Future works

The result has several avenues for further research. First, there may exist other inter-mediaries. Competition between intermediaries may occur and can be further studied.

Second, in our current study, the intermediary cannot influence the demand of customers.

In reality, intermediary might provide mechanism that affect customers’ valuation on the product such as advertisement, data technology, better services, etc. It is interesting how these options may influence the decisions of the players in the environment.

To highlight the impact of industry structure and product similarity on the pricing decisions, we omit demand uncertainty in our model; otherwise, the intermediary would have a strong incentive favoring the platform model. We also assume that there is no

information asymmetry in the market. If the product quality is hidden to consumers, quality risk may force the intermediary to choose the merchant model. Nevertheless, it would contribute more to the literature if we further study the joint impact of these factors.

Appendix A

Proofs of lemmas and propositions

Proof of Lemma 1. We first verify the concavity of the objective function. Since

O²π_I^{W W}(p₁, p₂) =

, and its leading principals are −2 < 0 and  will be necessary and sufficient for an optimal solution. Let’s omit the constraints for a while. The first-order condition gives us ^δπ_δp^{W WI}

1 = (1 − 2p^∗₁+ bp₂ + w₁) + b(p₂− w₂) = 0, i.e., p^∗₁ = ^1+2bp2+w₂ ^1−bw2. Similarly, we have p^∗₂ = ^1+2bp1+w₂ ^1−bw2. Solving the two equations leads to p^∗₁ = ^1+(1−b)w_2(1−b) ¹ and p^∗₂ = ^1+(1−b)w_2(1−b) ².

Based on the response of the intermediary, manufacturer i maximizes his profit π_i^{W W}(w_i) = (w_i − c_i)^1−wi+bw₂ ³⁻ⁱ, i = 1, 2. As π^{W W}_i (w_i) is concave (^{d2πW Wi}_dw2^(wi)

i

= −1 < 0), the first-order condition requires the optimal wholesale price w_i^∗to satisfy w_i^∗ = ^1+bw3−i₂ ^+ci, i = 1, 2, where c₁ = c and c₂ = 0. Solving the two equations for the two manufacturers results in w^∗₁ = ^(2+b)+2c_4−b2^1+bc2 and w₂^∗ = ^(2+b)+2c_4−b2^2+bc1. By plugging in w^∗₁ and w^∗₂ to the expressions of p^∗_i, π_i^{W W}, and π^{W W}_I , we may obtain the equilibrium retail prices and profits

given in the lemma. applying the first-order condition ^dπ_dp^RRi

i = 0 for manufacturer i and yield p^∗_i = ^1+bp3−i₂^+ci+r. Solving the two equations results in p^∗_i(r) = (2+b)(1+r)+2ci+bc3−i

(4−b²) , which is the optimal price responding to r. Note that whatever r is, p^∗₁(r) ≥ p^∗₂(r) owing to the fact that c₁ ≥ c₂.

Proof of Lemma 3. While our main focus is the intermediary’s profit, it can be obtained as a function of r by plugging in p^∗_i into her profit function. Afterwards, the in-termediary decides r to maximize her profit π_I^RR(r) = max_{r 2−b}^r (2 − 2(1 − b)r − (1 − b)c). there are two possible equilibrium profit functions in the platform model. That is,

π_I^RR =

where the former is the first-order condition (FOC) case and the latter is the boundary

case. We would discuss the two cases separately.

Taking account of the FOC case first, after tedious calculations we get π^RR_I − π_I^{W W} =

G

−16 < 0 ). Second, we can prove that the first-order condition point of T , ˙c, re-sides at the left side of ¯c, the point that distinguishes the FOC case and boundary case. More precisely, ¯c − ˙c = _−3b^2(2+b)2+b+6 − ^(4b3−14b2−2b+20)(2+b)

−U = _{−U (−3b}^(2+b)H2+b+6) ≥ 0, where H = −4b⁵+ 2b⁴+ 28b³− 26b²− 24b + 24 = (b − 1)(b + 1)(b − 2)(−4b²− 6b + 12) ≥ 0.

Proof of Proposition 2. To compare the system profits of the two models, we have to derive the system profit of each model first. For the merchant model part, according to Lemma 1, the system profit of the merchant model is

π^{W W}_s = π_I^{W W} + π₁^{W W} + π₂^{W W}

can derive

Now we are ready to compare the system profits.

π_s^RR− π^{W W}_s = (1−b)(8+16b−6b²−10b³+4b⁴)(c²₁+c²₂)−4(1−b)(4+8b−3b²−5b³+2b⁴)c1c2

Proof of Lemma 4. Applying backward induction, we start from the optimal prices.

First, since π_I^{W R} and π^{W R}₂ are concave ( ^d2_dp^{πIW R}2

1 = ^d2_dp^{πW R2}2

2 = −2 ), we get the optimal p_i by applying the first-order condition ^dπ_dp^IRR

1 = 0 and ^dπ_dp^2RR

2 = 0, and yield p^∗₁ = ^1+bp2+w₂ ^1+br and p^∗₂ = ^1+bp1₂^+c2+r. Solving the two equations results in p^∗₁ = (2+b)+3br+2w1+bc2

4−b² and

p^∗₂ = ^{(2+b)(3−b2)+(2+b}_4−b2^2)r+bw1+2c2.

