α 2 Bsw as given in Outcome 4. Substituting the optimal solutions in Eq. (D.36), the demand functions as well as the profit functions, we obtain the optimal solutions as given in the Proposition. □

Analysis for Section 7.1

First, by backward induction, we can obtain the following outcomes straightforwardly.

α i RPEC = − θ 2 + θ , π i RPEC = a 2 ( 2 − θ ) ( 2 + θ ) 16 ( 1 + θ ) , C S RPEC = a 2 ( 2 + θ ) 2 16 ( 1 + θ ) , S W RPEC = a 2 12 + 4 θ − θ 2 16 ( 1 + θ ) ,

where superscript RPEC denotes the case in which both firms adopt the RPE under quantity competition. Additionally, when both firms set consumer surplus as a performance indicator, we get the following outcomes.

α i CSC = 2 + 2 θ + θ 2 − ψ 2 ( 1 + θ ) , π i CSC = 2 a 2 ψ − θ ( 2 + θ ) ψ + 2 − θ 2 2 , C S CSC = 4 a 2 ( 1 + θ ) ψ + 2 − θ 2 2 , S W CSC = 4 a 2 ψ + 1 − θ − θ 2 ψ + 2 − θ 2 2 ,

where superscript CSC denotes the case in which both firms adopt the social performance under quantity competition.^[12] Additionally, ψ ≡ 4 + 8 θ + 4 θ 2 + θ 4 .

Based on the above result, we obtain.

α 2 RPEC − α 2 Ccs = ( 2 − θ ) 2 4 − A Ccs + 8 − A Ccs θ − 6 θ 2 − 6 θ 3 − 3 θ 4 θ 2 ( θ + 2 ) 12 − 5 θ 2 − 4 θ − 2 θ 3 > 0 , α 1 CSC − α 1 Ccs = 2 ψ + θ θ 3 − ψ θ − ψ − A Ccs 2 ( 1 + θ ) 2 − θ − θ 2 > 0 .

Next, we specify the equilibrium strategy in this case. Comparing the outcomes between this section and the previous section, we obtain the following outcomes.

π 1 RPEC − π 1 Ccs = a 2 ( 2 − θ ) θ ( θ 2 + 4 θ − 2 ) + A Ccs − 4 32 1 − θ 2 > 0 , π 2 CSC − π 2 Ccs = − a 2 θ θ 2 ( 3 − θ ) ψ + 2 θ 3 + 3 θ 3 + 2 ( 1 − θ − A Ccs ) + A Ccs + 2 ( A Ccs − 2 ψ ) 16 θ 2 1 − θ 2 < 0 .

Under the small θ, A ^Ccs > 4 is obtained, and θ(θ ² + 4θ − 2) + A ^Ccs − 4 > 0. On the other hand, under the large θ, we can get the large value in θ ² + 4θ − 2, and therefore θ(θ ² + 4θ − 2) + A ^Ccs − 4 > 0 is satisfied. Because we suppose the symmetric profit functions, we can confirm the optimal choice of the performance indicator by considering this comparison. We can confirm the following two facts from this analysis. First, when the rival uses the RPE, the firm can enhance the profit by the RPE as a performance evaluation system. Second, when the rival uses social performance, the firm can enhance the profit by the RPE as a performance evaluation system. This analysis implies that if the firm can choose the performance indicator, then both firms set the RPE as a performance evaluation system in the equilibrium under quantity competition. This outcome leads to Proposition 10.

Lastly, we consider the welfare effect of endogenous choice of performance evaluation systems. To analyze the welfare effect, we compare the asymmetric case with the outcome under the optimal choice in equilibrium.

C S Ccs − C S RPEC = a 2 4 A Ccs − 4 + θ θ − 2 A Ccs + θ θ ( θ ( θ + 4 ) − 14 ) + A Ccs − 4 + 28 − 2 A Ccs 64 θ 2 θ 2 − 1 > 0 , S W Ccs − S W RPEC = − a 2 4 A Ccs − 4 + θ 6 A Ccs + θ − 10 A Ccs + θ θ ( ( θ − 4 ) θ − 38 ) + A Ccs + 44 + 44 − 32 64 θ 2 θ 2 − 1 > 0 .

This result leads to Proposition 11.

Analysis for Section 7.2

By backward induction, we obtain the following profits for firm i, consumer surplus, and social welfare under both cases as follows.

