Relative profit maximization and Bertrand equilibrium with convex cost functions

Satoh, Atsuhiro; Tanaka, Yasuhito Working Paper Relative profit maximization and Bertrand equilibrium with convex cost functions Economics Discussion Papers, No. 2014-7 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Satoh, Atsuhiro; Tanaka, Yasuhito (2014) : Relative profit maximization and Bertrand equilibrium with convex cost functions, Economics Discussion Papers, No. 2014-7, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/92416 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Licensed under the Creative Commons License - Attribution 3.0 Discussion Paper No. 2014-7 | February 11, 2014 | http://www.economics-ejournal.org/economics/discussionpapers/2014-7 Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions Atsuhiro Satoh and Yasuhito Tanaka Abstract The authors study pure strategy Bertrand equilibria in a duopoly in which two firms produce a homogeneous good with convex cost functions, and they seek to maximize the weighted sum of their absolute and relative profits. They show that there exists a range of the equilibrium price in duopolistic equilibria. This range of the equilibrium price is narrower and lower than the range of the equilibrium price in duopolistic equilibria under pure absolute profit maximization, and the larger the weight on the relative profit, the narrower and lower the range of the equilibrium price. In this sense relative profit maximization is more aggressive than absolute profit maximization. JEL D43 L13 Keywords Bertrand equilibrium; convex cost function; relative profit maximization Authors Atsuhiro Satoh, Faculty of Economics, Osaka University of Commerce, Higashi-Osaka, Osaka, 570-8505, Japan, [email protected] Yasuhito Tanaka, Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto, 602-8580, Japan, [email protected] Citation Atsuhiro Satoh and Yasuhito Tanaka (2014). Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions. Economics Discussion Papers, No 2014-7, Kiel Institute for the World Economy. http:// www.economics-ejournal.org/economics/discussionpapers/2014-7 conomics Discussion Paper 1 Introduction Using a model developed by Dastidar (1995) we study pure strategy Bertrand equilibria in a duopoly in which two firms produce a homogeneous good with convex cost functions, and they seek to maximize the