Considering the interaction between manufacturer 2 and the intermediary, manufac-turer 1 sets w₁ to maximize his profit π₁^{W R} = (w₁ − c₁)^{(2+b)+b(b2−1)r+(b}_4−b2 ^2−2)w1+bc2. Due to the concavity of the profit function (^d2_dw^{πW R1}2

1

= ^−2(2−b_4−b2²⁾ < 0 as b ∈ [0, 1)), he opti-mally sets the wholesale price by applying the first-order condition ^{dπW R1} = 0 and derives

w^∗₁ = (2+b)−b(1−b²)r+(2−b²)c1+bc2

Proof of Lemma 5. Following Lemma 4, we first check the concavity of π_I^{W R}(r).

Since ^d2π_dr^{W RI} 2^(r) = ^−2b2_4(4−b^(1−b22^)(7−b)^{2 2)} + ^(1−b_2(4−b^2)(−16+6b2)(2−b²)²⁾ < 0, π_I^{W R}(r) is concave in r. There-fore, the first-order condition ^{dπIW R}_dr^(r) = 0 results to the optimal transaction fee r^∗ =

2(2+b)(8−3b−3b²+b³)−2b(1−b)(2−b²)c1+2(1−b)(−16+9b²−b⁴)c2) (64−26b²−3b⁴+b⁶)(1−b) .

We have to verify the non-negative demand constraints in the mixed model (WR).

Starting from the demand of product 1, since D^{W R}₁ = ^{(2+b)B0+(2−b}_2(4−b²2^)C)A^00c1+bD0c2 where A⁰ = D^{W R}₂ (r^∗) > 0, r^∗ is feasible considering the non-negative demand constraints.

Now we are able to compare p^{W R}₁ and p^{W R}₂ . First, since p^{W R}₁ −p^{W R}₂ = (2+b)B+(2−b²)Cc1+Dc2 optimal pi by applying the first-order condition ^dΠ_dp^RRi

i = 0 for manufacturer i and yield p^∗_i = ^{(1−φ)(1+bp}_2(1−φ)^3−i)+ci, where c₁ = c and c₂ = 0. Solving the two equations results to p^∗_i = (2+b)(1−φ)+2ci+bc3−i

(4−b²)(1−φ) . The equilibrium demands and the intermediary’s profit as a function of φ can then be obtained by plugging in p^∗_i into their formulas.

Proof of Lemma 7. Through several steps of arithmetic, we obtain h(φ) = −2(2 + b)²φ³+ 6(2 + b)²φ²+ (−6(2 + b)²− (2 + b)²bc + (3b²− 4)c²)φ + (2(2 + b)²+ (2 + b)²bc + monotonically decreases as φ goes up, and eventually reaches h(1) < 0. In other words, Π^RR_I (φ) is quasi-concave. Therefore, there exists a unique first-order solution ˜φ ∈ (0, 1) that satisfies ^dΠRRI_dφ^(φ) = 0.

Now we take the constraints into consideration. First, D2 = (2+b)(1−φ)+bc

(4−b²)(1−φ) ≥ 0. How-ever, D₁ = (2+b)(1−φ)+(−2+b²)c

(4−b²)(1−φ) ≥ 0 if c ≤ ^{(1−φ)(2+b)}_(2−b2 , i.e., φ ≤ ˆφ = 1 − ^(2−b_2+b^2)c. Therefore, if

φ ≤ ˆ˜ φ, ˜φ is optimal; if not, then ˆφ is optimal.

Proof of Proposition 3. When c = 0 and b = 0, ˜φ = max_φ^φ₂ = 1 = ˆφ. Therefore, we have φ^∗ = 2 and Π^RR_I = ¹₂. By plugging c = 0 and b = 0 into Lemma 1, we derive π_I^{W W} = ¹₈. The facts that Π^RR_I |_b=c=0 > π_I^{W W}|_b=c=0 and both profit functions are continuous result in our first conclusion regarding the existence of ˆb₁ ∈ (0, 1) and ˆ

c1 ∈ (0, 1). On the contrary, when c and b both approach 1, ˆφ approaches ²₃ and ˜φ approaches to a value around 0.81, which is greater than ˆφ. Therefore, φ^∗ = ˆφ = ²₃ and Π^RR_I = ⁸₃. In this case, π_I^{W W} approaches infinity. Again, because all functions are continuous, we obtain our second conclusion regarding the existence of ˆb2 ∈ (0, 1) and ˆ

c₂ ∈ (0, 1).

Proof of Proposition 4. Consider the case that c = ⁴₅. Through several steps of arith-metic, we obtain h( ˆφ)|_c=⁴ models can be conducted by investigating the sign of

g(b) = Π^RR_I (φ^∗)|_c=⁴ proposition then follows due to the continuity of all profit functions.

