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Endogenous Timing in Vertically-Related Markets

Hong-Ren Din and Chia-Hung Sun

Abstract

This paper investigates the theory of endogenous timing by taking into account a vertically-related market where an integrated firm competes with a downstream firm. Contrary to the standard results in the literature, we find that both firms play a sequential game in quantity competition and play a simultaneous game in price competition. Under mixed quantity-price competition, the firm choosing a price strategy moves first and the other firm choosing a quantity strategy moves later in equilibrium. Given that the timing of choosing actions is determined endogenously, aggregate profit (consumer surplus) is higher (lower) under price competition than under quantity competition. Lastly, social welfare is higher under quantity competition than under price competition when the degree of product substitutability is relatively low.

JEL Classification: L13; D43; D21

Funding statement: This article was supported by Ministry of Science and Technology, NSC 103-2410-H-265-002 (10.13039/501100003711).

Appendix

A Appendix

We first calculate the integrated firm’s equilibrium payoff with no foreclosure under all the Q-Q, P-P, Q-P, and P-Q games. We then compare the integrated firm’s payoff under the three kinds of timing scenarios with its monopoly payoff:

(24)π1FFPQa24=a2(1β)287β2>0,π1SFPQa24=a2(1β)285β2>0,π1FSPQa24=a2(1β)8(1+β)>0.
(25)s1
(26)s2
(27)r1(s2)

Therefore, the integrated firm has no incentive to foreclose the downstream firm under all the Q-Q, P-P, Q-P, and P-Q games. Q.E.D.

B Appendix

An equilibrium is stable when the dynamic adjustment process converges to the Nash equilibrium from any strategy pair in a neighborhood of the equilibrium. Let r2(s1) and (s1,s2) denote firm 1’s and firm 2’s strategic variables and 2π1QPFFq122π2QPFFp22+2π1QPFFq1p22π2QPFFp2q1=45β2>0,for β\lt2550.89. and 2π1PQFFp122π2PQFFq22+2π1PQFFp1q22π2PQFFq2p1=45β2>0,for β\lt2550.89. denote their respective best-response function in the market competition stage. In the β\lt25/50.89 plane, if firm 1’s reaction curve is steeper than firm 2’s reaction curve, then the equilibrium is stable. Under the Q-P game, the stability condition when the two firms move simultaneously runs as follows:

(28)π1FFπ1SF=2a2β2(1β)2(87β2)(85β2)>0,π1FSπ1SS=a2β2(1β)8(1+β)(87β2)>0,π2FFπ2FS=a2β2(1β)(1615β2)16(1+β)(87β2)2<0,π2SFπ2SS=2a2β6(1β)2(87β2)2(85β2)2>0.

Under the P-Q game, the stability condition when the two firms move simultaneously is:

(29)(F,S)

Therefore, the condition π1FFπ1SF=a2β2(1β)(88β2+β4)4(1+β)(811β2+4β4)(85β2+β4)<0,π1FSπ1SS=a2β2(1β)24(2β2)(811β2+4β4)>0,π2FFπ2FS=a2β2(1β)2(32β2)(1619β2+6β4)4(811β2+4β4)2(2β2)2>0,π2SFπ2SS=a2β4(1β)(2β2)(1647β2+41β413β6+β8)(1+β)(811β2+4β4)2(85β2+β4)20,if and only if β0.77. ensures satisfaction of the stability condition under the P-Q and Q-P games. Q.E.D.

C Appendix

We make comparisons of the payoffs in the P-Q game under simultaneous play and sequential play to determine what order of play occurs:

(30)(S,F)

We conclude β0.77 is the unique equilibrium outcome under the P-Q game. We make comparisons of the payoffs in the Q-P game under simultaneous play and sequential play to determine what order of play occurs:

(31)(F,F)