weighted sum of their absolute and relative profits instead of their absolute profits themselves. The relative profit of a firm is the difference between its absolute profit and the absolute profit of the rival firm. For analyses about relative profit maximization please see Gibbons and Murphy (1990), Lu (2011), Matsumura, Matsushima and Cato (2013), Schaffer (1989), Vega-Redondo (1997) and Miller and Pazgal (2001)1. We think that seeking for relative profit or utility is based on the nature of human. Even if a person earns a big money, if his brother/sister or close friend earns a bigger money than him, he is not sufficiently happy and may be disappointed. On the other hand, even if he is very poor, if his neighbor is more poor, he may be consoled by that fact. We show that there exists a range of the equilibrium price in duopolistic equilibria. This range of equilibrium price is narrower and lower than the range of the equilibrium price in duopolistic equilibria under pure absolute profit maximization2, and the larger the weight on the relative profit, the narrower and lower the range of the equilibrium price. In this sense relative profit maximization is more aggressive than absolute profit maximization. In another paper, Satoh and Tanaka (2013), we have shown a similar result in a case of linear demand functions and quadratic cost functions. In this paper we extend this result to a case of general demand functions and convex cost functions. 2 The model There are two firms, A and B. They produce a homogeneous good. The price of the good of Firm A is 𝑝𝐴and the price of the good of Firm B is 𝑝𝐵. The outputs of Firm A and B are denoted, respectively, by 𝑥𝐴and 𝑥𝐵. The firms set the prices of their goods, and consumers buy the good from the firm whose price is lower. Let 𝑝=min{𝑝𝐴,𝑝𝐵}. Consumers' demand is represented by the demand function 𝐷(𝑝). The cost functions of Firm A and B are 𝑐𝐴(𝑥𝐴)and 𝑐𝐵(𝑥𝐵). 1In Vega-Redondo (1997) it was shown that the equilibrium in a Cournot oligopoly with a homogeneous good under relative profit maximization is equivalent to the competitive equilibrium. But as shown in this paper the equilibrium in a Bertrand duopoly with a homogeneous good under relative profit maximization may not be equivalent to the competitive equilibrium 2Dastidar (1995) proved that there exists a range of the equilibrium price in duopolistic equilibria under absolute profit maximization. www.economics-ejournal.org 2 conomics Discussion Paper Similarly to the model in Dastidar (1995) we make the following assumptions. 