Quality Competition and Market-Share Leadership in Network Industries

Appendix A: The Demand in (11)–(14)

Solving x ₂ = 0 and x ₁ = 0, we obtain the two price boundaries ( p 1 ̲ and p 1 ̄ ) in (9) and (10), respectively. Accordingly, the monopoly Case (ii) exists when p 1 ≤ p 1 ̲ ensures x ₂ = 0 and p ₁ < q ₁ ensures x ₁ > 0; the monopoly Case (iii) exists when p 1 ≥ p 1 ̄ ensures x ₁ = 0 and p ₂ < q ₂ ensures x ₂ > 0; and Case (iv) exists for p ₁ > q ₁ and p ₂ > q ₂.

Moreover, if q 2 q 1 < ( 1 − α ) 2 , it must be p 1 ̲ < p 1 ̄ for p ₂ < q ₂; that is, the duopoly Case (i) exists when p 1 ̲ < p 1 < p 1 ̄ . The demands are thus summarized as in (11) and (12) where for p 1 ̲ < p 1 < p 1 ̄ , the duopoly Case (i) is the unique equilibrium outcome, where the demand x _i decreases with its own price p _i and increases with its rival’s p _j . Moreover, it can be shown that the total demand (x ₁ + x ₂) increases with p ₁ (but decreases with p ₂), which stems from the fact that the price effect on the demand for the low-quality product is stronger in the case of partial market coverage.

In contrast, if q 2 q 1 > ( 1 − α ) 2 , from (9) and (10), it must be p 1 ̄ < p 1 ̲ for p ₂ < q ₂; that is, the duopoly Case (i) exists when p 1 ̄ < p 1 < p 1 ̲ . The demands are thus summarized as in (13) and (14) where Cases (i)–(iii) may all emerge in the equilibrium for p 1 ̄ < p 1 < p 1 ̲ so there are multiple equilibria for different cases.^[18] According to the demand of the monopoly Cases (ii) and (iii), the demand for the product decrease with the product price. However, in the duopoly Case (i), the demand x _i increases with its own price p _i . This is because, for a large extent α of externalities with q 2 q 1 > ( 1 − α ) 2 , consumers mostly value a product’s network size when choosing between the two products. Thus, as the demand for a product increases, the value of the product increases significantly while that of the other product decreases, resulting in a higher willingness to pay of consumers for the former product.

Appendix B: Total Quality and q 2 q 1 < ( 1 − α ) 2 under Price Competition

1. Figure A1 shows that the equilibrium gives △qx = q ₁ x ₁ − q ₂ x ₂ > 0 for 0 < α < 0.47 under quality-then-price competition Because q ₁ > q ₂, we must have θq ₁ + αq ₁ x ₁ − (θq ₂ + αq ₂ x ₂) = θ(q ₁ − q ₂) + α(q ₁ x ₁ − q ₂ x ₂) > 0, which verifies that the total quality of product 1 is higher than that of product 2.

2. Figure A2 shows that these equilibrium qualities satisfy q 2 q 1 < ( 1 − α ) 2 for 0 ≤ α < 0.47.

Figure A1:

Total quality under price competition.

Figure A2:

Price competition: q 2 q 1 < ( 1 − α ) 2 .

Appendix C: Proof of Proposition 1

1. Applying the implicit theorem to (22), we derive

   (33) d r d α = − ∂ g B ( r , A ) ∂ α / ∂ g B ( r , A ) ∂ r d A d α = 2 56 A 6 − 192 A 7 − 10 A 4 r + 264 A 5 r + 192 A 7 r − 15 A 2 r 2 − 142 A 3 r 2 − 360 A 5 r 2 + 56 A 6 r 2 + 2 r 3 + 16 A r 3 + 206 A 3 r 3 − 10 A 4 r 3 − 32 A r 4 − 15 A 2 r 4 + 2 r 5 / − 4 A 5 + 88 A 6 + 48 A 8 − 20 A 3 r − 142 A 4 r − 240 A 6 r + 32 A 7 r + 12 A r 2 + 48 A 2 r 2 + 309 A 4 r 2 − 12 A 5 r 2 − 128 A 2 r 3 − 40 A 3 r 3 + 20 r 4 + 20 A r 4

