Economic Theory 7, 693 700 (200) Profitability of price quantity strategies in a duopoly with vertical product differentiation Yasuhito Tanaka Faculty of Law, Chuo University, 742-, Higashinakano, Hachioji, Tokyo, 92-0393, JAPAN (e-mail: yasuhito@tamacc.chuo-u.ac.jp) Received: April 23, 999; revised version: May 3, 2000 Summary. Using a model according to Mussa Rosen (978) Bonanno Haworth (998) we consider a sub-game perfect equilibrium of a two-stage game in a duopolistic industry in which the products of the firms are vertically differentiated. In the industry, there are a high quality firm a low quality firm. In the first stage of the game, the firms choose their strategic variables, price or quantity. In the second stage, they determine the levels of their strategic variables. We will show that, under an assumption about distribution of consumers preference, we obtain the result that is similar to Singh Vives (984) proposition (their Proposition 3) in the case of substitutes with nonlinear dem functions. That is, in the first stage of the game, a quantity strategy dominates a price strategy for both firms. Keywords Phrases: Price quantity strategies, Duopoly, Vertical product differentiation. JEL Classification Numbers: L3. Introduction Singh Vives (984) showed the following result. In a duopoly with (horizontally) differentiated products in which firms can choose a quantity or price strategy, if the products are substitutes the firms reaction functions in a Cournot game (a quantity game) are downward sloping those in a Bertr game (a price game) are upward sloping, some assumptions which ensure the existence of unique Cournot Bertr equilibria are satisfied, a quantity I would like to thank an anonymous referee for his very useful comments.
694 Y. Tanaka strategy dominates a price strategy, the Cournot equilibrium constitutes the sub-game perfect equilibrium of the two-stage game. In this paper, we consider a sub-game perfect equilibrium of a two-stage game in a duopolistic industry with vertical product differentiation. In the industry, there are a high quality firm a low quality firm. In the first stage of the game, the firms choose their strategic variables, price or quantity. In the second stage, they determine the levels of their strategic variables. In the next section, we present the model of this paper. In Section 3 4 we investigate the conditions for our model to satisfy the requirements for Singh Vives proposition in the case of substitutes with nonlinear dem functions (their Proposition 3), analyze a subgame perfect equilibrium of the game. We will show that, under an assumption about distribution of consumers preference, we obtain the result that is similar to Singh Vives Proposition 3. That is, in the first stage of the game, a quantity strategy dominates a price strategy for both firms. 2 The model We use a model of vertical product differentiation according to Mussa Rosen (978) Bonanno Haworth (998). There is a continuum of consumers with the same income, denoted by y, but different values of the taste parameter θ. Each consumer buys at most one unit of a product. If a consumer with parameter θ buys one unit of a product of quality k at price p, his utility is equal to y p+θk. If a consumer does not buy the product, his utility is equal to his income y. The parameter θ is distributed according to a smooth distribution function ρ = F(θ) in the interval 0 <θ 2. ρ denotes the probability that the taste parameter is smaller than θ. The size of consumers is normalized as one. There are two firms, Firm H (the high-quality firm) Firm L (the low-quality firm). Firm H sells a product of quality k H, Firm L sells a product of quality, with k H > > 0. k H are fixed. Let p i be the price charged by Firm i (i=h, L) q i be the output of Firm i. Let θ 0 be the value of θ for which the corresponding consumer is indifferent between buying nothing buying the low-quality product. Then θ 0 = p L. Cheng (985) presented a geometric analysis, Jéhiel Walliser (995) generalized an analysis by Singh Vives (984) to a general two person game. Klemperer (986) analyzed Nash equilibria of a one-stage game, not two-stage game, in which strategic variables are endogenously determined. Qin Stuart (997) considered a choice of strategic variables in a homogeneous oligopoly. 2 If we assume a uniform distribution like Bonanno Haworth (998), dem functions are linear.
