Quantty Precommtment and Cournot and Bertrand Models wth Complementary Goods Kazuhro Ohnsh 1 Insttute for Basc Economc Scence, Osaka, Japan Abstract Ths paper nestgates Cournot and Bertrand duopoly models wth complementary goods, where frms can enter nto lfetme employment contracts wth ther respecte workers as a strategc dece. The paper treats the followng four cases: Cournot competton wth strategc complements, Cournot competton wth strategc substtutes, Bertrand competton wth strategc substtutes and Bertrand competton wth strategc complements. The paper presents the equlbrum outcomes of the four cases. In addton, t s shown that lfetme employment s benefcal for frms n the cases wth strategc complements. Keywords: Cournot model, Bertrand model, complementary goods, lfetme employment Cte ths artcle: Ohnsh, K. (2017). Quantty Precommtment and Cournot and Bertrand Models wth Complementary Goods. Internatonal Journal of Management, Accountng and Economcs, 4(1), 17-29. Introducton Ths paper consders a market stuaton n whch frms are allowed to use lfetme employment as a strategc commtment dece (see Ohnsh, 2001, 2002). If a frm legally enters nto a lfetme employment contract wth ts workers, then ts wage cost snks and ts margnal cost decreases. Kreps and Schenkman (1983) use capacty nestment, whch functons ust as well as lfetme employment used n ths paper. They examne the subgame perfect equlbrum of prce-settng duopoly competton wth homogeneous goods, and show that the unque equlbrum concdes wth the Cournot soluton. Yn and Ng (1997) show that two-stage prce-settng duopoly competton wth substtute goods, followng a smultaneous 1 Correspondng author s emal: ohnsh@e.people.or.p 17
endogenous choce of capacty, yelds the Cournot soluton. Ohnsh (2006) nestgates the respecte equlbrum outcomes of a prce-settng lfetme-employment-contract game wth substtute goods and a prce-settng lfetme-employment-contract game wth complementary goods, and demonstrates that n each game, the equlbrum concdes wth the Bertrand outcome wth no lfetme employment. In addton, Ohnsh (2012) examnes the respecte equlbrum outcomes of the two games of a quantty-settng duopoly game wth substtute goods and a quantty-settng duopoly game wth complementary goods, and fnds that the ntroducton of lfetme employment nto the analyss of quantty-settng game wth complementary goods ncreases economc welfare. We study quantty-settng and prce-settng models wth complementary goods, where duopolsts can enter nto lfetme employment contracts wth ther respecte workers as a strategc commtment. We treat the followng four cases: Cournot competton wth strategc complements, Cournot competton wth strategc substtutes, Bertrand competton wth strategc substtutes, and Bertrand competton wth strategc complements. We present the subgame perfect equlbra when duopolsts are allowed to adopt lfetme employment by usng quantty-settng and prce-settng models. As a result of ths analyss, t s shown that lfetme employment s benefcal for duopolsts n the cases wth strategc complements. The Quantty-Settng Model and Its Equlbrum Outcomes In ths secton, we formulate the quantty-settng model wth complementary goods and dscuss ts equlbrum outcomes. There are two frms, desgnated frm 1 and frm 2. In the balance of ths paper, when and are used n an expresson, they represent 1 and 2 wth. Frm s proft s ( q, q ) p ( q, q ) q q (1) where p represents frm s nerse demand functon, q s frm s output, and s frm s constant margnal cost. The quantty-settng game s played n the followng sequence. In stage 1, each frm noncooperately decdes whether to use lfetme employment as a strategc commtment dece. If frm uses lfetme employment, then t chooses ts quantty q and enters nto a lfetme employment contract wth the workers necessary to achee q. In stage 2, each frm noncooperately chooses ts actual quantty q. At the end of stage 2, the market opens and frm sells ts actual output q. Therefore, frm s proft s shown as follows: ˆ ( q, q, q ) ( q, q ) f q q ( q, q ) ( q q ) r f q q (2) where r (0, ] represents frm s wage cost per unt of output. If frm chooses q and enters nto a lfetme employment contract wth all of the workers necessary to achee q, then ts margnal cost s affected by the lfetme employment contract. Hence, frm s 18
