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Price versus quantity in a mixed duopoly: The case of relative profit maximization
Economic Modelling 44 (2015) 37–43
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Price versus quantity in a mixed duopoly: The case of relative profit maximization☆ Yasuhiko Nakamura ⁎ College of Economics, Nihon University, Japan
a r t i c l e
i n f o
Article history: Accepted 19 September 2014 Available online xxxx Keywords: Quantity competition Price competition Mixed markets Differentiated products Relative profit maximization
a b s t r a c t This paper examines the endogenous choice problem of each firm's price or quantity contract in a mixed duopoly composed of one social welfare maximizing public firm and one relative profit-maximizing private firm. In this paper, we show that unless the degree of product differentiation and the degree of importance of the private firm's relative performance are low, the quantity competition can analytically become the unique equilibrium market structure on the basis of the dominant strategies of the public firm and the private firm when the degree of importance of firm 1's relative performance is sufficiently high. This result contrasts strikingly with that obtained in a standard mixed duopoly composed of one social welfare-maximizing public firm and one absolute profit maximizing private firm. In addition, even in the area wherein both the degree of product differentiation and the degree of the private firm's relative performance are sufficiently low, through numerical simulations, the quantity competition tends to become the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas the price competition tends to become the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The purpose of this paper is to explore the equilibrium result for the endogenous selection of strategic variables in a mixed duopoly with differentiated goods composed of one social welfare-maximizing public firm and one generalized relative profit-maximizing private firm. In their recent work, Matsumura and Ogawa (2012) found that in a standard mixed duopoly with differentiated goods in which the private firm maximizes its absolute profit, the dominant strategies of both the public firm and the private firm are price contracts, which imply that price-setting competition is observed in the equilibrium regardless of the relation between the goods the firms produce. In this paper, we check the equilibrium result for the endogenous selection of strategic variables by the public firm and the private firm against the introduction of relative profit maximization for the private firm whose objective function is a weighted sum of its absolute profit and the absolute profit of its rival public firm.
☆ We would like to thank an anonymous referee for her/his helpful comments and suggestions. We are grateful for the financial support of KAKENHI (25870113). All remaining errors are our own. ⁎ Corresponding author at: College of Economics, Nihon University, 1-3-2, Misaki-cho, Chiyoda-ku, Tokyo 101-8360, Japan. Tel./fax: +81 3 3219 3497. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econmod.2014.09.019 0264-9993/© 2014 Elsevier B.V. All rights reserved.
Many studies have conducted equilibrium analyses on mixed oligopoly (duopoly) in various economic contexts; however, most supposed quantity competition; an exception is Matsumura and Ogawa (2012).1 In the context of private oligopoly, Singh and Vives (1984), Klemperer and Meyer (1986), and others have indicated that in the real world economy, firms frequently choose to use either their price level or their output level as their strategic contract in market competition.2 In addition to the importance of the endogenous choice of each firm's strategic variable, recent works on both private oligopoly (duopoly) and mixed oligopoly (duopoly) have investigated the relative performance approach. In the real world economy, as indicated in Alchian (1950) and Vega-Redondo (1997) in the context of evolutionary stability and in Coats and Neilson (2005) in the context of laboratory and experimental issues, when firms' owners evaluate their managers' performances,
1 As described above, Matsumura and Ogawa (2012) showed the advantage of pricesetting competition in the standard mixed duopolistic market regardless of the relation between the goods, that is, substitutability or complementarity, by allowing the endogenous determination of each firm's strategic contract in the market. 2 In the private oligopolistic context, Tanaka (2001a,b), and Tasnádi (2006) checked the robustness of the results obtained in Singh and Vives (1984) and Klemperer and Meyer (1986) such that in a private duopoly market, quantity competition can be observed in the equilibrium when the products are substitutes whereas price competition can be observed in the equilibrium when the products are complements.
38
Y. Nakamura / Economic Modelling 44 (2015) 37–43
they use their relative profits rather than their absolute profits.3 Most recently, Matsumura and Matsushima (2012) investigated the relationship between the degree of competition and the stability of collusive behavior by introducing the element of relative performance into the objective functions of firms. In this paper, given the above-mentioned importance of the endogenous selection of each firm's strategic variable and relative profit as the objective function of the private firm, we take both elements into account in our analysis. In this paper, by introducing the degree of importance of the private firm's relative performance, we show that unless both the degree of product differentiation and the degree of importance of the private firm's relative performance are sufficiently low, the quantity competition can become the unique equilibrium market structure on the basis of the dominant strategies of the public firm and the private firm when the degree of importance of the private firm's relative performance is sufficiently high. On the other hand, unless both the degree of product differentiation and each firm's relative performance are sufficiently low, the price competition can be observed as the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low. Furthermore, through the simulation analysis, we find that the quantity competition can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas when the degree of importance of firm 1's relative performance is sufficiently low, the price competition can become the unique equilibrium market structure. In addition, we show that the highest social welfare is achieved in the quantity competition which can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas the highest social welfare is achieved in the price competition which can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low.4 Thus, the corresponding regulatory authority does not need to prescribe a strong policy from the viewpoint of social welfare as long as the degree of importance of the private firm's relative performance is sufficiently high or low. In sum, the advantage of the price competition under a standard mixed duopoly with differentiated goods as in Matsumura and Ogawa (2012) decreases with the introduction of the degree of importance of the private firm's relative performance. Thus, in a mixed duopoly in which the private firm maximizes its generalized relative profit, the fact that the quantity competition is supposed is not unnatural, different from the mixed market that a private firm maximizes its absolute profit. The remainder of this paper is organized as follows. In Section 2, we formulate a mixed duopolistic model with substitutable goods composed of a social welfare-maximizing public firm and a generalized relative profit-maximizing private firm in order to explore their endogenous selection of strategic contracts. In Section 3, we derive the subgame perfect Nash equilibrium in the model developed in Section 2 and give the intuition behind the results for the equilibrium market structures. Section 4 concludes with several remarks. In Appendix A, we provide the concrete values of the payoffs of both the public firm and the private firm on the basis of numerical analysis of the degree of product differentiation and the degree of importance of the private firm's relative performance when both the levels are sufficiently low.
(firm 1) whose objective function is a weighted sum of its own profit and firm 0's absolute profit. Firms 0 and 1 produce differentiated commodities for which the inverse demand functions are given by pi = a − qi − bqj, (i, j = 0, 1; i ≠ j), where pi and qi are firm i's price and output levels, respectively, a is the demand parameter that denotes the market size, and b ∈ (0, 1). A positive (negative) value of b indicates that the goods produced by firms 0 and 1 are substitutes (complements). We suppose that the marginal production costs of firms 0 and 1 are constant and identical. Let c denote firm i's marginal cost, (i = 0, 1). In addition, we assume that a N c in order to ensure all the equilibrium market outcomes. The objective function of firm 0, V0, is represented as follows: " # q2 þ 2bq0 q1 þ q21 −p0 q0 −p V 0 ¼ W ¼ ðp0 −cÞq0 þ ðp1 −cÞq1 þ aðq0 þ q1 Þ− 0 ; 1 q1 t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ producer surplus ¼ consumer surplus
which is equal to the total social welfare.5 On the other hand, firm 1 maximizes its generalized relative profit as follows: V1 = π1 − απ0, where πi denotes firm i's absolute profit which is equal to (pi − c)qi and α ∈ (0, 1) is the degree of importance of firm 1's relative performance, (i = 0, 1). The game runs as follows. In the first stage, firms 0 and 1 simultaneously choose whether to adopt price contracts or quantity contracts. In the second stage, after observing the choice of the strategic contract by the rival firm in the first stage, firms 0 and 1 simultaneously and independently choose either their price or quantity in the market, on the basis of their decisions in the first stage. 3. Equilibrium analysis of the strategic variable selection game In this section, we investigate the four possible subgames which are classified on the basis of each firm's strategic variable selected in the first stage: (i) firms 0 and 1 choose price contracts (p–p game); (ii) firms 0 and 1 choose quantity contracts (q–q game); firm 0 chooses a price contract while firm 1 chooses a quantity contract (p–q game); (iv) firm 0 chooses a quantity contract while firm 1 chooses a price contract (q–p game). Similar to Matsumrua and Ogawa (2012), we assume that the solutions in the four subgames are interior—that is, the equilibrium prices and quantities of both firms are strictly positive. 3.1. p–p game For the price-setting competition, the reaction functions of firms 0 and 1 are given as follows: p0 ðp1 ; b; α Þ ¼ c−bc þ bp1 ; p1 ðp0 ; b; α Þ ¼ ½a−ab þ c−bp0 ð1−α Þ þ bcαÞ=2; yielding 2c þ b½a−ab−cð1−bα Þ a−ab þ c þ bc½1−bð1−α Þ a−c pp pp ; p1 ¼ ; q0 ¼ ; 2 1 þ b 2−b 2−b2 ð1−α Þ ð1−α Þ 2 2 2 ð a−c Þ 1 þ b α ð 1−b Þ ð a−c Þ 1 þ b2 α ð1−bÞbða−cÞ pp pp pp π0 ¼ ; π1 ¼ q1 ¼ h i2 ; 2 ð1 þ bÞ 2−b2 ð1−α Þ ð1 þ bÞ 2−b2 ð1−α Þ ð1 þ bÞ 2−b ð1−α Þ h i 2 2 3 4 ða−cÞ 5−b−b ð3−6α Þ þ b −2b ð1−α Þα pp : CS ¼ h i2 2 2ð1 þ bÞ 2−b ð1−α Þ pp
p0 ¼
2. Model We formulate a mixed duopolistic model with differentiated goods that is composed of one social welfare-maximizing public firm (firm 0) and one generalized relative profit-maximizing private firm
4 Note that when the degree of importance of the private firm's relative performance is medium-sized, no equilibrium market structure exits.
5 Similar to Matsumura and Okamura (2010) and Nakamura and Saito (2013), we regard social welfare as the sum of consumer surplus and the profits of firms 0 and 1 rather than the sum of consumer surplus and the relative profits of firms 0 and 1. Although the firms' CEOs emphasize their relative profit rather than their original profits because relatively good performance increases current and future incomes, we consider such an idea to simply imply an income transfer. This is why we suppose social welfare to be the sum of consumer surplus and the profits of firms 0 and 1.