1. 𝐷(𝑝)is continuous and twice continuously differentiable. 2. There exists finite positive numbers 𝑝𝑚𝑎𝑥 and 𝑥𝑚𝑎𝑥 such that 𝐷(𝑝𝑚𝑎𝑥)=0 and 𝐷(0)=𝑥𝑚𝑎𝑥, and 𝐷′(𝑝)<0for 0≤𝑝≤𝑝𝑚𝑎𝑥. 3. 𝑐(𝑥𝐴)and 𝑐(𝑥𝐵)are continuous, twice continuously differentiable and strictly convex. Further we assume that there is no fixed cost and two firms have the same cost function. Thus, 𝑐𝑖(0)=0for 𝑖∈{𝐴,𝐵}. If 𝑝𝐴=𝑝𝐵, each firm acquires a half of the demand, and two firms constitute a duopoly. Thus, if 𝑝𝐴=𝑝𝐵, we have 𝑥𝐴=𝑥𝐵=1 2𝐷(𝑝). On the other hand if 𝑝𝐴<𝑝𝐵(or 𝑝𝐵<𝑝𝐴) Firm A (or Firm B) acquires total demand, and it becomes a monopolist. If 𝑝𝐴<𝑝𝐵, the absolute profit of Firm A is 𝜋𝑀 𝐴(𝑝)=𝑝𝐷(𝑝)−𝑐𝐴(𝐷(𝑝)). 𝑀indicates monopoly. Of course the profit of Firm B is zero. Similarly if 𝑝𝐵<𝑝𝐴, we have 𝜋𝑀 𝐵(𝑝)=𝑝𝐷(𝑝)−𝑐𝐵(𝐷(𝑝)). The profit of Firm A is zero. On the other hand, if 𝑝𝐴=𝑝𝐵, the absolute profits of Firm A and B are 𝜋𝐷 𝐴(𝑝)=1 2𝑝𝐷(𝑝)−𝑐(𝑥𝐴),𝑥𝐴=1 2𝐷(𝑝), and 𝜋𝐷 𝐵(𝑝)=1 2𝑝𝐷(𝑝)−𝑐(𝑥𝐵),𝑥𝐵=1 2𝐷(𝑝). 𝐷indicates duopoly. In this case 𝑝=𝑝𝐴=𝑝𝐵. The objective of Firm A is the weighted sum of its absolute profit and its relative profit. In a duopoly it is expressed as follows. Π𝐷 𝐴=(1−𝛼)𝜋𝐷 𝐴+𝛼(𝜋𝐷 𝐴−𝜋𝐷 𝐵)=𝜋𝐷 𝐴−𝛼𝜋𝐷 𝐵, and the objective of Firm B is Π𝐷 𝐵=(1−𝛼)𝜋𝐷 𝐵+𝛼(𝜋𝐷 𝐵−𝜋𝐷 𝐴)=𝜋𝐷 𝐵−𝛼𝜋𝐷 𝐴, www.economics-ejournal.org 3 conomics Discussion Paper where0<𝛼<1. Call a firm in a duopoly a duopolist. Since, at a duopolistic equilibrium 𝜋𝐷 𝐴=𝜋𝐷 𝐵, we have Π𝐷 𝐴=Π𝐷 𝐵=(1−𝛼)𝜋𝐷 𝐴. We assume max 𝑝𝜋𝑀 𝑖(𝑝)>0, and max 𝑝𝜋𝐷 𝑖(𝑝)>0for 𝑖∈{𝐴,𝐵}. In a monopoly the absolute profit of a firm other than the monopolist is zero. Thus, the absolute profit and the relative profit of the monopolist are equal, and the objective of the monopolist is its absolute profit, that is, if Firm A is a monopolist, Π𝑀 𝐴=𝜋𝑀 𝐴, and if Firm B is a monopolist, Π𝑀 𝐵=𝜋𝑀 𝐵. Without loss of generality we assume 𝑝𝐴≤𝑝𝐵. 3 Preliminary results According to Dastidar (1995) for 𝑖∈{𝐴,𝐵}we define 𝑝𝑖such that 𝜋𝐷 𝑖( 𝑝𝑖)=0, 𝑝𝑖such that 𝜋𝑀 𝑖( 𝑝𝑖)=0, 𝑝𝑖such that 𝜋𝐷 𝑖( 𝑝𝑖)=𝜋𝑀 𝑖( 𝑝𝑖). By symmetry of the model 𝑝𝐴= 𝑝𝐵,𝑝𝐴= 𝑝𝐵and 𝑝𝐴= 𝑝𝐵. So, we denote them, respectively, by 𝑝,𝑝and 𝑝. In Dastidar (1995) the following results have been proved. 