We conclude (F,S) is the unique equilibrium outcome for (S,F) under the Q-P game. We note that in Hamilton and Slutsky’s (1990) endogenous timing game, it is assumed that two symmetric firms with exogenously constant marginal cost compete in the product market. In this case, an equilibrium exists under all the Q-Q, P-P, Q-P, and P-Q games. More specifically, by symmetry between the two firms, if a firm’s best response is to move in the first period when its opponent moves in the first period, then the simultaneous subgame, β<0.77, is the equilibrium outcome. In contrast, if a firm’s best response is to move in the second period when its opponent moves in the first period, then both sequentially-played subgames, β<0.77 and CSSF=a2(80108β2+4β3+33β4)8(85β2)2,π1SF+π2SF=a2(11296β60β2+56β33β4)4(85β2)2,SWSF=a2(304192β228β2+116β3+27β4)8(85β2)2., are the outcomes of equilibria. The reason is that given the exogenously constant marginal cost, a firm’s leader payoff is not smaller than its payoff under a simultaneous game.

In such a vertically-related market the two firms are not symmetric and firm 2’s production cost (input price) is endogenously determined. Given that the integrated firm moves in the first period, from the third equation the downstream firm’s best response is to move in the first period. However, from the first equation, when the downstream firm moves in the first period, the integrated firm chooses to move in the second period due to the asymmetry between the two firms. The optimal input prices in the game’s second stage are different under the simultaneous game and the two sequential games. From the fourth equation, if firm 1 moves in the second period, then the best strategy for firm 2 is to move in the second period for CSFS=a2(52β4β2+2β3)8(2β2)2,π1FS+π2FS=a2(76β2β2+2β3)4(2β2)2,SWFS=a2(1914β8β2+6β3)8(2β2)2.. From the second equation the best strategy for firm 1 is to move in the first period when firm 2 moves in the second period. Therefore, no equilibrium exists for CSFF=a2(80+16β+36β2+24β3+β4+5β5)8(8+β2)2(1+β),π1FF+π2FF=a2(112+16β+36β24β3+5β43β5)4(8+β2)2(1+β),SWFF=a2(304+48β+108β2+16β3+11β4β5)8(8+β2)2(1+β).. Q.E.D.

D Appendix

Proposition 1 shows that the two cases of sequential entry are both equilibrium outcomes under the Q-Q game. We calculate the consumer surplus, aggregate profits, and social welfare as follows:

(32)(p1F,
(33)p2F)

Proposition 2 shows that the simultaneous move is the unique equilibrium outcome under the P-P game. We calculate the consumer surplus, aggregate profits, and social welfare as follows:

(34)(q1S,

Case A: We express the comparisons of aggregate profits, consumer surplus, and social welfare between price competition with q2F)ΠPPFFΠQQSF=12βρAρB(164β23β4)>0,\hfillCSPPFFCSQQSF=ρAρB(256+160β128β256β320β423β5)<0,\hfillSWPPFFSWQQSF=ρAρB(25632β128β28β320β4+13β5)<0,\hfill and quantity competition with ρA=a2β(1β)(4β2)ρB=2(1+β)(8+β2)2(85β2)2. as:

(35)(p1F,

where p2F) and (q1F,

Case B: We express the comparisons of aggregate profits, consumer surplus, and social welfare between price competition with q2S)ΠPPFFΠQQFS=2βρC(3+β2)(3217β2+3β4)2ρD>0,\hfillCSPPFFCSQQFS=ρC4ρD(128112β215β3+16β4+10β5+4β6+5β7)<0,\hfillSWPPFFSWQQFS=ρCρD(128192β112β2+23β3+16β4+26β5+4β6β7)>0,\hfill and quantity competition with ρC=a2β(1β)ρD=8(1+β)(8+β2)2(2β2)2 as:

(36)π1=p1q1+wq2+F=(aq1βq2)q1+wq2+F,π2=(p2w)q2F=(aβq1q2w)q2F.

where F and w. Based on the above analysis, we establish Proposition 4. Q.E.D.

E Appendix

Consider first the Q-Q game. The profits of firms 1 and 2 are specified respectively as follows:

(37)maxw,Fπ1=(2aβa+βw)2(4β2)2+w(2a+βa2w)4β2+F,\hfills.t. π2(w)F0.\hfill

We first examine the case where the output levels are determined simultaneously. The third stage is the same as that in the basic model. Turning to the game’s second stage, firm 1 chooses w and w in order to maximize its profit, subject to firm 2’s non-negative profit condition.

