Home Endogenous Merger Under Product Differentiation and Price Competition: The Role of High-Quality Producer
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Endogenous Merger Under Product Differentiation and Price Competition: The Role of High-Quality Producer

  • Neelanjan Sen ORCID logo EMAIL logo , Uday Bhanu Sinha ORCID logo and Pattuvanmar Veettil Vaishnav
Published/Copyright: October 16, 2025
The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy

Abstract

This paper studies the profitability of a bilateral merger under price competition in the presence of horizontal and vertical product differentiation. One firm produces a high-quality product and the other two firms produce a low-quality product. We show that i) a merger between a high-quality producer and a low-quality producer or ii) a merger between two low-quality producers is always profitable. After the merger of two firms consumer surplus always decreases, but welfare sometimes increases. However, the total surplus for the merging firms is higher when a high-quality producer merges with a low-quality producer than when two low-quality producers merge. We also consider the endogenous merger selection process and show that i) if the quality difference (net of cost) is very high, then the high-quality producer will merge with a low-quality producer and ii) if the quality difference (net of cost) is very low, then the high-quality producer will bribe the low-quality producers to merge. Interestingly, we show that if the quality difference (net of cost) is moderate, then the high-quality producer a) will bribe other firms to merge if it has high bargaining power in the merger negotiation process; b) otherwise will use the profit-sharing scheme and merge with a low-quality producer.

Keywords: merger; product differentiation; quality; price competition
JEL Classifications: L24; L13; D43; D12
Corresponding author: Neelanjan Sen, Madras School of Economics, Gandhi Mandapam Road, Chennai, 600025, Tamil Nadu, India, E-mail: 

The authors thank Rajit Biswas and Anirban Kar for their valuable suggestions and comments. Neelanjan Sen acknowledges the hospitality received from the Centre for Development Economics, Delhi School of Economics and the Institute of Economic Growth, New Delhi during his visit in September 2024. We thank the three referees and the Editor of the journal, Till Requate, for very helpful comments and suggestions on an earlier version of the paper. Their valuable comments and suggestions have greatly helped us to significantly improve the paper. The usual disclaimer applies.

  1. Author contributions: Neelanjan Sen; Uday Bhanu Sinha & P.V. Vaishnav: Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization.

  2. Conflict of interest: The authors declare that there is no conflict of interest.

  3. Research funding: No funding is received from any sources.

  4. Data availability: Data sharing not applicable – the article describes entirely theoretical research.

Appendices

A Profitability of Merger

A.1 Type A L Merger

Firm 1 and firm 2 will merge in zone C of Figure 1 as Π 12 > Π 1 B + Π 2 B or

(37) 4 ( 3 γ + 2 ) 2 ( 4 + 4 γ − 3 γ 2 ) − γ ( 2 + 3 γ ) θ × ( 4 + 8 γ + γ 2 − 2 γ 3 ) − γ ( 6 + 7 γ − 2 γ 2 ) θ   + ( 4 + 6 γ − γ 2 − 2 γ 3 ) θ − γ ( 4 + 5 γ − 2 γ 2 ) ( 4 + 2 γ − 5 γ 2 ) θ − γ 2 > 16 ( 2 + 2 γ − γ 2 ) 2 ( 2 + 3 γ − γ 2 ) ( 1 + γ ) − 2 γ θ ( 1 + γ ) 2 2   + 2 θ ( 1 + γ − γ 2 ) ( 1 + γ ) − γ ( 1 + γ ) 2 2 .

A.2 Type A L0 Merger

The merger of firm 1 and firm 2 is profitable in zone D of Figure 1 as Π 12 0 > Π 1 B + Π 2 B or

(38) 1 ( 1 − γ 2 ) ( 2 − γ 2 ) − γ θ ( 4 − γ 2 ) 2 > 1 + γ ( 1 − γ ) ( 2 γ + 1 ) ( 2 + 3 γ − γ 2 ) − 2 γ ( 1 + γ ) θ 2 ( 3 γ + 2 ) 2 + 2 ( 1 + γ − γ 2 ) θ − γ ( 1 + γ ) 2 ( 3 γ + 2 ) 2 .