   where A = 1 − α and r satisfy Eq. (22). Applying Buchberger’s algorithm to solve (22) and d r d α = 0 simultaneously, we compute the Gröbner bases, one of which is the polynomial with a single variable α. We then verify that no solution of α satisfies d r d α = 0 for 0 ≤ α < 2 3 . Accordingly, we can readily show d r d α < 0 for 0 ≤ α < 2 3 , because substituting any value of 0 ≤ α < 2 3 into (22) and (33) results in d r d α < 0 .

   Using (33), (21), and q ₂ = rq ₁ and applying the implicit theorem, we derive the following:

   d q 1 d α = ∂ q 1 ∂ r d r d α + ∂ q 1 ∂ A d A d α ≤ 0 , for 0 ≤ α ≤ 0.5364 .   ∂ q 1 ∂ r d r d α + ∂ q 1 ∂ A d A d α > 0 , for 0.5364 < α < 2 3 .   d q 2 d α = ∂ q 2 ∂ r d r d α + ∂ q 2 ∂ A d A d α < 0 , for 0 ≤ α < 2 3

   in which we apply Buchberger’s algorithm to solve (22) and d q 1 d α = 0 simultaneously, and compute the Gröbner bases, one of which is the polynomial with one variable α. Then, we then verify that a unique solution α = 0.5364 satisfies d q 1 d α = 0 for 0 ≤ α < 2 3 , and d q 1 d α ⋚ 0 for α ⋚ 0.5364. Similarly, applying Buchberger’s algorithm to solve (22) and d q 2 d α = 0 simultaneously, we verify that no solution of α satisfies d q 2 d α = 0 for 0 ≤ α < 2 3 and d q 2 d α < 0 , for 0 ≤ α < 2 3 .

2. When then firms compete in price, their network sizes and profits are, respectively,

   (34) x 1 = 4 A 5 − 3 A 3 ( 1 + 4 A 2 ) r + A 3 ( 11 + 4 A ) r 2 − 2 A ( 1 + A ) r 3 4 A 5 + A 3 r − 24 A 6 r − 2 A r 2 + 38 A 4 r 2 − 19 A 2 r 3 + 2 r 4

   (35) x 2 = 2 A r ( 2 A 3 − 4 A 4 + 4 A 2 r − r 2 + A ( 1 − 2 r ) ) 4 A 5 + A 3 r − 24 A 6 r − 2 A r 2 + 38 A 4 r 2 − 19 A 2 r 3 + 2 r 4

   and

   (36) π 1 = 2 Φ 4 A 5 − 3 A 3 ( 1 + 4 A 2 ) r + A 3 ( 11 + 4 A ) r 2 − 2 A ( 1 + A ) r 3 2 4 A 5 + A 3 r − 24 A 6 r − 2 A r 2 + 38 A 4 r 2 − 19 A 2 r 3 + 2 r 4 3

   (37) π 2 = 8 Φ A 2 r 3 2 A 3 − 4 A 4 + 4 A 2 r − r 2 + A ( 1 − 2 r ) 2 4 A 5 + A 3 r − 24 A 6 r − 2 A r 2 + 38 A 4 r 2 − 19 A 2 r 3 + 2 r 4 3

   where Φ = A 2 − r 4 A 4 ( 1 − 2 A ) + A 2 ( 1 + 10 A ) r + ( 2 + 5 A ) r 2 .

   Similarly, using (33)–(35) and applying the implicit theorem and Buchberger’s algorithm, we derive

   d x 1 d α = ∂ x 1 ∂ r d r d α + ∂ x 1 ∂ A d A d α > 0 , for 0 ≤ α < 2 3 . d x 2 d α = ∂ x 2 ∂ r d r d α + ∂ x 2 ∂ A d A d α > 0 , for 0 ≤ α < 0.1943 .   ∂ x 2 ∂ r d r d α + ∂ x 2 ∂ A d A d α ≤ 0 , for 0.1943 ≤ α < 2 3 .  

