Supplier Encroachment as an Enhancement or a Hindrance to Nonlinear Pricing

Authors


Abstract

The objective of this study was to extend existing understanding of supplier encroachment to contexts in which there is information asymmetry and the supplier can use nonlinear pricing. Prior research has shown that supplier encroachment can mitigate double marginalization and thus benefit both the supplier and the reseller. However, under symmetric information, this benefit disappears if the supplier can use nonlinear pricing. In our model, the reseller observes the true market size while the supplier knows only the prior distribution, that is, a seemingly ideal setting for implementing mechanism design through nonlinear pricing. We first show that, because encroachment capability enables the supplier to make an ex post output decision, it fundamentally alters the structure of the optimal nonlinear pricing policy. In addition to the usual downward distortion effect, where the reseller may purchase less than the efficient quantity, we also have the possibility for upward distortion. Thus, under asymmetric information and nonlinear pricing, supplier encroachment has two opposing effects. On one hand, the ability to shift sales to the direct channel allows the supplier to reduce information rents with less sacrifice of efficiency; but on the other hand, by introducing the possibility of her own opportunistic behavior, it can result in upward distortion of the quantities sold through the reselling channel, which is a new source of inefficiency. Depending upon the relative efficiency of the reselling channel and the demand distribution, either of these two effects may dominate and the supplier's ability to encroach may either benefit or hurt both the supplier and the reseller.

1 Introduction

In practice, firms may sell their products through both intermediaries (reselling channels) and their own direct channels. For instance, electronic product makers may sell their products through third-party retail stores as well as their own stores or websites (e.g., Apple, Sony, Microsoft); apparel and fashion accessory makers may sell their products through independent retailers as well as their own factory outlets (e.g., Coach, Nike, Adidas); airlines and hotels sell tickets and rooms through both travel agencies and their websites (e.g., American Airlines, Hilton). This is not just limited to the large branded companies. More small and local firms also start to use direct sales besides the traditional distribution channels for various products (Blank 2013, Reisinger 2012).

There are a number of reasons why a firm might introduce its own direct channel in addition to relying on resellers (a practice that is often referred to as “supplier encroachment” in the literature), including: obtaining more direct feedbacks from consumers, increasing market size, and obtaining an additional source of leverage with respect to resellers. Our focus is on the last one. In particular, a subtle benefit is revealed in the literature that supplier encroachment can mitigate double marginalization in the reselling channel and result in a “win–win” outcome even if the products sold through the two channels are perfect substitutes. In reaching such a conclusion, most studies assume that a wholesale price only contract (i.e., a linear contract) is adopted in the reselling channel and information is complete. However, this benefit will disappear if the supplier can use a nonlinear price scheme to coordinate the channel. Nonlinear pricing is not uncommon in practice. For instance, many firms offer quantity discounts to their customers. Nonlinear pricing has also been extensively studied in the economics, marketing, and operations literatures (e.g., Ha 2001, Jeuland and Shugan 1983, Spence 1977).

In this study, we show that, although a supplier that can use nonlinear pricing gains nothing from the ability to encroach under symmetric demand information, this is not necessarily the case when the reseller has private demand information. Resellers often have access to better demand information than do their suppliers. Not only do they have access to data on a wider range of products, they often possess greater capability in interpreting whatever data are available as a result of their business focus on market mediation. Moreover, these informational advantages may persist even when suppliers operate their own direct channels. While the supplier encroachment literature has explored various factors, the interplay of nonlinear pricing and asymmetric information has not been investigated.

In our model, the supplier's ability to encroach endows her with the option to sell through her own direct channel as well as through a reseller (or both). To focus on how this ability affects her leverage with respect to the reseller, we assume that the products sold through the direct and indirect channels are perfect substitutes. To represent the fact that production must occur before the market clears, we assume that the two firms compete in quantities rather than prices, that is, Cournot competition. In addition to having an informational advantage, we also assume that the reseller has an efficiency advantage that allows him to sell the product at a lower cost per unit than can the supplier. The sequence of events is as follows: The supplier acts as a Stackelberg leader by announcing a take-it-or-leave-it menu of quantity–price pairs (henceforth referred to as contracts) to the reseller. The reseller responds by choosing one of the price and quantity pairs from the menu. If the supplier has the infrastructure that allows her to have the option to encroach, then she determines her direct selling quantity. Finally, the market clears.

We first show that in the absence of information asymmetry, it is never beneficial for the supplier to possess the ability to encroach upon the reseller when she has the ability to use a nonlinear price scheme to contract with the reseller. With nonlinear pricing, the supplier can already capture the entire supply chain surplus for the first-best quantity while enjoying the efficiency of the reselling channel. Consequently, the only effect of her developing the ability to encroach is that it introduces the potential for her own opportunism, which interferes with her implementing the first-best solution. However, in the presence of information asymmetry, the supplier's ability to encroach is a double-edged sword for the supplier. Although the supplier's ability to encroach upon the reselling channel can reduce the efficiency loss that is required to reduce the information rents surrendered to the reseller, it also creates the possibility of her own opportunistic behavior, which will distort the reseller's order quantity. In contrast to the classical mechanism design problems where distortion occurs only in the less favorable states, in our context, the supplier's potential opportunism can result in distortion for even the most favorable state. Moreover, because the distortion of quantities has implications for both the magnitude of information rents as well as for the supply chain efficiency, the supplier and the reseller may either benefit from, or be hurt by, the supplier's ability to encroach, depending on the reseller's cost advantage in the selling process and the prior distribution of the market size. Specifically, we find that the supplier is always better off with the ability to encroach when the reseller's cost advantage in the selling process is small. When the reseller's cost advantage is intermediate, encroachment capability can be either beneficial or detrimental for the supplier, depending on the prior distribution of the market size. For the reseller, the supplier's ability to encroach always makes him worse off when his cost advantage is small and makes him better off when his cost advantage is intermediate. (When the reseller's cost advantage is sufficiently large, the supplier would never exercise her ability to encroach.) We reveal regions where the supplier's ability to encroach can lead to either “win–lose” or “lose–lose” outcomes for the two parties. These findings are robust to either a discretely or continuously distributed market size.

Hence, this study complements the existing literature on supplier encroachment. We demonstrate that in the presence of information asymmetry, supplier encroachment capability can be helpful even if the supplier can implement a nonlinear price scheme. Moreover, the effects of the supplier's encroachment capability on the two parties’ profits are not monotonous, and depend critically on the reseller's selling advantage and the information structure.

The remainder of our paper is organized as follows. Section 'Literature Review' reviews the related literature. In section 'The Model', we describe the model. We analyze a base case with perfect information in section 'Base Case with Perfect Information' and our model under asymmetric information in section 'Analysis with Asymmetric Information'. We present two extensions of our model in section 'Extensions' and conclude in section 'Conclusion and Discussion'.

