Optimal Contract Under Endogenous Platform Services: Implications of Tax and Advertising

Abstract

The paper evaluates the contracting problem between a platform and a seller under information asymmetry where the seller holds private information about his/her cost for product quality. Price per product is influenced by seller’s product quality and platform’s service quality. Cost-sharing contract is more desirable as it induces a higher level of qualities and generates higher profit for the platform compared to revenue-sharing contract. The product quality and platform’s service quality vary negatively with the ad-valorem tax imposed on price of the product. We then introduce advertising in our model and observe that the level of advertising is lower under information asymmetry.

JEL Classifications: D86, L21, M37

Keywords

Platform contracting asymmetric information advertising ad-valorem tax

Introduction

The present era is marked by an exponential growth of the e-commerce sector. The e-commerce sales reached $ 4.28 trillion in 2020, and it is expected to grow to $ 5.4 trillion in 2022.¹ The global website visits grew substantially from 16.07 billion in January 2020 to 22 billion in June 2020.² Moreover, the Covid-19 pandemic led retail e-commerce revenue to grow at a striking rate of 25%.³ The growing platform market is subject to several disputes over sales proceeds sharing between platform providers and sellers who continue to argue over getting unfair share of revenue (Sur et al., 2019). Thus, contract design between them has been one of the most important business decisions in the space of digital market (Sur et al., 2019).

We emphasize the contract design between a platform and a seller targeting the customer pool through the intermediary platform where price per product that a consumer pays is influenced by seller’s product quality and platform’s service quality. Under complete information, a simple contract can help the platform to squeeze out all the first-best profit, giving the seller only his/her reservation value. However, under information asymmetry when the seller holds the private information about the cost for product quality, the efficient seller can achieve higher profit than the reservation level at the cost of the platform. Next, we introduce the ad valorem tax in our model where the government levies per unit tax on price paid by a consumer. Sensitivity analysis emphasizes the effect of tax on product quality, service quality and profit of the platform.

The rest of the study proceeds as follows. The detailed review of the literature along with the major contributions of our study to the literature has been provided in the second section. We establish the basic framework with two possible kinds of sellers in the third section. The fourth section explores two common forms of contract structure and examines the effect of tax on contract variables for both types of contracts. The fifth section compares the findings obtained for two contracts. The sixth section introduces the effect of advertising on our analytical model. Ultimately, the seventh section draws the conclusion of the paper.

Review of Literature

There exists a gamut of literature concerning the contracting problem of retail platform and supply chain networks. Ma et al. (2013) regarded a supply chain in their study where demand is affected by retailer’s marketing effort and manufacturer’s quality effort. The paper discussed different kinds of contracts and concluded that the two-part tariff contract with both kinds of cost-sharing (CS) (marketing cost and quality cost) by the members of supply networks coordinates the members well and increases the total profit. Zhang et al. (2019) discussed two types of contracts, revenue-sharing (RS) and fixed fee contracts in platform-manufacturer setting and figured that fixed fee contract leads to higher quality compared to RS. Roger and Vasconcelos (2014) studied the moral hazard for two-sided platform in infinite horizon structure and found that registration fees along with transaction fees alleviate moral hazard. Ma et al. (2017) analysed the optimal contract in the presence of information asymmetry for supply chain where the manufacturer influences the demand by investing on corporate social responsibility. Mukhopadhyay et al. (2008) studied the contracting problem with information asymmetry for mixed networks where manufacturer not only sells through retailer but it also directly targets the consumer pool. Babich et al. (2012) explored the buyback contract problem with one supplier and one retailer who has private information about the state of demand.

The contribution of our study is that we discuss a crucial topic regarding the coordination problem between two agents of the platform and derive the optimal choices for the platform and the seller for the two most commonly adopted agreements, RS and CS contracts under asymmetric information, an issue that has not been received exhaustive and in-depth research attention in literature pertaining to online platforms. In addition to this, we keep the service quality of platform as an endogenous choice under contracting decisions which has not previously been studied in the existing literature on online platforms, to the best of our knowledge.

Our study contributes to the literature on contract design with the following results. The key finding obtained is that the optimal contract in presence of information asymmetry induces high type seller to supply product quality less than the first-best quality when service quality is endogenous. This result sets a major departure from the standard one-sided model which states the high type seller to provide first-best efficient quality in presence of information asymmetry. Additionally, the platform supplies lower level of service quality compared to first-best quality in presence of asymmetry. We then compare the optimal values of contract variables for two contracts and conclude that CS contract is more desirable as it induces higher level of product quality, service quality and platform’s profit compared to RS contract. Comparative static analysis shows the product quality, service quality and profit of the platform are negatively related to tax rate. A numerical analysis based on certain parameter values has been conducted to validate the analytical findings. Finally, we introduce advertising in our model and observe that platform uses more advertising signals in complete information case when it is certain about the type of the seller.

The Model

Consider a digital marketplace with one intermediary platform and one seller. The seller produces a product with quality ‘q’ at a unit cost of β and enters into a contract with the monopoly platform in order to sell the product to final consumer at a unit price P against a payment of a part of his/her revenue, α to the platform. The seller gives a fraction of his/her revenue to platform to be able to trade on it. Moreover, the platform is providing services to all the buyers and the service quality of platform is represented by ‘s’. The services offered by the platform to buyers can be considered as a composite measure for daily service level settled by platform like return and refund policy, replacement policy, delivery speed etc. All these come under the category of platform’s services. We assume, per unit price is a function of product quality, q and service quality, s; $P = f (q, s)$ and in particular, for simplicity of analysis, we specify the following constant elasticity functional form as:

P (q_{i}, s) = {γsq}_{i}^{λ} where 0 < λ < 1; \frac{\partial P}{\partial q_{i}} > 0; \frac{\partial^{2} P}{\partial q_{i}^{2}} < 0

(1)

This may be interpreted as consumers’ willingness to pay per unit of product where $γ$ signifies a parameter that relies on components not included in our framework and $λ$ captures the price elasticity measuring the responsiveness of the price of the product to its product quality.⁴ In addition, we assume that government levies ad valorem tax, t on product price. Studies by Kind et al. (2008, 2009), Belleflamme and Toulemonde (2018) and Bajo-Buenestado and Kinateder (2019) have dealt with the issues related to the effect of ad valorem taxes. Following these pioneering studies, we also emphasize the impact of ad valorem taxes on product quality, service quality and platform’s profit.