A.3 Type B Merger

Moreover, merger is profitable in zone C and zone D of Figure 1 as Π 23 > Π 2 B + Π 3 B or

(39) 2 ( 1 + γ − γ 2 ) θ − γ ( 1 + γ ) 2 4 ( 2 + 2 γ − γ 2 ) 2 > ( 1 + γ ) 2 ( 1 + γ − γ 2 ) θ − γ ( 1 + γ ) 2 4 ( 3 γ + 2 ) 2 .

B Welfare Analysis

B.1 Bertrand Nash Equilibrium

Consumer surplus in the pre-merger stage is

C S B = 1 2 q 1 B 2 + q 2 B 2 + q 3 B 2 + 2 γ q 1 B q 2 B + 2 γ q 1 B q 3 B + 2 γ q 2 B q 3 B .

Since q 2 B = q 3 B ; we have C S B = 1 2 q 1 B 2 + ( 2 + 2 γ ) ( q 2 B ) 2 + 4 γ q 1 B q 2 B . Thus,

(40) C S B = ( 1 + γ ) 2 8 ( 2 + 3 γ ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 + ( 2 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2   + 4 γ ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) .

The industry profit in the pre-merger stage is I P B = Π 1 B + Π 2 B + Π 3 B or

(41) I P B = 1 + γ 4 ( 2 + 3 γ ) 2 ( 1 − γ ) ( 2 γ + 1 ) ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2   + 2 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 .

Welfare is the sum of consumer surplus and industry profit. Therefore, in the pre-merger stage welfare is W B = CS B  + IP B or

(42) W B = ( 1 + γ ) 2 8 ( 2 + 3 γ ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 + ( 2 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 + 4 γ ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) + 1 + γ 4 ( 2 + 3 γ ) 2 ( 1 − γ ) ( 2 γ + 1 ) × ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2   + 2 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 .

B.2 Merger Between Firm 1 and Firm 2

B.2.1 Good 2 is Produced

Consumer surplus in the post-merger stage such that the merged firm (firm 1 and firm 2) produce good 1 and good 2 is

C S 12 = 1 2 q 1 2 + q 2 2 + q 3 2 + 2 γ q 1 q 2 + 2 γ q 1 q 3 + 2 γ q 2 q 3

or

(43) C S 12 = 1 32 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) 2 + ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) 2 + 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) 2 + 2 γ ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) × ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) + 2 γ ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) × 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) + 2 γ ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) × 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) .

After comparing we observe that in zone C of Figure 1 consumer surplus will fall post-merger as CS 12 < CS B .

The industry profit in the post-merger stage such that the merged firm (firm 1 and firm 2) produce good 1 and good 2 is IP 12 = Π12 + Π3 or

(44) I P 12 = 1 16 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) ( 1 + 2 γ ) × ( 4 + 4 γ − 3 γ 2 ) ( α 1 − c 1 ) − γ ( 2 + 3 γ ) ( α 2 − c 2 ) × ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) + ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) × ( 4 + 2 γ − 5 γ 2 ) ( α 2 − c 2 ) − γ 2 ( α 1 − c 1 ) + 1 + γ ( 1 − γ ) ( 1 + 2 γ ) 2 ( 2 + γ − 2 γ 2 ) ( α 2 − c 2 ) − 2 γ ( α 1 − c 1 ) 4 ( 2 + 2 γ − γ 2 ) 2 .

Welfare is the sum of consumer surplus and industry profit. Therefore, in the pre-merger stage welfare is W 12 = CS 12 + IP 12 or

(45) W 12 = 1 32 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) 2 + ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) 2 + 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) 2 + 2 γ ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) × ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) + 2 γ ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) × 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) + 2 γ ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 )   × 2 ( 2 + γ − 2 γ 2 ) ( 1 + γ ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) + 1 16 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) ( 1 + 2 γ ) × ( 4 + 4 γ − 3 γ 2 ) ( α 1 − c 1 ) − γ ( 2 + 3 γ ) ( α 2 − c 2 ) × ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 ) + ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 ) × ( 4 + 2 γ − 5 γ 2 ) ( α 2 − c 2 ) − γ 2 ( α 1 − c 1 ) + 1 + γ ( 1 − γ ) ( 1 + 2 γ ) 2 ( 2 + γ − 2 γ 2 ) ( α 2 − c 2 ) − 2 γ ( α 1 − c 1 ) 4 ( 2 + 2 γ − γ 2 ) 2 .

