Industrial Technology Boundary, Product Quality Choice, and Market Segmentation

Haoxing Ma

doi:10.1515/bejeap-2022-0304

Published by De Gruyter February 2, 2024

Industrial Technology Boundary, Product Quality Choice, and Market Segmentation

Haoxing Ma

From the journal The B.E. Journal of Economic Analysis & Policy

https://doi.org/10.1515/bejeap-2022-0304

Showing a limited preview of this publication:

Abstract

This paper studies how firms in a duopoly market choose product qualities when facing two types of consumers: high-end consumers value quality more than low-end consumers. Firms’ highest possible quality (referred to as industrial technology boundary) is determined by an industrial common technology. I consider price competition and show that in equilibrium, an increase in the technology boundary can induce a decrease in the equilibrium quality of one firm. In this case, the firms enlarge their quality difference, triggering a market segmentation. In this market segmentation, the firm with a lower quality does not serve the high-end consumers and obtains higher profits from the low-end consumers, whereas the firm with a higher quality supplies both types of consumers and obtains higher profits as well. This market segmentation causes additional mismatch costs for high-end consumers, therefore lowering both consumer and social surplus.

Keywords: quality choice; market segmentation; pricing strategy; social surplus

JEL Classification: L11; L13; L22; M21

Corresponding author: Haoxing Ma, Osaka School of International Public Policy, Osaka University, Machikaneyama 1-31, Toyonaka, Osaka, 560-0043, Japan, E-mail: haoxing.ma@gmail.com

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 22J12790

Acknowledgments

I am indebted to co-editor Till Requate and two anonymous referees for insightful comments and suggestions. I would like to express my sincere thanks to Takeshi Murooka for instructions on this project. Besides, I appreciate valuable comments and suggestions from Junichiro Ishida, Noriaki Matsushima, and Cong Pan, the participants in seminars in Osaka School of International Public Policy, at the 2023 Asia Meeting of the Econometric Society, and at the Japanese Economic Association 2023 Spring Meeting. Financial support from the Japan Society for the Promotion of Science (22J12790) is acknowledged. Needless to say, all the remaining errors are mine.

Research funding: This work was supported by Japan Society for the Promotion of Science.

Appendix

Proof of Lemma 1

(1) The BB equilibrium

Under BB, firms solve

max p 0 1 2 ( x ̂ H ( p 0 , p 1 ) + x ̂ L ( p 0 , p 1 ) ) p 0 , max p 1 1 − 1 2 ( x ̂ H ( p 0 , p 1 ) + x ̂ L ( p 0 , p 1 ) ) p 1 ,

subject to p ₁ − p ₀ ∈ (−t, t − Δ^s), which has an interior solution p 0 * B B , p 1 * B B = ( 6 t + Δ s 6 , 6 t − Δ s 6 ) with π 0 * B B , π 1 * B B = ( ( 6 t + Δ s ) 2 72 t , ( 6 t − Δ s ) 2 72 t ) if and only if 0 ≤ Δ s < 3 2 t .

Given p 0 = p 0 * B B , if firm 1 induces 0B, it solves max p 1 1 2 1 − x ̂ L p 0 * B B , p 1 p 1 , subject to t − Δ s ≤ p 1 − p 0 * B B < t , which has an interior solution p 1 = 12 t + Δ s 12 with profits ( 12 t + Δ s ) 2 576 t when Δ s ∈ [ 12 t 11 , 2 t ) and a corner solution p 1 = 12 t − 5 Δ s 6 with profits Δ s ( 12 t − 5 Δ s ) 24 t when Δ s ∈ [ 0 , 12 t 11 ) . The profits under the interior solution are weakly less than π 1 * B B if 12 t 11 ≤ Δ s ≤ 12 ( 5 − 3 2 ) 7 t , while the profits under the corner solution are always strictly less than π 1 * B B when Δ s ∈ [ 0 , 12 t 11 ) . It is straightforward to check that neither firm will have an incentive to deviate to other situations, 00, B1, and 11, and that no situation other than BB will happen in an equilibrium. See the Supplementary Material for detailed derivations.

(2) The mixed equilibrium

Between the range of Lemma (1) and (3) when Δ s ∈ ( Δ ̲ , 5 t 3 ) , there exists a mixed-strategy equilibrium (p ₀, σ ₁) constructed as follow,

Firm 0 takes a pure strategy p ₀;
Firm 1 mixes between (1) p 1 h such that x ̂ H > 1 and 0 < x ̂ L < 1 and (2) p 1 l such that 0 < x ̂ H < 1 and 0 < x ̂ L < 1 (denoted as σ ₁). Let β be the probability that p 1 h is chosen, and 1 − β be the probability that p 1 l is chosen.