The profit of the platform is influenced by the quality choice made by the seller. If choices are easily verifiable, the agreement between the seller and the platform would be pretty straightforward. However, when there is no way for platform to notice the seller’s choices then a contract must be designed in a way that elicits seller to choose the right action. Depending on the unit cost for quality of product, the following two types of seller can exist: efficient seller (h type) and inefficient seller (l type). An efficient seller can produce a product in a cost-effective way than the inefficient seller and thus incurs lower unit cost for producing the similar quality of product compared to the inefficient seller. Let, $β_{i}$ be the unit cost for producing a product with quality $q_{i}$ by the ith type seller, $i є (h, l)$ . By our assumption, $β_{l} > β_{h}$ if $q_{i} = \bar{q}, i є (h, l)$ . Thus, the efficient seller bears lower unit cost compared to inefficient low-type seller when each seller produces a product with $\bar{q}$ level of quality. Further we assume, $P r o b (β_{h}) = ϕ \in (0, 1)$ . While the seller certainly knows its individual type, the platform possesses only a subjective evaluation about the occurrence of two possible states. In this form of setting, the platform whose objective is to maximize its expected profit $(Π^{p})$ , proposes a menu of contract that induces the seller to truthfully declare its type through the selection of agreement from the menu. The platform proposes two sets of contract to seller, specified by $(α_{i}, s)$ for each type $i$ , $i є (h, l)$ , where $α_{i}$ be the fraction of the revenue the seller ‘i’ pays. The platform determines these two revenue fees together with service quality by maximizing its expected net profit. We answer this platform-seller problem as a game of principal-agent framework. The sequence of the game is as follows: First, the platform takes the lead by offering the menu of contracts to a seller of the particular type, i. Finally, the seller chooses a particular contract along with the product quality and thus, type i is revealed through the selection of the contract from the menu. Following Ma et al. (2013), Tsao and Sheen (2012), Zhang et al. (2019), we examine two common forms of contract: RS Contract and CS Contract, however, with information asymmetry.

Contract Design

Revenue-Sharing Contract

We discuss the form of contract where seller shares a portion of his/her revenue with platform indexed as α. When the platform exerts services with ‘s’ level of quality and seller of type i produces the product with quality $q_{i}$ , net profit of the seller ‘i’ per product sold can be expressed as,

Π_{i}^{s} (q, α, β) = (1 - α_{i}) (1 - t) γ s q_{i}^{λ} - \frac{β_{i} q_{i}^{2}}{2}

where $\frac{β_{i} q_{i}^{2}}{2}$ is the lump sum quadratic cost for producing a product with quality, $q_{i}$ .⁵

When β Is Observable (First-Best Case)

Optimal Contracting Problem.

A contract between a platform and a seller where both possess the complete knowledge of type of seller can directly state RS rate conditional on β. The contracting problem of the platform can be described as,

\underset{t_{h}, \pm_{1}, s}{\underset{︸}{Max}} Π^{p} = {ΦΠ}_{h}^{P} (\pm_{h}, q_{h} (\pm_{h}, s), s) + (1 - Φ) Π_{1}^{p} (\pm_{1}, q_{1} (\pm_{1}, s), s)

= ϕ [α_{h} (1 - t) γ s {(q_{h} (α_{h}, s))}^{λ} - \frac{s^{2}}{2}] + (1 - ϕ) [α_{1} (1 - t) γ s {(q_{1} (α_{1}, s))}^{λ} - \frac{η s^{2}}{2}]

(2)

{Subject to IR}_{h} : Π_{h}^{s} = (1 - α_{h}) (1 - t) γ {sqq}_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} \geq 0

(3a)

{IR}_{1} : Π_{1}^{s} = (1 - α_{1}) (1 - t) γ s {sq}_{1}^{λ} - \frac{β_{1} q_{1}^{2}}{2} \geq 0

(3b)

where the share of the revenue per product sold that platform receives from seller of type i is amounted to $[α_{i} (1 - t) γ s ({q_{i} (., .))}^{λ}]$ . Here, $q_{i} (α_{i}, s)$ qi (αi, s) is decided by the ith seller. The platform incurs a lump sum quadratic cost for providing service quality. When inefficient seller supplies a product there remains a high possibility of return or replacement of the product. In which case the platform has to incur a higher service quality cost. So, we assume $η > 1$ .

Given the optimal quality chosen by sellers ${(q}_{h}^{*}, q_{l}^{*})$ , for any answer $(α_{h}^{*}, α_{l}^{*} {, s}^{*})$ to the above optimization problem, reservation utility or individual rationality constraints must bind; or else the platform could higher its share of the revenue fee $(α_{i})$ and still compel the seller to accept the agreement. Thus, IR conditions imply that seller’s profit is equalized across types and the profit is equal to the reservation profit in each state. Therefore, the platform’s profit is maximized with respect to $(α_{h}, α_{l}, s)$ given that the individual rationality constraint for each type is satisfied, and we obtain the optimal values given below,

s_{R}^{*} = {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}, α_{h_{R}}^{*} = (1 - \frac{λ}{2}), α_{l_{R}}^{*} = (1 - \frac{λ}{2})

Hence, optimal product quality levels chosen by sellers are,

\begin{array}{l} q_{1 R}^{*} = {(\frac{(1 - t) γ λ}{2_{1}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}} \\ q_{h R}^{*} = {(\frac{(1 - t) γ λ}{2_{h}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {({(1 - t)}^{3})}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}} \end{array}

In the optimum ${q_{h}}_{R}^{*} > {q_{l}}_{R}^{*}$ must satisfy which gives us the result similar to the contract problem in one-sided market. The type specific equilibrium requires for cost-efficient h-type seller to exert higher product quality than the l-type as unit cost for same level of effort is lower for type h than l-type. Both types of sellers share same amount of revenue with platform. The first-best results can be taken as benchmark since this serves as ideal scenario with full observable situation.