2 Literature Review

Manufacturers selling to multiple channels have been widely observed in practice (Nair and Pleasance 2005). The findings in the academic literature on supplier encroachment are, however, divided. There are studies that show supplier encroachment reduces the incentive of the resellers to promote the manufacturers’ products and dilutes brand image (Fein and Anderson 1997, Frazier and Lassar 1996). However, there is also a stream of research that shows supplier encroachment can improve the system efficiency by alleviating double marginalization. Specifically, Chiang et al. (2003) demonstrate that a supplier's threat to launch and sell through her direct channel can lower the reseller's selling price, while Cattani et al. (2006) and Arya et al. (2007) demonstrate, based on price and quantity competition models that supplier encroachment can motivate the supplier to lower her wholesale price in the reselling channel. As shown by Tsay and Agrawal (2004), the result that the launch of a direct channel can mitigate double marginalization holds even in a context where the supplier and the reseller can exert sales efforts to promote the demand.

The above literature generally assumes that the supplier can use only a linear price scheme to contract with the reseller. As noted by Arya et al. (2007), if the supplier can alternatively use a nonlinear price scheme, then double marginalization would be completely resolved under the optimal supply contract, and encroachment could not provide strict gains for the manufacturer. While their observation is, of course, correct when both firms have symmetric market size information, we demonstrate that it may not hold when the reseller is endowed with private market size information. Specifically, under nonlinear pricing, encroachment generates a force that pushes the reseller's order quantities upward, and in general, we no longer observe the usual “efficiency at the top” that one normally expects for nonlinear pricing problems.

Related to our work, there exist a few studies that explore the incentive of information sharing with different supply chain structures under asymmetric information. For instance, Li (2002) and Zhang (2002) investigate information sharing in a setting where a central supplier sells to multiple competing resellers that have better demand information, while Ha and Tong (2008) and Ha et al. (2011) focus on a setting with two competing supply chains and explore the incentive of each reseller to share information with his supplier. Different from the above studies where the resellers have the same information advantage, Anand and Goyal (2009) and Kong et al. (2012) investigate a one supplier-two competing reseller setting where the incumbent reseller has better information than the other parties. Anand and Goyal (2009) show that the supplier's incentive to leak the information learned from the incumbent reseller to an entrant reseller may block information sharing in the supply chain, while Kong et al. (2012) analyze a revenue sharing scheme to resolve information leakage. Finally, Guo and Iyer (2010) and Guo et al. (2011) investigate the effect of strategic ex post information sharing in a vertical supply chain where a party, either the supplier or the reseller, is able to acquire advanced information.

To our best knowledge, Li et al. (2014) is the only study to examine the impact of information asymmetry in a setting in which the supplier has encroachment capability. In that study, the supplier is assumed to use linear pricing, and it is shown that, as a consequence of the signaling game that the reseller initiates in response to a linear wholesale price, the supplier's ability to encroach can either mitigate or amplify double marginalization in the presence of information asymmetry with a wholesale price only contract. Of course, when the supplier can implement nonlinear pricing, double marginalization is no longer a concern, and that is why our focus is on investigating how encroachment capability affects the information rents and the efficiency of the pricing menu offered by the supplier. Under nonlinear pricing, the supplier's development of encroachment capability can have two opposing effects: By allowing the supplier to sell through the direct channel, it allows the supplier to reduce information rents with less sacrifice of sales volume. However, because the supplier's ability to encroach creates the potential for her own opportunism, it can also result in upward distortion of the quantities sold through the reselling channel for the best realization of market size. This upward distortion is in contrast to the usual efficiency at the top that we expect in screening contracts, and it is also distinct from the downward distortion that we observe under linear wholesale pricing.

3 The Model

We consider a supplier (she) that can sell her product either through a reseller (he), her direct channel, or both. To focus attention on the coordinating role of supplier encroachment, we assume that the products sold through the two channels are perfect substitutes, and thus adding a direct channel would not affect the total market size. This eliminates the possibility that a direct channel would allow the product to reach a broader set of consumers, which tends to favor the use of the direct channel. Specifically, we assume that the total consumer demand follows a linear, downward sloping function, P = a − Q, where a represents the market size, Q is the total number of units of the product deployed for sale in the channels, and P is the market clearing price. To incorporate the notion of information asymmetry, we further assume that the market size a is, ex ante, random, which can be either large (a = aH) with probability λH = λ or small (a = aL) with probability λL = 1 − λ, where aH > aL > 0. Denote by λ = [λH,λL] the vector of these two probabilities for high and low demands. This simple, two-point distribution of demand facilitates the demonstration of our main results. However, to confirm that our qualitative results do not depend upon the two-point distribution, we extend our analysis to a setting with a continuously distributed a in section EC.2.1 of our online Appendix.

As in Arya et al. (2007), we assume that, because the supplier is less efficient in retail operations than the reseller, her per unit selling cost is c higher than that for the reseller. Such a premium can arise as a result of the supplier needing to pay higher transportation cost to ship items directly to consumers while the reseller can take advantage of bulk shipping to transport the items in bulk to a traditional retail location. To simplify the presentation of our results, we normalize the selling cost for the reseller to zero, and the selling cost of the supplier to c.

Finally, the supplier is the Stackelberg game leader who can provide a “take-it-or-leave-it” offer with a menu of contracts to the reseller. Without loss of generality, we assume the reseller's reservation profit is zero.

Figure 1 details the sequence of events in our model. First, the supplier designs a menu of contracts, math formula, where w(ai) is the per unit wholesale price and qR(ai) is the corresponding quantity in a contract. That is, if the reseller chooses one specific contract i, then he obtains qR(ai) units and pays the supplier w(ai) per unit. Second, the reseller observes a = aH or a = aL and chooses one contract from the offer and the contract is executed immediately. Third, based on the contract chosen by the reseller, the supplier then stocks quantity, qS(ai), of the product that she will sell through her direct channel. Lastly, the market clearing price P is realized according to P = ai − (qR(ai) + qS(ai)) for i ∈ {HL}, and the two parties obtain their final profits.

Figure 1.

The Timeline of the Model

Note that the assumption that the two firms determine their stocking quantities sequentially, that is, the reseller makes his ordering decision before the supplier determines her own stocking quantity, reflects the reality that the supplier typically has no means of making a credible commitment to not adjust her own stocking quantity in response to the order placed by the reseller. If the supplier does have such capability, then she can specify her own stocking quantity for each quantity–price pair in the menu of contracts that she offers to the reseller. Because this gives the supplier an additional degree of freedom in design of the menu of contracts without introducing the possibility of opportunism, it is intuitive that this would benefit the supplier and hurt the reseller.

4 Base Case with Perfect Information

In this section, we analyze a base case with perfect information. Specifically, the supplier and the reseller both know perfectly the realization of the market size, ai, i = HL. This analysis provides a useful benchmark with which to compare the results we will reveal later for asymmetric information.

We first derive the solution for the case where the supplier does not have the option to encroach upon the reselling channel. As the information is complete, the supplier will offer only one contract, math formula, corresponding to the realization of the market size, ai. If the reseller takes this contract, his sales revenue will be (ai − qR(ai))qR(ai), which is maximized at math formula. It is straightforward now that the optimal contract math formula is equal to math formula for either realization of the market size. Under such a contract, the reseller obtains zero profit, while the supplier captures the entire surplus, math formula. Notice that this contract achieves the largest possible surplus of the system.