After comparing we observe that post-merger welfare always decreases or W 12 < W B .

B.2.2 Good 2 Not Produced

Since q 2 = 0, consumer surplus in the post-merger stage such that the merged firm (firm 1 and firm 2) produce good 1 but not good 2 is

C S 0 = 1 2 q 1 0 2 + q 3 0 2 + 2 γ q 1 0 q 3 0

or

(46) C S 0 = 1 2 ( 1 − γ 2 ) 2 ( 4 − γ 2 ) 2 ( 2 − γ 2 ) ( α 1 − c 1 ) − γ ( α 2 − c 2 ) 2 + ( 2 − γ 2 ) ( α 2 − c 2 ) − γ ( α 1 − c 1 ) 2 + 2 γ ( 2 − γ 2 ) ( α 1 − c 1 ) − γ ( α 2 − c 2 ) ( 2 − γ 2 ) ( α 2 − c 2 ) − γ ( α 1 − c 1 )

After comparing we observe that in zone D of Figure 1 consumer surplus will decrease post-merger as CS 0 < CS B .

The industry profit in the post-merger stage such that the merged firm (firm 1 and firm 2) produce good 1 but not good 2 is

I P 0 = Π 12 0 + Π 3 0

or

(47) I P 0 = 1 ( 1 − γ 2 ) 2 ( 4 − γ 2 ) 2 ( 2 − γ 2 ) ( α 1 − c 1 ) − γ ( α 2 − c 2 ) 2 + ( 2 − γ 2 ) ( α 2 − c 2 ) − γ ( α 1 − c 1 ) 2 .

Therefore, welfare is W 0 = CS 0 + IP 0 or

(48) W 0 = 1 2 ( 1 − γ 2 ) 2 ( 4 − γ 2 ) 2 3 ( 2 − γ 2 ) ( α 1 − c 1 ) − γ ( α 2 − c 2 ) 2 + 3 ( 2 − γ 2 ) ( α 2 − c 2 ) − γ ( α 1 − c 1 ) 2 + 2 γ ( 2 − γ 2 ) ( α 1 − c 1 ) − γ ( α 2 − c 2 ) ( 2 − γ 2 ) ( α 2 − c 2 ) − γ ( α 1 − c 1 ) .

After comparing we observe that in zone D 1 of Figure 2 welfare will increase post-merger as W 0 > W B .

B.3 Merger Between Firm 2 and Firm 3

Consumer surplus in the post-merger stage is

C S 23 = 1 2 q 1 2 + q 2 2 + q 3 2 + 2 γ q 1 q 2 + 2 γ q 1 q 3 + 2 γ q 2 q 3 .

Since q 2 = q 3; we have C S 23 = 1 2 q 1 2 + ( 2 + 2 γ ) q 2 2 + 4 γ q 1 q 2 . Thus,

(49) C S 23 = 1 8 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × 2 ( 1 + γ − γ 2 ) ( 1 + γ ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 + ( 2 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 + 4 γ 2 ( 1 + γ − γ 2 ) ( 1 + γ ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) .

After comparing we observe that in zone C and zone D of Figure 1 consumer surplus will fall post-merger as CS 23 < CS B .

The industry profit in the post-merger stage is IP 23 = Π1 + Π23 or

(50) I P 23 = ( 1 + γ ) ( 1 − γ ) ( 1 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( α 2 − c 2 ) 2 ( 2 + 2 γ − γ 2 ) 2 + 2 4 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) ( 1 + 2 γ ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 .

Therefore, welfare is

(51) W 23 = 1 8 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) 2 ( 2 γ + 1 ) 2 × 2 ( 1 + γ − γ 2 ) ( 1 + γ ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 + ( 2 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 + 4 γ 2 ( 1 + γ − γ 2 ) ( 1 + γ ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) + ( 1 + γ ) ( 1 − γ ) ( 1 + 2 γ ) 2 ( 1 + γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( α 2 − c 2 ) 2 ( 2 + 2 γ − γ 2 ) 2 + 2 4 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) ( 1 + 2 γ ) × 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 .

