# 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.

**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:

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

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

Therefore, the condition

## 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:

We conclude

We conclude

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

## 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:

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:

**Case A**: We express the comparisons of aggregate profits, consumer surplus, and social welfare between price competition with

where

**Case B**: We express the comparisons of aggregate profits, consumer surplus, and social welfare between price competition with

where

## E Appendix

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

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

Differentiating firm 1’s profit with respect to

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

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:

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:

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:

Based on the above analysis, we establish the following three equilibrium outcomes:

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:

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

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

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

It follows that the equilibrium outcomes are given by

# 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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