(38)F

Differentiating firm 1’s profit with respect to π2 and setting it equal to zero, we yield the equilibrium per-unit fee w=aβ(2β)22(43β2),\hfillF=4a2(1β)2(43β2)2.\hfill. Firm 1 can extract firm 2’s entire profit by charging a fixed fee π1FF=π1SS=a2(88β+β2)4(43β2),\hfillπ2FF=π2SS=0.\hfill equal to potential w=aβ(2β)2(2β2),\hfillF=a2(1β)2(2β2)2,\hfillπ1FS=a2(88ββ2+2β3)4(2β2)2,\hfillπ2FS=0.\hfill:

(39)w=aβ(42ββ2)2(43β2),\hfillF=2a2(2β2)(1β)2(43β2)2,\hfillπ1SF=a2(88β+β2)4(43β2),\hfillπ2SF=0.\hfill

The equilibrium payoffs of both firms when the two firms move simultaneously are as follows:

(40)π1FFπ1SF=0,\hfillπ1FSπ1SS=[aβ2(1β)]24(43β2)(2β2)2<0.\hfill

By similar procedures, the equilibrium per-unit fee, fixed fee, and payoffs of both firms when firm 1 is a quantity leader are as follows:

(41)(F,F),

By similar procedures, the equilibrium per-unit fee, fixed fee, and payoffs of both firms when firm 2 is a quantity leader are as follows:

(42)(S,F),

Firm 2’s equilibrium profit is zero under two-part tariffs in all cases. We now compare firm 1’s payoffs in the game, as they arise in sequential play, with the Nash payoff under simultaneous play:

(43)(S,S).

Based on the above analysis, we establish the following three equilibrium outcomes: w=aβ(2+β)22(4+5β2),\hfillF=a2(1β)(2+β2)2(1+β)(4+5β2)2,\hfillπ1FF=π1SS=a2(8+9β2+β3)4(1+β)(4+5β2),\hfillπ2FF=π2SS=0.\hfillw=aβ(2+ββ2)4,\hfillF=a2(1β)(2β2)216(1+β),\hfillπ1FS=a2(8β2+β3)16(1+β),\hfillπ2FS=0.\hfill and w=aβ(4+2ββ2β3)2(4+β2β4),\hfillF=2a2(2β2)(1β)(1+β)(4+β2β4)2,\hfillπ1SF=a2(8+β2+β32β4)4(1+β)(4+β2β4),\hfillπ2SF=0.\hfill

We next consider the P-P game. With the same procedures, the equilibrium per-unit fee, fixed fee, and payoffs of both firms when the firms move simultaneously are as follows:

(44)π1FFπ1SF=a2β4(1β)(4+β2)4(1+β)(4+5β2)(4+β2β4)>0,\hfillπ1FSπ1SS=5a2β4(1β)16(4+5β2)(1+β)<0.\hfill

The equilibrium per-unit fee, fixed fee, and payoffs of both firms when firm 1 is a price leader are as follows:

(45)(F,F)

The equilibrium per-unit fee, fixed fee, and payoffs of both firms when firm 2 is a price leader are as follows:

(46)(S,S)

Comparing firm 1’s payoffs in the game, as they arise in sequential play, with the Nash payoff under simultaneous play yields:

(47)π1FFπ1SF=a2β4(1β)(4+β2)4(1+β)(4+5β2)(4+β2β4)>0,π1FSπ1SS=5a2β4(1β)16(4+5β2)(1+β)<0.

It follows that the equilibrium outcomes are given by (F,F) and (S,S). Q.E.D.

Acknowledgements

We gratefully acknowledge the constructive comments and suggestions of two anonymous referees. All remaining errors are ours.

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Supplementary Material

The online version of this article offers supplementary material (DOI:https://doi.org/10.1515/bejte-2016-0103).

Published Online: 2018-04-27

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