After comparing we observe that post-merger welfare always decreases or W 23 > W B in zones D and zones C 1 of Figure 3 (see the main text), however welfare decreases in zone C 0 of Figure 3, where zones C 1 and zones C 0 of Figure 3 comprise zone C of Figure 1.

C Surplus from Different Types of Merger

The surplus if firm 2 and firm 1 merge (Type A L ) such that good 2 is produced (i.e. in zone C of Figure 1), using Equations (7) and (11), is

(52) Π 12 − Π 1 B + Π 2 B = 1 Z ( 4 + 4 γ − 3 γ 2 ) ( α 1 − c 1 ) − γ ( 2 + 3 γ ) ( α 2 − c 2 )   × ( 4 + 8 γ + γ 2 − 2 γ 3 ) ( α 1 − c 1 ) − γ ( 6 + 7 γ − 2 γ 2 ) ( α 2 − c 2 )   + ( 4 + 6 γ − γ 2 − 2 γ 3 ) ( α 2 − c 2 ) − γ ( 4 + 5 γ − 2 γ 2 ) ( α 1 − c 1 )     × ( 4 + 2 γ − 5 γ 2 ) ( α 2 − c 2 ) − γ 2 ( α 1 − c 1 )   − 1 + γ ( 1 − γ ) ( 2 γ + 1 ) ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 ( 3 γ + 2 ) 2   − 1 + γ ( 1 − γ ) ( 2 γ + 1 ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 ( 3 γ + 2 ) 2 ,

where Z = 16 ( 2 + 2 γ − γ 2 ) 2 ( 1 − γ ) ( 1 + 2 γ ) . On the other hand, the surplus if firm 2 and firm 1 merge (Type A L0) such that good 2 is not produced (i.e. in zone D of Figure 1) using Equations (7) and (18), is

(53) Π 12 0 − Π 1 B + Π 2 B = 1 + γ ( 1 − γ ) ( 1 + 2 γ ) ( 2 + 4 γ + γ 2 ) ( α 1 − c 1 ) − γ ( 1 + γ ) ( α 2 − c 2 ) ( 4 + 8 γ + 3 γ 2 ) 2 − 1 + γ ( 1 − γ ) ( 2 γ + 1 ) ( 2 + 3 γ − γ 2 ) ( α 1 − c 1 ) − 2 γ ( 1 + γ ) ( α 2 − c 2 ) 2 ( 3 γ + 2 ) 2 − 1 + γ ( 1 − γ ) ( 2 γ + 1 ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 ( 3 γ + 2 ) 2 .

The surplus if firm 2 and firm 3 merge (Type B), i.e. both in zone C and zone D of Figure 1, using Equations (7) and (23), is

(54) Π 23 − Π 2 B + Π 3 B = 2 Z 4 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − 2 γ ( 1 + γ ) ( α 1 − c 1 ) 2 − 2 ( 1 + γ ) ( 1 − γ ) ( 2 γ + 1 ) 2 ( 1 + γ − γ 2 ) ( α 2 − c 2 ) − γ ( 1 + γ ) ( α 1 − c 1 ) 2 ( 3 γ + 2 ) 2 .

  1. For zone C of Figure 1, using Equations (52) and (54), we observe that Π 12 − Π 1 B + Π 2 B > Π 23 − Π 2 B + Π 3 B . Thus, the Type A L merger generates a higher surplus than the Type B merger.

  2. For zone D of Figure 1, using Equations (53) and (54), we observe that Π 12 0 − Π 1 B + Π 2 B > Π 23 − Π 2 B + Π 3 B . Hence, the Type A L0 merger generates a higher surplus than the Type B merger.

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Received: 2024-10-25
Accepted: 2025-09-18
Published Online: 2025-10-16

© 2025 Walter de Gruyter GmbH, Berlin/Boston

https://doi.org/10.1515/bejeap-2024-0370
Keywords for this article
merger; product differentiation; quality; price competition
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