1. There exists a unique 𝑝in [0,𝑝𝑚𝑎𝑥). (Lemma 1 in Dastidar (1995)) 2. There exists a unique 𝑝in [0,𝑝𝑚𝑎𝑥). (Lemma 4 in Dastidar (1995)) 3. There exists a unique 𝑝in [0,𝑝𝑚𝑎𝑥). (Lemma 5 in Dastidar (1995)) www.economics-ejournal.org 4 conomics Discussion Paper 4. 𝑝< 𝑝< 𝑝. (Lemma 6 in Dastidar (1995)) Now we define another critical price 𝑝∗ 𝑖by Π𝐷 𝑖(𝑝∗ 𝑖)=Π𝑀 𝑖(𝑝∗ 𝑖). Also by symmetry we have 𝑝∗ 𝐴=𝑝∗ 𝐵, and so denote them by 𝑝∗. We show the following lemmas. Lemma 1. There exists a unique 𝑝∗in [0,𝑝𝑚𝑎𝑥), and 𝑝<𝑝∗< 𝑝 for 0<𝛼<1. Proof. Note that Π𝐷 𝑖(𝑝)=(1−𝛼)𝜋𝐷 𝑖(𝑝), and Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)=𝜋𝑀 𝑖(𝑝)−(1−𝛼)𝜋𝐷 𝑖(𝑝). If 𝑝= 𝑝,𝜋𝑀 𝑖(𝑝)−𝜋𝐷 𝑖(𝑝)=0, and so Π𝑀 𝑖( 𝑝)−Π𝐷 𝑖( 𝑝)=𝛼𝜋𝐷 𝑖( 𝑝). If 𝑝= 𝑝,𝜋𝑀 𝑖(𝑝)=0, and so Π𝑀 𝑖( 𝑝)−Π𝐷 𝑖( 𝑝)=−(1−𝛼)𝜋𝐷 𝑖( 𝑝). Now 𝜕𝜋𝐷 𝑖(𝑝) 𝜕𝑝 =1 2{𝐷(𝑝)+𝐷′(𝑝)[𝑝−𝑐′ 𝑖(1 2𝐷(𝑝))]}.(1) When 𝜋𝐷 𝑖(𝑝)≤0, we have −𝑐𝑖(1 2𝐷(𝑝))≤−1 2𝑝𝐷(𝑝). Since 𝑐𝑖(⋅)is strictly convex, 𝑐𝑖(1 2𝐷(𝑝))−𝑐𝑖(0)=𝑐𝑖(1 2𝐷(𝑝))<1 2𝐷(𝑝)𝑐′ 𝑖(1 2𝐷(𝑝)), or −𝑐𝑖(1 2𝐷(𝑝))>−1 2𝐷(𝑝)𝑐′ 𝑖(1 2𝐷(𝑝)). This means −1 2𝐷(𝑝)𝑐′ 𝑖(1 2𝐷(𝑝))<−1 2𝑝𝐷(𝑝). www.economics-ejournal.org 5 conomics Discussion Paper Therefore, 𝑝<𝑐′ 𝑖(1 2𝐷(𝑝)). Since 𝐷′(𝑝)<0, from (1) we find that when 𝜋𝐷 𝑖(𝑝)≤0,𝜕𝜋𝐷 𝑖(𝑝) 𝜕𝑝 >0holds. Thus, the continuity of 𝜋𝐷 𝑖(𝑝)and the uniqueness of 𝑝means that 𝜋𝐷 𝑖( 𝑝)>0, and so Π𝑀 𝑖( 𝑝)−Π𝐷 𝑖( 𝑝)>0for 𝛼>0because 𝑝< 𝑝. Similarly Π𝑀 𝑖( 𝑝)−Π𝐷 𝑖( 𝑝)=−(1−𝛼)𝜋𝐷( 𝑝)<0for 0<𝛼<1because 𝑝< 𝑝. Therefore, by the continuity of 𝜋𝑀 𝑖(𝑝) and 𝜋𝐷 𝑖(𝑝) there exists 𝑝∗such that Π𝑀 𝑖(𝑝∗)−Π𝐷 𝑖(𝑝∗)=0between 𝑝and 𝑝, that is, 𝑝<𝑝∗< 𝑝. We show uniqueness of 𝑝∗. Note that Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)=𝑝𝐷(𝑝)−𝑐𝑖(𝐷(𝑝))−(1−𝛼)[1 2𝑝𝐷(𝑝)−𝑐𝑖(1 2𝐷(𝑝))] =1+𝛼 2𝑝𝐷(𝑝)−𝑐𝑖(𝐷(𝑝))+(1−𝛼)𝑐𝑖(1 2𝐷(𝑝)). Now 𝜕 𝜕𝑝[Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)]=1+𝛼 2𝐷(𝑝)+𝐷′(𝑝){1+𝛼 2[𝑝−𝑐′ 𝑖(𝐷(𝑝))](2) +1−𝛼 2[𝑐′ 𝑖(1 2𝐷(𝑝))−𝑐′ 𝑖(𝐷(𝑝))]} When Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)≤0, we have −𝑐𝑖(𝐷(𝑝))+(1−𝛼)𝑐𝑖(1 2𝐷(𝑝))≤−1+𝛼 2𝑝𝐷(𝑝). Since 𝑐𝑖(⋅)is strictly convex, 𝑐𝑖(𝐷(𝑝))−𝑐𝑖(0)=𝑐𝑖(𝐷(𝑝))<𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝)), 𝑐𝑖(𝐷(𝑝))−𝑐𝑖(1 2𝐷(𝑝))<1 2𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝)), or −𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝))<−𝑐𝑖(𝐷(𝑝)), −1 2𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝))<−𝑐𝑖(𝐷(𝑝))+𝑐𝑖(1 2𝐷(𝑝)), From them −1+𝛼 2𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝))<−𝑐𝑖(𝐷(𝑝))+(1−𝛼)𝑐𝑖(1 2𝐷(𝑝)). It means 𝑝<𝑐′ 𝑖(𝐷(𝑝)). www.economics-ejournal.org 6 conomics Discussion Paper Also we have 𝑐′ 𝑖(1 2𝐷(𝑝))<𝑐′ 𝑖(𝐷(𝑝)). Since 𝐷′(𝑝)<0, from (2) we find that when Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)≤0, 𝜕 𝜕𝑝[Π𝑀 𝑖(𝑝)−Π𝐷 𝑖(𝑝)]>0holds. Since Π𝑀 𝑖(𝑝)and Π𝐷 𝑖(𝑝)are continuously differentiable, this fact implies that 𝑝∗is unique. Lemma 2. 