margnal cost has a dscontnuty at q = q. Gen q, frm soles ts proft maxmzaton problem wth respect to q. If frm s margnal cost of output equals, then a usual way to defne ts quantty reacton functon s R ( q ) arg max ( q, q ) (3) { q 0} and f frm s margnal cost of output equals r, then ts quantty reacton functon s llustrated by R ( q ) arg max[ ( q, q ) rq ] (4) r { q 0} Therefore, f frm chooses q and prodes lfetme employment, then we can defne ts quantty best response as follows: R ( q ) f q q r ( ) f R ( q ) f q q R q q q q (5) The Cournot-Nash equlbrum can be defned as quantty leels (q 1 C, q 2 C ) where q C R (q C ). It s assumed that there exsts a unque Cournot-Nash equlbrum n 0 < q <. Moreoer, the followng assumptons are added. Assumpton 1 (dfferentablty): p (q, q ) s twce contnuously dfferentable wth p / q < 0 (downward-slopng demand) and p / q > 0 (complementary goods). Assumpton 2 (concaty of proft functon): 2 π / q 2 < 0. Assumpton 3 (stablty): If (R (q ), q ) 2, then 0 < R '(q ) < 1. ++ These are standard assumptons n Cournot duopoly games except complementary goods. Throughout ths paper, our soluton concept s subgame perfecton. As usual, we apply backward nducton. We present here the followng lemmas. emma 1: If frm chooses q and uses lfetme employment as a strategc commtment dece, then at equlbrum q = q. 19
Proof: See Ohnsh (2015, emma 1). emma 2: The proson of lfetme employment by frm ncreases ts proftmaxmzng quantty. Proof: See Ohnsh (2015, emma 2). We now explore the followng two cases. Case 1: R / q > 0 Case 2: R / q < 0 We treat these cases n turn. Case 1 Ths s the case of strategc complements n whch goods are complements. Fgure 1 depcts both frms reacton cures. q 2 R 2 r π 1 q 2 C R 2 π 2 0 q 1 Fgure 1 The Cournot game wth strategc complements 20
In ths fgure, R represents frm s reacton cure when ts margnal cost of output r equals, R s frm s reacton cure when ts margnal cost of output equals r, and π s frm s soproft cure extendng through pont. Both R and R are upward slopng. If nether frm uses lfetme employment as a strategc commtment dece, then the unque soluton s at C. Assume that only frm 2 prodes lfetme employment to ts workers. From emma 2, we can understand that the proson of lfetme employment by frm ncreases ts optmal quantty and shfts ts quantty reacton cure to the rght. Therefore, f frm 2 chooses q 2 and prodes lfetme employment to ts workers, then ts margnal cost of output has a dscontnuty at q 2 = q 2 and ts quantty reacton cure becomes the thck knked lne as drawn n ths fgure. The soluton s determned n a Cournot-Nash fashon. The quantty reacton cures cross at. From ths fgure, we can understand that frm 2 s proft s hgher at pont than pont C. The soluton of ths case s characterzed n the followng proposton. Proposton 1: In the quantty-settng game wth p / q > 0 and R / q > 0, there exsts a unque subgame perfect Nash equlbrum n whch at least one frm uses lfetme employment as a strategc commtment dece. At the equlbrum, both frms obtan hgher profts than n the Cournot game wth no lfetme employment. Proof: See Ohnsh (2012, Propostons 3 and 4). Proposton 1 mples that each duopolst can mproe ts proft by usng lfetme employment as a strategc dece. 21
q 2 π 2 C π 1 r R 2 Case 2 0 q 1 q 1 Fgure 2 The Cournot game wth strategc substtutes We next analyse the case of strategc substtutes n whch goods are complements. Fgure 2 depcts both frms reacton cures. Both R and R r slope downwards. If nether frm prodes lfetme employment to ts workers, then the unque Cournot equlbrum occurs at pont C. emma 2 states that the proson of lfetme employment by frm ncreases ts optmal quantty and shfts ts quantty reacton cure to the rght. Therefore, f frm 1 chooses q 1 and prodes lfetme employment to ts workers, then ts margnal cost of output has a dscontnuty at q 1 = q 1 and ts quantty reacton cure becomes the thck knked lne as drawn n ths fgure. The frms choose quanttes n a Cournot fashon. The quantty reacton cures cross at pont. From ths fgure, we can understand that frm 1 s proft s lower at pont than pont C. We now characterze the equlbrum of Case 2 n the followng proposton. Proposton 2: In the quantty-settng game wth p / q > 0 and R / q < 0, there exsts a unque subgame perfect Nash equlbrum that concdes wth the Cournot output wth no lfetme employment. 22