Now, we derive the solution for the case where the supplier has the ability to encroach upon the reseller. We apply backward induction. After the reseller takes the contract, math formula, the supplier determines her direct selling quantity by solving

display math

which yields the optimal direct selling quantity

display math(1)

Notice that this quantity is a function of both the demand parameter, ai, and the quantity sold by the reseller, qR(ai). Of course, the reseller anticipates this direct selling quantity when he determines how to respond to the quantity–price pair offered by the supplier. Hence, when the supplier designs the contract, she must consider how her own subsequent incentive to encroach will affect the reseller's participation condition. Specifically, the supplier solves:

display math(2)

Let us denote by math formula and wPI(ai) the optimal solution to the supplier's optimization problem that is defined in Equation (2), and let math formula be the supplier's equilibrium direct selling quantity.

Proposition 1. With perfect information of the market size ai, i = HL, and the option of encroachment, the supplier's optimal contract offer and her direct selling quantity follow:

ScenarioswPI(ai) math formula math formula
math formula math formula 2c math formula
math formula c aic0
math formula math formula math formula 0

It is intuitive that when the supplier's selling cost is sufficiently large (math formula), the supplier will not use her direct channel, that is, encroachment is not a practical option for the supplier. It is straightforward that the optimal contract under such a scenario follows: math formula, which achieves the maximum surplus of the system. When the supplier's selling cost is intermediate or small, the supplier's incentive to encroach plays a role. Recall that when information is complete, the supplier always obtains the entire supply chain surplus under the optimal contract. However, in the presence of the ability to encroach, the supplier may fall victim to her own potential opportunism. That is, in anticipation of the supplier's ex-post encroachment, the reseller may be unwilling to take the efficient contract, math formula, to procure math formula and pay math formula, since doing so would lead to a negative profit for himself. To mitigate this effect, the supplier must either reduce the per unit wholesale price or increase the quantity that is targeted at a reseller who observes a given market size. Note that these two actions have different effects. Reducing the per unit wholesale price would compensate the reseller for the lower retail price that he will receive as a consequence of the supplier's direct sales, whereas increasing the quantity would reduce the supplier's ex-post incentive to sell through her direct channel. We can observe from Proposition 1 that the total output (math formula) is always greater than the first-best (efficient) quantity, math formula, for all math formula. That is, the maximum supply chain surplus is not achieved in the presence of the supplier's ability to encroach. Proposition 2 formalizes this finding.

Proposition 2. With perfect information of the market size ai, i = HL, the supplier (and also the supply chain) is never better off with the option of encroachment. In particular, the supplier is strictly worse off when math formula. The reseller is indifferent as he always obtains zero profit with or without supplier encroachment capability.

Because of the fact that, with complete information, the supplier can use nonlinear pricing to simultaneously achieve the first-best solution and to extract the full surplus from the reseller, encroachment does not add anything beneficial for the supplier or for the supply chain. In fact, because the ability to encroach creates the unavoidable possibility of supplier opportunism, it can only be detrimental. This result contrasts those revealed in the literature based on a linear price scheme where supplier encroachment can alleviate double marginalization and thus benefit the supplier and the supply chain.

5 Analysis with Asymmetric Information

In this section, we analyze our model with a binary distribution of the market size; that is, a = aH (aL) with probability λH (λL), ex ante (where λH = 1 − λL = λ). In the absence of supplier encroachment capability, it is reasonable to assume that the reseller has access to better demand information than does the supplier. To represent this, we assume that the reseller observes the true realization of market size, while the supplier knows only the prior distribution at the time that she proposes the pricing policy. When the supplier develops the ability to encroach, this will change the strategic interactions that she has with the reseller. In addition, it may also give her access to her own demand information. To disentangle these two effects, for most of our analysis, we assume that encroachment capability does not alter the information that is available to the supplier before she announces her price policy. However, in section EC.2.2 of our online Appendix, we confirm that our results are robust with respect to the possibility that, because encroachment capability puts the supplier in direct contact with end consumers, it also provides her with an independent signal about demand.

Let us begin by deriving the solution for the case where the supplier does not have the ability to encroach.

5.1 Without the Option of Encroachment

With asymmetric information, the supplier can implement nonlinear pricing through a menu of contracts, math formula, one targeting the large market size and the other targeting the small market size. Without encroachment, the supplier's problem can be formulated as follows:

display math(3)

The supplier designs the contracts to maximize her expected profit by satisfying the reseller's individual rationality (IR) and incentive compatibility (IC) constraints. From the revelation principle, in the optimal solution, the reseller will self-select the menu-option that corresponds to the true demand parameter, and we can represent the reseller's expected profit as follows:

display math(4)

where math formula and wN(ai) for i ∈ {LH} denote the optimal solution to Equation (3). We derive the following proposition from solving Equation (3).

Proposition 3. Without the option of encroachment, the optimal menu of contracts under asymmetric information follows:

ScenariosSmall market sizeLarge market size
wN(aL) math formula wN(aH) math formula
math formula math formula math formula math formula math formula
math formula 00 math formula math formula

Corollary 1. The above menu of contracts induces the first-best (efficient) quantity math formula to be sold when the realized market size is large, and it induces a less than the efficient quantity to be sold when the realized market size is small.

The two properties in the above corollary are quite standard in mechanism design settings (see, e.g., Moorthy 1984) and are often referred to as “efficiency at the top” and “downward distortion,” respectively. This can be observed from the fact that math formula and math formula for any λ.

Notice that in our problem, when math formula (or identically aL > λaH), the possible profit that the supplier can achieve from the small market size is significant and thus it is beneficial for the supplier to offer positive quantities for both of the two market sizes. However, to induce the reseller observing the large market size to choose the optimal order quantity math formula, the supplier has to downward distort the order quantity targeting the small market size by math formula from the efficient level math formula. The supplier can capture the entire supply chain surplus when the market size is small, but she has to surrender some information rents to the reseller when the market size is large. Specifically, the information rents (or the expected profit the reseller obtains) are math formula.

When math formula (or identically aL ≤ λaH), the information rents become sufficiently large so that the supplier prefers to avoid them by foregoing all sales when the market size is small. Consequently, the supplier's pricing policy induces a positive order quantity from only the reseller who observes the large market size. Note that, in this case, the information rents for the reseller are zero. Although the supplier extracts the full supply chain surplus conditional on the market size being large, neither firm earns anything when the market size is small.

5.2 With the Option of Encroachment

In many principal-agent settings with asymmetric information (such as the one analyzed in the above), it is possible to rely on the revelation principle in which there exists an optimal menu of contracts such that the principal learns the agent's true type from his choice of contract. However, in our setting with encroachment, because the supplier's ex-post output decision may depend upon the information that she obtains from the reseller's choice of contract, it is possible that the optimal contract will not induce each retailer to reveal his type. Consequently, we must consider two types of contract menus that the supplier can offer: a pooling menu, in which the reseller is offered a single quantity–price pair and accepts it regardless of the observed market size; and a separating menu, in which the reseller chooses a distinct quantity–price pair for each observed market size.