   Finally, using (33), (36), and (37) and applying the implicit theorem and Buchberger’s algorithm, we can show that for 0 ≤ α < 2 3 ,

   d π 1 d α = ∂ π 1 ∂ r d r d α + ∂ π 1 ∂ A d A d α > 0 ; d π 2 d α = ∂ π 2 ∂ r d r d α + ∂ π 2 ∂ A d A d α < 0 .

Appendix D: Effect of the Higher Quality q ₁ on the Price p ₁ and the Demand x ₁

In the following, we show that the higher q ₁ increases the price p ₁, and it is unfavorable for the high-quality firm to enlarge its network size x ₁.

Differentiating (15) with respect to q ₁ yields

where we define

By differentiating h(q ₁, q ₂, α) with respect with q ₁, we obtain

That is, h(q ₁, q ₂, α) increases with q ₁ for q 1 > q 2 ( 1 − α ) 2 . We then substitute the lowest q ₁ into h(q ₁, q ₂, α), which yields

for 0 < α < 1 and q ₂ > 0. Thus, we may conclude that h(q ₁, q ₂, α) > 0, i.e., d p 1 d q 1 > 0 , for q 2 q 1 < ( 1 − α ) 2 .

In addition, differentiating (16) with respect to q ₁, it can be easily shown that

because it must be q 1 2 ( 1 − α ) 2 − q 1 q 2 > 0 and q 1 2 ( 1 − α ) 2 + 3 − q 2 ( 1 − α ) − 2 α > 0 . That is, d p 2 d q 1 > 0 for q 2 q 1 < ( 1 − α ) 2 .

To sum up, it is shown that d p 1 d q 1 > 0 and d p 2 d q 1 > 0 for q 2 q 1 < ( 1 − α ) 2 . That is, raising quality q ₁ increases both prices p ₁ and p ₂. This is because raising the intrinsic quality q ₁ increases the product’s unit cost, and it also enlarges vertical product differentiation which reduces competition. Both effects leads to higher prices p ₁ and p ₂.

Then, we further decompose the direct and indirect effects of raising q ₁ on the network size x ₁:

where d p 1 d q 1 > 0 and d p 2 d q 1 > 0 for q 2 q 1 < ( 1 − α ) 2 as mentioned previously. Moreover, it must be ∂ x 1 ∂ p 1 < 0 , ∂ x 1 ∂ p 2 > 0 and ∂ x 1 ∂ q 1 > 0 according to the demands in (7) and (8). The total derivative in (38) indicates that raising the intrinsic quality q ₁ has a negative effect on the expansion of its network size because it results in a higher price p ₁ which is unfavorable for an expansion in its network size x ₁, as shown in the first term ( ∂ x 1 ∂ p 1 d p 1 d q 1 ).

Appendix E: Total Quality and q 2 q 1 < ( 1 − α ) 2 under Quantity Competition

1. Figure A3 shows that the equilibrium gives △qx = q ₁ x ₁ − q ₂ x ₂ > 0 for 0 < α < 0.47 under quality-then-quantity competition. Because q ₁ > q ₂, we must have θq ₁ + αq ₁ x ₁ − (θq ₂ + αq ₂ x ₂) = θ(q ₁ − q ₂) + α(q ₁ x ₁ − q ₂ x ₂) > 0, which verifies that the total quality of product 1 is higher than that of product 2.

2. Figure A4 shows that these equilibrium qualities satisfy q 2 q 1 < ( 1 − α ) 2 for α < 0.107, while they conform to q 2 q 1 ≥ ( 1 − α ) 2 for α ≥ 0.107.

Figure A3:

Total quality under quantity competition.

Figure A4:

Quantity competition: q 2 q 1 < ( 1 − α ) 2 and q 2 q 1 ≥ ( 1 − α ) 2 .