Proof: Frst of all, we proe that f frm prodes lfetme employment to ts workers, then frm s proft s lower than ts Cournot proft wth no lfetme employment. emma 2 states that the proson of lfetme employment by frm ncreases ts proftmaxmzng output. From Assumpton 1, f we dfferentate π wth respect to q, then t ge us π / q = ( p / q ) q > 0. Hence, frm s proft exceeds ts Cournot proft wth no lfetme employment. Increasng frm s output ncreases frm s amount demanded because of complementary goods. Frm s proft ncreases een at ts Cournot output. In frm s optmal strategy, ts output decreases because of strategc substtutes. Decreasng frm s output decreases frm s amount demanded because of complementary goods. From Assumpton 1, f we dfferentate π wth respect to q, then t ge us π / q = ( p / q ) q > 0. Decreasng frm s output decreases frm s proft. Therefore, nether frm has an ncente to unlaterally deate from the equlbrum wth no lfetme employment. Thus the proposton s ald. QED Proposton 2 ndcates that nether duopolst can mproe ts proft by prodng lfetme employment. Thus, the unque outcome concdes wth the Cournot soluton wth no lfetme employment. The Prce-Settng Model and Its Equlbrum Outcomes In ths secton, we formulate the prce-settng model wth complementary goods and dscuss ts equlbrum outcomes. Frm s proft s ( p, p ) p q ( p, p ) q ( p, p ) (6) where p represents frm s prce, and q s frm s demand functon. The prce-settng game runs as follows. In stage 1, each frm noncooperately decdes whether to prode lfetme employment to ts workers. If frm uses lfetme employment as a strategc dece, then t chooses q and enters nto a lfetme employment contract wth the workers necessary to achee q. In stage 2, each frm noncooperately chooses ts prce p. At the end of stage 2, the market opens and frm sells ts actual quantty q (p, p ). Therefore, frm s proft changes as follows: ˆ ( q, p, p ) ( p, p ) f q ( p, p ) q ( p, p ) ( q q ( p, p )) r f q ( p, p ) q (7) Gen p, frm must sole ts proft maxmzaton problem wth respect to p. If frm s margnal cost of output equals, then a usual way to defne ts prce reacton functon s 23
R ( p ) arg max ( p, p ) (8) { p 0} and f frm s margnal cost of output equals r, then ts prce reacton functon s R ( p ) arg max[ ( p, p ) rq ( p, p )] (9) r { p 0} Therefore, f frm chooses q and enters nto a lfetme employment contract wth ts workers, then we can defne ts prce best response changes as follows: R ( p ) f q ( p, p ) q r ( ) f (, ) R ( p ) f q ( p, p ) q R p q q p p q (10) The Bertrand-Nash equlbrum can be defned as prce leels (p 1 B, p 2 B ) where p B R (q B ). It s assumed that there s a unque Bertrand-Nash equlbrum n 0 < p <. Moreoer, the followng assumptons are ntroduced. Assumpton 4 (dfferentablty): q (p, p ) s twce contnuously dfferentable wth q / p < 0 (downward-slopng demand) and q / p < 0 (complementary goods). Assumpton 5 (concaty of proft functon): 2 π / p 2 < 0. Assumpton 6 (stablty): If (R (p ), p ) 2, then 0 < R '(p ) < 1. ++ These are standard assumptons n Bertrand games except complementary goods. We proe here the followng lemma. emma 3: The proson of lfetme employment by frm lowers ts proftmaxmzng prce. Proof: From (7), we understand that the proson of lfetme employment by frm neer ncreases ts margnal cost of output. If frm s margnal cost of output equals, then the frst-order condton for proft maxmzaton s q q q p 0 p p (11) Howeer, f frm s margnal cost of output equals r, the frst-order condton for proft maxmzaton s 24
q q q q p r 0 p p p (12) where r > 0 and q / p < 0 (from Assumpton 1). Hence, the sgn of q + p ( q / p ) ( q / p ) needs to be plus n order to satsfy (12). Ths completes the proof. QED We handle the followng two cases. Case 3: R / p < 0 Case 4: R / p > 0 We now examne these two cases n turn. Case 3 We explore the case of strategc substtutes n whch goods are complements. Fgure 3 depcts both frms reacton cures. Both R and R r are downward slopng. Assume that frm 2 unlaterally uses lfetme employment as a strategc commtment dece. From emma 4, we can understand that the proson of lfetme employment by frm lowers ts optmal prce and shfts ts prce reacton cure to the left. Therefore, f frm 2 chooses q 2 and prodes lfetme employment to ts workers, then ts margnal cost of output has a dscontnuty at q 2 = q 2 and ts prce reacton cure becomes the thck knked lne as drawn n ths fgure. The prce reacton cures cross at pont. From ths fgure, we see that frm 2 s proft s lower at pont than pont B. 25