To formalize this, let math formula be the maximum profit that the supplier can earn under encroachment conditional upon her using a pooling menu that causes the reseller to select a single quantity–price pair regardless of the observed market size, and let math formula be the maximum profit that the supplier can earn under encroachment conditional upon her using a separating menu that causes the reseller to select a distinct quantity–price pair for each observed market size.

If the supplier offers a pooling menu, then she will not learn the market size from the reseller's response. Consequently, the supplier's output quantity will be as folllows:

display math(5)

and the conditionally optimal pooling contract can be identified as the solution to the following:

display math(6)

Alternatively, if the supplier offers a separating menu, then she will learn the true market size from the reseller's response, and her own optimal output quantity will be tailored to each market size, that is:

display math(7)

and the conditionally optimal separating menu can be identified as the solution to the following:

display math(8)

Note that, in both the pooling and separating menu design problems, the reseller's IR and IC constraints incorporate the supplier's direct selling quantities. However, because the reseller's type is revealed only in the separating menu, it is only there that the supplier can tailor his quantity to the realized market size. We derive the following proposition from solving Equations (6) and (8).

Proposition 4. With the option of encroachment, the optimal separating menu of contracts dominates the optimal pooling menu for the supplier, that is, math formula. The optimal separating menu of contracts contains:

(a) Small market size
 Scenariosmath formula
math formulamath formulamath formula
math formulaaL − c
math formulamath formula
math formulac  ∈  (0,∞)0
(b) Large market size
 Scenariosmath formula
 math formula2c
 math formulaaH − c
 math formulamath formula

with math formula and math formula. The supplier's direct selling quantity is math formula, i ∈ {HL}.

Under the above optimal separating menu of contracts, the reseller's expected profit is as follows:

display math(9)

Corollary 2. When the supplier has encroachment capability, the optimal nonlinear menu of contracts may no longer induce the reseller who observes the large market size to order the efficient quantity, math formula, that is, the optimal menu may lack the efficiency at the top property.

The above corollary highlights the fact that the reseller's willingness to pay for any given quantity is adversely affected by the supplier's own incentive to behave opportunistically after the reseller accepts the contract. Consequently, for the large market size, the supplier may no longer offer the efficient quantity math formula.

This result is driven by the fact that the supplier cannot precommit to her own output quantity, and her ex-post optimal quantity response is a function of the quantity that she sells to the reseller. In particular, it can be verified that math formula is the value of qR(aH) that maximizes the total supply chain profit conditional upon the supplier selling math formula through her direct channel. Thus, although math formula is conditionally efficient, it is not absolutely efficient, that is, it differs from the first-best solution. It can be confirmed that once we incorporate the functional form of the supplier's ex post optimal direct selling quantity response into the reseller's utility function, which forms the basis for the IR and IC constraints, we continue to have the single crossing property in which a reseller's preference for a larger quantity is increasing in the size of market that he observes. In addition, the supplier's objective function is separable and concave in the quantities offered. Because of this structure, the solution to the mechanism design problem defined in Equation (8) does have the efficiency at the top and downward distortion properties relative to this conditional optimization problem, but it may not have either of these properties relative to the first-best solution.

From Proposition 4, we can notice that when the supplier's selling cost is low (math formula), she sets math formula to less than the efficient quantity, and partially compensates by relying on her own direct channel.1 But when her selling cost increases to the range, math formula, she sets math formula above the efficient quantity to credibly commit to limiting her own subsequent sales through her direct channel. For the small market size, the supplier's choice of math formula still involves the trade-off between information rents (math formula) and wholesale revenue from the small market, but her ability to encroach gives her the ability to generate sales revenue from the small market without increasing information rents, especially when her selling cost is low. When math formula, similar to the case without the ability of encroachment, the supplier does not induce the reseller to sell anything to the small market. However, for math formula, different from the case without the option of encroachment, the supplier will induce the reseller to order a positive quantity for the small market only if the reseller's cost advantage is sufficiently large, that is, when math formula. In such a scenario, the value that the supplier can obtain by selling through the reselling channel under the small market size outweighs the information rents that will be introduced. Otherwise, the supplier will prefer to forego any sales through the reselling channel and rely entirely upon her direct channel, when the market size is small. By doing so, the supplier saves the information rents that would otherwise be surrendered to the reseller under the contract targeting the large market size while still capturing some amount of sales through her direct channel. That is, in contrast to the complete information setting, supplier encroachment can be helpful from the perspective of reducing the information rents paid to the reseller under asymmetric information.

Hence, supplier encroachment can have two distinct effects on the supplier's profit under a nonlinear price scheme with asymmetric information. On the one hand, it can help reduce the information rents surrendered to the reseller; on the other hand, it can worsen the ordering distortion in the reselling channel. Below, we characterize when having the option to encroach benefits or hurts the supplier (i.e., comparing the supplier's profit under the decisions given by Propositions 3 and 4). We divide the comparison into two cases with math formula and math formula since the structures of the contracts differ significantly.

Proposition 5. When math formula, if math formula, then there exists one threshold math formula such that encroachment capability makes the supplier strictly better off when math formula, weakly worse off when math formula, and has no effect when math formula. If math formulamath formula, then there exist two thresholds math formula and math formula such that encroachment capability makes the supplier strictly better off when math formula and math formula, weakly worse off when math formula, and has no effect when c ≥ aL.

When math formula, the supplier sells through the reseller only if the market size is large, regardless whether she has the option to encroach. It is apparent that, conditional upon the market size being small, the supplier can gain extra profit by having the option to sell directly. This gain, however, decreases as the supplier's direct selling cost (c) increases. On the other hand, conditional upon the market size being large, the supplier's ability to encroach can reduce her profit due to the ordering distortion in the reselling channel. Notice that this distortion loss is minimal when the supplier's direct selling cost is either small or large, while it can be significant when the supplier's selling cost is intermediate. As a result, it is intuitive that the supplier is better off by having the option to encroach when c is sufficiently small, and that there exists some intermediate range of c for which she is worse off. However, as c approaches math formula, she may be either better off or worse off. Specifically, when math formula, there exists a range, math formula, for which the supplier is also better off by having the ability to sell directly. In this range, the extra profit the supplier obtains by selling directly for the small market size outweighs the cost of upward distortion for the large market size. In contrast, when math formula, the supplier's direct selling quantity for the small market size would be zero when c is close to math formula and thus such a better-off region will not appear. Finally, for any math formula, the supplier will never sell directly as it is too costly. Consequently, she does not need to upward distort the reselling quantity for the large market size. Hence, the supplier is indifferent toward encroachment when math formula.