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Note that the superscript pp is used to denote the equilibrium market outcomes for the price-setting competition. The values of the objective functions for public firm 0 and private firm 1 are given as follows:
yielding pq
2
pp
ða−cÞ 7 þ b−b2 ð7−8α Þ−b3 þ 2b4 ð1−α Þ ; h i2 2 2ð1 þ bÞ 2−b ð1−α Þ h i 2 ð1−bÞða−cÞ 1−2bα þ b2 α þ b3 ð1−α Þα pp pp ¼ π1 −απ0 ¼ : h i2 2 ð1 þ bÞ 2−b ð1−α Þ
V0 ¼ W
pp
V1
2
i
pp
¼
3.2. q–q game
a−c ð2 þ bÞða−cÞ 1 pq pq ; q0 ¼ ; p1 ¼ ða−ab þ bc þ cÞ; 2 þ 2b 2ð1 þ bÞ 2 2 2 ð1−bÞða−cÞ ð5 þ 3bÞða−cÞ pq ; CS ¼ : ¼ 4ð1 þ bÞ 8ð1 þ bÞ pq
p0 ¼ c; pq
h
π0
q1 ¼
pq
q0 ðq1 ; b; α Þ ¼ a−c−bq1 ;
q1 ðq0 ; b; α Þ ¼ ½a−c−bq0 ð1−α Þ=2;
pq
π0 ¼ 0;
Note that the superscript pq is used to denote the equilibrium market outcomes for the market competition wherein the strategic contract of firm 0 is its price level while the strategic contract of firm 1 is its output level. The values of the objective functions of firms 0 and 1 are given as follows:
V0 ¼ W
For the quantity-setting competition, the reaction functions of firms 0 and 1 are given as follows:
39
pq
2
¼
ð7 þ bÞða−cÞ ; 8ð1 þ bÞ
pq
pq
pq
V 1 ¼ π1 −απ0 ¼
2
ð1−bÞða−cÞ : 4ð1 þ bÞ
3.5. q–p game For the market competition wherein the strategic contract of firm 0 is its output level while the strategic contract of firm 1 is its price level, the reaction functions of firms 0 and 1 are given as follows:
yielding
qq
q0 ¼
qq
p1 ¼ qq
qq
2−ð1−α Þb2
q0 ðp1 ; b; α Þ ¼ ða−cÞ=ð1 þ bÞ; ;
qq
q1 ¼
ða−cÞ½1−ð1−α Þb ; 2−ð1−α Þb2
2 að1−bÞð1−αbÞ þ c 1 þ b−b þ αb 2−ð1−α Þb
2
;
qq
π0 ¼ 0;
ð1−bÞða−cÞ ½1−ð1−α Þbð1−αbÞ ; h i 2 2 2−ð1−α Þb
h i 2 ða−cÞ 5−2bð1−α Þ−b2 4−2α−α 2 þ 2b3 ð1−α Þ ¼ : h i2 2 2 2−b ð1−α Þ
3.3. p–q game Note that the superscript qq is used to denote the equilibrium market outcomes for the quantity-setting competition. The values of the objective functions of firms 0 and 1 are given as follows:
qq
V0 ¼ W qq
qq
h i 2 2 2 2 3 ða−cÞ 7−2bð3−α Þ−b 2−4α þ α þ 2b ð1−α Þ ¼ ; h i2 2 2 2−b ð1−α Þ
qq
qq
V 1 ¼ π 0 −απ1 ¼
2
ð1−bÞða−cÞ ½1−bð1−α Þð1−bα Þ : h i2 2 2−b ð1−α Þ
3.4. p–q game For the market competition wherein the strategic contract of firm 0 is its price level while the strategic contract of firm 1 is its output level, the reaction functions of firms 0 and 1 are given as follows: p0 ðq1 ; b; α Þ ¼ c;
p1 ðq0 ; b; α Þ ¼ ½a þ c−bq0 ð1 þ α Þ=2;
qq
p0 ¼ c;
2
π1 ¼
CS
2−b2 ða−cÞ
q1 ðp0 ; b; α Þ 2 ¼ ½að1−bÞ−c−bcα þ bp0 ð1 þ α Þ=2 1−b ;
yielding a b−b2 α þ c 2 þ b þ b2 α a−c a þ c þ 2bc−abα þ bcα qp qp ; p1 ¼ ; p0 ¼ ; 1þb 2 þ 2b 2ð1 þ bÞ 2 2 ða−cÞð1 þ bα Þ bða−cÞ ð1−bα Þ ða−cÞ ð1−bα Þð1 þ bα Þ qp qp qp ; π0 ¼ ; π1 ¼ ; q1 ¼ 2 2 2ð1 þ bÞ 2ð1 þ bÞ i 4ð1 þ bÞ h 2 2 ða−cÞ 5 þ 2bð2 þ α Þ þ b α ð4 þ α Þ qp : CS ¼ 2 8ð1 þ bÞ qp
q0 ¼
Note that the superscript qp is used to denote the equilibrium market outcomes for the market competition wherein the strategic contract of firm 0 is its output level while the strategic contract of firm 1 is its price level. The values of the objective functions of firms 0 and 1 are given as follows:
qp V0
¼W
qp
¼
h i 2 2 2 ða−cÞ 7 þ 2bð4 þ α Þ−b α 2
8ð1 þ bÞ
;
qp
V 1 ¼ π 1 −απ 0 ¼
2
ða−cÞ ð1−bα Þ 2
4ð1 þ bÞ
2
:
3.6. Equilibrium market structures We derive the market structure(s) in the subgame perfect Nash equilibrium by endogenizing the strategic contracts of firms 0 and 1 in the first stage. In this subsection, we determine whether the market structures for the q–q, p–q, and q–p games can be observed in the equilibrium; this depends on both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α.6 The first stage of the game is summarized in Table 1. Given the above consequences for the four subgames in the second stage, we now discuss each
6 In Matsumura and Ogawa (2012), in a mixed duopoly with differentiated goods wherein the private firm is an absolute profit-maximizer, the p–p game is observed to be the unique market equilibrium market structure; in addition, the social welfare in the p–p game is higher than that in the q–q game in the cases of both substitutable goods and complementary goods.
40
Y. Nakamura / Economic Modelling 44 (2015) 37–43
1.0
Table 1 The first-stage game. Firm 0/firm 1
Price
Quantity
Price Quantity
pp (Vpp 0 , V1 ) qp (Vqp 0 , V1 )
pq (Vpq 0 , V1 ) qq (Vqq 0 , V1 )
firm's choice of a strategic contract in the first stage. From easy calculations, when b ∈ (0, 1), we have b−2α þ bα þ b2 α−b2 α 2 4−3b2 −2bα þ b2 α þ b3 α−b3 α 2 z b0 ¼ 2 2 2 2 8ð1 þ bÞ 2−b þ b α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 þ b þ b2 þ 4−4b−3b2 þ 6b3 þ b4 ; ⇔ α≤ N α0 ≡ 2b2 2 z ð2−bÞð1−bÞbða−cÞ ð2 þ b−bα Þðb−2α−bα Þ z pq qq b0 ⇔ α bα ≡ b ; V 0 −V 0 ¼ 2 0 2 2 2þb 8ð1 þ bÞ 2−b þ b α 2 3 2 3 2 2 3 2 ð1−bÞbða−cÞ 4b−b −8α þ 4b α þ 2b α−4b α −b α z pp pq b0 V 1 −V 1 ¼ 2 2 2 4ð1 þ bÞ 2−b þ b α qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 3 −4 þ 2b þ b þ 2 2 2−2b2 þ b3 þ b4 ⇔ α≤ ; 2 N α1 ≡ h b ð4 þ bÞ i 2 2 2 3 2 4 b ð a−c Þ ð 1−bα Þ 4b−8α þ 4b ð 2−α Þα−b 3−2α−α −b ð1−α Þ α z qp qq b0 V 1 −V 1 ¼ 2 2 2 2 4ð1 þ bÞ 2−b þ b α 2 3 2 b4 ð1 þ bÞ −8 þ 8b þ b2 z 1 2 3 4 1=3 þA 5 ⇔ α b α 1 ≡ 4 4−4b þ b þ 2b þ 1=3 3b A
0.8
0
III
I
0.6
1
II
2
bða−cÞ
pp qp V 0 −V 0
0.4
1
IV 0
0.2
V 0.0
0.0
0.2
0.4
0.6
0.8
1.0
b Fig. 1. Areas I to V for the optimal strategic contracts of firms 0 and 1.
The above results for areas (I)–(V) in Fig. 1 are summarized as Proposition 1.
where 6
7
8
9
10
11
12
A ¼ 80b þ 12b −132b þ 7b þ 57b −15b −b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 2 þ 6b ð1 þ bÞ 6 32−112b þ 158b2 −120b3 þ 54b4 −11b5 −b6 : Fig. 1 shows the results for the above calculations. We state the ranking orders of the payoffs of firms 0 and 1 for the five areas shown. In Fig. 1, the threshold value of α as the function of the degree of product differentiation b between areas (I) and (II), between (II) and (III), between (III) and (IV), and between (IV) and (V) are α 0 , α 1 , α 1 , and α 0, respectively. Except for the case wherein both the degree of product differentiation and the degree of importance of firm 1's relative performance are sufficiently low, from Fig. 1, α 0 ≥ α 1 ≥ ha 1 ≥ α 0 .7 A concise analysis of the comparison of each firm's payoff when its rival firm's strategic contract is fixed provides the following results. qp
pp
qq
pq
pq
pp
qq
qp
• In area (I), when α ≥ α 0, V0 ≥ V0 , V0 N V0 , V1 N V1 , and V1 N V1 are satisfied, implying that the q–q game is observed to be the unique equilibrium market structure. pp qp qq pq pq pp • In area (II), when α 0 N α ≥ α 1 , V0 N V0 , V0 N V0 , V1 N V1 , and qq qp V1 ≥ V1 are satisfied, implying that the q–q game is observed to be the unique equilibrium market structure. pp qp qq pq pq pp • In area (III), when α 1 N α ≥ α 1 , V0 N V0 , V0 N V0 , V1 ≥ V1 , and qq V1qp N V1 are satisfied, implying that no equilibrium market structure exists for any α∈ðα 1 ; α 1 Þ, whereas the p–p game is observed to be the equilibrium market structure when α ¼ α 1 . pp qp qq pq pp pq • In area (IV), when α 1 N α ≥ α 0 , V0 N V0 , V0 ≥ V0 , V1 N V1 , and qq qp V1 N V1 are satisfied, implying that the p–p game is observed to be the unique market structure. pp qp pq qq pp pq qp qq • In area (V), when α 0 N α, V0 N V0 , V0 N V0 , V1 N V1 , and V1 N V1 are satisfied, implying that the p–p game is observed to be the unique market structure. 7 Owing to overly complex calculations, when both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, the ranking order of α 0 , α 1 , α 1 , and α 0 cannot be obtained. However, the purpose of this paper is to confirm whether or not the market structures for the q–q, p–q, and q–p games can be observed. First, we consider the case wherein both the degree of product differentiation, b, and each firm's relative performance are not sufficiently low that α 0 ≥ α 1 ≥ α 1 ≥ α 0 is satisfied.
