Accounting Letters

Article in Press
The Accounting and Economic Association of Japan
Volume 1, Issue 12024
pp. XX-XX

Weight Placed on the Rival's Profit under
the Relative Performance Evaluation
with Cournot-Bertrand Competition andSequential Decision of the Weight
Jumpei Hamamura
Momoyama Gakuin University
Submitted: September 6, 2023; Revised: September 30, 2023; Accepted November 9, 2023;
Available online XXXX XX, 2023
Abstract
In this note, assuming a duopoly market, we revisit the weight placed on the rival's profit when two firms
choose weights sequentially under Cournot-Bertrand competition. Based on the managerial delegation game,prior relative performance evaluation studies mainly focus on the optimal weight placed on the rival's profit anddemonstrate that the sign of weights depends on economic environments. This study shows that if firms decide
the quantity in a product market under Cournot-Bertrand competition, then the owner sets the positive weighton the rival's profit in managerial compensation contracts.
Keywords: relative performance evaluation; sequential game; Cournot-Bertrand competition; managerial delegation; managerial compensation contractI would like to thank Yasuhiro Ohta (Editor), and an anonymous reviewer for their helpful comments. This work was partially
supported by the JSPS KAKENHI Grant Number JP21K13409, and The Japanese Association of Management Accounting(Study Group in 2022). The usual disclaimers apply.
Corresponding Author: Jumpei Hamamura, 1-1, Manabino, Izumi-city, Osaka 594-1198, Japan.
© 2023 The Accounting and Economic Association of JapanAll rights reserved.

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Accounting Letters, Volume 1, Issue 1 (2024)

1 Introduction
Several managerial compensation practices indicate the use of the relative performance evaluation(RPE). For example, in Japan, Mitsubishi Materials Corporation explicitly states the use of theRPE to compensate the manager.＊ Under the managerial delegation, it is important to consider
optimal weights on peer performances because its incentive system may enhance the firm-wide
performance by inducing the manager's behavior in the managerial accounting practice.
In RPE research, Hamamura (2021) demonstrates that the disadvantaged firm sets a positive
weight on the rival's profit under quantity competition. This is one of the important results because seminal works show that if firms face quantity competition in the product market, thenowners choose the negative weight for strategic substitutes (Aggarwal and Samwick 1999; Fumas1992). Additionally, focusing on industrial profits, several firms cooperate to facilitate the productmarket in practice. In RPE research, as a disadvantaged firm, Hamamura (2021) considers twocases, the inefficient marginal cost firm and the follower. Actually, in practice, we can observe
many examples of both cases.
Following Hamamura (2021), Hamamura (2022) analyzes the optimal level of weights underasymmetric costs and Cournot-Bertrand competition. In practice, it is difficult to distinguish whichfirms set the price or quantity in a product market, and therefore, in several cases, firms face
Cournot-Bertrand competition in a product market. However, Hamamura (2022) does not examine
the other disadvantage case. In other words, previous studies do not consider the sequential gameunder Cournot-Bertrand competition to specify the optimal weight placed on the rival's profit under accounting-based RPEs. Therefore, supposing RPEs to compensate managers, we revisit theoptimal weight placed on the rival's profit when two firms choose weights sequentially under
Cournot-Bertrand competition.
Let us summarize our results. If the leader in the sequential game decides the quantity and thefollower decides the price, the leader sets the positive weight, and the follower sets the negative
weight on the rival's profit. On the other hand, if the leader in the sequential game decides theprice and the follower decides the quantity, then the leader sets the negative weight, and the follower sets the positive weight on the rival's profit under the RPE.

＊We can observe this information on managerial compensation contracts in the Integration Report of Mitsubishi Materials Corporation. URL: https://ssl4.eir-parts.net/doc/5711/ir̲material̲for̲fiscal̲ym7/121391/00.pdf (Last accessed on September 4,2023)†Xu and Matsumura (2022) also considers Cournot-Bertrand competition with convex costs and RPEs. However, they do notconsider the sequential game. Additionally, because, while Hamamura (2021) supposes quantity competition, our paper examines Cournot-Bertrand competition, we do not call the follower as a disadvantaged firm in this paper.