We depict a numerical demonstration of the results of Proposition 5 in Figure 2. In this experiment, math formula. Recall from Propositions 3 and 4 that this implies that, regardless of whether the supplier has encroachment capability, she sells nothing to the reseller when demand is low and she extracts the full surplus from the reseller, that is, math formula. In plot (a), we have math formula, while in plot (b), we have math formula. There is one threshold c in plot (a) that divides the regions where the supplier is better off and worse off by encroachment; and in plot (b), in addition to the pattern in plot (a), the supplier can also be better off when c is close to math formula.

Figure 2.

The Effects of Supplier Encroachment and Asymmetry Information on the Supply Chain Parties’ Profits when math formula. The Parameters are: λ = 0.8, (aL,aH) = (1,2.2) in the Left Plot and (aL,aH) = (1,1.4) in the Right Plot

The comparison becomes relatively more involved when math formula under which the supplier sells through the reseller for both market sizes. We derive the following proposition.

Proposition 6. When math formula, the effect of encroachment capability upon the supplier can be characterized according to three thresholds: math formula, such that encroachment capability makes the supplier strictly better off when math formula or math formula, weakly worse off when math formula or math formula, and has no effect when math formula.

First, notice that supplier encroachment makes the supplier better off when c is small (math formula), worse off when c is relatively large (close to but smaller than math formula), and indifferent when c exceeds math formula. In particular, when c is small, the direct channel is nearly as efficient as the reselling channel. In such a scenario, having the option to sell directly allows the supplier to reduce the quantity that she sells through the reseller for the small market size. Because the information rents are linearly increasing in qR(aL), this reduces the information rents available to the reseller while losing little in the selling process, which benefits the supplier. However, as the direct channel becomes less efficient, the direct channel can lead to upward distortion of the quantities for both the large and small market sizes. This means that the direct channel not only introduces inefficient upward distortion for the large market size, it also increases the information rents. In particular, there exists a region close to math formula, where the supplier is worse off by having the option to encroach. In this region, the direct channel is relatively inefficient and is used only sparingly. The costs associated with the increment of information rents and the upward distortion exceed the benefit from direct sales. Obviously, as c continues to increase, the supplier becomes less and less efficient relative to the reseller, and the potential impact of the direct channel eventually vanishes. Indeed, when c is greater than math formula, the supplier does not use the direct channel and is indifferent toward the option to encroach.

Second, it is interesting to notice that in the middle range of c, supplier encroachment can be first detrimental for the supplier (math formula) and then become beneficial (math formula), as c increases. Such a result can arise because both the benefit from reducing the information rents and gaining direct sales and the cost of upward distortion are not monotone in c. In fact, for the large market ize, as c increases from zero, the cost of upward distortion caused by encroachment first increases from zero to some large value and then gradually decreases to zero. On the other hand, for the small market size, as c increases from zero, supplier encroachment first reduces the information rents and, at the same time, gains the supplier some extra profit from the direct sales. However, as c increases, these benefits decrease, and when c reaches a critical level, upward distortion in the reselling quantity may appear, and this increases the information rents. The above effects make it possible that in the middle range of c, supplier encroachment can first reduce the supplier's profit and then increase her profit relative to what it would be without encroachment. Note that two or three of these thresholds may coincide with each other depending on the prior distribution of the market sizes.

We demonstrate the results of Proposition 6 numerically in Figure 3. We can observe from the left plot that when λ = 0.58, the three thresholds are indistinguishable; as c increases, encroachment capability first makes the supplier better off, then worse off, and eventually has no effect. In contrast, in the right plot where λ = 0.65, there are three distinct thresholds below math formula; as c increases, encroachment capability first makes the supplier better off, then worse off, then better off (again), then worse off (again), and eventually has no effect.

Figure 3.

The Effects of Supplier Encroachment and Asymmetry Information on the Supply Chain Parties’ Profits when math formula . The Parameters are: aL = 1 and aH = 1.4

While the above discussion is from the supplier's perspective, the effects of supplier encroachment on the reseller's profit can be easily understood through the impact that is upon information rents and quantity distortion. The comparison between the gain and the loss for the reseller is, in fact, much simpler.

Proposition 7. When math formula, the reseller always obtains zero profit with or without supplier encroachment. When math formula, there exists one threshold math formula such that the reseller is weakly worse off when math formula, strictly better off when math formula, and indifferent when math formula, with supplier encroachment compared to without.

When math formula, the supplier does not sell through the reseller for the small market size, with or without supplier encroachment, which yields zero information rent to the reseller. Hence, the reseller's profit does not depend upon whether the supplier has the option to encroach. When math formula, the supplier sells through the reseller when the market size is small, which yields positive information rents to the reseller. Note that the information rents increase linearly in the order quantity qR(aL) targeting the small market size. We can thus find a threshold math formula such that this ordering quantity with supplier encroachment is smaller than that without when math formula, while it is the reverse when math formula due to upward distortion. For any math formula, the supplier does not use the direct channel and thus the reseller is indifferent whether the supplier has or has not the option to encroach. The results of Proposition 7 are depicted in Figures 2 and 3.

To have a full comparison of the parties’ profits with and without supplier encroachment, we generate Figure 4 that shows the regions where the supplier and the reseller are better off, worse off, or indifferent by supplier encroachment with respect to c and λ.

Figure 4.

Illustration of the Regions where the Supplier and the Reseller are Better off, Worse off, and Indifferent by Supplier Encroachment. The Parameters are: aL = 1 and aH = 1.4

It is of interest to compare these results to those of Arya et al. (2007), who study supplier encroachment under linear wholesale pricing and symmetric demand information. Recall that they find that the supplier is weakly better off with encroachment capability, and that there exists an intermediate range of cost advantage for the reseller for which he too is better off. However, they acknowledge that the benefits of encroachment disappear when the supplier can implement nonlinear pricing. In contrast, we have demonstrated that, when the information is asymmetric, the ability to encroach can either enhance or hinder a supplier's nonlinear pricing policy, depending upon the distribution of the demand parameter and her cost disadvantage.

Finally, it is natural to question how supplier encroachment affects the total supply chain surplus in our context. While it is challenging to assess the effects analytically, we can clearly observe from Figures 2 and 3 that the total supply chain surplus can be either increased or reduced by supplier encroachment for different parameters. The underlying intuition is similar to what we have explained for the results from the perspectives of the supplier and the reseller.

6 Extensions

We now discuss two extensions to our original model. In the first, to test the robustness of our model, we allow for the possibility that the reseller can freely dispose of units that he orders, so that he does not necessarily sell everything if it is ex-post suboptimal to do so. In the second extension, we consider how the supplier's encroachment capability will affect the reseller's willingness to acquire private demand information.

6.1 Free Disposal by the Reseller

It is easy to notice that without supplier encroachment, the reseller will not withhold any units if he orders optimally. In the following, we discuss the case in the presence of encroachment where the supplier offers a separating menu of contracts. The timeline of the game will be: First, the supplier offers a menu of contracts math formula to the reseller who then selects one to order. Second, based on the reseller's contract choice, the supplier prepares a direct selling quantity qS(ai). Finally, the reseller and the supplier simultaneously determine the quantities to sell and dispose of any units they withhold.