α i RPEB = θ 2 − θ , π i PREB = a 2 ( 2 − θ ) ( 2 + θ ) 16 ( 1 + θ ) , C S RPEB = a 2 ( 2 + θ ) 2 16 ( 1 + θ ) , S W RPEB = a 2 12 + 4 θ − θ 2 16 ( 1 + θ ) ,

where superscript RPEB denotes the case in which both firms adopt the RPE under price competition. Additionally, when both firms set consumer surplus as a performance indicator, we get the following outcomes.

α i CSB = 1 − θ 2 − 1 − θ 2 1 − θ 2 , π i CSB = a 2 1 − θ 2 1 + θ + 1 − θ 2 2 , C S CSB = a 2 2 1 + 1 − θ 2 , S W CSB = a 2 1 + θ + 2 1 − θ 2 1 + θ + 1 − θ 2 2 ,

where superscript CSB indicates the case in which both firms adopt social performance as a performance indicator under price competition.^[13]

Based on the above result, we obtain.

α 2 RPEB − α 2 Bcs = 2 θ 2 + A Bcs − 4 − 2 θ θ 2 ( 2 + θ ) + θ 2 − θ > 0 , α 1 CSB − α 1 Bcs = θ 3 + A Bcs − 2 θ 1 − θ 2 − 4 1 − θ 2 2 ( 2 + θ ) 1 − θ 2 > 0 .

Next, we specify the equilibrium strategy in this case. Comparing the outcomes between this section and the previous section, we obtain the following outcomes.

π 1 RPEB − π 1 Bcs = a 2 ( 2 + θ ) 4 − A Bcs + 2 θ − 2 θ 2 − θ 3 32 ( 1 + θ ) 2 > 0 , π 2 CSB − π 2 Bcs = − a 2 2 A Bcs − θ θ 3 + A Bcs − 2 θ − 8 1 − θ 2 16 θ 2 ( 1 + θ ) < 0 .

Because we suppose the symmetric profit functions, we can confirm the optimal choice of the performance indicator by considering this comparison. We can confirm the following two facts from this analysis. First, when the rival uses the RPE, the firm can enhance the profit by the RPE as a performance evaluation system. Second, when the rival uses social performance, the firm can enhance the profit by the RPE as a performance evaluation system. This analysis implies that if the firm can choose the performance indicator, then both firms set the RPE as a performance evaluation system in the equilibrium under quantity competition. This outcome leads to Proposition 12.

C S Bcs − C S RPEB = a 2 4 4 − A Bcs − θ 6 − 8 θ 5 − 14 θ 4 − 4 + A Bcs θ 3 + 4 5 − A Bcs θ 2 + 32 − 6 A Bcs θ 64 θ 2 ( θ + 1 ) 2 > 0 , S W Bcs − S W RPEB = a 2 4 4 − A Bcs − θ 6 + 10 θ 4 − 4 + A Bcs θ 3 − 4 7 − A Bcs θ 2 − 2 A Bcs θ 64 θ 2 ( θ + 1 ) 2 > 0 .

This result leads to Proposition 13.

Proof of Proposition A.2

Comparing the contract parameter chosen by Firm 1 under CS and SW, we have

α 1 Bcs − α 1 Bsw = 16 − 2 ( A Bcs + A Bsw ) + θ ( A Bcs − A Bsw ) − 16 θ 2 + 2 θ 3 + 6 θ 4 2 ( 4 − 5 θ 2 + θ 4 ) > 0 ,

implying that α 1 Bcs > α 1 Bsw .

Similarly, comparing the contract parameter chosen by Firm 2 under CS and SW, we have

α 2 Bcs − α 2 Bsw = 4 ( 8 − 3 A Bcs + A Bsw ) + 4 θ ( 8 − A Bcs + A Bsw ) + 2 θ 4 − 4 θ 5 + 2 θ 6 − 2 θ 3 ( 4 + A Bcs ) − θ 2 ( 16 − 5 A Bcs − A Bsw ) θ 2 ( 2 + θ ) ( 12 + 4 θ − 5 θ 2 + 2 θ 3 ) > 0 ,

implying that α 2 Bcs > α 2 Bsw . □

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Received: 2023-04-28

Accepted: 2024-03-07

Published Online: 2024-04-16

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Articles in the same Issue

https://doi.org/10.1515/bejeap-2023-0135

Keywords for this article

managerial delegation; price competition; quantity competition; relative performance; social performance