Price quantity strategies vertical differentiation 695 Let θ be the value of θ for which the corresponding consumer is indifferent between buying the low-quality product the high-quality one. Then θ = p H p L. We assume 0 <θ 0 <θ <. Accordingly, the direct dem functions are given by q H = h H (p H, p L )= F, () pl q L = h L (p H, p L )=F F. (2) We have 0 < q L < 0 < q H <. The unit cost for Firm H is c H that for Firm L is c L, with c H > c L > 0. There is no fixed cost. From () (2) we obtain the inverse dem functions as follows, p H = f H (q H, q L )=G( q H )+ G( q H q L ), p L = f L (q H, q L )= G( q H q L ), where G(ρ) is the inverse function of F(θ). We have G (ρ) = F (θ) > 0, G (ρ) = F (θ) [F (θ)] 2. Since 0 < G( q H q L ) < G( q H q L ) < G( q H ), we have 0 < p H < k H 0 < p L <. We assume Assumption. F(θ) satisfies the following relation for 0 <θ, F (θ) < F (θ), k H or equivalently G (ρ) < G (ρ). k H This means that F(θ) is not so concave or convex.
696 Y. Tanaka 3 The Singh Vives proposition the equilibria in the second stage The Singh Vives proposition in the case of substitutes is stated as follows. The Singh Vives Proposition In a duopoly with differentiated products in which firms can choose a quantity or price strategy, if the following conditions are satisfied, a quantity strategy dominates a price strategy.. The products are substitutes, the reaction functions in a Cournot game (a quantity game) are downward sloping the reaction functions in a Bertr game (a price game) are upward sloping. 2. Some assumptions (their Assumption 2) which ensure the uniqueness of the Cournot equilibrium the Bertr equilibrium are satisfied. From the dem functions we obtain h H = h L = F > 0. p L p H Also from the inverse dem functions we obtain f H = f L = G ( q H q L )= < 0. q L q H F pl These mean that the products of Firm H Firm L are substitutes. Next, we consider the conditions for profit maximization for the firms. When one of the firms chooses a price (respectively quantity) strategy, the other firm determines its price or quantity given the rival firm s price (respectively quantity). We call the latter firm a price taking (respectively quantity taking) firm or a price taker (respectively quantity taker). When Firm L chooses a price strategy, Firm H is a price taker its profit is [ ] π H = F (p H c H ). The first order second order conditions for Firm H are π H = F p H c H F =0, (3) p H p 2 H = [ 2F + p H c H F ] < 0. (4) is When Firm H chooses a price strategy, Firm L is a price taker its profit [ ] pl π L = F F (p L c L ).
Price quantity strategies vertical differentiation 697 The first order second order conditions for Firm L are [ π ph L p L pl p L = F F (p L c L ) F + ] F pl =0, (5) [ = 2 F + ] F pl pl 2 [ (p L c L ) 2 F + ] kl 2 F pl < 0. (6) Similarly, the first order second order conditions for Firm H as a quantity taker are π H q H = G( q H )+ G( q H q L ) [G ( q H ) + G ( q H q L )]q H c H =0, (7) q 2 H = 2[G ( q H )+ G ( q H q L )] +[G ( q H )+ G ( q H q L )]q H < 0. (8) The first order second order conditions for Firm L as a quantity taker are π L = G( q H q L ) G ( q H q L )q L c L =0, (9) q L ql 2 = [2G ( q H q L ) G ( q H q L )q L ] < 0. (0) From Assumption we find that (4), (6), (8) (0) globally (for 0 < p L <, 0 < p H < k H,0< q L < 0 < q H < ) hold. Now we can show Lemma. The Bertr reaction functions are upward sloping, the Cournot reaction functions are downward sloping. Proof. See Appendix A. And Lemma 2. 2 π i pi 2 + 2 π i p i p j < 0 for 0 < p L <, 0 < p H < k H, i = H, L, j /= i, () 2 π i qi 2 + 2 π i q i q j < 0 for 0 < q L <, 0 < q H <, i = H, L, j /= i. (2)