Proposition 1. Unless both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, for any value of product differentiation, b, when the degree of importance of firm 1's relative performance is sufficiently high, the q–q can become the equilibrium market structure, whereas when it is sufficiently low, the p–p game can become the equilibrium market structure. Note that when the degree of importance of firm 1's relative performance, α, is medium-sized, no equilibrium market structures exist under the pure strategies of the strategic contracts of firms 0 and 1. In Proposition 1, the intuition behind the results when the degree of importance of firm 1's relative performance, α, is sufficiently low is the same as that given in Matsumura and Ogawa (2012) since firm 1's behavior is similar to that of an absolute profit-maximizer. Thus, hereafter, we focus on the case wherein the degree of importance of firm 1's relative performance is not low. For the case wherein the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are not sufficiently low, we discuss the intuition behind Proposition 1, taking into account the fact that α 0 ≥ α 1 ≥ α 1 ≥ α 0 is satisfied in Fig. 1. We focus on the situation wherein the degree of product differentiation, b, is not sufficiently low and the degree of importance of firm 1's relative performance, α, is sufficiently high such that the q–q game can be observed as the unique market structure. This is different from a mixed duopoly wherein a private firm is an absolute profit maximizer à la Matsumura and Ogawa (2012). First, when firm 1 chooses a price contract, we consider the optimal contract of firm 0. Firm 0 attempts to induce the aggressive behavior of firm 1 in order to enhance social welfare, which is equal to its payoff. When firm 1 chooses a price contract, it is more difficult for firm 0 to control the quantity level of firm 1 since the price level of firm 1 becomes increasingly independent of the price level of firm 0 as the degree of importance of firm 1's relative performance, α, becomes higher if firm 0 chooses a price contract. If firm 0 chooses a quantity contract, the quantity level of firm 0 is negatively associated with the price level of firm 1 irrespective of α. Thus, when α is sufficiently high, firm 0 attempts to increase social welfare by choosing a quantity contract rather than a price contract. Next, we consider the optimal strategic contract of firm 0 in the case wherein firm 1 chooses a quantity contract. As the degree of importance
Y. Nakamura / Economic Modelling 44 (2015) 37–43 Table 2 Firm 0's payoff when b = 0.01 among the four games. pp
qp
41
Table 4 Firm 0's payoff when b = 0.05 among the four games. pq
qq
pp
qp
pq
qq
α
V0
V0
V0
V0
α
V0
V0
V0
V0
0.05 0.01 0.001 0.0001
0.867575 0.867574 0.867574 0.867574
0.867685 0.867587 0.867564 0.867562
0.867574 0.867574 0.867574 0.867574
0.867685 0.867587 0.867564 0.867562
0.1 0.05 0.01 0.001
0.839314 0.8393 0.839289 0.839286
0.840133 0.839568 0.839116 0.839014
0.839286 0.839286 0.839286 0.839286
0.840161 0.839583 0.839119 0.839014
Table 3 Firm 1's payoff when b = 0.01 among the four games. pp
qp
Table 5 Firm 1's payoff when b = 0.05 among the four games. pq
qq
α
V1
V1
V1
V1
0.05 0.01 0.001 0.0001
0.244829 0.245025 0.245069 0.245074
0.244829 0.245025 0.245069 0.245074
0.24505 0.24505 0.24505 0.24505
0.245049 0.24505 0.24505 0.24505
of firm 1's relative performance, α, becomes higher, the strategic substitutability between q0 and q1 in the reaction function of firm 1 is weaker if firm 1 chooses a quantity contract. Thus, when α is sufficiently high, firm 0 can induce relatively aggressive behavior by firm 1 more easily in the q–q game. Therefore, when α is sufficiently high, it is optimal for firm 0 to choose a quantity contract if firm 1 chooses a quantity contract. Second, we consider the optimal strategy of firm 1 given a strategic contract for firm 0. When a strategic contract for firm 0 is fixed, the optimal contract for firm 1 is determined mainly by the following two factors: (1) its payoff is equal to its generalized relative profit, π1 − απ0 and (2) as α increases, the market competition in the p–p, q–q, and q–p games becomes more intense.8 Let us consider the case in which firm 0 chooses its price contract. From easy calculations, although pp pq pp pq pp both p0 N p0 and p1 N p1 hold for any α ∈ [0, 1], the values of p0 − pq pp pq p0 and p1 − p1 decrease with the increase in α since the market competition in the p–p game becomes more intense as α increases, which pp pq pq implies that π1 − π1 decreases as α increases. Taking π0 = 0 on the pq pq pq basis of p0 = c into account, when α is sufficiently high, V1 = π1 − pq pp pp pp απ0 N π1 − απ0 = V1 can be satisfied, implying that the optimal strategic contract for firm 1 is a quantity contract. Similar to the case wherein the strategic contract of firm 1 is a price contract, when firm 0 chooses its quantity contract, although qp qq qp qq p 0 N p 0 = c and p1 N p 1 hold for any α ∈ [0, 1], the differences qp qq qp qq between p0 and p0 and between p1 and p1 are sufficiently low when qp qq α is sufficiently high. Thus, the difference between π1 and π1 also decreases as pha increases. Therefore, when α is sufficiently high, qq qq qq qp qp qp V1 = π1 − α π0 N π1 − α π0 = V1 is satisfied. Of course, these facts concerning the optimal strategic contract of firm 1 result from the introduction of the degree of importance of firm 1's relative performance, α. In sum, when the degree of importance of firm 1's relative performance, α, is sufficiently high, quantity contracts can become the dominant strategies for both firms 0 and 1. Therefore, when α is sufficiently high, the q–q game can become supported by the dominant strategic contracts of both firms 0 and 1 unless the degrees of both product differentiation and each firm's relative performance are low. Finally, through numerical calculation concerning the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, particularly when both are sufficiently low, we derive the equilibrium market structures. More precisely, given sufficiently low levels of α and b, we consider the following three cases: (i) α = 0.05, 0.01, 0.001, and 0.0001 when b = 0.01; (ii) α = 0.1, 0.05, 0.01, and 0.001 when b = 0.05; and (iii) α = 0.2, 0.1, 0.05, and 0.01 when b = 0.1.9 As the data in Tables 2–7 shows for all three
qq
pq
pq
8 Except for p0 , p0 , and p1 such that the price levels do not depend on α as the value of α increases, we obtain the result that each firm's price level decreases for any degree of product differentiation, b, in the p–p game, the q–q game, and the q–p game. 9 In Appendix A, we provide the values of the payoffs of firms 0 and 1 in the four games for the above three cases from Tables 2–7.
pp
qp
pq
qq
α
V1
V1
V1
V1
0.1 0.05 0.01 0.001
0.224492 0.225625 0.226531 0.226734
0.224495 0.225625 0.226531 0.226735
0.22619 0.22619 0.22619 0.22619
0.226187 0.22619 0.22619 0.22619
cases, b = 0.01, b = 0.5, and b = 0.1, the quantity competition can be observed in the equilibrium on the basis of the dominant strategies of firms 0 and 1 when α is sufficiently high, whereas the price competition can be observed in the equilibrium on the basis of the strategic contracts of firms 0 and 1 when α is sufficiently low.10 The above results lead to the following proposition. Proposition 2. In the case wherein both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, the quantity competition can become the equilibrium market structure when α is high relative to b, whereas the price competition can become the equilibrium market structure when α is low relative to b. Proposition 2 states that similar to the case wherein both α and b are relatively high, in the situation wherein both α and b are sufficiently low, quantity competition can be observed in the equilibrium when α is high relative to b, whereas price competition can be observed in the equilibrium when α is low relative to b. However, analogous to the case wherein both α and b are relatively high, in the situation wherein both α and b are sufficiently low, we cannot determine whether or not any other market structure beyond the xtitq–q and p–p games can become the equilibrium market structure(s) even through the simulation of the levels of both α and b. Next we clarify whether or not the equilibrium market structure coincides with the optimal market structure from the viewpoint of social welfare.11 Comparing the equilibrium social welfare of the four games, we obtain the result in the following proposition concerning whether the highest social welfare is achieved in the quantity competition or in the price competition, which can be observed in the equilibrium.12 n pffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 Proposition 3. When α Nmax 1− 1−b =b; α 0 , the highest social welfare is achieved in the quantity competition. On the other hand, n
when αbmin
pffiffiffiffiffiffiffiffiffiffiffiffiffi o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 3 4 2 1− 1−b =b; −2 þ b þ b þ 4−4b−3b þ 6b þ b =2b ,
the highest social welfare is achieved in the price competition. Proposition 3 states that in the situation wherein both α and b are not low, it is unnecessary for the regulatory authority to prescribe any strong policy from the viewpoint of social welfare, as long as the degree of importance of the relative performance of the private firm is sufficiently 10
More precisely, we obtain the results for the equilibrium market structure in the three
8 > > ½b ¼ 0:01 : equilibrium market structure > > > > < cases as follows: > ½b ¼ 0:05 : equilibrium market structure > > > > > : ½b ¼ 0:1 : equilibrium market structure
quantity competition when α ¼ 0:05 when α ¼ 0:001
price competition quantity competition when α ¼ 0:1 when α ¼ 0:01
price competition quantity competition when α ¼ 0:2 price competition when α ¼ 0:05
and 0:01; and 0:0001; and 0:05; and 0:001; and 0:1; and 0:01:
11 Even although public firm 0 is a social welfare-maximizer, the equilibrium market structure does not always become the optimal market structure from the viewpoint of social welfare. 12 In Appendix A, concrete calculations of the equilibrium social welfare of the four games are given.