When β Is Not Observable by Platform (Second-Best Case)

Optimal Contracting Problem

In this subsection, we consider the case where platform cannot differentiate the type of seller. If platform proposes an agreement similar to the first-best case in presence of information asymmetries, the seller of type h will end up choosing the contract intended for cost-inefficient seller without revealing his/her true type.⁶ In that case the platform must form a contract in a way that elicits seller to reveal his/her true type voluntarily. For this purpose, the platform trusts on the revelation principle (Myerson, 1979), which ensures that the seller reveals his/her true type ‘i’ by choosing the contract designed for his/her type since by doing so, he/she will earn higher profit. Therefore, the revelation principle guarantees that the announced type of the seller is his/her true type. The essence of revelation principle with its truthful revelation property is captured by introducing an additional constraint, Incentive Compatibility (IC) Constraint in our model which assures a seller chooses the particular contract assigned to his/her true type. The characterization of platform’s problem which is to maximize its expected profit subject to IR and IC constraints is outlined as,

\underset{α_{h}, α_{1}, s}{\underset{︸}{Max}} Π^{p} = ϕ [α_{h} (1 - t) {γsq}_{h}^{λ} - \frac{s^{2}}{2}] + (1 - ϕ) [α_{1} (1 - t) {γsq}_{1}^{λ} - \frac{{ηs}^{2}}{2}] where η > 1

(4)

{Subject to IR}_{h} : Π_{h}^{s} = (1 - α_{h}) (1 - t) {γsqq}_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} \geq 0

(5a)

{IR}_{1} : Π_{1}^{s} = (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{1} q_{1}^{2}}{2} \geq 0

(5b)

{IC}_{h} : (1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} \geq (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{h} q_{1}^{2}}{2}

(5c)

{IC}_{l} : (1 - α_{l}) (1 - t) {γsq}_{l}^{λ} - \frac{β_{l} q_{l}^{2}}{2} \geq (1 - α_{h}) (1 - t) {γsq}_{h}^{λ} - \frac{β_{l} q_{h}^{2}}{2}

(5d)

Constraints (5a) and (5b) that represent the individual rationality or participation constraints for the seller of type h and l respectively indicate a seller of type i must receive his/her reservation profit if he/she accepts the contract. Constraints (5c) and (5d) which constitute the IC or truth-telling constraints for type h and l seller respectively induce the seller to select that particular contract menu entitled to his/her true state. Let us consider constraint ${I C}_{h}$ in (5c). Seller of type h will obtain a profit of $(1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2}$ if he/she reveals his/her true type, but it is $(1 - α_{l}) (1 - t) γ s q_{l}^{λ} - \frac{β_{h} q_{l}^{2}}{2}$ if he/she instead declares his/her type to be ‘l’. Thus he/she reveals his/her true type if $(1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} \geq (1 - α_{l}) (1 - t) γ s q_{l}^{λ} - \frac{β_{h} q_{l}^{2}}{2}$ . Constraint represented by (5d) follows the similar ground. Note that full information contract is not incentive compatible as it does not satisfy the Constraint (5c). We begin to solve the platform’s optimization problem by identifying the above four constraints in its contract problem. In the process of identification of constraints, we lay down two claims which will be crucial for solving the optimization problem.

Claim 1: If ${I R}_{l}$ and ${I C}_{h}$ are binding then ${I R}_{h}$ is satisfied and non-binding. Thus the constraint ${I R}_{h}$ can be ignored while solving the optimization problem.

Claim 2: If ${I C}_{h}$ is binding then ${I C}_{l}$ is satisfied and non-binding.⁷

Therefore, the set of optimal contracts obtained by ignoring the constraints ${I R}_{h}$ and ${I C}_{l}$ in the optimization problem will yield the identical menu of contracts derived by taking into account all the constraints. We proceed to answer the optimization problem by ignoring the two constraints ${I R}_{h}$ and ${I C}_{l}$ as indicated by the above two claims. The reduced optimization problem of the platform which becomes maximization of expected profit subject to the constraints ${I R}_{l}$ and ${I C}_{h}$ is described as follows,

\underset{α_{h}, α_{1}, s}{\underset{︸}{Max}} Π^{p} = ϕ [α_{h} (1 - t) γs {(q_{h} (α_{h}, s))}^{λ} - \frac{s^{2}}{2}] + (1 - ϕ) [α_{1} (1 - t)γs {(q_{1} (α_{1}, s))}^{λ} - \frac{η s^{2}}{2}]

(4)

Subject to {IR}_{1} : Π_{1}^{s} = (1 - α_{1}) (1 - t) γ {sq}_{1}^{λ} - \frac{β_{1} q_{1}^{2}}{2} = 0

(5b)

{IC}_{h} : (1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} = (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{h} q_{1}^{2}}{2}

(5c)

where $q_{i} (α_{i}, s)$ is determined by the seller through the truthful revelation of his/her type. The optimal decision of the above optimization problem is,

s_{R}^{'} = {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} {ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + {(1 - ϕ)}^{\frac{2}{2 - λ}} {(\frac{1}{(β_{1} - ϕ β_{h})})}^{\frac{λ}{2 - λ}})}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}

q_{h_{R}^{'}}^{'} = {(\frac{(1 - t) γ λ}{β_{h}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + {(1 - ϕ)}^{\frac{2}{2 - λ}} {(\frac{1}{(β_{1} - ϕ β_{h})})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}}

q_{l}^{'} = {(\frac{(1 - ϕ) (1 - t) γ λ}{(β_{1} - ϕ β_{h})})}^{\frac{1}{2 - λ}} [\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} {{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + {(1 - ϕ)}^{\frac{2}{2 - λ} {(\frac{1}{(β_{1} - ϕ β_{h})})}^{\frac{λ}{2 - λ}}})}^{\frac{1}{2 (1 - λ)}}}{(ϕ + (1 - ϕ) η)}

α_{h_{R}^{'}}^{'} = 1 - \frac{λ}{2} - \frac{λ (β_{1} - β_{h}) {(β_{h})}^{\frac{λ}{2 - λ}}}{2} {(\frac{1 - ϕ}{(β_{1} - ϕ β_{h})})}^{\frac{2}{2 - λ}}; α_{l_{R}}^{'} = 1 - \frac{β_{1} λ (1 - ϕ)}{2 (β_{1} - ϕ β_{h})}

The equilibrium contract dependent on type still induces h-type seller to exert higher product quality than the l-type as unit cost for producing similar level of quality is lower for type h than l-type. Thus the equilibrium quality-fee bundle for high type comprises of higher product quality and lower RS fee relative to the bundle for low type.⁸ The intuition is that a cost-effective seller (here, h type) will give a lower proportion of its revenue to platform and exert higher product quality and therefore, can expect a higher price per product sold. All these forces in turn contribute to earn higher profits by both the platform and the seller. This is a crucial insight of our study. To complete the investigation on two state framework under RS contract, it can be examined that both the constraints ${I R}_{h}$ and ${I C}_{l}$ which are dropped in analysing the optimal contract, are satisfied by the optimal values of contract variables.

