Integration of Complementary Multiproduct Firms

Hao Wang

doi:10.1515/rle-2020-0058

Article

Integration of Complementary Multiproduct Firms

Hao Wang

Published/Copyright: November 4, 2021

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From the journal Review of Law & Economics Volume 17 Issue 3

Abstract

Two firms offer product series from which multiple complementary pairs are formed. The firms engage in a price- or quantity-choosing game in the market. It is found that the integration of the two firms may not necessarily lower the equilibrium prices because it precludes “indirect competition” in the market. Therefore, the integration, which may appear as a vertical integration, could be an antitrust concern even in the absence of exclusionary purpose.

Keywords: antitrust; complement products; integration; multiproduct

JEL Codes: L1; L4

Corresponding author: Hao Wang, China Center for Economic Research, National School of Development, Peking University, Beijing 100871, China, E-mail: hwang@nsd.pku.edu.cn

Funding source: Ministry of Education of China

Award Identifier / Grant number: 16JJD790002

Appendix A. Market Outcomes of the Quantity-Choosing Games

In the case of a quantity-choosing game, the (symmetric and linear) inverse demands for the products can be represented by:

(16) p e q e , q E , q f , q F = B − η q e + κ q E − λ q f − μ q F ,

(17) p E q e , q E , q f , q F = B − η q E + κ q e − λ q F − μ q f ,

(18) p f q e , q E , q f , q F = B − η q f + κ q F − λ q e − μ q E ,

(19) p F q e , q E , q f , q F = B − η q F + κ q f − λ q E − μ q e ,

with B > 0, η ≥ 0, κ ≥ 0, and λ ≥ 0, but the sign of μ not definite. The costs of production are still assumed to be zero. The product relationships are shown in Figure 2.

Figure 2:

Product relationships (quantity-choosing game).

It can be checked that integration of the upstream firm U and downstream firm D lowers the equilibrium prices (i.e., p i U = p i D > p i I ) if and only if κ − μ > 0. The previous proposition still holds. Again, the integration strictly increases the industrial profits as long as κ − μ ≠ 0. In the parallel horizontal integration, the profits of the firms are

(20) π U = q e p e q e , q E , q f , q F + q f p f q e , q E , q f , q F ,

(21) π D = q E p E q e , q E , q f , q F + q F p F q e , q E , q f , q F .

The equilibrium outcome is

(22) q e U = q E D = q f U = q F D = B 2 η − κ + 2 λ + μ ,

(23) p e U = p E D = p f U = p F D = η + λ B 2 η − κ + 2 λ + μ ,

(24) π U = π D = 2 η + λ B 2 2 η − κ + 2 λ + μ 2 .

If the two firms integrate into a monopolist, the merged firm’s profits are

(25) π I = q e p e + q E p E + q f p f + q F p F .

The equilibrium outcome is

(26) q e I = q E I = q f I = q F I = B 2 η − 2 κ + 2 λ + 2 μ ,

(27) p e I = p E I = p f I = p F I = B 2 ,

(28) π I = B 2 η − κ + λ + μ .

Therefore, the integration lowers the equilibrium prices (i.e., p i U = p i D > p i I ) if and only if κ − μ > 0, and the integration strictly increases industrial profits as long as κ − μ ≠ 0.

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Received: 2020-11-04

Revised: 2021-09-16

Accepted: 2021-09-16

Published Online: 2021-11-04

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Articles in the same Issue

https://doi.org/10.1515/rle-2020-0058

Keywords for this article

antitrust; complement products; integration; multiproduct