42
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Table 6 Firm 0's payoff when b = 0.1 among the four games. pp
qp
pq
qq
α
V0
V0
V0
V0
0.2 0.1 0.05 0.01
0.807026 0.806923 0.806872 0.806831
0.809876 0.807841 0.806816 0.805992
0.806818 0.806818 0.806818 0.806818
0.810078 0.807945 0.80687 0.806005
high or sufficiently low. However, when α is medium-sized, no equilibrium market structure exists under the class of the pure strategy of strategic contracts for both the public firm and the private firm. Therefore, in such a situation, the regulatory authority may be cautious given the non-coincidence between the achieving market structure in the equilibrium and the optimal market structure from the viewpoint of social welfare. 4. Conclusion This paper investigated each firm's endogenous choice of a strategic contract in a mixed duopoly with differentiated goods composed of one social welfare-maximizing public firm and one generalized relative profit-maximizing private firm. More precisely, we investigated the situation in which both the public firm and private firm endogenously choose either of a quantity contract or a price contract in the market competition in the fashion of Singh and Vives (1984) and Matsumura and Ogawa (2012) when the objective of the private firm is to maximize the difference between the weighted sum of its absolute profit and the public firm's absolute profit. In contrast to the results for a standard mixed duopoly composed of one social welfare-maximizing public firm and one absolute profit-maximizing private firm in Matsumura and Ogawa (2012), we showed that when the degree of importance of the private firm's relative performance is sufficiently high, quantity competition can be observed as the equilibrium market structure on the basis of the quantity contracts of both the public firm and the private firm in areas in which both the degree of product differentiation and the degree of importance of the private firm's relative performance are relatively high. The result that the quantity competition can be observed in the equilibrium when the degree of importance of the private firm's relative performance is relatively high is supported by the dominant strategies concerning the strategic contracts by the public firm and the private firm. Thus, in the mixed duopoly with substitutable goods in which the private firm maximizes its generalized relative profit, the advantage of the price competition in Matsumura and Ogawa (2012) decreases with the introduction of the degree of importance of the private firm's relative performance. The results of this paper have certain limitations. Because of the high difficulty of analytical calculations, when the degree of product differentiation and the degree of the private firm's relative performance are sufficiently low, we cannot fully characterize the equilibrium market structures. Instead, using numerical examples for the degree of product differentiation and the degree of importance of the private firm's relative performance, we showed that quantity competition tends to be observed as the unique market structure when the degree of the private firm's relative performance is
sufficiently high, whereas the price competition tends to be observed as the unique market structure when the degree of the private firm's relative performance is sufficiently low, even in the case wherein both the degree of product differentiation and the degree of importance of the private firm's relative performance are sufficiently low. We found that for any value of the degree of product differentiation, quantity competition can be observed as the unique equilibrium market structure when the degree of importance of firm 1's relative performance is sufficiently high, whereas price competition can be observed as the unique market structure when each firm's relative performance is sufficiently low. On the other hand, it is unknown whether or not there exist levels of the degree of product differentiation and the degree of importance of the private firm's relative performance such that the p–q game and the q–p game, in which the strategic contracts of the public firm and the private firm are different from each other can be observed as the equilibrium market structures. In our future research, we will determine whether or not the above asymmetric market structures with respect to the strategic contracts of the public firm and the private firm can become the equilibrium market structures. Finally, we mention several relevant topics for future studies. First, the influence of the change of the cost functions of the public firm and the private firm (for example, the quadratic and symmetric cost functions of both firms) on the equilibrium market structures should be investigated throughout this paper, we used horizontal product differentiation without each firm's vertical structure. Wang and Wang (2009) compared the equilibrium market outcomes in the four games on the basis of each firm's strategic contract by using a vertical differentiation model in the context of a private duopoly. In addition, Wang and Wang (2010) investigated the vertically related market in the context of a managerial private oligopoly with the separation between ownership and management within each firm. Thus, we should merge the approaches of Wang and Wang (2009) and Wang and Wang (2010) to consider the endogenous selection of strategic contracts in the context of a mixed oligopoly with or without separation between ownership and management in each firm with a vertically related market in the model with vertical product differentiation. Since there are no works on the endogenous selection problem of the strategic contracts in the context of the mixed oligopoly using a vertically related market model with or without the vertically related structure, such studies would be interesting and meaningful. Appendix A. Comparison of the equilibrium social welfare among the four games Here, we give the results for the comparison of the equilibrium social welfare, which is equal to the payoff of firm 0, among the four games. We focus on the situations with the highest social welfare in either the quantity competition or the price competition, both of which can become the equilibrium market structure. From direct calculations of whether or not the highest social welfare is achieved in the q–q game, we obtain the following result:
qq
pp
qq
pq
qq
qp
W −W W −W
W −W Table 7 Firm 1's payoff when b = 0.1 among the four games. pp
qp
pq
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð−1 þ bÞbða−cÞ b−2α þ bα 2 z z b 0 ⇔ α b 1− 1−b ; 2 2 2 b 2ð1 þ bÞ 2−b þ b α z ð2−bÞð1−bÞb½2 þ bð1−α Þ½bð−1 þ α Þ þ 2α z b 0 ⇔ α b b=ð2 þ bÞ ≡ α ; ¼ h i2 0 2 8ð1 þ bÞ 2−b ð1−α Þ 2 2 2 2 2 2 2 b ða−cÞ ð1−bα Þ b þ 4α−2b α þ b α ¼ N0: 2 2 2 2 8ð1 þ bÞ 2−b þ b α ¼
qq
α
V1
V1
V1
V1
0.2 0.1 0.05 0.01
0.198389 0.202496 0.204551 0.206195
0.19843 0.2025 0.204551 0.206199
0.204545 0.204545 0.204545 0.204545
0.204493 0.204539 0.204545 0.204542
Thus, we obtain the result that the highest equilibrium social welfare pffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 is achieved in the q–q game when α Nmax 1− 1−b = b; α 0 ≡ b= o ð2 þ bÞ .
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Moreover, we can calculate the condition of α with respect to b such that the highest equilibrium social welfare is achieved in the p–p game as follows: h ih i 2 2 3 2 bða−cÞ 4−b ð3−α Þ−2bα þ b ð1−α Þα 2α−b ð1−α Þα−bð1 þ α Þ z b0 h i2 2 2 8ð1 þ bÞ 2−b ð1−α Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ⇔ α≤ þ 6b3 þ b4 =2b ; N −2 þ b þ b þ 4−4b−3b h i 2 2 2 2 ð1−bÞb ða−cÞ b ð1−α Þ þ 4α pp pq N0; W −W ¼ h i2 2 8ð1 þ bÞ 2−b ð1−α Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ð1−bÞbða−cÞ b−2α þ bα z pp qq b 0⇔ α ≤ 1− 1−b : W −W ¼ 2 N 2 2 b 2ð1 þ bÞ 2−b þ b α pp
W −W
qp
¼−
Thus, we find that the highest equilibrium social welfare is achieved n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 4 −2 þ b þ b þ 4−4b−3b þ 6b þ b =
in the p–p game when αbmin pffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 2 2b ; 1− 1−b =b .
From Table 2, when b = 0.01, it is a dominant strategy for firm 0 to chose a quantity contract when α = 0.05 and α = 0.01, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.001 and α = 0.0001. From Table 3, when b = 0.01, it is a dominant strategy for firm 1 to choose a quantity contract when α = 0.01 and α = 0.05, whereas it is a dominant strategy for firm 1 to choose a price contract when α = 0.001 and α = 0.0001. Therefore, when b = 0.01, quantity competition can become the equilibrium market structure on the basis of the dominant strategies of both firms 0 and 1 when α = 0.01, whereas price competition can become the equilibrium market structure on the basis of the dominant strategies of firms 0 and 1 when α = 0.001 and α = 0.0001. From Table 4, when b = 0.05, it is a dominant strategy for firm 0 to choose a quantity contract when α = 0.1 and α = 0.05, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.01 and α = 0.001. From Table 5, when b = 0.05, it is a dominant strategy for firm 1 to choose its quantity contract when α = 0.1 and α = 0.05, whereas it is a dominant strategy for firm 1 to choose its price contract when α = 0.01 and α = 0.001. Therefore, when b = 0.05, quantity competition can become the equilibrium market structure on the basis of the strategic contracts of
43
both firms 0 and 1 when α = 0.1 and α = 0.05, whereas price competition can become the equilibrium market structure on the basis of the strategic contracts of firms 0 and 1 when α = 0.01 and α = 0.001. From Table 6, when b = 0.1, it is a dominant strategy for firm 0 to choose a quantity contract when α = 0.2 and α = 0.1, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.05 and α = 0.01. From Table 7, when b = 0.1, it is a dominant strategy for firm 1 to choose a quantity contract when α = 0.2 and α = 0.1, whereas it is a dominant strategy for firm 1 to choose a price contract when α = 0.05. Therefore, when b = 0.1, quantity competition can become the equilibrium market structure based on the strategic contracts of firms 0 and 1 when α = 0.2 and α = 0.1, whereas price competition can become the equilibrium market structure on the basis of the strategic contracts of both firms 0 and 1 when α = 0.05 and α = 0.01. References Alchian, A.A., 1950. Uncertainty, evolution, and economic theory. J. Polit. Econ. 57, 211–221. Coats, J.C., Neilson, W.S., 2005. Beliefs about other-regarding preferences in a sequential public goods game. Econ. Inq. 43, 614–622. Klemperer, P., Meyer, M., 1986. Price competition vs. quantity competition: the role of uncertainty. RAND J. Econ. 17, 618–638. Matsumura, T., Matsushima, N., 2012. Competitiveness and stability of collusive behavior. Bull. Econ. Res. 64, s22–s31. Matsumura, T., Ogawa, A., 2012. Price versus quantity in a mixed duopoly. Econ. Lett. 116, 174–177. Matsumura, T. and Okamura, M. (2010) “Competition and Privatization Policy: The Relative Performance Approach” mimeo. Nakamura, Y., Saito, M., 2013. Capacity choice in a mixed duopoly: the relative performance approach. Theor. Econ. Lett. 3, 124–133. Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND J. Econ. 15, 546–554. Tanaka, Y., 2001a. Profitability of price and quantity strategies in an oligopoly. J. Math. Econ. 35, 409–418. Tanaka, Y., 2001b. Profitability of price and quantity strategies in a duopoly with vertical product differentiation. Economic Theory 17, 639–700. Tasnádi, A., 2006. Price vs. quantity in oligopoly games. Int. J. Ind. Organ. 24, 541–554. Vega-Redondo, F., 1997. The evolution of Walrasian behavior. Econometrica 65, 375–384. Wang, Y.C., Wang, L.F.S., 2009. Equivalence of competition mode in a vertically differentiated duopoly with delegation. South Afr. J. Econ. 77, 577–590. Wang, Y.C., Wang, L.F.S., 2010. Input pricing and market share delegation in a vertically related markets: is the timing order relevant? Int. J. Econ. Bus. 17, 207–221.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Price versus quantity in a mixed duopoly: The case of relative profit maximization☆ Yasuhiko Nakamura ⁎ College of Economics, Nihon University, Japan
a r t i c l e
i n f o
Article history: Accepted 19 September 2014 Available online xxxx Keywords: Quantity competition Price competition Mixed markets Differentiated products Relative profit maximization
a b s t r a c t This paper examines the endogenous choice problem of each firm's price or quantity contract in a mixed duopoly composed of one social welfare maximizing public firm and one relative profit-maximizing private firm. In this paper, we show that unless the degree of product differentiation and the degree of importance of the private firm's relative performance are low, the quantity competition can analytically become the unique equilibrium market structure on the basis of the dominant strategies of the public firm and the private firm when the degree of importance of firm 1's relative performance is sufficiently high. This result contrasts strikingly with that obtained in a standard mixed duopoly composed of one social welfare-maximizing public firm and one absolute profit maximizing private firm. In addition, even in the area wherein both the degree of product differentiation and the degree of the private firm's relative performance are sufficiently low, through numerical simulations, the quantity competition tends to become the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas the price competition tends to become the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The purpose of this paper is to explore the equilibrium result for the endogenous selection of strategic variables in a mixed duopoly with differentiated goods composed of one social welfare-maximizing public firm and one generalized relative profit-maximizing private firm. In their recent work, Matsumura and Ogawa (2012) found that in a standard mixed duopoly with differentiated goods in which the private firm maximizes its absolute profit, the dominant strategies of both the public firm and the private firm are price contracts, which imply that price-setting competition is observed in the equilibrium regardless of the relation between the goods the firms produce. In this paper, we check the equilibrium result for the endogenous selection of strategic variables by the public firm and the private firm against the introduction of relative profit maximization for the private firm whose objective function is a weighted sum of its absolute profit and the absolute profit of its rival public firm.