Comparing Two Regimes Under Revenue-Sharing Model

In this subsection, we compare the results obtained under two regimes of RS contract. We validate the theoretical findings by employing a numerical study with the parameter values as: $β_{l} = 2$ , $β_{h} = 1$ , $ϕ = 0.5$ , $λ = 0.3$ , $γ = 1$ , $η = 2$ , $θ = 0.6$ . In the presence of complete information, the platform can exactly know the type of the seller and it can offer the exact contract $(α_{i}, s)$ if it believes the seller to be of type ‘i’. In that case the platform can extract all its first-best profit for itself using a simple contract, leaving the seller to have the reservation profit. In case of hidden information two state structure, binding ${I R}_{l}$ constraint indicates the l type seller will earn no additional profit over his reservation profit (i.e.., $Π_{l}^{s} = 0$ ). However, the fact that ${I R}_{h}$ is satisfied and non-binding, signifies that the h type seller will realize additional profit greater than his reservation profit (that is, $Π_{h}^{s} > 0$ ). The expected profit earned by h type seller, defined as $ϕ Π_{h}^{s}$ , which is also known as information rent, is strictly greater than his reservation profit and the platform requires to pay this expected rent to h type seller to obtain the hidden information regarding the type. The information rent, in our model, is precisely obtained as $ϕ \frac{{(β}_{l} - {β_{h}) q}_{l}^{2}}{2}$ . Thus, with asymmetric information, the platform fails to derive all the first-best profit (that is $Π_{P R}^{*} > Π_{P R}^{'}$ ) because of the information rent which helps the high type seller to achieve higher surplus more than the reservation profit. The straight lines AB and CD represent the profit level of the platform for first-best and second-best model respectively in Figure 1. We find the profit level falls throughout for both cases with increase in the level of tax. We observe the line AB lies above the line CD for any particular value of tax. Thus the profit level achieved by platform with full information, $Π_{P R}^{*}$ is higher than the profit attained with information asymmetry, $Π_{P R}^{'}$ for the same level of tax rate.

Figure 1.

Note: For color usage in this figure, please refer to the web version of the article.

With platform has to spend the information rent in presence of information asymmetry, it invests less on service quality than in the first-best regime. Thus, we obtain a strictly lower level of service quality when platform is uncertain about the type of seller. Therefore, $s_{R}^{*} > s_{R}^{'}$ . The curve, S_R^* depicting the service quality level with full information case lies above the service quality curve, S_R^ʼ under incomplete information case in Figure 2.

Figure 2.

Note: For color usage in this figure, please refer to the web version of the article.

We find an interesting observation regarding the quality of the product sold. If service quality is exogenously given, then the high type seller is provided with the efficient first-best product quality and the optimum bundle for low type seller contains less quality in presence of information asymmetry than under complete information. Thus, the results derived with exogenous service quality are similar to the one-sided market outcomes. However, when we make the service quality endogenous to the system, the findings get altered. With endogenous service quality, the profit maximizing level of product quality for h type seller is strictly smaller under incomplete information case than in first-best regime, that is, ${q_{h}}_{R}^{*} > {q_{h}}_{R}^{'}$ . So, we observe a major deviation in incomplete information model with endogenous service quality from the results obtained under standard one-sided model. The interpretation of this result is simple. We notice the product quality of each type of seller is an increasing function of service quality. Since when platform improves the service performance, the price per unit of product increases, thereby increasing the revenue of the seller. This will act as an incentive for seller for upgrading the product quality. Thus, when platform sets service quality lower under incomplete information model, it will demotivate the high type seller to upgrade the product quality compared to full information case. The optimum bundle for h type seller comprises less quality and less revenue fee paid to platform and for low type seller, the equilibrium bundle consists of less quality and more revenue payment than first-best bundle. To mitigate the cost of asymmetric information and to obtain the hidden information, the contract requires the h type seller to pay lower revenue share to platform and supply lower quality. By employing the identical parameter values: $β_{l} = 2$ , $β_{h} = 1$ , $ϕ = 0.5$ , $λ = 0.3$ , $γ = 1$ , $η = 2$ , $θ = 0.6$ , we compare the product quality levels supplied by high type and low type seller under two regimes respectively in Figures 3 and 4. It is evident that both types of seller provide superior quality products when there is no hidden information. The next proposition outlines the results obtained by comparing the two regimes under RS agreement.

Figure 3.

Note: For color usage in this figure, please refer to the web version of the article.

Figure 4.

Note: For color usage in this figure, please refer to the web version of the article.

Proposition 1: In information asymmetric (i.e., when the platform fails to identify a seller of type ‘i’) two state framework with one intermediary platform, the optimal type contingent equilibrium sets the product qualities for both type of sellers below the first-best amount (i.e., ${q_{h}}_{R}^{*} > {q_{h}}_{R}^{'}$ &; ${q_{l}}_{R}^{*} > {q_{l}}_{R}^{'}$ ) and service quality is settled lower than the first-best service quality level as well. Additionally, the seller gains a surplus in excess of the reservation profit if $β = β_{h}$ and no additional surplus if $β = β_{l}$ . Since the platform bears the cost of hidden information in the form of information rent, the expected second-best platform’s profit is strictly lesser than the first-best profit achieved when type being full observable.

A fundamental and most important point that arises from our study is that the optimal product quality for each type of seller and service quality of platform in the second-best framework are necessarily distorted from the full observable level.

Cost-Sharing (CS) Contract

We analyse a CS model where seller not only offers a portion of his revenue to platform but platform also shares a fraction of product quality cost of seller.⁹ The fraction of product quality cost that seller bears is indexed as θ while the platform shares the remaining fraction, $(1 - θ)$ of the cost. Following Ma et al. (2013), Tsao and Sheen (2012) we also assume that $θ$ is exogenously given.