A straightforward analysis of a simultaneous move Cournot competition can reveal that without any constraint, the optimal selling quantities of the reseller and the supplier should be math formula and math formula, respectively, for each market size. Certainly, in our context, the reseller cannot sell more than what he orders, that is, his selling quantity will be math formula. Therefore, comparing with the results in section 'With the Option of Encroachment', we can notice that the free disposal option will have an impact on the reseller's selling quantity when c is intermediate.

Proposition 8. With the reseller's option of free disposal, the optimal separating menu of contracts under asymmetric information and supplier encroachment contains the following:

Small market size
 Scenariosmath formula
 math formulamath formula
math formulamath formulamath formula
 math formulaaL − c
 math formulamath formula
math formulac  ∈  (0,∞)0
(b) Large market size
 Scenariosmath formula
 math formula2c
 math formulamath formula
 math formulamath formula

with math formula and math formula. The supplier's direct selling quantity is math formula, i ∈ {HL}.

Comparing with Proposition 4, we can verify that the supplier will be worse off when the reseller has the option of free disposal than without. The supplier now faces more constraints when optimizing the menu of contracts. Intuitively, free disposal limits the reseller's commitment power on his selling quantity, which enhances the supplier's ex-post encroaching incentive. From Figure 5, we can observe that when math formula, the supplier's profit with free disposal is significantly lower than that without. Note that the reseller is also (weakly) worse off by the option of free disposal as his order quantity for the small market size becomes smaller.

Figure 5.

Demonstration of the Impact of the Reseller's Free Disposal Option. In this Example, λ = 0.8, aH = 1.4, and aL = 1. The Dash-Dotted Curve: the Supplier's Profit without the Option of Encroachment; The Dashed Curve: the Supplier's Profit with the Option of Encroachment but without the Reseller's Option of Free Disposal; The Solid Curve: the Supplier's Profit with the Option of Encroachment and the Reseller's Option of Free Disposal

It is worth noting that, because the reseller incurs no cost for obtaining the product other than the wholesale price, upward distortion will not arise in our current model if the reseller has the option of free disposal and cannot commit to selling everything that he orders. However, if the reseller incurs positive handling costs, then upward distortion can still be present. For example, suppose the reseller has a traditional, bricks and mortar, operation so that he incurs a positive per unit logistics handling cost cR as soon as he orders and takes delivery, and the supplier sells through a direct online channel. Because the supplier's online channel requires her to ship units that she sells to individual consumers, her logistics costs are not only higher, that is, cS > cR, they are also incurred at the point of sale rather than at the time that the units are produced. As long as we assume that the reseller can commit to selling everything that he orders, that is, no free disposal, the reseller's (supplier's) cost can be normalized to zero (c = cS − cR), and this is exactly what we have done in the base model. However, when we allow for the reseller to have the free disposal option, his positive logistics cost will play an interesting role. Notice that with this logistics cost, the efficient quantity for the reseller to order and sell under complete information will become math formula. However, in the case with asymmetric information, at the second stage when the supplier and the reseller simultaneously choose their amounts to sell, this logistics cost is already sunk and will not play a role in the reseller's decision. As a result, if the reseller is not constrained by his order quantity, then he will sell math formula when the market size is large. Clearly, supplier encroachment can still induce the reseller to sell more than his efficient quantity if math formula, or equivalently, aH < 5cR + 2c.

6.2 Downstream Information Acquisition

In the above analysis, we have assumed that the reseller knows the true market size. Here, we investigate how downstream information acquisition affects the two parties’ profits in the presence of supplier encroachment. To do so, we assume that without acquiring any information, the reseller has the same information set as does the supplier (i.e., both of them use the prior distribution of the market size, Pr(a = aH) = λH and Pr(a = aL) = λL with λH = 1 − λL = λ, to make their contracting and stocking decisions). Further, we assume that information acquisition is costless.

When neither firm observes the market size before the reseller orders, the supplier will offer one contract to the reseller that depends only on the expected market size. Specifically, let (w(μ),qR(μ)) denote the contract where μ = λaH + (1 − λ)aL is the expected market size. Because the reseller's order quantity conveys no information about demand, after the reseller accepts the contract, the supplier's direct selling quantity will be:

display math

Consequently, the supplier's nonlinear pricing problem can be stated as:

display math

The solution follows the same format as that in Proposition 1 with ai replaced by μ.

Comparing the two parties’ profits with and without downstream information acquisition, we obtain the following proposition.

Proposition 9.

  1. For the reseller, when math formula, he is indifferent toward acquiring information (since he always obtains zero expected profit); when math formula, there exists a threshold math formula such that he is better off by acquiring information when math formula and indifferent when math formula.
  2. For the supplier, there exist math formula, math formula and math formulasuch that she is strictly better off by downstream information acquisition when math formula, or math formula and math formula.

For the reseller, without information acquisition, he will always obtain zero expected profit under the supplier's optimal nonlinear contract. In contrast, as revealed in Proposition 7, under some parameters, the reseller is able to obtain a positive expected profit when he has private information of the market size. Hence, the reseller is always (weakly) better off by acquiring the market information.

For the supplier, however, the reseller acquiring the market information can be a double-edged sword. On one hand, the supplier can screen out the market information and thus make more accurate contracting and direct selling decisions; on the other hand, the supplier will have to surrender some information rents to the reseller. Proposition 9(ii) provides two sufficient conditions for when downstream information acquisition benefits the supplier. In particular, when c is sufficiently small (math formula), the direct channel will be very efficient, which limits the information rents paid to the reseller. In such a scenario, downstream information acquisition always benefits the supplier. When c is sufficiently large (math formula) which prevents encroachment, downstream information acquisition benefits the supplier if the market size is very likely to be large (math formula). In such a scenario, the gain from accurately contracting outweighs the information rents paid to the reseller. When these conditions do not hold, downstream information acquisition may make the supplier strictly worse off. We demonstrate the possible outcomes in Figure 6.

Figure 6.

Illustration of the Regions where Downstream Information Acquisition Benefits or Hurts the Supplier and the Reseller in the Presence of Supplier Encroachment. The Parameters are: aL = 1 and aH = 2.5

7 Conclusion and Discussion

The main contribution of our paper is to identify the complex trade-offs that are involved when a supplier develops encroachment capability in contexts where resellers have private demand information and nonlinear pricing can be implemented. Although encroachment capability provides the supplier with a more refined mechanism for managing information rents, it also introduces the possibility of her own opportunism, which can lead to inefficient distortions in the quantities sold through the reselling channel. As a consequence of these complex interactions, it is possible for either/neither the supplier or/nor the reseller to benefit from encroachment.

In particular, we find that with some parameters, supplier encroachment can reduce the amount of efficiency that the supplier must sacrifice to reduce information rents received by the reseller. On the other hand, we also find that, because supplier encroachment capability allows the supplier to make an ex-post output decision, it may cause inefficient distortions in the quantities sold through the reselling channel. Specifically, the supplier's own potential to behave opportunistically in determining her own direct selling quantity can cause inefficient distortions in the quantities sold through the reselling channel.