Now I verify if (p ₀, σ ₁) constitutes a mixed Nash equilibrium.

First, write down firm 0’s expected profit from (p ₀, σ ₁),

π 0 = p 0 ⋅ 1 2 β ⋅ 1 + ( 1 − β ) p 1 l − p 0 + t + Δ s 2 t + p 0 ⋅ 1 2 β p 1 h − p 0 + t 2 t + ( 1 − β ) p 1 l − p 0 + t 2 t .

When choosing p 1 h , firm 1 obtains π 1 h = p 1 h 1 2 ⋅ 0 + 1 2 1 − p 1 h − p 0 + t 2 t ; when choosing p 1 l , firm 1 obtains π 1 l = p 1 l 1 2 1 − p 1 l − p 0 + t + Δ s 2 t + 1 2 1 − p 1 l − p 0 + t 2 t . Thus, firm 1 obtains the following expected profit from (p ₀, σ ₁),

π 1 = β π 1 h + ( 1 − β ) π 1 l = β ⋅ p 1 h 1 2 ⋅ 0 + 1 2 1 − p 1 h − p 0 + t 2 t + ( 1 − β ) ⋅ p 1 l 1 2 1 − p 1 l − p 0 + t + Δ s 2 t + 1 2 1 − p 1 l − p 0 + t 2 t .

Then, I solve for p ₀, p 1 l and p 1 h respectively by first-order conditions,

p 0 * = t ( 6 + β ) + Δ s − β Δ s 6 − 3 β ;

p 1 h * = 2 t ( β − 6 ) + ( β − 1 ) Δ s 6 ( β − 2 ) ;

p 1 l * = 4 t ( β − 6 ) − ( β − 4 ) Δ s 12 ( β − 2 ) .

Substituting p 0 * , p 1 h * and p 1 l * into the expected profits, we can obtain

π 1 h * = ( 2 t β − 12 t − Δ s + β Δ s ) 2 144 t ( β − 2 ) 2 ;

π 1 l * = ( 4 t β − 24 t + 4 Δ s − β Δ s ) 2 288 t ( β − 2 ) 2 .

If (p ₀, σ ₁) is a mixed Nash equilibrium, firm 1 should be indifferent between choosing p 1 h and p 1 l . That is

π 1 h * = π 1 l * .

By π 1 h * = π 1 l * , we can solve for β and obtain

β = 24 ( 2 − 1 ) t − 2 ( 1 + 2 2 ) Δ s ( 4 2 − 4 ) t − ( 2 + 2 ) Δ s .

Substituting β into π ₀ and π ₁, I obtain π 0 mix , π 1 mix = ( Δ s − 8 t ) ( 2 ( 2 − 1 ) t − 2 Δ s ) 2 8 ( 2 − 1 ) ( 4 ( 2 − 1 ) t − ( 2 + 2 ) Δ s ) ) t , ( Δ s ) 2 32 ( 2 − 1 ) 2 t .

To verify if (p ₀, σ ₁) constitutes a mixed Nash equilibrium, I examine if β ∈ (0, 1) under the assumptions on t and Δ^s i.e. t ∈ (0, 1) and Δ^s ∈ (0, 2t). We can find that β ∈ (0, 1) if

12 ( 5 − 3 2 ) 7 t < Δ s < 10 ( 2 − 2 ) 3 t .

Recall that the range gap between Lemma 1 (1) and (3) is

12 ( 5 − 3 2 ) 7 t < Δ s < Δ ̄ .

Since Δ ̄ < 10 ( 2 − 2 ) 3 t , I confirm that β ∈ (0, 1) is true for 12 ( 5 − 3 2 ) 7 t < Δ s < Δ ̄ .

Therefore, I confirm that (p ₀, σ ₁) constitutes a mixed-strategy Nash equilibrium. By substituting the value of β in (2) back into (1), I confirm that the following mixed Nash equilibrium exists when 12 ( 5 − 3 2 ) 7 t < Δ s < Δ ̄ :

Firm 0 takes a pure strategy p 0 = 2 + 2 2 Δ s − t ;
Firm 1 takes p 1 h = 2 + 2 4 Δ s with probability β = 24 ( 2 − 1 ) t − 2 ( 1 + 2 2 ) Δ s ( 4 2 − 4 ) t − ( 2 + 2 ) Δ s and takes p 1 l = 1 + 2 4 Δ s with probability 1 − β = 20 ( 2 − 1 ) t − 3 2 Δ s ( 4 2 − 4 ) t − ( 2 + 2 ) Δ s .