When β Is Observable (First-Best Case)

Optimal Contracting Problem

Under full information, the platform knows the specific type of seller and thus it proposes the definite contract allotted to the specific type. The platform’s optimal contract problem is illustrated as,

\underset{α_{h}, α_{1}, s}{\underset{︸}{Max}} Π^{p} = Φ [α_{h} (1 - t) γ s q_{h}^{λ} - \frac{s^{2}}{2} - \frac{(1 - θ) β_{h} q_{h}^{2}}{2}] + (1 - ϕ) [α_{1} (1 - t) {γsq}_{1}^{λ} - \frac{η s^{2}}{2} - \frac{(1 - θ) β_{1} q_{1}^{2}}{2}]

(6)

{Subject to IR}_{h} : Π_{h}^{s} = (1 - α_{h}) (1 - t) γ {sqq}_{h}^{λ} - \frac{θ β_{h} q_{h}^{2}}{2} \geq 0

(7a)

{IR}_{1} : Π_{1}^{s} = (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{θ β_{1} q_{1}^{2}}{2} \geq 0

(7b)

Equation (6) describes the expected profit of the platform under complete information model of CS contract. As discussed earlier, the platform in this case not only receives α portion of revenue from seller but incurs $(1 - θ)$ fraction of product quality cost as well. We already know for any result $(α_{h}^{*}, q_{h}^{*}, α_{l}^{*}, q_{l}^{*} {, s}^{*})$ to the above contracting problem, IR constraints must bind. We obtain,

S_{C}^{*} = {[\frac{λ \frac{λ}{2 - λ} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ^{+ (1 - ϕ) η)}}]}^{\frac{(2 - λ)}{2 (1 - λ)}}, α_{h_{C}}^{*} = (1 - \frac{θ λ}{2}), α_{l_{C}}^{*} = (1 - \frac{θ λ}{2})

q_{IC}^{*} = {(\frac{(1 - t) γ λ}{β_{1}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}}

q_{h_{C}}^{*} = {(\frac{(1 - t) γ λ}{β_{h}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{ϕ {(\frac{1}{β_{h}})}^{\frac{λ}{2 - λ}} + (1 - ϕ) {(\frac{1}{β_{1}})}^{\frac{λ}{2 - λ}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}}

The equilibrium bundle for high type seller under full information symmetry case of CS agreement comprises higher product quality compared to the bundle for low type.

When β Is Not Observable by Platform (Second-Best Case)

Optimal Contracting Problem

In this subsection, we interpret the optimal contract for the case where the platform cannot exactly know the type of the seller. As explained earlier for the case of RS contract, the high type seller may imitate the low type as he has an incentive to choose the contract pair designated to the low type seller. Thus a contract should be designed in a way that compel seller to truthfully reveal his true type when information asymmetry is present. We begin our analysis by only considering constraints ${I R}_{l}$ and ${I C}_{h}$ and abandoning constraints ${I R}_{h}$ and ${I C}_{l}$ . Hence the platform’s problem can be expressed as,

\underset{α_{h}, α_{1}, s}{\underset{︸}{M a x}} Π^{p} = ϕ [α_{h} (1 - t) γ s {(q_{h} (α_{h}, s))}^{λ} - \frac{s^{2}}{2} - \frac{(1 - θ) β_{h} {(q_{h} (α_{h}, s))}^{2}}{2}] + (1 - ϕ) [α_{1} (1 - t) γ s {(q_{1} (α_{1}, s))}^{λ} - \frac{η s^{2}}{2} - \frac{(1 - θ) β_{1} {(q_{1} (α_{1}, s))}^{2}}{2}]

(8)

Subject to {IR}_{1} : Π_{1}^{s} = (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{θ β_{1} q_{1}^{2}}{2} \geq 0

(9a)

{IC}_{h} : (1 - α_{h}) (1 - t) γ {sqq}_{h}^{λ} - \frac{θ β_{h} q_{h}^{2}}{2} \geq (1 - α_{l}) (1 - t) γ {sqq}_{1}^{λ} - \frac{θ β_{h} q_{1}^{2}}{2}

(9b)

The optimal solution of the above optimization problem can be expressed as $(s_{C}^{'}, {α_{h}}_{C}^{'}, {α_{l}}_{C}^{'}, {q_{h}}_{c}^{'}$ , ${q_{l}}_{c}^{'})$ , where

s_{C}^{'} = {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{\frac{ϕ}{β_{h} \frac{λ}{2 - λ}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}

q_{h_{c}^{'}}^{'} = {(\frac{(1 - t) γ λ}{β_{h}})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{\frac{ϕ}{β_{h} \frac{λ}{2 - λ}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}}

q_{l}^{'} = {(\frac{(1 - ϕ) (1 - t) γ λ}{(θ ϕ (β_{l} - β_{h}) + (1 - ϕ) β_{l})})}^{\frac{1}{2 - λ}} {[\frac{λ^{\frac{λ}{2 - λ}} {((1 - t) γ)}^{\frac{2}{2 - λ}} \{\frac{ϕ}{β_{h} \frac{λ}{2 - λ}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{1}{2 (1 - λ)}}

α_{h_{c}^{'}} = 1 - \frac{θ λ}{2} - \frac{θ λ (β_{1} - β_{h}) β_{h^{- 1}}^{\frac{λ}{2 - λ}}}{2} {(\frac{1 - ϕ}{(θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})})}^{\frac{2}{2 - λ}} α_{h_{c}^{'}} = 1 - \frac{β_{1} λ θ (1 - ϕ)}{2 (θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})}

The equilibrium bundle for high type comprises higher product quality and lower RS fee relative to the bundle for low type. It can be easily verified that both the constraints ${I R}_{h}$ and ${I C}_{l}$ which are not considered in analysing the optimal contract, are fulfilled by the optimal values of contract variables.

We examine the contract values $(α_{i}, s)$ under CS contract and produce the effect of change of CS parameter, θ on service quality and product quality. Platform’s service quality and product quality of each seller fall unambiguously with the CS fraction of seller, θ $(\frac{\partial s_{C}^{'}}{\partial θ} < 0, \frac{\partial {q_{h}}_{c}^{'}}{\partial θ} < 0, \frac{\partial {q_{l}}_{c}^{'}}{\partial θ} < 0)$ . When the seller bears a greater share of cost of product quality (as θ rises), his/her level of product quality provision becomes lower and this in turn reduces the price per product and thus the revenue of the seller. Lower profit of seller generating through the higher θ adversely affects the platform’s level of service quality as well. The numerical analysis with $β_{l} = 2$ , $β_{h} = 1$ , $ϕ = 0.5$ , $λ = 0.3$ , $γ = 1$ , $η = 2$ , $t = 0.1, θ Є [0.01, 0.95]$ validates the findings of falling level of service quality and product quality with θ under CS asymmetric contract design. Figure 5 describes the effect of CS parameter on model variables under CS contract with information asymmetry.

Figure 5.

Note: For color usage in this figure, please refer to the web version of the article.