From a practical perspective, our results clearly refute existing results that suggest that supplier encroachment would have no impact when a supplier can use nonlinear pricing. Indeed, we have shown that, if the supplier's direct selling channel is sufficiently efficient, then she can always benefit from developing encroachment capability, even if she is using nonlinear pricing. (For specific parameters, she may also benefit when her direct channel is at intermediate levels of efficiency.) Yet our results also highlight the dark-side of encroachment; there exists a moderate range of direct channel efficiency for which the supplier's ability to encroach renders both the supplier and the supply chain worse off.

In presenting our analysis, we have tried to simplify our model as much as possible to highlight the trade-offs that encroachment creates for the supplier between her enhanced ability to control information rents and the introduction of potential opportunism. However, in sections 'Free Disposal by the Reseller' and 'Downstream Information Acquisition', we consider two extensions of our base model. In the first, we confirm that our main results continue to hold, if the reseller can freely dispose of units that he acquires from the supplier when it is not optimal for him to release them all to the market. In the latter extension, we analyze how the supplier's encroachment capability affects the reseller's willingness to acquire information about the market size. Interestingly, although the reseller always at least weakly prefers to acquire information, the supplier may either benefit from or be hurt by downstream information acquisition.

Of course, when the supplier develops a direct channel to provide her with encroachment capability, her direct interactions with the market may provide her with a source of information that is independent from the reseller. In the online Appendix, we model this as a noisy signal of the true market size, and we show that the independent source of information has both a direct effect and an indirect effect. The direct effect is that the supplier can tailor her pricing menu according to the signal that she receives, and this helps to reduce information rents. In the extreme case where the signal is perfectly accurate, the information rents are eliminated. The indirect effect arises from the fact that, because the signal affects the price–quantity pairs that are offered to the reseller, it indirectly influences the supplier's direct selling quantity. Thus, the accuracy of the demand signal indirectly affects the supplier's direct selling quantity in spite of the fact that she is fully informed of the market size (via the reseller's order) at the time that she determines the quantity to sell directly. As a consequence of these two effects, the supplier may or may not benefit from the independent demand signal. More interesting is the observation that, for certain parameters (relatively high values of both c and λ), the reseller benefits from the supplier's development of encroachment capability only if it results in the supplier obtaining an independent source of demand information. Finally, our online Appendix also includes a demonstration that our qualitative results continue to hold when the market size follows a continuous (uniform) distribution.

Appendix

Proof of Proposition 1. For each market size ai, the reseller's participation constraint follows: w(ai) = ai − qR(ai) − qS(ai). We first assume math formula. Plugging w(ai) and math formula into Equation (2), we have the supplier's optimization problem as: math formula. The first-order condition yields the optimal unbounded reselling quantity math formula. Notice that in order for qS(ai) > 0, we need math formula, that is, the condition math formula. When this inequality does not hold, we have math formula, and the supplier's optimization problem is simply math formula, which yields math formula. Clearly, given math formula, in order for the supplier's direct selling quantity to be zero, we need math formula, that is, the condition math formula. For the rest of the parameter space math formula, the optimal reselling quantity follows the corner solution: math formula. With math formula, the optimal wholesale price and equilibrium direct selling quantity can easily be obtained.

Proof of Proposition 2. This result follows directly from Proposition 1. One can easily verify that when math formula, the total output (math formula) with supplier encroachment is larger than the efficient total output math formula. Hence, the total supply chain surplus with encroachment is lower than that without encroachment. The supplier's profit always equals the supply chain surplus as she can use nonlinear pricing to capture the entire supply chain surplus with perfect information.

Proof of Proposition 3. The classical mechanism design principle asserts that: there exists an optimal solution in which the two binding constraints are the reseller's individual rationality constraint for the small market size and his downward incentive comparability constraint. From these two binding constraints, we can obtain:

display math

Substituting these expressions for w(aL) and w(aH) into the objective function of Equation (3), we are left with an unconstrained objective function with only two variables, qR(aL) and qR(aH), and it is separable and concave. The result in Proposition 3 follows from applying the first-order conditions.

Proof of Proposition 4. In this proof, we solve the optimal separating menu of contracts. The comparison between the optimal separating menu of contracts and the optimal pooling contract is provided in the online appendix.

To solve the optimal separating menu of contracts, notice that once we incorporate the functional form of the supplier's ex-post optimal direct selling quantity response into the reseller's utility function, which forms the basis for the IR and IC constraints, we continue to have the single crossing property in which a reseller's preference for a larger quantity is increasing in the size of market that he observes. In addition, the supplier's objective function is separable and concave in the quantities offered. Therefore, this problem is a classical mechanism design problem. At optimum, the reseller's IR constraint for the small market size must be binding, that is, w(aL) = aL − qR(aL) − qS(aL). The reseller's IC constraint for the large market size must satisfy:

display math

Thus, the optimal wholesale price for the large market size satisfies:

display math

We can now substitute the above expressions into the supplier's objective function to obtain:

display math

Notice that the objective is separable and we can derive the optimal qR(aL) and qR(aH) separately. (i) We first optimize qR(aH). Suppose math formula is positive (i.e., when qR(aH) < aH − c). We can maximize the term:

display math

which has the first-order condition as:

display math

Therefore, the unbounded optimal quantity is:

display math

Notice that when math formula, math formula.

Now, suppose math formula is zero (i.e., when qR(aH) ≥ aH − c). We can maximize the term:

display math

The first-order condition yields the optimal quantity

display math

It is clear that when math formula, math formula.

Therefore, when math formula, the optimal quantity is math formula; when math formula, math formula; and when math formula, math formula.

(ii) We use a similar procedure to optimize qR(aL). Suppose math formula is positive (i.e., when qR(aL) < aL − c). We can maximize the term:

display math

The first-order condition is

display math

which yields the optimal quantity

display math

Notice that when math formula, math formula.

Now, suppose math formula is zero (i.e., when qR(aL) ≥ aL − c). We can maximize the term:

display math

The first-order condition yields the optimal quantity

display math

It is clear that when math formula, math formula.

Therefore, if math formula (i.e., aL > λaH), then, when math formula, the optimal quantity is math formula; when math formula, math formula; and when math formula, math formula.

In contrast, if math formula (i.e., aL ≤ λaH), then math formula. Therefore, when c ≥ aL, both the optimal directing selling and the optimal reselling quantities equal zero for the small market size. When c < aL, we can notice that math formula. Therefore, the optimal reselling quantity is zero. The other results follow immediately.