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2 ModelFirms 1 and 2 engage in product market competition. The profit function of firm
, whererepresents the market price,quantity of firm . We assume

is the marginal cost, andisdenotes thefor simplicity. Owners compensate managers using prof-

it-based RPEs. Thus, we assume(1)
whereis the payoff function of firm 's manager, andrepresents the weight assigned to therival's profit. Managers decide on the quantity or price in a duopoly market to maximize Eqs. (1).
Owners choose the level of

to maximize their profits,.

The timeline of events is as follows. First, the owner of Firm 1 decides the level ofSecond, the owner of Firm 2 decides the level ofin Stage 1.

in Stage 2. Lastly, the manager chooses thequantity or price in Stage 3.

3 Results
We identify the sub-game perfect Nash equilibrium (SPNE) by backward induction, considering
two cases: (i) Firm 1 decides the quantity, and Firm 2 decides the price, and (ii) Firm 1 decides
the price, and Firm 2 decides the quantity.
3.1 Firm 1 Decides the Quantity, and Firm 2 Decides the Price
We suppose the following demand functions. ‡whererepresents the constant greater than zero,tution, and superscript

denotes the degree of product substi-indicates the case in which Firm 1 decides the quantity and Firm 2

decides the price in the product market.
In Stage 3, from the first-order condition (FOC), we obtain the following best response functions.§‡

Following earlier studies (e.g., Singh and Vives 1984), we define the utility function of a representative consumer as
whereand. Using this utility function and the budget constraint, we. From this analysis, we obtain inverse demand functions which are represented as. Rearranging the inverse demand function, we obtain the demand functions under Cournot-Bertrand competition. Tremblay and Tremblay (2011) is a typical paper to derive this type of the demand functions.
§We can confirm that the second order condition is satisfied in this case because we getand.
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Accounting Letters, Volume 1, Issue 1 (2024)

One can notice that, under,increases asincreases, anddecreases asincreases.

This outcome indicates that it is important to consider the rival's decision variable in a productmarket for strategic relationships. In other words,
tegic substitutes tois strategic complements tois stra-. When we examine the weight placed on the rival's profit, its strategic rela-tionship plays an important role. Additionally, if Firm 1 rises
On the other hand, if Firm 2 enhances
the incentive to raise, and, then, thendecreases andincreases.

increases andincreases. Therefore, Firm 1 has, and Firm 2 has the incentive to reduce. Using the above outcomes, weobtain the following outcomes in this stage.

Next, we analyze the optimal decision by the follower in Stage 2. From the FOC, Firm 2 sets thefollowing weight.

Because, if Firm 1 raises

then Firm 2 also enhances
the large,. Therefore, to enhance Firm 2's price for tacit collusion, Firm 1 sets

for strategic complements. Using this outcome, we identify

and summarize theresult as follows.
Result 1-–4–

Jumpei Hamamura, Weight Placed on the Rival's Profit under the Relative Performance Evaluation with CournotBertrand Competition and Sequential Decision of the Weight

We discuss the sign of. Because we assume, we obtain

andstraightforwardly. We summarize this result in the following proposition.
Proposition 1'Classical studies (Aggarwal and Samwick 1999; Fumas 1992) demonstrate that when the firm
decides the strategic complementary variable in the product market, the firm sets the positiveweight. On the other hand, when the firm decides the strategic substitute variable in the productmarket, the firm sets the negative weight. Our result may indicate that, under Cournot-Bertrand
competition, it does not play an important role in determining whether the sign of the weight ispositive or negative.
3.2 Firm 1 Decides the Price, and Firm 2 Decides the Quantity
Next, we consider the case in which Firm 1 (leader) decides the price, and Firm 2 (follower) decides the quantity. In this case, we suppose the following demand functions.