Comparing Two Regimes Under Cost-Sharing Contract

In this subsection, we present the comparison between the two regimes under CS contract and perform a numerical analysis with parameter values $β_{l} = 2$ , $β_{h} = 1$ , $ϕ = 0.5$ , $λ = 0.3$ , $γ = 1$ , $η = 2$ , $θ = 0.6$ . In the case of incomplete information, platform carries the cost of hidden information in the form of information rent which has to be paid to high type seller in order to reveal his/her true type. Thus, platform fails to achieve the first-best profit in presence of information asymmetry and second-best profit is strictly lower than the profit obtained with complete information. Figure 6 depicts the profit of the platform under two regimes. The straight lines AʼBʼ and CʼDʼ represent the profit line of platform for full information and incomplete information regime respectively. It is evident from the diagram that the platform obtains lower level of profit under incomplete information case, $Π_{P C}^{'}$ than under first-best case, $Π_{P C}^{*}$ .

Figure 6.

Note: For color usage in this figure, please refer to the web version of the article.

Since the platform has to forgo a part of its profit to acquire the hidden information regarding the true type of the seller under incomplete information case, it chooses to invest less on service quality and that’s why service quality with asymmetric information $(s_{C}^{'})$ is lower than the first-best quality $(s_{C}^{*})$ . Thus, $s_{C}^{*} > s_{C}^{'}$ . Figure 7 displays two curves, S_C^* and S_C^ʼ indicating the level of service quality under complete &; incomplete information respectively. It shows that platform provides higher level of service quality with no hidden information than in the case of incomplete information.

Figure 7.

Note: For color usage in this figure, please refer to the web version of the article.

A result identical with one-sided market has been derived in incomplete information model of online platform when service quality is exogenously specified (say, $s = \bar{s}$ ). In which case, the optimal level of product quality for each type of seller under second-best contract is set at, $q_{{hc}^{'} | s = \bar{s}} = {(\frac{(1 - t) γλ \bar{s}}{2_{h}})}^{\frac{1}{2 - λ}}$ $q_{1 c^{'} | |s = \bar{s}} = {(\frac{(1 - ϕ) (1 - t) γλ \bar{s}}{[, ϕ (β_{l} - β_{h}) + {(1 - ϕ)}_{1}^{2}]})}^{\frac{1}{2 - λ}}$ Evidently, ${q_{h}}_{c}^{'}_{|s = \bar{s}} = {(\frac{(1 - t) γ λ \bar{s}}{β_{h}})}^{\frac{1}{2 - λ}} = {q_{h}}_{C}^{*}_{|s = \bar{s}}$ $q_{1 c^{'} | s = \bar{s}} = {(\frac{(1 - ϕ) (1 - t) γλ \bar{s}}{[θ ϕ (β_{l} - β_{h}) + (1 - ϕ) β_{l}]})}^{\frac{1}{2 - λ}} < {(\frac{(1 - t) γλ \bar{s}}{β_{l}})}^{\frac{1}{2 - λ}}$ .

Thus, the high type seller is required to supply the optimum first-best level and low type seller supplies a lower level of product quality with incomplete information compared to the first-best regime when service quality is exogenously given. However, the outcomes modify when we allow quality of service to vary in our model. With the endogenous level of service quality, the optimum level of product quality for h type seller under information asymmetry is significantly lower than the first-best quality. So, we notice a significant departure under information asymmetry regime from one sided market outcome. The equilibrium contract-pair for h type seller contains lower product quality and lower revenue payment to platform than the first-best outcome and l type seller is required to provide smaller level of product quality and higher amount of revenue to platform in case of incomplete information compared to the first best outcome. Both the Figures 8 and 9 show that each type of seller provides higher level of product quality under full information regime than the information asymmetry case. Thus, ${q_{h}}_{C}^{*} > {q_{h}}_{C}^{'}$ &; ${q_{l}}_{C}^{*} > {q_{l}}_{C}^{'}$ . All the comparative results described above have been summarized in the next proposition.

Figure 8.

Note: For color usage in this figure, please refer to the web version of the article.

Figure 9.

Note: For color usage in this figure, please refer to the web version of the article.

Proposition 2: In case of CS agreement, optimal contract equilibrium in the presence of information asymmetry requires each type of seller to provide lower level of product quality than the full observable level (i.e., ${q_{h}}_{C}^{*} > {q_{h}}_{C}^{'}$ &; ${q_{l}}_{C}^{*} > {q_{l}}_{C}^{'}$ ) and the service quality of platform is also set below the first-best quality ( $s_{C}^{*} > s_{C}^{'}$ ). Both the product quality of each type of seller and service quality of platform in the presence of incomplete information downgrade from the first-best level. Platform realizes lower level of profit in the presence of the hidden information than with full information ( $Π_{P C}^{*} > Π_{P C}^{'}$ ).

Sensitivity Analysis: Revenue-Sharing and Cost-Sharing Contracts

Let us now find the effect of tax on service quality, product quality and profit of platform for both the RS and CS contracts. Proposition 3 outlines the results obtained using comparative static exercise.

Proposition 3

The product quality supplied by each type of seller deteriorates as ad valorem tax on product price rises under each information regime of both contracts.

The optimum service quality falls unambiguously as tax on product price increases.

The equilibrium level of profit of the platform decreases with tax imposed on product price.

The intuition behind Proposition 3 is pretty straightforward. As ad valorem tax increases, the seller who receives $(1 - t) P$ per product sold after imposition of tax, obtains lower level of revenue per product. Fall in revenue works as disincentive for upgrading product quality for either type of seller and thus he invests less on improving product quality.

With revenue reduces with tax rate, seller now transfers lower level of his/her revenue to platform. Thus platform spends less on improving its service performance in each regime. Moreover, lower revenue earned per product by seller with rise in tax leads to lower payment of revenue to platform which in turn reduces the profit of the platform. The result for each information regime for either form of contract is summarized in Table 1 (see Appendix B for Table 1).

Figures 1 and 6 present the graphical illustration of the relationship between tax and profit level of platform under two regimes of RS and CS contracts, respectively. Both figures support the theoretical finding of negative association between tax and profit of platform as profit lines fall with tax rate. Figures 2 and 7 display the negative relationship between tax rate and service quality of platform under two contract regimes. Both Figures 3 and 4 validate the negative association between tax and product quality of each seller type diagrammatically and Figures 8 and 9 establish that product qualities vary adversely with tax rate.