698 Y. Tanaka Firm L Price Quantity Firm H Price πh B, πb L πh P, πq L Quantity π Q H, πp L πh C, πc L Table The first stage game Proof. See Appendix B. () (2) are similar to Assumption 2 in Singh Vives (984). They ensure that the Bertr reaction functions the Cournot reaction functions are well behaved, the abolute values of whose slopes are less than, there exist unique Bertr Cournot equilibria (Friedman, 977, 983). The four equilibrium configurations in the second stage of the game are as follows.. The Cournot equilibrium. Both firms are quantity takers. 2. The Bertr equilibrium. Both firms are price takers. 3. Firm H chooses a price strategy, Firm L chooses a quantity strategy. In this case Firm H is a quantity taker, Firm L is a price taker. 4. Firm H chooses a quantity strategy, Firm L chooses a price strategy. In this case Firm H is a price taker, Firm L is a quantity taker. Denote the profit of Firm H in these four cases by, respectively, πh C, πb H, πp H π Q H, denote the profit of Firm L in these four cases by, respectively, πc L, πl B, πq L πp L. Then we can show Proposition. π P H <π C H,π P L <π C L,π Q H >πb H, π Q L >πb L. Proof. Similar to the proof of Proposition 3 in Singh Vives (984). 4 Price or quantity: The first stage Next we consider the firms choice of strategic variables in the first stage of the game. The game is depicted in Table. From Proposition we have π P H <πc H,πP L <πc L,πQ H >πb H, πq L >πb L. Then we obtain the following result. Proposition 2. A quantity strategy is dominant for both firms, both firms choose a quantity strategy in the first stage of the game. Therefore the Cournot equilibrium constitutes the subgame perfect equilibrium of the two-stage game.
Price quantity strategies vertical differentiation 699 Appendices A Proof of Lemma This lemma is equivalent to the following inequalities. 2 [ π H = F + p ] H c H F > 0, (3) p H p L p L p H = [ F p L c L F ] > 0, (4) q H q L = [ G ( q H q L )+G ( q H q L )q H ] < 0, (5) = [ G ( q H q L )+G ( q H q L )q L ] < 0. (6) q L q H (3) (4) are derived from p L c L < < k H, p H c H < k H Assumption. (5) (6) are derived from 0 < q H <, 0 < q L < Assumption. B Proof of Lemma 2 From (4) (3) ph 2 From (6) (4) p 2 H 2 π H p H p L < 0, + 2 π H p H p L = F p 2 L 2 π L p L p H < 0, < 0. p 2 L + 2 π L p L p H = [ [ = k H F ( ph p L k H ) pl cl F pl kl 2 k H F ( ph p L [F ( pl ) k H ) + 2 F ( pl )] + F ( pl )] ] + pl cl F pl < 0.
700 Y. Tanaka From (8) (5) q 2 H + 2 π H q H q L < 0, q 2 H 2 π H q H q L = [2G ( q H )+ G ( q H q L )] +G ( q H )q H = [G ( q H )+ G ( q H q L )] [G ( q H ) G ( q H )q H ] < 0. From (0) (6) q 2 L q 2 L + 2 π L q L q H < 0, 2 π L q L q H = G ( q H q L ) < 0. References Bonanno, G. Haworth, B.: Intensity of competition the choice between product process innovation. International Journal of Industrial Organization 6, 495 50 (998) Cheng, L.: Comparing Bertr Cournot equilibria: A geometric approach. R Journal of Economics 6, 46 47 (985) Friedman, J. W.: Oligopoly the theory of games. Amsterdam: North-Holl 977 Friedman, J. W.: Oligopoly theory. Cambridge: Cambridge University Press 983 Jéhiel, P. Walliser, B.: How to Select a Dual Nash Equilibrium. Games Economic Behavior 0, 333 354 (995) Klemperer, P. Meyer, M.: Price competition vs. quantity competition: The role of uncertainty. R Journal of Economics 7, 68 638 (986) Mussa, M. Rosen, S.: Monopoly product quality. Journal of Economic Theory 8, 30 56 (978) Qin, C.-Z. Stuart, C.: Bertr versus Cournot revisited. Economic Theory 0, 497 507 (997) Singh, N. Vives, X.: Price quantity competition in a differentiated duopoly. R Journal of Economics 5, 546 554 (984) Vives, X.: On the efficiency of Bertr Cournot equilibria with Product differentiation. Journal of Economic Theory 36, 66 75 (985)