where superscript

represents the case in which Firm 1 decides the price and Firm 2 decides

the quantity in the product market.
In Stage 3, considering the FOC, we obtain

Compared with the previous analysis, this is a reversal outcome, and the effect ofon firms isdifferent. In other words, the strategic effects are different from the previous case.
Using the above outcome, we explore the optimal

Because, underin Stage 2. From the FOC, we get, we obtain, we can confirm that, contrarily to the previous section,

has a negative impact on. This outcome implies that, to reduce Firm 2's quantity for credible threaten, Firm 1 sets thesmallfor a strategic substitute. We identify

and summarize the result as follows.

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Result 2--

Using the above outcomes, we specify the sing ofgetandstraightforwardly because. Under the positive conditions, we can. From this outcome, weconduct the following proposition.
Proposition 2'This is a just reversal outcome of Proposition 1. In other words, the decision timing does notplay an important role, and the strategic relationship has a significant effect on the weight.

4 Discussion
This note documents the optimal weight placed on the rival's profit under the RPE with the sequential game and Cournot-Bertrand competition. If the leader decides the quantity, the leader
sets the positive weight, and the follower sets the negative weight on the rival's profit. In contrast,
if the leader decides the price, the leader sets the negative weight, and the follower sets the positive weight on the rival's profit.
According to Hamamura (2021, 2022), one can anticipate that we obtain the analogies of Hamamura (2022). That is, depending on parameters, both firms set the positive weight placed on therival's profit in a specific case. However, our result demonstrates that, in our assumption, one firm
only sets the positive weight. This result implies that while the strategic effect has a significant
role, the decision timing does not affect the weight. In particular, while, according to Hamamura–6–

Jumpei Hamamura, Weight Placed on the Rival's Profit under the Relative Performance Evaluation with CournotBertrand Competition and Sequential Decision of the Weight(2022), the difference in marginal costs is the most important factor to decide the weight, our paper does not consider the continuous variable. In other words, the difference between firms is represented as two discrete variables, and the effect of its difference is fixed in our model. One canconsider the reason for differences between our study and Hamamura (2022).
Our analysis may help to understand the managerial compensation contract in practice as amain contribution. According to Hamamura and Inoue (2023), empirical results suggest that Japanese firms set peer compensation which depends on peer profits as a benchmark. This means that
if the rival earns a large profit, then the manager obtains a large compensation. However, several
firms apply the RPE to reward the manager. Our model explains this practice which is the asymmetric rewarding system.
In future research, one may consider the asymmetric performance evaluation system proposed
by Hamamura and Ramani (2023), because, recently, many firms consider social performance toevaluate the manager in practice.
References
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Theory and evidence.

54(6): 1999-2043.

Fumas, V. S. 1992. Relative performance evaluation of management: The effects on industrial competition and risk sharing.

-10(3): 473-489.
Hamamura, J. 2021. What level do disadvantaged firms weight rivals' profits in relative performance evaluations under quantitycompetition?

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Hamamura, J. 2022. Weight assigned to a rival's profit by an advantaged firm in relative performance evaluation with
Cournot-Bertrand competition.

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Hamamura, J., and K. Inoue. 2023.
. Available at SSRN: ID 4398675. DOI: https://dx.doi.
org/10.2139/ssrn.4398675.
Hamamura, J., and V. Ramani. 2023. Social performance vs. relative performance evaluation, asymmetric costs, and quantity
competition under managerial delegation.

44(3): 1706-1719.

Singh, N., and X. Vives. 1984. Price and quantity competition in a differentiated duopoly.

15(4): 546-554.
Tremblay, C. H., and V. J. Tremblay. 2011. The Cournot-Bertrand model and the degree of product differentiation.
111(3): 233-235.
Xu, L., and T. Matsumura. 2022. Welfare-reducing price competition under relative performance delegation with convex costs.
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