Comparison Between Revenue-Sharing and Cost-Sharing Contracts

We note that optimal product quality provided by each type of seller and service quality of platform are greater for CS contract compared to the RS agreement in presence of information asymmetry. Thus, ${q_{h}}_{C}^{'} > {q_{h}}_{R}^{'}$ , ${q_{l}}_{C}^{'} > {q_{l}}_{R}^{'}$ &; $s_{C}^{'} > s_{R}^{'}$ . Moreover, the platform attains higher profit in CS contract compared to RS under incomplete information model, $Π_{P C}^{'} > Π_{P R}^{'}$ . Here, we formulate our Proposition 4.

Proposition 4: CS contract pronounces each type of seller to provide higher product quality and platform to offer better services than RS agreement in presence of incomplete information.

Proposition 4 reveals that platform and each type of seller under RS contract actually underinvests on quality relative to CS contract in presence of information asymmetry. Moreover, by persuading the seller to accept CS contract, platform earns higher profit compared to RS contract. It may be possible that information asymmetry is mitigated to some extent in CS contract when platform not only shares revenue of seller but bears a part of product quality cost of seller as well. Platform can have a better grasp about the type of seller under the CS model as sharing the cost burden helps platform to realize the cost structure of each type of seller. The CS contract is more preferable as it induces the members of online platform to exert better performance. Due to the presence of CS element, seller enables to uplift its product quality provision. Better provision of product quality raises the price per product, which in turn increases the profit earned by both the platform and the seller. Intermediary provides improved services to its users as well under CS contract. Our model derives a crucial facet of contract design by comparing these two forms of contract.

Extension: Introduction of Advertising in the Model

Suppose the platform introduces advertising (defined as ‘a’) to expand the audience base and boost the sales volume. Advertisement helps platform to drive up sales in many ways by generating brand and product awareness among customers, informing them about promotional offers, daily offers, etc. We let, $γ = γ (a)$ where $γ^{'} (a) > 0$ . For sending advertising signal, platform incurs a cost and let that be equal to ‘a’ under both contracts. When the platform sends ‘a’ level of advertising to its users, the final form of platform’s profit is written under first-best case of RS structure as,

Π_{PR}^{*} = \frac{(1 - λ) {\{(1 - t) γ (a)\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{2 {(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{β_{1}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} - a

Differentiating the above profit equation with respect to advertising level, a and making that equal to zero, we attain,

\frac{{dΠ}_{PR}^{*}}{da} = \frac{(1 - λ) {\{(1 - t)\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{2 {(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{β_{1}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} \frac{2}{1 - λ} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) - 1 = 0

Or, \frac{{\{(1 - t)\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{β_{1}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) = 1

(10)

By solving the above first order condition, we get the advertising level under information symmetry case ( ${a^{*}}_{R})$ . The profit of platform under second-best case is as follows:

Π_{PR}^{'} = \frac{(1 - λ) {(1 - t) γ (a)}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{2 {(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h^{\frac{λ}{2 - λ}}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(β_{1} - ϕ β_{h})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} - a

From the FOC for profit maximization with respect to advertisement, a, we have,

\frac{{dΠ}_{PR}^{'}}{da} = \frac{(1 - λ) {(1 - t)}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{2 {(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(β_{l} - ϕ β_{h})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} \frac{2}{1 - λ} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) - 1 = 0

O r, \frac{{(1 - t)}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h} \frac{λ}{2 - λ}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(β_{1} - ϕ β_{h})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) = 1

(11)

By solving (11), we get the advertising level under information asymmetry case ( ${a^{'}}_{R})$ .

Comparing the first order conditions for profit maximization of platform under first-best case (Equation (10)) and second-best case (Equation (11)) under the RS model, we conclude that the advertising level under first-best case, ${a^{*}}_{R}$ is higher than the level obtained under second-best case, ${a^{'}}_{R}$ . The intuition is straightforward. When platform is absolutely sure about the seller’s type, it will send more advertising signal compared to when it is uncertain about the nature of the seller.

Similarly, first-order conditions for both symmetry and asymmetry cases under the CS model are as follows:

\frac{{(1 - t)}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h^{2}}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{β_{1}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) = 1

(12)

\frac{{(1 - t)}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{\frac{ϕ}{β_{h}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ (β_{1} - β_{h}) + (1 - ϕ) β_{1})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}} γ {(a)}^{\frac{1 + λ}{1 - λ}} γ^{'} (a) = 1

(13)

By comparing these two conditions (equations (12) and (13)), we infer that platform will deliver higher amount of advertising under information symmetry regime (since ${a^{*}}_{C} > {a^{'}}_{C}$ ) for CS contract. Moreover, the platform will provide higher level of advertising under CS contract compared to RS regime for incomplete information case as ${a^{'}}_{C} > {a^{'}}_{R}$ .

Conclusion

The existing literature pertaining to platforms has paid little attention to contracting relationship concerning the members of platform and impact of taxation under information asymmetry. The present study discusses the issue of contract design between an online platform and a seller when the willingness to pay per product is influenced by the platform’s service quality and quality of product supplied by seller. The model discusses the contracting problem for the case when the platform is not informed about the type of seller, which can be of either high type or low type based on unit cost of product quality. To analyse this issue of contracting, we explore two common forms of agreements: RS and CS. An interesting observation shows that at optimum, high type seller is required to provide less product quality under incomplete information compared to full information for both contracts when service quality is endogenously determined. Most importantly, with hidden information, platform achieves lower profit compared to the first-best profit as the platform gives up a portion of its profit in order to extract the hidden information about the type from the seller. The comparison between two contracts yields that CS contract secures more profit for platform by inducing every member of platform to supply greater quality relative to RS contract. The study then derives the effect of ad valorem tax on contract variables and obtains that product quality, service quality and platform’s profit fall unambiguously with increase of ad valorem tax rate.

We then introduce advertising in our model and find that the platform will invest more on advertising when it is fully informed about the type of the seller rather when it does not know the type. For the future analysis, we will examine the contract design for competitive platform setting and find how competitive setting changes the results found for monopoly platform.