It is straightforward to check that the above p 1 h is a price such that x ̂ H > 1 and 0 < x ̂ L < 1 , and p 1 l is a price such that 0 < x ̂ H < 1 and 0 < x ̂ L < 1 .

(3) The 0B equilibrium

Under 0B, firms solve

max p 0 1 2 ( 1 + x ̂ L ( p 0 , p 1 ) ) p 0 , max p 1 1 2 ( 1 − x ̂ L ( p 0 , p 1 ) ) p 1 ,

subject to p ₁ − p ₀ ∈ [t − Δ^s, t), which has an interior solution p 0 * 0 B , p 1 * 0 B = ( 7 t 3 , 5 t 3 ) with π 0 * 0 B , π 1 * 0 B = ( 49 t 36 , 25 t 36 ) if and only if Δ ̄ ≤ Δ s < 2 t .

Given p 0 = p 0 * 0 B , if firm 1 induces B1, it solves max p 1 1 2 ( ( 1 − x ̂ A p 0 * 0 B , p 1 ) + 1 ) p 1 , subject to − t − Δ s < p 1 − p 0 * 0 B ≤ − t , which always has a corner solution p 1 = 4 t 3 with profits 4 t − Δ s 3 that are less than π 1 * 0 B if Δ ̄ ≤ Δ s < 2 t . It is straightforward to check that neither firm will have an incentive to deviate to other situations, 00, BB, and 11, and that no situation other than BB will happen in an equilibrium. See the Supplementary Material for this. ■

Proof of Proposition 1

Proposition 1-(1)

When Δ ∈ [ 0 , Δ ̲ ] , only the BB equilibrium in Lemma 1-(1) can be realized. In this equilibrium, firm 0’s profit increases in Δ^s, whereas firm 1’s profit decreases in it. Given firm 0 chooses the highest quality level s 0 = s ̲ + Δ , firm 1’s profits decrease if it deviates from choosing the same level s 1 = s ̲ + Δ . Also, given firm 1 chooses the highest level s 1 = s ̲ + Δ , firm 0 cannot further increase its quality from s 0 = s ̲ + Δ .

When Δ ∈ ( Δ ̲ , Δ ̂ ] , both the BB equilibrium in Lemma 1-(1) and the mixed equilibrium in Lemma 1-(2) can be realized. Since firm 1’s profit decreases in Δ^s when Δ s ∈ [ 0 , Δ ̲ ] and then increases in Δ^s when Δ s ∈ ( Δ ̲ , Δ ] , I only need to compare the two local maxima π 1 * B B Δ s = 0 and π 1 mix Δ s = Δ . As π 1 * B B Δ s = 0 ≥ π 1 mix Δ s = Δ for Δ ∈ ( Δ ̲ , Δ ̂ ] , given firm 0’s quality level s 0 = s ̲ + Δ , firm 1 prefers to following the same quality strategy s 1 = s ̲ + Δ as above to have the BB equilibrium. Since I have assumed Δ^S = s ₀ − s ₁ ≥ 0, firm 0 will follow with the same quality level s 0 = s ̲ + Δ .

Proposition 1-(2)

When Δ ∈ ( Δ ̂ , Δ ̄ ) , both the BB equilibrium in Lemma 1-(1) and the mixed equilibrium in Lemma 1-(2) can be realized. Since π 1 * B B Δ s = 0 < π 1 mix Δ s = Δ , firm 1 prefers to having the largest Δ^s. In the mixed equilibrium, both firms’ profits increase in Δ^s so that Δ^s = Δ maximizes both firms’ profits. Then, by assumption Δ^s = s ₀ − s ₁ ≥ 0, firm 0 chooses the highest quality level s 0 = s ̲ + Δ while firm 1 chooses the lowest quality level s 1 = s ̲ .