𝑝∗is decreasing with respect to 𝛼. Proof. 𝑝∗satisfies Π𝑀 𝑖(𝑝∗)−Π𝐷 𝑖(𝑝∗)=𝑝∗𝐷(𝑝∗)−𝑐𝑖(𝐷(𝑝∗))−(1−𝛼)[1 2𝑝∗𝐷(𝑝∗)−𝑐𝑖(1 2𝐷(𝑝∗))]=0. Differentiating this with respect to 𝛼, (𝜕 𝜕𝑝[Π𝑀 𝑖(𝑝∗)−Π𝐷 𝑖(𝑝∗)])|𝑝=𝑝∗𝑑𝑝∗ 𝑑𝛼 =−[1 2𝑝∗𝐷(𝑝∗)−𝑐𝑖(1 2𝐷(𝑝∗))]. This is negative because 𝜋𝐷 𝑖(𝑝∗)=1 2𝑝∗𝐷(𝑝∗)−𝑐𝑖(1 2𝐷(𝑝∗))>0for 𝑝<𝑝∗. 4 Pure strategy Bertrand equilibrium We verify the following result. Lemma 3. For 𝑝> 𝑝we have Π𝑀 𝑖(𝑝)=𝜋𝑀 𝑖(𝑝)>0, and for 𝑝< 𝑝 Π𝑀 𝑖(𝑝)=𝜋𝑀 𝑖(𝑝)<0. Proof. Now 𝜕𝜋𝑀 𝑖(𝑝) 𝜕𝑝 ={𝐷(𝑝)+𝐷′(𝑝)[𝑝−𝑐′ 𝑖(𝐷(𝑝))]}.(3) When 𝜋𝑀 𝑖(𝑝)≤0, we have −𝑐𝑖(𝐷(𝑝))≤−𝑝𝐷(𝑝). www.economics-ejournal.org 7 conomics Discussion Paper Since 𝑐𝑖(⋅)is strictly convex, 𝑐𝑖(𝐷(𝑝))<𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝)), or −𝑐𝑖(𝐷(𝑝))>−𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝)). This means −𝐷(𝑝)𝑐′ 𝑖(𝐷(𝑝))<−𝑝𝐷(𝑝). Therefore, 𝑝<𝑐′ 𝑖(𝐷(𝑝)). Since 𝐷′(𝑝)<0, from (3) we find that when 𝜋𝑀 𝑖(𝑝)≤0,𝜕𝜋𝑀 𝑖(𝑝) 𝜕𝑝 >0holds. Thus, the continuity of 𝜋𝑀 𝑖(𝑝)and the uniqueness of 𝑝means that 𝜋𝑀 𝑖(𝑝)>0for 𝑝> 𝑝 and 𝜋𝑀 𝑖(𝑝)<0for 𝑝< 𝑝. First we show non-existence of monopolistic equilibrium. Theorem 1. There is no monopolistic equilibrium. Proof. A monopolistic equilibrium is an equilibrium where Firm A is the monopolist. Suppose that 𝑝𝐴<𝑝𝐵and 𝑝𝐴> 𝑝. Then, Firm B can set 𝑝𝐵slightly lower than 𝑝𝐴and earn the positive profit. If 𝑝𝐴<𝑝𝐵and 𝑝𝐴= 𝑝, Firm A can set 𝑝𝐴 slightly higher than 𝑝but lower than 𝑝𝐵and earn the positive profit, or Firm B can set 𝑝𝐵=𝑝𝐴and earn the positive profit in a duopoly ( 𝑝< 𝑝). Of course 𝑝𝐴< 𝑝is not profitable for Firm A. Next we show Theorem 2. There exists a range of the equilibrium price [ 𝑝,𝑝∗]in a duopoly. Proof. 1. Suppose 𝑝𝐴=𝑝𝐵=𝑝and 𝑝∗<𝑝< 𝑝. The relative profits of the firms are zero. Firm B (or A) can set 𝑝𝐵(or 𝑝𝐴) slightly lower then 𝑝, and earn the positive absolute profit as a monopolist. Although that absolute profit is smaller than its absolute profit in a duopolistic equilibrium (because 𝑝< 𝑝), its relative profit is positive and it is equal to its absolute profit because the profit of the rival firm is zero, and we have Π𝑀 𝐴>Π𝐷 𝐴. . www.economics-ejournal.org 8