Proof of Lemma 8. Applying backward induction, we start from the optimal prices.

First, since π_i^ERR is concave ( ^d2π_dp^iERR2 i

= −2 ) where i = 1, 2, we get the optimal p_i by applying the first-order condition ^dπ_dp^RRi

i = 0 and yield p^∗_i = ^1+bp3−i₂^+ci+ri. Solving the two equations results in p^∗_i(r₁, r₂) = ^(2+b)+2(ri+c_4−b^i)+b(r₂ ^{3−i+c3−i)}, which is the optimal price responding to r.

Considering the interaction between manufacturers, the intermediary sets r1 and r2

to maximize his profit

π_I^ERR(r1, r2) = maxr1,r2r1D1+ r2D2

= max_r1,r2(r₁+ r₂) − (r₁− br₂)p^∗₁(r) − (r₂− br₂)p^∗₂(r)

= max_r1,r2 ^{((2+b)+bc2−(2−b2)(c1+r1))r1+((2+b)+bc}_4−b2 ^{1−(2−b2)(c2+r2))r2+2br1r2}. We first check the concavity of the profit function, since

O²π_I^ERR(r₁, r₂) = _4−b¹2

given b ∈ [0, 1), π^ERR_I (r1, r2) is negative semi-definite. Therefore, the intermediary op-timally sets the transaction fees by applying the first-order condition. Let’s omit the constraints for a while. The first-order condition gives us ^δπ_δr^IERR

1 = 0, which implies that r2 = −^{(2+b)+bc2−(2−b2)(c1+2r1)}. Similarly, we have r1 = −^{(2+b)+bc2−(2−b2)(c1+2r2)}. Solving the

two equations leads to r₁^∗ = _2(1−b)¹ −¹₂c₁ and r^∗₂ = _2(1−b)¹ −¹₂c₂. Furthermore, we can derive

Proof of Proposition 5. We may show that the intermediary can well simulate the product prices in the mixed model by operating the transaction fees in the enhanced plat-form model. According to Lemma 4, we have p^{W R}₁ (r) = ^{(2+b)(3−b2)+b(5−2b}_(4−b²2^)r+(2−b)(2−b²)^{2)c1+b(3−b2)c2}

and p^{W R}₂ (r) = (2+b)(4+b−2b²)+(8−b²−b⁴)r+b(2−b²)c1+(8−3b²)c2

2(4−b²)(2−b²) . In contrast, according to Lemma 8, we have p^ERR_i (r1, r2) = ^(2+b)+2(ri+c_4−b^i)+b(r2 ^{3−i+c3−i)}, where i = 1, 2. Solving the two equa-tions p^ERR₁ = p^{W R}₁ and p^ERR₂ = p^{W R}₂ leads to r₁ = ^(2+b)+(3−b_2(2−b^{2)br−(2−b}2) ^2)c1+bc2 and r₂ = r.

In addition, we have to prove that the profit of the intermediary is higher in the enhanced platform model than that in the mixed model. Since the product prices are the same, the demands of both products are also the same. Consequently, we only have to compare the margin profit of each products. For the product 2 part, the mar-gin profits are the same in both models, i.e., r₂ = r. For the product 1, the margin profit is r₁ in the enhanced platform model and the margin profit is p^{W R}₁ − w₁^{W R} in the mixed model. The difference between the margin profits in two models is D(r) = r₁ − (p^{W R}₁ − w₁^{W R}) = ^(2+b)+(3−b_2(2−b^{2)br−(2−b}2) ^2)c1+bc2 − (^{(2+b)(3−b2)+b(5−2b}_(4−b²2^)r+(2−b)(2−b²)^{2)c1+b(3−b2)c2} −

(2+b)−(1−b²)br+(2−b²)c1+bc2

2 ) = (2+b)−b(1−b²)r−(2−b²)c1+bc2

2 2 .

It can be observed that D(r) is decreasing in r. Therefore, if we can find a r⁰ greater than the optimal transaction fee r^{W R} such that D(r⁰) ≥ 0, we show that D(r^{W R}) ≥ 0.

By plugging r⁰ = 2(2+b)(8−3b−3b²+b³)

(1−b)(62−29b) ≥ 2(2+b)(8−3b−3b²+b³)−2b(1−b)(2−b²)c1+2(1−b)(−16+9b²−b⁴)c2

(64−26b²−3b⁴+b⁶)(1−b) = r^{W R} into D(r), we have D(r⁰) = (2+b)(64−45b−10b²+12b³+4b⁴−2b⁵)+(64−29b)(bc2−(2−b²)c1)

(4−b²)(2−b²)(64−29b) . It can be prove that D(r⁰) ≥ 0, since D(r⁰) is increasing in c₂ but decreasing in c₁, and D(r⁰)|_c1=1,c2=0= ^{b(32−b−15b}_(4−b2)(2−b^22+20b)(64−29b)^3−4b5) ≥ 0 for all b ∈ [0, 1), c₁ ∈ [0, 1] and c₂ ∈ [0, 1].

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