☆ We would like to thank an anonymous referee for her/his helpful comments and suggestions. We are grateful for the financial support of KAKENHI (25870113). All remaining errors are our own. ⁎ Corresponding author at: College of Economics, Nihon University, 1-3-2, Misaki-cho, Chiyoda-ku, Tokyo 101-8360, Japan. Tel./fax: +81 3 3219 3497. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econmod.2014.09.019 0264-9993/© 2014 Elsevier B.V. All rights reserved.
Many studies have conducted equilibrium analyses on mixed oligopoly (duopoly) in various economic contexts; however, most supposed quantity competition; an exception is Matsumura and Ogawa (2012).1 In the context of private oligopoly, Singh and Vives (1984), Klemperer and Meyer (1986), and others have indicated that in the real world economy, firms frequently choose to use either their price level or their output level as their strategic contract in market competition.2 In addition to the importance of the endogenous choice of each firm's strategic variable, recent works on both private oligopoly (duopoly) and mixed oligopoly (duopoly) have investigated the relative performance approach. In the real world economy, as indicated in Alchian (1950) and Vega-Redondo (1997) in the context of evolutionary stability and in Coats and Neilson (2005) in the context of laboratory and experimental issues, when firms' owners evaluate their managers' performances,
1 As described above, Matsumura and Ogawa (2012) showed the advantage of pricesetting competition in the standard mixed duopolistic market regardless of the relation between the goods, that is, substitutability or complementarity, by allowing the endogenous determination of each firm's strategic contract in the market. 2 In the private oligopolistic context, Tanaka (2001a,b), and Tasnádi (2006) checked the robustness of the results obtained in Singh and Vives (1984) and Klemperer and Meyer (1986) such that in a private duopoly market, quantity competition can be observed in the equilibrium when the products are substitutes whereas price competition can be observed in the equilibrium when the products are complements.
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Y. Nakamura / Economic Modelling 44 (2015) 37–43
they use their relative profits rather than their absolute profits.3 Most recently, Matsumura and Matsushima (2012) investigated the relationship between the degree of competition and the stability of collusive behavior by introducing the element of relative performance into the objective functions of firms. In this paper, given the above-mentioned importance of the endogenous selection of each firm's strategic variable and relative profit as the objective function of the private firm, we take both elements into account in our analysis. In this paper, by introducing the degree of importance of the private firm's relative performance, we show that unless both the degree of product differentiation and the degree of importance of the private firm's relative performance are sufficiently low, the quantity competition can become the unique equilibrium market structure on the basis of the dominant strategies of the public firm and the private firm when the degree of importance of the private firm's relative performance is sufficiently high. On the other hand, unless both the degree of product differentiation and each firm's relative performance are sufficiently low, the price competition can be observed as the unique equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low. Furthermore, through the simulation analysis, we find that the quantity competition can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas when the degree of importance of firm 1's relative performance is sufficiently low, the price competition can become the unique equilibrium market structure. In addition, we show that the highest social welfare is achieved in the quantity competition which can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently high, whereas the highest social welfare is achieved in the price competition which can become the equilibrium market structure when the degree of importance of the private firm's relative performance is sufficiently low.4 Thus, the corresponding regulatory authority does not need to prescribe a strong policy from the viewpoint of social welfare as long as the degree of importance of the private firm's relative performance is sufficiently high or low. In sum, the advantage of the price competition under a standard mixed duopoly with differentiated goods as in Matsumura and Ogawa (2012) decreases with the introduction of the degree of importance of the private firm's relative performance. Thus, in a mixed duopoly in which the private firm maximizes its generalized relative profit, the fact that the quantity competition is supposed is not unnatural, different from the mixed market that a private firm maximizes its absolute profit. The remainder of this paper is organized as follows. In Section 2, we formulate a mixed duopolistic model with substitutable goods composed of a social welfare-maximizing public firm and a generalized relative profit-maximizing private firm in order to explore their endogenous selection of strategic contracts. In Section 3, we derive the subgame perfect Nash equilibrium in the model developed in Section 2 and give the intuition behind the results for the equilibrium market structures. Section 4 concludes with several remarks. In Appendix A, we provide the concrete values of the payoffs of both the public firm and the private firm on the basis of numerical analysis of the degree of product differentiation and the degree of importance of the private firm's relative performance when both the levels are sufficiently low.
(firm 1) whose objective function is a weighted sum of its own profit and firm 0's absolute profit. Firms 0 and 1 produce differentiated commodities for which the inverse demand functions are given by pi = a − qi − bqj, (i, j = 0, 1; i ≠ j), where pi and qi are firm i's price and output levels, respectively, a is the demand parameter that denotes the market size, and b ∈ (0, 1). A positive (negative) value of b indicates that the goods produced by firms 0 and 1 are substitutes (complements). We suppose that the marginal production costs of firms 0 and 1 are constant and identical. Let c denote firm i's marginal cost, (i = 0, 1). In addition, we assume that a N c in order to ensure all the equilibrium market outcomes. The objective function of firm 0, V0, is represented as follows: " # q2 þ 2bq0 q1 þ q21 −p0 q0 −p V 0 ¼ W ¼ ðp0 −cÞq0 þ ðp1 −cÞq1 þ aðq0 þ q1 Þ− 0 ; 1 q1 t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ producer surplus ¼ consumer surplus
which is equal to the total social welfare.5 On the other hand, firm 1 maximizes its generalized relative profit as follows: V1 = π1 − απ0, where πi denotes firm i's absolute profit which is equal to (pi − c)qi and α ∈ (0, 1) is the degree of importance of firm 1's relative performance, (i = 0, 1). The game runs as follows. In the first stage, firms 0 and 1 simultaneously choose whether to adopt price contracts or quantity contracts. In the second stage, after observing the choice of the strategic contract by the rival firm in the first stage, firms 0 and 1 simultaneously and independently choose either their price or quantity in the market, on the basis of their decisions in the first stage. 3. Equilibrium analysis of the strategic variable selection game In this section, we investigate the four possible subgames which are classified on the basis of each firm's strategic variable selected in the first stage: (i) firms 0 and 1 choose price contracts (p–p game); (ii) firms 0 and 1 choose quantity contracts (q–q game); firm 0 chooses a price contract while firm 1 chooses a quantity contract (p–q game); (iv) firm 0 chooses a quantity contract while firm 1 chooses a price contract (q–p game). Similar to Matsumrua and Ogawa (2012), we assume that the solutions in the four subgames are interior—that is, the equilibrium prices and quantities of both firms are strictly positive. 3.1. p–p game For the price-setting competition, the reaction functions of firms 0 and 1 are given as follows: p0 ðp1 ; b; α Þ ¼ c−bc þ bp1 ; p1 ðp0 ; b; α Þ ¼ ½a−ab þ c−bp0 ð1−α Þ þ bcαÞ=2; yielding 2c þ b½a−ab−cð1−bα Þ a−ab þ c þ bc½1−bð1−α Þ a−c pp pp ; p1 ¼ ; q0 ¼ ; 2 1 þ b 2−b 2−b2 ð1−α Þ ð1−α Þ 2 2 2 ð a−c Þ 1 þ b α ð 1−b Þ ð a−c Þ 1 þ b2 α ð1−bÞbða−cÞ pp pp pp π0 ¼ ; π1 ¼ q1 ¼ h i2 ; 2 ð1 þ bÞ 2−b2 ð1−α Þ ð1 þ bÞ 2−b2 ð1−α Þ ð1 þ bÞ 2−b ð1−α Þ h i 2 2 3 4 ða−cÞ 5−b−b ð3−6α Þ þ b −2b ð1−α Þα pp : CS ¼ h i2 2 2ð1 þ bÞ 2−b ð1−α Þ pp
p0 ¼
2. Model We formulate a mixed duopolistic model with differentiated goods that is composed of one social welfare-maximizing public firm (firm 0) and one generalized relative profit-maximizing private firm
4 Note that when the degree of importance of the private firm's relative performance is medium-sized, no equilibrium market structure exits.
5 Similar to Matsumura and Okamura (2010) and Nakamura and Saito (2013), we regard social welfare as the sum of consumer surplus and the profits of firms 0 and 1 rather than the sum of consumer surplus and the relative profits of firms 0 and 1. Although the firms' CEOs emphasize their relative profit rather than their original profits because relatively good performance increases current and future incomes, we consider such an idea to simply imply an income transfer. This is why we suppose social welfare to be the sum of consumer surplus and the profits of firms 0 and 1.
