In the platform model, there exists other mechanism for the intermediary to cooperate with manufacturers. Instead of transaction fee, it is also possible for the intermediary to share profit by a revenue sharing proportion φ. In this section we would discuss the profitability of revenue sharing in the platform model.
5.2.1 Equilibrium analysis
First of all, we have to find out the equilibrium under the platform model. The interme-diary sets the revenue sharing proportion φ ∈ [0, 1] to maximize her profit
ΠRRI (φ) = φ(p1D1+ p2D2).
Then the two manufacturers simultaneously set their retail prices by solving
ΠRRi = max
pi
((1 − φ)pi− ci)Di, i = 1, 2.
All these optimization problems are subject to the demand non-negativity constraints D1 ≥ 0 and D2 ≥ 0. In Lemma 6, we first characterize the manufacturers’ equilibrium decisions given the revenue sharing proportion φ.
Lemma 6. Given the revenue sharing proportion φ, the equilibrium retail prices in the platform model are p∗1 = (2+b)(1−φ)+2c
Lemma 6 summarizes the equilibrium prices of products in response to the revenue sharing proportion φ from the manufactures’ perspective. Both products’ prices increase in the revenue sharing proportion since the revenue sharing proportion acts as sales cost. Furthermore, similar to the merchant model, manufacturer 2 benefits from his lower production cost. With the cost advantage, he sets a lower price and attracts more consumers. With the manufacturers’ responses in mind, the intermediary looks for φ to maximize ΠRRI (φ) derived in Lemma 6. Though the complicated structure of ΠRRI (φ) makes it impossible to have a closed-form expression for the intermediary’s optimal φ,
in the next lemma we present some analytical properties regarding the intermediary’s problem.
Lemma 7. The function ΠRRI (φ) is quasi-concave in φ ∈ [0, 1]. Therefore, there exists a unique ˜φ ∈ [0, 1] that satisfies dΠRRIdr(φ)|φ= ˜φ = 0. The intermediary’s optimal revenue
Lemma 7 states the existence and uniqueness of the optimal revenue sharing propor-tion φ∗ to maximize the profit of the intermediary in the platform model. There are two special values of φ mentioned in Lemma 7. On the one hand, ˜φ is the ideal optimal φ which satisfies the first-order condition of the quasi-concave profit function. ˆφ, on the other hand, is the cap of the feasible region of φ derived from the demand constraints.
Therefore, the first-order point ˜φ is the optimal feasible revenue sharing proportion for the intermediary when it does not reach the cap ˆφ. Once ˜φ violates the cap, ˆφ is the optimal feasible revenue sharing proportion.
As aforementioned, the product prices increase in the revenue sharing proportion.
While the revenue sharing proportion represents a tool for the intermediary to extract revenue from manufacturers, once it goes beyond the cap, manufacturer 1 earns nothing and stops raising its product price. As a result, manufacturer 2 would also stop raising its product price. Consequently, the intermediary no longer benefits from raising the proportion once it goes beyond the cap.
By utilizing Lemmas 6 and 7, we are able to numerically derive the equilibrium of the market in the platform model. Given the values of b and c, the intermediary can first
numerically search for the first-order point ˜φ and compare it with the upper bound ˆφ to find the optimal φ∗. Substituting φ∗ into Lemma 6 then gives us the equilibrium retail prices, product demands, and manufacturers’ profits.
5.2.2 Optimal model selection
As Lemma 7 shows, there is no closed-form expression for φ∗ in the platform model. As it is hard to derive the profit of the intermediary, we do a numerical study to obtain some intuitions first. For each combination of b ∈ [0, 1) and c ∈ [0, 1), we numerically find φ∗ and the associated platform’s profit ΠRRI under the merchant model. We then compare that with the platform’s profit πIW W under the merchant model, which may be calculated by the closed-form formula we derived. Figure 5.2 is a visualization of our result.
A first look at Figure 5.2 will give us the following idea: When b or c is large, the merchant model is better; on the contrary, when b and c are both small, the platform model is better. This idea is analytically proved in the following proposition.
Proposition 3. There exist two cut-off values ˆb1 ∈ (0, 1) and ˆc1 ∈ (0, 1) such that for all (b, c) < (ˆb1, ˆc1), we have πW WI < ΠRRI . On the contrary, there exist ˆb2 ∈ (ˆb1, 1) and ˆ
c2 ∈ (ˆc1, 1) such that for all (b, c) > (ˆb2, ˆc2), we have πW WI > ΠRRI .
We find that the intermediary prefers the merchant model when b goes up but prefers the platform model when b goes down. The phenomenon is quite different to the plat-form model with transaction fee, where the platplat-form model is preferred as b increases.
To explain the contradiction, we should take a closer look at the difference between the two revenue sharing mechanisms. The transaction fee is a fixed cost from the perspective
Figure 5.2: Comparison of pure models under revenue sharing
of manufacturers. In contrast, the revenue sharing mechanism dynamically shares more profit as the products’ prices increase. As higher product prices hurt demands, manu-facturers are more willing to increase price in the case of transaction fee comparing to revenue sharing.
Now that when b goes up the demands of products goes up, which results to the incentive of raising up the products’ prices. Because of the different tendency of raising prices, the equilibrium product prices in the case of implementing transaction fee is higher than that using revenue sharing. The system profit and the intermediary’s profit are thus lower in the platform model implementing revenue sharing.
If we look at Figure 5.2 more deeply, we would obtain an interesting observation at the top-left corner. There is a region of moderate c (roughly between 0.75 and 0.85) such that the impact of b on the optimal model is non-monotone. When b is either small or large, the intermediary prefers the merchant model; in contrast, the platform model is more advantageous when b stays in the medium. We may again analytically confirm this observation.
Proposition 4. There exist ˜c1 ∈ (0, 1) and ˜c2 ∈ (˜c1, 1) such that for all c ∈ [˜c1, ˜c2], there exist ˜b1 ∈ (0, 1) and ˜b2 ∈ (˜b1, 1) such that πW WI > πIRR if b < ˜b1 or b > ˜b2 but πW WI < πIRR if b ∈ (˜b1, ˜b2).
While c is large, the impact of the cost difference dominates the selection of models. In other words, with a huge gap between products at the production side, the intermediary has no choice but to adopt the merchant model. Similarly, when c is small, the platform model becomes dominant. When c is moderate, the impact of the competition intensity enlarges. When b is large, the restriction of price from revenue sharing hurts the profit of the intermediary. Therefore, taking control of product prices is a solution to avoid low prices. However, when b is small, the two products are quite different at the consumer side, a situation that is quite similar to the case that c is large. The disadvantage of relying on only one single revenue sharing product is too significant for the intermediary.
The merchant model is thus preferred.