Appendix F: Proof of Proposition 2

1. Applying the implicit theorem to (32), we derive, for 0 ≤ α ≤ 1 2 ,

   (39) d r ̃ d α = − ∂ g C ( r ̃ , A ) ∂ α / ∂ g C ( r ̃ , A ) ∂ r ̃ d A d α = 4 r ̃ { 4 α [ 35 − 12 α ( 3 − α ) ] − 45 } − 4 r ̃ 3 − 48 r ̃ 2 ( 1 − α ) α + 48 ( 1 − α ) 2 ( 3 −   4 α ) − 3 r ̃ 2 ( 3 − 4 α ) + 2 r ̃ [ 7 − 8 α 2 ( 3 − 2 α ) ] − 4 ( 1 − α ) { 2 α [ 17 − 6 α ( 3 − α ) ] − 11 }

   where r ̃ satisfies Eq. (32). Applying Buchberger’s algorithm to solve (32) and d r ̃ d α = 0 simultaneously, we compute the Gröbner bases, one of which is the polynomial with one variable α. We then verify that no solution of α satisfies d r ̃ d α = 0 for 0 ≤ α ≤ 1 2 . Accordingly, we can readily show d r ̃ d α > 0 for 0 ≤ α ≤ 1 2 because substituting any value of 0 ≤ α < 2 3 into (32) and (39) results in d r ̃ d α > 0 .

   Similarly, using (39), (27), and (28) and applying the implicit theorem and Buchberger’s algorithm, we can derive, for 0 ≤ α ≤ 1 2 ,

   d q 1 d α = ∂ q 1 ∂ r ̃ d r ̃ d α + ∂ q 1 ∂ A d A d α < 0 d q 2 d α = ∂ q 2 ∂ r ̃ d r ̃ d α + ∂ q 2 ∂ A d A d α > 0 .

2. When the firms compete in quantity, their network sizes and profits are, respectively,

   (40) x 1 = A 2 ( 4 − 12 r ̃ ) + r ̃ − r ̃ 2 + 4 A r ̃ 2 r ̃ + 2 A ( 2 A − 12 A 2 r ̃ + r ̃ ) ,

   (41) x 2 = 4 A r ̃ 1 − 2 A r ̃ + 2 A ( 2 A − 12 A 2 r ̃ + r ̃ ) ,

   and

   (42) π 1 = 2 A 4 A 2 + r ̃ 1 − 2 A A 2 ( 4 − 12 r ̃ ) + r ̃ − r ̃ 2 + 4 A r ̃ 2 2 r ̃ + 2 A ( 2 A − 12 A 2 r ̃ + r ̃ ) 3  ,

   (43) π 2 = 32 A 3 r ̃ 3 4 A 2 + r ̃ 1 − 2 A 3 r ̃ + 2 A ( 2 A − 12 A 2 r ̃ + r ̃ ) 3 .

   Again, using (39)–(41) and applying the implicit theorem and Buchberger’s algorithm, we can derive, for 0 ≤ α ≤ 1 2 ,

   d x 1 d α = ∂ x 1 ∂ r ̃ d r ̃ d α + ∂ x 1 ∂ A d A d α > 0 ,  d x 2 d α = ∂ x 2 ∂ r ̃ d r ̃ d α + ∂ x 2 ∂ A d A d α > 0 .

   Finally, using (33), (42), and (43) and applying the implicit theorem and Buchberger’s algorithm, we derive

   d π 1 d α = ∂ π 1 ∂ r ̃ d r ̃ d α + ∂ π 1 ∂ A d A d α > 0 , for 0 ≤ α < 0.3654 .   ∂ π 1 ∂ r ̃ d r ̃ d α + ∂ π 1 ∂ A d A d α ≤ 0 , for 0.3654 ≤ α ≤ 1 2 .   d π 2 d α = ∂ π 2 ∂ r ̃ d r ̃ d α + ∂ π 2 ∂ A d A d α > 0 , for 0 ≤ α < 0.3773 .   ∂ π 2 ∂ r ̃ d r ̃ d α + ∂ π 2 ∂ A d A d α ≤ 0 , for 0.3773 ≤ α ≤ 1 2 .