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Note that the superscript pp is used to denote the equilibrium market outcomes for the price-setting competition. The values of the objective functions for public firm 0 and private firm 1 are given as follows:
yielding pq
2
pp
ða−cÞ 7 þ b−b2 ð7−8α Þ−b3 þ 2b4 ð1−α Þ ; h i2 2 2ð1 þ bÞ 2−b ð1−α Þ h i 2 ð1−bÞða−cÞ 1−2bα þ b2 α þ b3 ð1−α Þα pp pp ¼ π1 −απ0 ¼ : h i2 2 ð1 þ bÞ 2−b ð1−α Þ
V0 ¼ W
pp
V1
2
i
pp
¼
3.2. q–q game
a−c ð2 þ bÞða−cÞ 1 pq pq ; q0 ¼ ; p1 ¼ ða−ab þ bc þ cÞ; 2 þ 2b 2ð1 þ bÞ 2 2 2 ð1−bÞða−cÞ ð5 þ 3bÞða−cÞ pq ; CS ¼ : ¼ 4ð1 þ bÞ 8ð1 þ bÞ pq
p0 ¼ c; pq
h
π0
q1 ¼
pq
q0 ðq1 ; b; α Þ ¼ a−c−bq1 ;
q1 ðq0 ; b; α Þ ¼ ½a−c−bq0 ð1−α Þ=2;
pq
π0 ¼ 0;
Note that the superscript pq is used to denote the equilibrium market outcomes for the market competition wherein the strategic contract of firm 0 is its price level while the strategic contract of firm 1 is its output level. The values of the objective functions of firms 0 and 1 are given as follows:
V0 ¼ W
For the quantity-setting competition, the reaction functions of firms 0 and 1 are given as follows:
39
pq
2
¼
ð7 þ bÞða−cÞ ; 8ð1 þ bÞ
pq
pq
pq
V 1 ¼ π1 −απ0 ¼
2
ð1−bÞða−cÞ : 4ð1 þ bÞ
3.5. q–p game For the market competition wherein the strategic contract of firm 0 is its output level while the strategic contract of firm 1 is its price level, the reaction functions of firms 0 and 1 are given as follows:
yielding
q0 ¼
p1 ¼ qq
2−ð1−α Þb2
q0 ðp1 ; b; α Þ ¼ ða−cÞ=ð1 þ bÞ; ;
q1 ¼
ða−cÞ½1−ð1−α Þb ; 2−ð1−α Þb2
2 að1−bÞð1−αbÞ þ c 1 þ b−b þ αb 2−ð1−α Þb
2
;
π0 ¼ 0;
ð1−bÞða−cÞ ½1−ð1−α Þbð1−αbÞ ; h i 2 2 2−ð1−α Þb
h i 2 ða−cÞ 5−2bð1−α Þ−b2 4−2α−α 2 þ 2b3 ð1−α Þ ¼ : h i2 2 2 2−b ð1−α Þ
3.3. p–q game Note that the superscript qq is used to denote the equilibrium market outcomes for the quantity-setting competition. The values of the objective functions of firms 0 and 1 are given as follows:
V0 ¼ W qq
h i 2 2 2 2 3 ða−cÞ 7−2bð3−α Þ−b 2−4α þ α þ 2b ð1−α Þ ¼ ; h i2 2 2 2−b ð1−α Þ
V 1 ¼ π 0 −απ1 ¼
2
ð1−bÞða−cÞ ½1−bð1−α Þð1−bα Þ : h i2 2 2−b ð1−α Þ
3.4. p–q game For the market competition wherein the strategic contract of firm 0 is its price level while the strategic contract of firm 1 is its output level, the reaction functions of firms 0 and 1 are given as follows: p0 ðq1 ; b; α Þ ¼ c;
p1 ðq0 ; b; α Þ ¼ ½a þ c−bq0 ð1 þ α Þ=2;
p0 ¼ c;
2
π1 ¼
CS
2−b2 ða−cÞ
q1 ðp0 ; b; α Þ 2 ¼ ½að1−bÞ−c−bcα þ bp0 ð1 þ α Þ=2 1−b ;
yielding a b−b2 α þ c 2 þ b þ b2 α a−c a þ c þ 2bc−abα þ bcα qp qp ; p1 ¼ ; p0 ¼ ; 1þb 2 þ 2b 2ð1 þ bÞ 2 2 ða−cÞð1 þ bα Þ bða−cÞ ð1−bα Þ ða−cÞ ð1−bα Þð1 þ bα Þ qp qp qp ; π0 ¼ ; π1 ¼ ; q1 ¼ 2 2 2ð1 þ bÞ 2ð1 þ bÞ i 4ð1 þ bÞ h 2 2 ða−cÞ 5 þ 2bð2 þ α Þ þ b α ð4 þ α Þ qp : CS ¼ 2 8ð1 þ bÞ qp
q0 ¼
Note that the superscript qp is used to denote the equilibrium market outcomes for the market competition wherein the strategic contract of firm 0 is its output level while the strategic contract of firm 1 is its price level. The values of the objective functions of firms 0 and 1 are given as follows:
qp V0
¼W
qp
¼
h i 2 2 2 ða−cÞ 7 þ 2bð4 þ α Þ−b α 2
8ð1 þ bÞ
;
qp
V 1 ¼ π 1 −απ 0 ¼
2
ða−cÞ ð1−bα Þ 2
4ð1 þ bÞ
2
:
3.6. Equilibrium market structures We derive the market structure(s) in the subgame perfect Nash equilibrium by endogenizing the strategic contracts of firms 0 and 1 in the first stage. In this subsection, we determine whether the market structures for the q–q, p–q, and q–p games can be observed in the equilibrium; this depends on both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α.6 The first stage of the game is summarized in Table 1. Given the above consequences for the four subgames in the second stage, we now discuss each
6 In Matsumura and Ogawa (2012), in a mixed duopoly with differentiated goods wherein the private firm is an absolute profit-maximizer, the p–p game is observed to be the unique market equilibrium market structure; in addition, the social welfare in the p–p game is higher than that in the q–q game in the cases of both substitutable goods and complementary goods.
40
Y. Nakamura / Economic Modelling 44 (2015) 37–43
1.0
Table 1 The first-stage game. Firm 0/firm 1
Price
Quantity
Price Quantity
pp (Vpp 0 , V1 ) qp (Vqp 0 , V1 )
pq (Vpq 0 , V1 ) qq (Vqq 0 , V1 )
firm's choice of a strategic contract in the first stage. From easy calculations, when b ∈ (0, 1), we have b−2α þ bα þ b2 α−b2 α 2 4−3b2 −2bα þ b2 α þ b3 α−b3 α 2 z b0 ¼ 2 2 2 2 8ð1 þ bÞ 2−b þ b α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 þ b þ b2 þ 4−4b−3b2 þ 6b3 þ b4 ; ⇔ α≤ N α0 ≡ 2b2 2 z ð2−bÞð1−bÞbða−cÞ ð2 þ b−bα Þðb−2α−bα Þ z pq qq b0 ⇔ α bα ≡ b ; V 0 −V 0 ¼ 2 0 2 2 2þb 8ð1 þ bÞ 2−b þ b α 2 3 2 3 2 2 3 2 ð1−bÞbða−cÞ 4b−b −8α þ 4b α þ 2b α−4b α −b α z pp pq b0 V 1 −V 1 ¼ 2 2 2 4ð1 þ bÞ 2−b þ b α qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 3 −4 þ 2b þ b þ 2 2 2−2b2 þ b3 þ b4 ⇔ α≤ ; 2 N α1 ≡ h b ð4 þ bÞ i 2 2 2 3 2 4 b ð a−c Þ ð 1−bα Þ 4b−8α þ 4b ð 2−α Þα−b 3−2α−α −b ð1−α Þ α z qp qq b0 V 1 −V 1 ¼ 2 2 2 2 4ð1 þ bÞ 2−b þ b α 2 3 2 b4 ð1 þ bÞ −8 þ 8b þ b2 z 1 2 3 4 1=3 þA 5 ⇔ α b α 1 ≡ 4 4−4b þ b þ 2b þ 1=3 3b A
0.8
0
III
I
0.6
1
II
2
bða−cÞ
pp qp V 0 −V 0
0.4
1
IV 0
0.2
V 0.0
0.0
0.2
0.4
0.6
0.8
1.0
b Fig. 1. Areas I to V for the optimal strategic contracts of firms 0 and 1.
The above results for areas (I)–(V) in Fig. 1 are summarized as Proposition 1.
where 6
7
8
9
10
11
12
A ¼ 80b þ 12b −132b þ 7b þ 57b −15b −b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 2 þ 6b ð1 þ bÞ 6 32−112b þ 158b2 −120b3 þ 54b4 −11b5 −b6 : Fig. 1 shows the results for the above calculations. We state the ranking orders of the payoffs of firms 0 and 1 for the five areas shown. In Fig. 1, the threshold value of α as the function of the degree of product differentiation b between areas (I) and (II), between (II) and (III), between (III) and (IV), and between (IV) and (V) are α 0 , α 1 , α 1 , and α 0, respectively. Except for the case wherein both the degree of product differentiation and the degree of importance of firm 1's relative performance are sufficiently low, from Fig. 1, α 0 ≥ α 1 ≥ ha 1 ≥ α 0 .7 A concise analysis of the comparison of each firm's payoff when its rival firm's strategic contract is fixed provides the following results. qp
pp
pq
pq
pp
qp
• In area (I), when α ≥ α 0, V0 ≥ V0 , V0 N V0 , V1 N V1 , and V1 N V1 are satisfied, implying that the q–q game is observed to be the unique equilibrium market structure. pp qp qq pq pq pp • In area (II), when α 0 N α ≥ α 1 , V0 N V0 , V0 N V0 , V1 N V1 , and qq qp V1 ≥ V1 are satisfied, implying that the q–q game is observed to be the unique equilibrium market structure. pp qp qq pq pq pp • In area (III), when α 1 N α ≥ α 1 , V0 N V0 , V0 N V0 , V1 ≥ V1 , and qq V1qp N V1 are satisfied, implying that no equilibrium market structure exists for any α∈ðα 1 ; α 1 Þ, whereas the p–p game is observed to be the equilibrium market structure when α ¼ α 1 . pp qp qq pq pp pq • In area (IV), when α 1 N α ≥ α 0 , V0 N V0 , V0 ≥ V0 , V1 N V1 , and qq qp V1 N V1 are satisfied, implying that the p–p game is observed to be the unique market structure. pp qp pq qq pp pq qp qq • In area (V), when α 0 N α, V0 N V0 , V0 N V0 , V1 N V1 , and V1 N V1 are satisfied, implying that the p–p game is observed to be the unique market structure. 7 Owing to overly complex calculations, when both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, the ranking order of α 0 , α 1 , α 1 , and α 0 cannot be obtained. However, the purpose of this paper is to confirm whether or not the market structures for the q–q, p–q, and q–p games can be observed. First, we consider the case wherein both the degree of product differentiation, b, and each firm's relative performance are not sufficiently low that α 0 ≥ α 1 ≥ α 1 ≥ α 0 is satisfied.