p 2 R 2 π 2 B π 1 r q 2 = q 2 0 p 1 Fgure 3 The Bertrand game wth strategc substtutes The soluton of Case 3 s characterzed n the followng proposton. Proposton 3: In the prce-settng game wth q / p < 0 and R / p < 0, there exsts a unque subgame perfect Nash equlbrum that concdes wth the Bertrand soluton wth no lfetme employment. Proof: See Ohnsh (2006, emma 3 and Proposton 2). Proposton 3 mples that there are no frms that uses lfetme employment as a strategc dece. That s, nether duopolst s able to ncrease ts proft by prodng lfetme employment. Therefore, the unque equlbrum concdes wth the Bertrand soluton wth no lfetme employment. Case 4 Case 4 s the case of strategc complements n whch goods are complements. Fgure r 4 depcts both frms reacton cures. Both R and R slope upwards. If nether frm prodes lfetme employment, then the unque soluton remans at B. 26
Assume that only frm 1 prodes lfetme employment to ts workers. From emma 3, we see that the proson of lfetme employment by frm lowers ts optmal prce and shfts ts prce reacton cure to the left. Therefore, f frm 1 chooses q 1 and prodes lfetme employment, then ts margnal cost of output has a dscontnuty at q 1 = q 1 and ts prce reacton cure becomes the thck knked lne as drawn n ths fgure. The prce reacton cures cross at. The frms choose prces n a Bertrand-Nash fashon. From ths fgure, we can understand that frm 1 s proft s hgher at pont than pont B. p 2 r π 2 R 2 B π 1 q 1 = q 1 0 p 1 Fgure 4 The Bertrand game wth strategc complements We present here the followng lemma. emma 4: In the prce-settng game wth q / p < 0 and R / p > 0, f one frm unlaterally uses lfetme employment as a strategc dece, then each frm s proft exceeds ts Bertrand proft wth no lfetme employment. Proof: Assume that frm unlaterally uses lfetme employment as a strategc dece. emma 3 states that the proson of lfetme employment by frm lowers ts proftmaxmzng prce. Decreasng frm s prce ncreases frm s amount demanded because of complementary goods. Frm s proft ncreases een at ts Bertrand prce. Frm s optmal strategy must yeld at least ths proft. Hence, frm s proft exceeds ts Bertrand proft wth no lfetme employment. In frm s optmal strategy, ts prce drops because 27
of strategc complements. Decreasng frm s prce ncreases frm s amount demanded because of complementary goods. If q = q, then decreasng frm s prce ncreases frm s proft. emma 1 states that at equlbrum q = q. Thus, frm s proft also exceeds ts Bertrand proft wth no lfetme employment. QED We now present the equlbrum of Case 4 n the followng proposton. Proposton 4: In the prce-settng game wth q / p < 0 and R / p > 0, there exsts a unque subgame perfect Nash equlbrum n whch at least one frm uses lfetme employment as a strategc commtment dece. At the equlbrum, both frms enoy hgher profts than n the Bertrand game wth no lfetme employment. Proof: emma 4 states that the unlateral lfetme employment soluton generates a hgher proft for each frm than at the Bertrand soluton wth no lfetme employment. Hence, there exsts no equlbrum n whch nether frm prodes lfetme employment because cyclng of choces s mpossble. From emma 4, the rest of the results hold f only one frm uses lfetme employment as a strategc commtment dece. There s a blateral lfetme employment equlbrum f the blateral lfetme employment profts exceed the unlateral lfetme employment profts. Thus, Proposton 4 follows. QED Proposton 4 mples that lfetme employment s benefcal for both duopolsts. Concluson We hae presented the equlbrum outcomes of quantty-settng and prce-settng models wth complementary goods. In addton, t has been shown that lfetme employment can be benefcal for frms n the games wth strategc complements. If a game s played n strategc complements, then at least one frm prodes lfetme employment n a non-cooperate soluton. fetme employment enables both frms to get more n a non-cooperate game. Therefore, we can say that t facltates tact colluson. As a result of ths analyss, we fnd that tact colluson s facltated n the games wth strategc complements. References Cooper, T. E. (1986). Most-faored-customer prcng and tact colluson. Rand Journal of Economcs, 17, 377-388. Fredman, J. W. (1977). Olgopoly and the Theory of Games, Amsterdam: North- Holland. Kreps, D. M., & Schenkman, J. A. (1983). Quantty precommtment and Bertrand competton yeld Cournot outcomes. Bell Journal of Economcs, 14, 326-337. 28
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