Proof of Proposition 5. Without encroachment, the optimal contract follows Proposition 3, based on which we can derive the supplier's expected profit as:

display math

When aL ≤ λaH, the profit reduces to math formula. With encroachment, the optimal solution of the supplier's problem follows Proposition 4 and the supplier's expected profit follows:

display math(A1)

When aL ≤ λaH, the above profit can be written as:

display math(A2)

When aL > c, we have the first derivative as:

display math(A3)

When aL ≤ c, we have the first derivative as:

display math(A4)

From the upper branch of Equation (A2), we can observe that when math formula, if aL ≤ c, then math formula, while if aL > c, then math formula and is decreasing in c. This observation implies that if math formula, then we always have math formula for any math formula. However, if math formula, then there exists an interval math formula in which math formula is larger than math formula but decreases to math formula as c approaches aL. Hence, we divide the analysis into two cases with (i) math formula and (ii) math formula, respectively.

(i) For the case in which math formula, we must show that math formula when c → 0, that math formula is first decreasing, and then increasing in c, and that math formula at math formula.

From the expression for math formula shown in Equation (A2), we can see that, when c → 0, we have math formula. In addition, we can see from Equation (A4) that math formula when math formula. Therefore, math formula is increasing in c for math formula and math formula when math formula.

It remains to be shown that math formula is decreasing and then increasing over the range math formula. From Equation (A3), we can see that for c sufficiently small, math formula. In addition, we can observe from both Equations (A4) and (A3) that math formula is increasing in c for all math formula.

For the range, math formula, we need to consider two possibilities: First, if math formula, then we have that math formula and math formula for all math formula. Alternatively, if math formula, then we have math formula in the range of math formula , while we have math formula for all math formula.

If 5λ ≤ 1, then math formula is nondecreasing in c for math formula. By assumption, we have aL ≤ λaH, which implies that math formula at the point math formula. It follows that math formula for all math formula.

If 5λ > 1, then math formula is decreasing in c for math formula. However, because math formula is continuous at c = aL for math formula, and math formula for math formula, it follows that we must have math formula for math formula as well.

Combining the analysis for the above two situations, we can conclude that there exists one threshold math formula such that math formula when math formula, math formula when math formula, and math formula when math formula. (ii) The analysis for the case with math formula is similar to the above, except that now we can have another threshold math formula such that math formula when math formula.

Proof of Proposition 6. First, it is clear from Proposition 4 that when math formula, the supplier never sells directly even if she has the option to encroach, which suggests math formula.

Second, we can show that the supplier's expected profit with encroachment is increasing in c when math formula. There can be two cases:

  1. If math formula (i.e., math formula), then when math formula, the supplier's expected profit (according to Equation (A1)) is:
    display math
    Taking the first derivative: math formula when math formula. Hence, math formula increases in c when math formula.
  2. If math formula (i.e., math formula), then when math formula, the supplier's expected profit is:
    display math
    which is obviously increasing in c when math formula.

Hence, math formula when math formula and math formula approaches math formula as c increases to math formula.

Third, when math formula, we can easily show that the supplier's expected profit math formula under encroachment is decreasing in c. Notice from Proposition 4 that math formula under encroachment when math formula . Thus,

display math

when math formula. By taking the first derivative of math formula, it can be confirmed that math formula decreases in c when math formula. Further, notice that when c goes to zero, math formula goes to math formula which is larger than math formula.

Combining the above results, we assert that there must exist a threshold math formula such that when math formula, math formula, and a threshold math formula such that when math formula, math formula. Furthermore, we can assert by comparing math formula and math formula according to the equilibrium solutions of Propositions 3 and 4 that there exists at most one more threshold math formula at which math formula, and these three thresholds may coincide with each other (the detailed comparison is long but mainly algebraic, which is thus omitted). Hence, the full comparison can be characterized by three thresholds: when math formula, math formula; when math formula, math formula; when math formula, math formula; when math formula, math formula; and when math formula, math formula.

Proof of Proposition 7. Without encroachment, the optimal contract follows Proposition 3, based on which we can derive the reseller's expected profit as:

display math

With encroachment, the optimal solution of the supplier's problem follows Proposition 4 and we can formulate the reseller's expected profit as:

display math

Proposition 4 reveals that if math formula, math formula for any c. Thus, the reseller is always indifferent with or without encroachment.

If math formula, math formula when math formula, so the reseller is indifferent with or without encroachment in this region. When math formula, math formula, which suggests that the reseller gains a larger profit under encroachment. When math formula, math formula decreases and approaches zero as c decreases. Hence, there exists a threshold math formula such that the reseller is worse off when math formula, better off when math formula, and indifferent when math formula with encroachment compared to without.

Proof of Proposition 8. An analysis of a simultaneous move Cournot competition can reveal that without any constraint, the optimal selling quantities of the reseller and the supplier should be math formula and math formula, respectively, for each market size. Certainly, in our context, the reseller cannot sell more than what he orders, that is, his selling quantity will be math formula.

As in the proof of Proposition 4, the reseller's individual rationality constraint for the small market size must be binding at optimum and thus we have w(aL) = aL − qR(aL) − qS(aL). The reseller's incentive compatibility constraint for the large market size must also bind at optimum and thus satisfy the following:

display math

Thus, the optimal wholesale price for the large market size satisfies the following:

display math

We can now substitute the above expressions into the supplier's objective function to obtain the following:

display math

Notice that in the proof of Proposition 4, without the constraint math formula, the optimal quantity math formula is an interior solution to an optimization with a concave objective function. Therefore, with the constraint math formula , the optimal quantity is math formula. In summary, the option of free disposal will lower the optimal reselling quantity to math formula for the large market size when math formula. For the small market size, the option of free disposal will lower the optimal reselling quantity to math formula when math formula.

Proof of Proposition 9.

  1. For the reseller, it is obvious that when math formula, his profit is always zero either with or without information acquisition, and thus he is indifferent. When math formula, without information acquisition, the reseller obtains zero profit; with information acquisition, he can obtain a positive profit when c is larger than math formula and zero otherwise. Therefore, there exists a threshold math formula such that the reseller is better off by information acquisition when math formula and indifferent when math formula.
  2. For the supplier, we first derive her profit in the case of no information acquisition. When math formula, we have:
    display math
    when math formula, we have: math formula; and when math formula, we have: math formula.

(ii-a) We show that there exists math formula such that math formula when math formula. Notice that when math formula (which implies that math formula and math formula), we can derive:

display math

Clearly, math formula if math formula. Hence, there exists a threshold math formula such that math formula when math formula.

(ii-b) We show that there exist math formula and math formula such that math formula when math formula and math formula. Notice that when math formula (which implies math formula), the supplier never sells through her direct channel either with or without downstream information acquisition. we compare math formula and math formula for math formula and math formula, respectively.

  1. When math formula, we have
    display math
    which is positive when math formula.
  2. When math formula, we have
    display math

Let math formula. We have

display math

As a result, U is convex, and U > 0 when math formula .

Combining cases (1) and (2), we confirm that there exist math formula and math formula such that math formula when math formula and math formula. It is worth noting that the above conditions are all sufficient conditions. To derive the sufficient and necessary conditions is technically challenging.

Note

  1. 1

    Our solution requires that c > 0. Notice that in the limiting case of c = 0 , the quantity for the reseller converges to math formula, which would preclude the supplier from obtaining the market information from the reseller's order.

Ancillary