Proposition 1. Unless both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, for any value of product differentiation, b, when the degree of importance of firm 1's relative performance is sufficiently high, the q–q can become the equilibrium market structure, whereas when it is sufficiently low, the p–p game can become the equilibrium market structure. Note that when the degree of importance of firm 1's relative performance, α, is medium-sized, no equilibrium market structures exist under the pure strategies of the strategic contracts of firms 0 and 1. In Proposition 1, the intuition behind the results when the degree of importance of firm 1's relative performance, α, is sufficiently low is the same as that given in Matsumura and Ogawa (2012) since firm 1's behavior is similar to that of an absolute profit-maximizer. Thus, hereafter, we focus on the case wherein the degree of importance of firm 1's relative performance is not low. For the case wherein the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are not sufficiently low, we discuss the intuition behind Proposition 1, taking into account the fact that α 0 ≥ α 1 ≥ α 1 ≥ α 0 is satisfied in Fig. 1. We focus on the situation wherein the degree of product differentiation, b, is not sufficiently low and the degree of importance of firm 1's relative performance, α, is sufficiently high such that the q–q game can be observed as the unique market structure. This is different from a mixed duopoly wherein a private firm is an absolute profit maximizer à la Matsumura and Ogawa (2012). First, when firm 1 chooses a price contract, we consider the optimal contract of firm 0. Firm 0 attempts to induce the aggressive behavior of firm 1 in order to enhance social welfare, which is equal to its payoff. When firm 1 chooses a price contract, it is more difficult for firm 0 to control the quantity level of firm 1 since the price level of firm 1 becomes increasingly independent of the price level of firm 0 as the degree of importance of firm 1's relative performance, α, becomes higher if firm 0 chooses a price contract. If firm 0 chooses a quantity contract, the quantity level of firm 0 is negatively associated with the price level of firm 1 irrespective of α. Thus, when α is sufficiently high, firm 0 attempts to increase social welfare by choosing a quantity contract rather than a price contract. Next, we consider the optimal strategic contract of firm 0 in the case wherein firm 1 chooses a quantity contract. As the degree of importance
Y. Nakamura / Economic Modelling 44 (2015) 37–43 Table 2 Firm 0's payoff when b = 0.01 among the four games. pp
qp
41
Table 4 Firm 0's payoff when b = 0.05 among the four games. pq
pp
qp
pq
α
V0
V0
V0
V0
α
V0
V0
V0
V0
0.05 0.01 0.001 0.0001
0.867575 0.867574 0.867574 0.867574
0.867685 0.867587 0.867564 0.867562
0.867574 0.867574 0.867574 0.867574
0.867685 0.867587 0.867564 0.867562
0.1 0.05 0.01 0.001
0.839314 0.8393 0.839289 0.839286
0.840133 0.839568 0.839116 0.839014
0.839286 0.839286 0.839286 0.839286
0.840161 0.839583 0.839119 0.839014
Table 3 Firm 1's payoff when b = 0.01 among the four games. pp
qp
Table 5 Firm 1's payoff when b = 0.05 among the four games. pq
α
V1
V1
V1
V1
0.05 0.01 0.001 0.0001
0.244829 0.245025 0.245069 0.245074
0.244829 0.245025 0.245069 0.245074
0.24505 0.24505 0.24505 0.24505
0.245049 0.24505 0.24505 0.24505
of firm 1's relative performance, α, becomes higher, the strategic substitutability between q0 and q1 in the reaction function of firm 1 is weaker if firm 1 chooses a quantity contract. Thus, when α is sufficiently high, firm 0 can induce relatively aggressive behavior by firm 1 more easily in the q–q game. Therefore, when α is sufficiently high, it is optimal for firm 0 to choose a quantity contract if firm 1 chooses a quantity contract. Second, we consider the optimal strategy of firm 1 given a strategic contract for firm 0. When a strategic contract for firm 0 is fixed, the optimal contract for firm 1 is determined mainly by the following two factors: (1) its payoff is equal to its generalized relative profit, π1 − απ0 and (2) as α increases, the market competition in the p–p, q–q, and q–p games becomes more intense.8 Let us consider the case in which firm 0 chooses its price contract. From easy calculations, although pp pq pp pq pp both p0 N p0 and p1 N p1 hold for any α ∈ [0, 1], the values of p0 − pq pp pq p0 and p1 − p1 decrease with the increase in α since the market competition in the p–p game becomes more intense as α increases, which pp pq pq implies that π1 − π1 decreases as α increases. Taking π0 = 0 on the pq pq pq basis of p0 = c into account, when α is sufficiently high, V1 = π1 − pq pp pp pp απ0 N π1 − απ0 = V1 can be satisfied, implying that the optimal strategic contract for firm 1 is a quantity contract. Similar to the case wherein the strategic contract of firm 1 is a price contract, when firm 0 chooses its quantity contract, although qp qq qp qq p 0 N p 0 = c and p1 N p 1 hold for any α ∈ [0, 1], the differences qp qq qp qq between p0 and p0 and between p1 and p1 are sufficiently low when qp qq α is sufficiently high. Thus, the difference between π1 and π1 also decreases as pha increases. Therefore, when α is sufficiently high, qq qq qq qp qp qp V1 = π1 − α π0 N π1 − α π0 = V1 is satisfied. Of course, these facts concerning the optimal strategic contract of firm 1 result from the introduction of the degree of importance of firm 1's relative performance, α. In sum, when the degree of importance of firm 1's relative performance, α, is sufficiently high, quantity contracts can become the dominant strategies for both firms 0 and 1. Therefore, when α is sufficiently high, the q–q game can become supported by the dominant strategic contracts of both firms 0 and 1 unless the degrees of both product differentiation and each firm's relative performance are low. Finally, through numerical calculation concerning the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, particularly when both are sufficiently low, we derive the equilibrium market structures. More precisely, given sufficiently low levels of α and b, we consider the following three cases: (i) α = 0.05, 0.01, 0.001, and 0.0001 when b = 0.01; (ii) α = 0.1, 0.05, 0.01, and 0.001 when b = 0.05; and (iii) α = 0.2, 0.1, 0.05, and 0.01 when b = 0.1.9 As the data in Tables 2–7 shows for all three
pq
pq
8 Except for p0 , p0 , and p1 such that the price levels do not depend on α as the value of α increases, we obtain the result that each firm's price level decreases for any degree of product differentiation, b, in the p–p game, the q–q game, and the q–p game. 9 In Appendix A, we provide the values of the payoffs of firms 0 and 1 in the four games for the above three cases from Tables 2–7.
pp
qp
pq
α
V1
V1
V1
V1
0.1 0.05 0.01 0.001
0.224492 0.225625 0.226531 0.226734
0.224495 0.225625 0.226531 0.226735
0.22619 0.22619 0.22619 0.22619
0.226187 0.22619 0.22619 0.22619
cases, b = 0.01, b = 0.5, and b = 0.1, the quantity competition can be observed in the equilibrium on the basis of the dominant strategies of firms 0 and 1 when α is sufficiently high, whereas the price competition can be observed in the equilibrium on the basis of the strategic contracts of firms 0 and 1 when α is sufficiently low.10 The above results lead to the following proposition. Proposition 2. In the case wherein both the degree of product differentiation, b, and the degree of importance of firm 1's relative performance, α, are sufficiently low, the quantity competition can become the equilibrium market structure when α is high relative to b, whereas the price competition can become the equilibrium market structure when α is low relative to b. Proposition 2 states that similar to the case wherein both α and b are relatively high, in the situation wherein both α and b are sufficiently low, quantity competition can be observed in the equilibrium when α is high relative to b, whereas price competition can be observed in the equilibrium when α is low relative to b. However, analogous to the case wherein both α and b are relatively high, in the situation wherein both α and b are sufficiently low, we cannot determine whether or not any other market structure beyond the xtitq–q and p–p games can become the equilibrium market structure(s) even through the simulation of the levels of both α and b. Next we clarify whether or not the equilibrium market structure coincides with the optimal market structure from the viewpoint of social welfare.11 Comparing the equilibrium social welfare of the four games, we obtain the result in the following proposition concerning whether the highest social welfare is achieved in the quantity competition or in the price competition, which can be observed in the equilibrium.12 n pffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 Proposition 3. When α Nmax 1− 1−b =b; α 0 , the highest social welfare is achieved in the quantity competition. On the other hand, n
when αbmin
pffiffiffiffiffiffiffiffiffiffiffiffiffi o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 3 4 2 1− 1−b =b; −2 þ b þ b þ 4−4b−3b þ 6b þ b =2b ,
the highest social welfare is achieved in the price competition. Proposition 3 states that in the situation wherein both α and b are not low, it is unnecessary for the regulatory authority to prescribe any strong policy from the viewpoint of social welfare, as long as the degree of importance of the relative performance of the private firm is sufficiently 10
More precisely, we obtain the results for the equilibrium market structure in the three
8 > > ½b ¼ 0:01 : equilibrium market structure > > > > < cases as follows: > ½b ¼ 0:05 : equilibrium market structure > > > > > : ½b ¼ 0:1 : equilibrium market structure
quantity competition when α ¼ 0:05 when α ¼ 0:001
price competition quantity competition when α ¼ 0:1 when α ¼ 0:01
price competition quantity competition when α ¼ 0:2 price competition when α ¼ 0:05
and 0:01; and 0:0001; and 0:05; and 0:001; and 0:1; and 0:01:
11 Even although public firm 0 is a social welfare-maximizer, the equilibrium market structure does not always become the optimal market structure from the viewpoint of social welfare. 12 In Appendix A, concrete calculations of the equilibrium social welfare of the four games are given.