Appendix

A. Proofs of Claim 1 and Claim 2

Proof of Claim 1: When the condition ${I C}_{h}$ binds then,

(1 - α_{h}) (1 - t) {γsq}_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} = (1 - α_{l}) (1 - t) {γsq}_{l}^{λ} - \frac{β_{h} q_{l}^{2}}{2}

(i)

(1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{h} q_{1}^{2}}{2} > (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{1} q_{1}^{2}}{2} Since β_{1} > β_{h}

(ii)

From (i) and (ii), we have, $(1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} > (1 - α_{l}) (1 - t) γ s q_{l}^{λ} - \frac{β_{l} q_{l}^{2}}{2}$

Thus, $(1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} > 0$ (Since if ${I R}_{l}$ binds: $(1 - α_{l}) γ s q_{l}^{λ} - \frac{β_{l} q_{l}^{2}}{2} = 0)$

Proof of Claim 2: When the condition ${I C}_{h}$ binds then,

(1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{h} q_{h}^{2}}{2} = (1 - α_{1}) (1 - t) γ s q_{1}^{λ} - \frac{β_{h} q_{1}^{2}}{2}

Or, $\frac{β_{h} (q_{h}^{2} - q_{l}^{2})}{2} = (1 - α_{h}) (1 - t) γ s q_{h}^{λ} - (1 - α_{l}) (1 - t) γ s q_{l}^{λ}$ (iii)

We know, $\frac{β_{l} (q_{h}^{2} - q_{l}^{2})}{2} > \frac{β_{h} (q_{h}^{2} - q_{l}^{2})}{2} = (1 - α_{h}) (1 - t) γ s q_{h}^{λ} - (1 - α_{l}) (1 - t) γ s q_{l}^{λ}$ (using (iii))

Or, $(1 - α_{l}) (1 - t) γ s q_{l}^{λ} - \frac{β_{l} q_{l}^{2}}{2} > (1 - α_{h}) (1 - t) γ s q_{h}^{λ} - \frac{β_{l} q_{h}^{2}}{2}$ (proved)

Table B1.

Summary of Results Under Sensitivity Analysis.

Contract Type Information Regime		Revenue Sharing Contract	Cost Sharing Contract
Information symmetry	$\frac{\partial s^{*}}{\partial t}$	$(-) \frac{{(1 - t)}^{\frac{λ}{1 - λ}}}{1 - λ} {[\frac{λ^{\frac{λ}{2 - λ}} γ^{\frac{2}{2 - λ}} \{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{{β_{l}}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}$	$(-) \frac{{(1 - t)}^{\frac{λ}{1 - λ}}}{1 - λ} {[\frac{λ^{\frac{λ}{2 - λ}} γ^{\frac{2}{2 - λ}} \{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{{β_{l}}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}$
	$\frac{{\partial q}_{h}}{\partial t}$	$(-) \frac{{s^{}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{h}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{}}^{- 1} \frac{d s^{*}}{d t}]$	$(-) \frac{{s^{}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{h}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{}}^{- 1} \frac{d s^{*}}{d t}]$
	$\frac{{\partial q}_{l}}{\partial t}$	$(-) \frac{{s^{}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{l}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{}}^{- 1} \frac{d s^{*}}{d t}]$	$(-) \frac{{s^{}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{l}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{}}^{- 1} \frac{d s^{*}}{d t}]$
	$\frac{\partial π_{p}}{\partial t}$	$(-) \frac{{\{γ\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}} {\{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{{β_{l}}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{(1 - t)\}}^{\frac{1 + λ}{1 - λ}}$	$(-) \frac{{\{γ\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}} {\{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{(1 - ϕ)}{{β_{l}}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{(1 - t)\}}^{\frac{1 + λ}{1 - λ}}$
Information symmetry	$\frac{\partial s^{'}}{\partial t}$	$(-) \frac{{(1 - t)}^{\frac{λ}{1 - λ}}}{1 - λ} {[\frac{λ^{\frac{λ}{2 - λ}} γ^{\frac{2}{2 - λ}} \{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(β_{l} - ϕ β_{h})}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}$	$(-) \frac{{(1 - t)}^{\frac{λ}{1 - λ}}}{1 - λ} {[\frac{λ^{\frac{λ}{2 - λ}} γ^{\frac{2}{2 - λ}} \{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ {(β}_{l} - β_{h}) + (1 - ϕ) β_{l})}^{\frac{λ}{2 - λ}}}\}}{(ϕ + (1 - ϕ) η)}]}^{\frac{(2 - λ)}{2 (1 - λ)}}$
	$\frac{{\partial q}_{h}}{\partial t}$	$(-) \frac{{s^{'}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{h}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{'}}^{- 1} \frac{d s^{'}}{d t}]$	$(-) \frac{{s^{'}}^{\frac{1}{2 - λ}}}{2 - λ} {(\frac{γ λ}{β_{h}})}^{\frac{1}{2 - λ}} [{(1 - t)}^{\frac{- (1 - λ)}{2 - λ}} - {(1 - t)}^{\frac{1}{2 - λ}} {s^{'}}^{- 1} \frac{d s^{'}}{d t}]$
	$\frac{{\partial q}_{l}}{\partial t}$	$(-) \frac{{(\frac{(1 - ϕ) γ λ s^{'} (1 - t)}{(β_{l} - ϕ β_{h})})}^{\frac{1}{2 - λ}}}{2 - λ} [{(1 - t)}^{- 1} - {s^{'}}^{- 1} \frac{d s^{'}}{d t}]$	$(-) \frac{{(\frac{(1 - ϕ) γ λ (1 - t) s^{'}}{(θ ϕ {(β}_{l} - β_{h}) + (1 - ϕ) β_{l})})}^{\frac{1}{2 - λ}}}{2 - λ} [{(1 - t)}^{- 1} - {s^{'}}^{\frac{- (1 - λ)}{2 - λ}} \frac{d s^{'}}{d t}]$
	$\frac{\partial π_{p}}{\partial t}$	$(-) \frac{{\{γ\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}} {\{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(β_{l} - ϕ β_{h})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{(1 - t)\}}^{\frac{1 + λ}{1 - λ}}$	$(-) \frac{{\{γ\}}^{\frac{2}{1 - λ}} λ^{\frac{λ}{1 - λ}} {\{\frac{ϕ}{{β_{h}}^{\frac{λ}{2 - λ}}} + \frac{{(1 - ϕ)}^{\frac{2}{2 - λ}}}{{(θ ϕ {(β}_{l} - β_{h}) + (1 - ϕ) β_{l})}^{\frac{λ}{2 - λ}}}\}}^{\frac{2 - λ}{(1 - λ)}}}{{(ϕ + (1 - ϕ) η)}^{\frac{1}{(1 - λ)}}} {\{(1 - t)\}}^{\frac{1 + λ}{1 - λ}}$

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Notes

ORCID iD

Tanmoyee Banerjee (Chatterjee)

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