When Δ ∈ [ Δ ̄ , 2 t ) , all the three equilibria in Lemma-(1), -(2) and -(3) are all possible. When the 0B equilibrium exists ( Δ s ∈ [ Δ ̄ , 2 t ) ), π 0 * 0 B = 49 36 t , π 1 * 0 B = 25 36 t . Firm 1’s maximum profit is π 1 * B B Δ s = 0 in the BB equilibrium (for Δ s ∈ [ 0 , Δ ̲ ] ) and lim Δ s → Δ ̄ π 1 mix in the mixed equilibrium (for Δ s ∈ ( Δ ̲ , Δ ̄ ) ). It can be confirmed that π 1 * 0 B > π 1 * B B Δ s = 0 and π 1 * 0 B > lim Δ s → Δ ̄ π 1 mix . For firm 0, its maximum profit is π 0 * B B Δ s = Δ ̲ in the BB equilibrium (for Δ s ∈ [ 0 , Δ ̲ ] ) and lim Δ s → Δ ̄ π 0 mix in the mixed equilibrium (for Δ s ∈ ( Δ ̲ , Δ ̄ ) ). It can be confirmed that π 0 * 0 B > π 0 * B B Δ s = Δ ̲ and π 1 * 0 B > lim Δ s → Δ ̄ π 0 mix . Then, both firm 0 and firm 1 prefer to have the 0B equilibrium. Since both firms’ profits are independent of Δ^s in the 0B equilibrium, neither firm can get better off by deviating from ( s 0 , s 1 ) = ( s ̲ + Δ , s ̲ ) so that ( s 0 , s 1 ) = ( s ̲ + Δ , s ̲ ) constitute the equilibrium in the first stage.^[16] ■

Proof of Proposition 2

The welfare formulas are given by

C S = 1 2 ∫ 0 x ̂ H ( v + s 0 − x t − p 0 ) d x + ∫ x ̂ H 1 ( v + s 1 − ( 1 − x ) t − p 1 ) d x + 1 2 ∫ 0 x ̂ L ( v − x t − p 0 ) d x + ∫ x ̂ l 1 ( v − ( 1 − x ) t − p 1 ) d x , P S = π 0 + π 1 , S S = C S + P S = 1 2 ∫ 0 x ̂ H ( v + s 0 − x t ) d x + ∫ x ̂ H 1 ( v + s 1 − ( 1 − x ) t ) d x + 1 2 ∫ 0 x ̂ L ( v − x t ) d x + ∫ x ̂ l 1 ( v − ( 1 − x ) t ) d x .

I calculate and compare the social surplus and consumer surplus under the three cases in Lemma 1.

(1) The BB equilibrium

By Lemma 1-(1), in the BB equilibrium, the equilibrium locations of the indifferent consumers are x ̂ L = x ̂ H = 1 2 . By Proposition 1-(1), in the BB equilibrium, both firms choose the highest possible quality level i.e. s 0 = s 1 = s ̲ + Δ so that Δ^s = 0. Then the social surplus SS ^BB and consumer surplus CS ^BB under the BB equilibrium can be calculated as follow:

S S B B = 1 2 ∫ 0 1 2 ( v + s ̲ + Δ − x t ) d x + ∫ 1 2 1 ( v + s ̲ + Δ − ( 1 − x ) t ) d x + 1 2 ∫ 0 1 2 ( v − x t ) d x + ∫ 1 2 1 ( v − ( 1 − x ) t ) d x = 1 4 ( 2 s ̲ + 4 v + 2 Δ − t ) , C S B B = S S B B − π 0 * B B − π 1 * B B = S S B B − 1 2 t − 1 2 t = 1 4 ( 2 s ̲ + 4 v + 2 Δ − 5 t ) .

(2) The mixed equilibrium

By Lemma 1-(2), in this mixed equilibrium, firm 0 chooses p 0 mix = t ( 6 + β ) + Δ s − β Δ s 6 − 3 β , and firm 1 chooses p ̄ 1 m i x = 2 t ( 6 − β ) + ( 1 − β ) Δ s 6 ( 2 − β ) with probability β and p ̲ 1 mix = 4 t ( 6 − β ) − ( 4 − β ) Δ s 12 ( 2 − β ) with probability 1 − β, where β = 24 ( 2 − 1 ) t − 2 ( 1 + 2 2 ) Δ s 4 ( 2 − 1 ) t − ( 2 + 2 ) Δ s ) , π 0 mix , π 1 mix = ( Δ s − 8 t ) ( 2 ( 2 − 1 ) t − 2 Δ s ) 2 8 ( 2 − 1 ) ( 4 ( 2 − 1 ) t − ( 2 + 2 ) Δ s ) ) t , ( Δ s ) 2 32 ( 2 − 1 ) 2 t . Therefore, the equilibrium locations of indifferent consumers are: x ̂ H = 1 and x ̂ L = 1 − Δ 8 t − 4 2 t when firm 1 takes strategy p ̄ 1 mix ; x ̂ H = 1 + Δ − 2 Δ 8 t and x ̂ L = 1 − ( 3 + 2 ) Δ 8 t when firm 1 takes strategy p ̲ 1 mix . By Proposition 1-(2), firm 0 chooses the highest quality level s 0 = s ̲ + Δ whereas firm 1 chooses the lower level s 1 = s ̲ so that Δ^s = s ₀ − s ₁ = Δ. Then the social surplus SS ^mix and consumer surplus CS ^mix under the BB equilibrium can be calculated as follow:

S S mix = β 1 2 ∫ 0 1 ( v + s ̲ + Δ − x t ) d x + ∫ 1 1 ( v + s ̲ − ( 1 − x ) t ) d x + 1 2 ∫ 0 1 − Δ 8 t − 4 2 t ( v − x t ) d x + ∫ 1 − Δ 8 t − 4 2 t 1 ( v − ( 1 − x ) t ) d x + ( 1 − β ) 1 2 ∫ 0 1 + Δ − 2 Δ 8 t ( v + s ̲ + Δ − x t ) d x + ∫ 1 + Δ − 2 Δ 8 t 1 ( v + s ̲ − ( 1 − x ) t ) d x + 1 2 ∫ 0 1 − ( 3 + 2 ) Δ 8 t ( v − x t ) d x + ∫ 1 − ( 3 + 2 ) Δ 8 t 1 ( v − ( 1 − x ) t ) d x = 64 ( − 4 + 3 2 ) t 3 − 32 s ̲ t ( ( − 8 + 6 2 ) t − Δ ) + 7 ( 3 − 2 2 ) Δ 3 − 16 t 2 ( 8 ( − 4 + 3 2 ) v + ( − 30 + 23 2 ) Δ ) + 2 t Δ ( 32 v + ( − 112 + 89 2 ) Δ ) 64 t ( ( 8 − 6 2 ) t + Δ ) , C S mix = S S mix − π 0 * m i x − π 1 * m i x = S S mix − ( Δ s − 8 t ) ( 2 ( 2 − 1 ) t − 2 Δ ) 2 8 ( 2 − 1 ) ( 4 ( 2 − 1 ) t − ( 2 + 2 ) Δ ) t − ( Δ ) 2 32 ( 2 − 1 ) 2 t .

(3) The 0B equilibrium

By Lemma 1-(2), in the 0B equilibrium, p 0 * 0 B , p 1 * 0 B = 7 t 3 , 5 t 3 , π 0 * 0 B , π 1 * 0 B = 49 36 t , 25 36 t , so that the equilibrium location of the indifferent consumers are x ̄ H = 1 and x ̄ L = 1 6 . By Proposition 1-(1), in the 0B equilibrium, firm 0 chooses the highest quality level s 0 = s ̲ + Δ whereas firm 1 chooses the lower level s 1 = s ̲ .

S S 0 B = 1 2 ∫ 0 1 ( v + s ̲ + Δ − x t ) d x + ∫ 1 1 ( v + s ̲ − ( 1 − x ) t ) d x + 1 2 ∫ 0 1 6 ( v − x t ) d x + ∫ 1 6 1 ( v − ( 1 − x ) t ) d x = 1 72 ( 36 s ̲ − 31 t + 72 v + 36 Δ ) , C S 0 B = S S 0 B − π 0 * 0 B − π 1 * 0 B = S S 0 B − 49 36 t − 25 36 t = 1 72 ( 36 s ̲ − 179 t + 72 v + 36 Δ ) .

(4) Comparison of SS and CS

Since the social surplus increase in Δ for all the above three cases (i.e. SS ^BB′(Δ), SS ^mix′(Δ), SS ^0B′(Δ) > 0 with conditions on Δ), I only need to compare three local maxima S S B B | Δ = Δ ̂ , S S mix | Δ = Δ ̄ and SS ^0B|_Δ=2t. Substituting the value of Δ, I have S S B B | Δ = Δ ̂ = v + 1 2 s ̲ + ( 2 3 − 2 2 − 1 4 ) t , S S mix | Δ = Δ ̄ = v + 1 2 s ̲ + 289 2 − 65 576 ( 47 2 − 65 ) t and S S 0 B | Δ = 2 t = v + 1 2 s ̲ + 41 72 t . A simple comparison shows that S S B B | Δ = Δ ̂ > S S 0 B | Δ = 2 t > S S mix | Δ = Δ ̄ for any t ∈ (0, 1).

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/bejeap-2022-0304).

Received: 2022-05-04

Accepted: 2024-01-10

Published Online: 2024-02-02

Industrial Technology Boundary, Product Quality Choice, and Market Segmentation

Abstract

Acknowledgments

Appendix

Proof of Lemma 1

(1) The BB equilibrium

(2) The mixed equilibrium

(3) The 0B equilibrium

Proof of Proposition 1

Proposition 1-(1)

Proposition 1-(2)

Proof of Proposition 2

(1) The BB equilibrium

(2) The mixed equilibrium

(3) The 0B equilibrium

(4) Comparison of SS and CS

References

Supplementary Material

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