42
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Table 6 Firm 0's payoff when b = 0.1 among the four games. pp
qp
pq
α
V0
V0
V0
V0
0.2 0.1 0.05 0.01
0.807026 0.806923 0.806872 0.806831
0.809876 0.807841 0.806816 0.805992
0.806818 0.806818 0.806818 0.806818
0.810078 0.807945 0.80687 0.806005
high or sufficiently low. However, when α is medium-sized, no equilibrium market structure exists under the class of the pure strategy of strategic contracts for both the public firm and the private firm. Therefore, in such a situation, the regulatory authority may be cautious given the non-coincidence between the achieving market structure in the equilibrium and the optimal market structure from the viewpoint of social welfare. 4. Conclusion This paper investigated each firm's endogenous choice of a strategic contract in a mixed duopoly with differentiated goods composed of one social welfare-maximizing public firm and one generalized relative profit-maximizing private firm. More precisely, we investigated the situation in which both the public firm and private firm endogenously choose either of a quantity contract or a price contract in the market competition in the fashion of Singh and Vives (1984) and Matsumura and Ogawa (2012) when the objective of the private firm is to maximize the difference between the weighted sum of its absolute profit and the public firm's absolute profit. In contrast to the results for a standard mixed duopoly composed of one social welfare-maximizing public firm and one absolute profit-maximizing private firm in Matsumura and Ogawa (2012), we showed that when the degree of importance of the private firm's relative performance is sufficiently high, quantity competition can be observed as the equilibrium market structure on the basis of the quantity contracts of both the public firm and the private firm in areas in which both the degree of product differentiation and the degree of importance of the private firm's relative performance are relatively high. The result that the quantity competition can be observed in the equilibrium when the degree of importance of the private firm's relative performance is relatively high is supported by the dominant strategies concerning the strategic contracts by the public firm and the private firm. Thus, in the mixed duopoly with substitutable goods in which the private firm maximizes its generalized relative profit, the advantage of the price competition in Matsumura and Ogawa (2012) decreases with the introduction of the degree of importance of the private firm's relative performance. The results of this paper have certain limitations. Because of the high difficulty of analytical calculations, when the degree of product differentiation and the degree of the private firm's relative performance are sufficiently low, we cannot fully characterize the equilibrium market structures. Instead, using numerical examples for the degree of product differentiation and the degree of importance of the private firm's relative performance, we showed that quantity competition tends to be observed as the unique market structure when the degree of the private firm's relative performance is
sufficiently high, whereas the price competition tends to be observed as the unique market structure when the degree of the private firm's relative performance is sufficiently low, even in the case wherein both the degree of product differentiation and the degree of importance of the private firm's relative performance are sufficiently low. We found that for any value of the degree of product differentiation, quantity competition can be observed as the unique equilibrium market structure when the degree of importance of firm 1's relative performance is sufficiently high, whereas price competition can be observed as the unique market structure when each firm's relative performance is sufficiently low. On the other hand, it is unknown whether or not there exist levels of the degree of product differentiation and the degree of importance of the private firm's relative performance such that the p–q game and the q–p game, in which the strategic contracts of the public firm and the private firm are different from each other can be observed as the equilibrium market structures. In our future research, we will determine whether or not the above asymmetric market structures with respect to the strategic contracts of the public firm and the private firm can become the equilibrium market structures. Finally, we mention several relevant topics for future studies. First, the influence of the change of the cost functions of the public firm and the private firm (for example, the quadratic and symmetric cost functions of both firms) on the equilibrium market structures should be investigated throughout this paper, we used horizontal product differentiation without each firm's vertical structure. Wang and Wang (2009) compared the equilibrium market outcomes in the four games on the basis of each firm's strategic contract by using a vertical differentiation model in the context of a private duopoly. In addition, Wang and Wang (2010) investigated the vertically related market in the context of a managerial private oligopoly with the separation between ownership and management within each firm. Thus, we should merge the approaches of Wang and Wang (2009) and Wang and Wang (2010) to consider the endogenous selection of strategic contracts in the context of a mixed oligopoly with or without separation between ownership and management in each firm with a vertically related market in the model with vertical product differentiation. Since there are no works on the endogenous selection problem of the strategic contracts in the context of the mixed oligopoly using a vertically related market model with or without the vertically related structure, such studies would be interesting and meaningful. Appendix A. Comparison of the equilibrium social welfare among the four games Here, we give the results for the comparison of the equilibrium social welfare, which is equal to the payoff of firm 0, among the four games. We focus on the situations with the highest social welfare in either the quantity competition or the price competition, both of which can become the equilibrium market structure. From direct calculations of whether or not the highest social welfare is achieved in the q–q game, we obtain the following result:
pp
pq
qp
W −W W −W
W −W Table 7 Firm 1's payoff when b = 0.1 among the four games. pp
qp
pq
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð−1 þ bÞbða−cÞ b−2α þ bα 2 z z b 0 ⇔ α b 1− 1−b ; 2 2 2 b 2ð1 þ bÞ 2−b þ b α z ð2−bÞð1−bÞb½2 þ bð1−α Þ½bð−1 þ α Þ þ 2α z b 0 ⇔ α b b=ð2 þ bÞ ≡ α ; ¼ h i2 0 2 8ð1 þ bÞ 2−b ð1−α Þ 2 2 2 2 2 2 2 b ða−cÞ ð1−bα Þ b þ 4α−2b α þ b α ¼ N0: 2 2 2 2 8ð1 þ bÞ 2−b þ b α ¼
α
V1
V1
V1
V1
0.2 0.1 0.05 0.01
0.198389 0.202496 0.204551 0.206195
0.19843 0.2025 0.204551 0.206199
0.204545 0.204545 0.204545 0.204545
0.204493 0.204539 0.204545 0.204542
Thus, we obtain the result that the highest equilibrium social welfare pffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 is achieved in the q–q game when α Nmax 1− 1−b = b; α 0 ≡ b= o ð2 þ bÞ .
Y. Nakamura / Economic Modelling 44 (2015) 37–43
Moreover, we can calculate the condition of α with respect to b such that the highest equilibrium social welfare is achieved in the p–p game as follows: h ih i 2 2 3 2 bða−cÞ 4−b ð3−α Þ−2bα þ b ð1−α Þα 2α−b ð1−α Þα−bð1 þ α Þ z b0 h i2 2 2 8ð1 þ bÞ 2−b ð1−α Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ⇔ α≤ þ 6b3 þ b4 =2b ; N −2 þ b þ b þ 4−4b−3b h i 2 2 2 2 ð1−bÞb ða−cÞ b ð1−α Þ þ 4α pp pq N0; W −W ¼ h i2 2 8ð1 þ bÞ 2−b ð1−α Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ð1−bÞbða−cÞ b−2α þ bα z pp qq b 0⇔ α ≤ 1− 1−b : W −W ¼ 2 N 2 2 b 2ð1 þ bÞ 2−b þ b α pp
W −W
qp
¼−
Thus, we find that the highest equilibrium social welfare is achieved n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 4 −2 þ b þ b þ 4−4b−3b þ 6b þ b =
in the p–p game when αbmin pffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 2 2b ; 1− 1−b =b .
From Table 2, when b = 0.01, it is a dominant strategy for firm 0 to chose a quantity contract when α = 0.05 and α = 0.01, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.001 and α = 0.0001. From Table 3, when b = 0.01, it is a dominant strategy for firm 1 to choose a quantity contract when α = 0.01 and α = 0.05, whereas it is a dominant strategy for firm 1 to choose a price contract when α = 0.001 and α = 0.0001. Therefore, when b = 0.01, quantity competition can become the equilibrium market structure on the basis of the dominant strategies of both firms 0 and 1 when α = 0.01, whereas price competition can become the equilibrium market structure on the basis of the dominant strategies of firms 0 and 1 when α = 0.001 and α = 0.0001. From Table 4, when b = 0.05, it is a dominant strategy for firm 0 to choose a quantity contract when α = 0.1 and α = 0.05, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.01 and α = 0.001. From Table 5, when b = 0.05, it is a dominant strategy for firm 1 to choose its quantity contract when α = 0.1 and α = 0.05, whereas it is a dominant strategy for firm 1 to choose its price contract when α = 0.01 and α = 0.001. Therefore, when b = 0.05, quantity competition can become the equilibrium market structure on the basis of the strategic contracts of
43
both firms 0 and 1 when α = 0.1 and α = 0.05, whereas price competition can become the equilibrium market structure on the basis of the strategic contracts of firms 0 and 1 when α = 0.01 and α = 0.001. From Table 6, when b = 0.1, it is a dominant strategy for firm 0 to choose a quantity contract when α = 0.2 and α = 0.1, whereas it is a dominant strategy for firm 0 to choose a price contract when α = 0.05 and α = 0.01. From Table 7, when b = 0.1, it is a dominant strategy for firm 1 to choose a quantity contract when α = 0.2 and α = 0.1, whereas it is a dominant strategy for firm 1 to choose a price contract when α = 0.05. Therefore, when b = 0.1, quantity competition can become the equilibrium market structure based on the strategic contracts of firms 0 and 1 when α = 0.2 and α = 0.1, whereas price competition can become the equilibrium market structure on the basis of the strategic contracts of both firms 0 and 1 when α = 0.05 and α = 0.01. References Alchian, A.A., 1950. Uncertainty, evolution, and economic theory. J. Polit. Econ. 57, 211–221. Coats, J.C., Neilson, W.S., 2005. Beliefs about other-regarding preferences in a sequential public goods game. Econ. Inq. 43, 614–622. Klemperer, P., Meyer, M., 1986. Price competition vs. quantity competition: the role of uncertainty. RAND J. Econ. 17, 618–638. Matsumura, T., Matsushima, N., 2012. Competitiveness and stability of collusive behavior. Bull. Econ. Res. 64, s22–s31. Matsumura, T., Ogawa, A., 2012. Price versus quantity in a mixed duopoly. Econ. Lett. 116, 174–177. Matsumura, T. and Okamura, M. (2010) “Competition and Privatization Policy: The Relative Performance Approach” mimeo. Nakamura, Y., Saito, M., 2013. Capacity choice in a mixed duopoly: the relative performance approach. Theor. Econ. Lett. 3, 124–133. Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND J. Econ. 15, 546–554. Tanaka, Y., 2001a. Profitability of price and quantity strategies in an oligopoly. J. Math. Econ. 35, 409–418. Tanaka, Y., 2001b. Profitability of price and quantity strategies in a duopoly with vertical product differentiation. Economic Theory 17, 639–700. Tasnádi, A., 2006. Price vs. quantity in oligopoly games. Int. J. Ind. Organ. 24, 541–554. Vega-Redondo, F., 1997. The evolution of Walrasian behavior. Econometrica 65, 375–384. Wang, Y.C., Wang, L.F.S., 2009. Equivalence of competition mode in a vertically differentiated duopoly with delegation. South Afr. J. Econ. 77, 577–590. Wang, Y.C., Wang, L.F.S., 2010. Input pricing and market share delegation in a vertically related markets: is the timing order relevant? Int. J. Econ. Bus. 17, 207–221.