Competitive multiproduct contracting under multinomial logit demand

Abstract

Despite the prevalence of multiproduct firms in many industries, the supply chain contracting literature has predominantly focused on problems where an upstream player offers only one product. This paper studies multiproduct contract design for competing manufacturers each with several products to sell via a common retailer. We consider two contracting schemes under multinomial logit demand (MNL): individual contracting, in which the manufacturers set a wholesale price for each of their products separately; and aggregate contracting, in which the manufacturers set the price as a function of the order quantities for their own products. We find that under individual contracting it is optimal for each manufacturer to offer the full range of his products to the retailer and set an equal wholesale margin for these products. This suggests that the classic result of an equal‐margin policy for the retailer also holds true for the manufacturers. Further, there is an equilibrium characterized by a system of nonlinear equations that can be solved using the standard bisection search method. Under aggregate contracting, a simple cost‐plus contract is optimal for the manufacturers. In equilibrium, each manufacturer charges a markup equal to his marginal contribution to the supply chain, and the supply chain profit is maximized. Comparing these two contracting schemes, we find among other results that the retailer prefers aggregate contracting when there is a large number of products from each manufacturer or when consumer heterogeneity is low, while the opposite is true for the manufacturers. In addition, both the retailer and the manufacturers prefer aggregate contracting when there are few manufacturers in the supply chain. It is also found that as the number of manufacturers or the number of products from each manufacturer increases, the supply chain efficiency loss induced by individual contracting is reduced. Finally, we extend our model to a hybrid setting in which one manufacturer adopts aggregate contracting and the other adopts individual contracting, and demonstrate to the manufacturers the value of aggregate contracting. These results provide useful guidance on optimal contract design for multiproduct firms.

INTRODUCTION

Multiproduct firms are prevalent in many industries. Even what seems to be the single product of a firm can differ in colors, styles and sizes. In this paper, we consider a supply chain in which multiple manufacturers each sell several products via a common retailer. Examples of this abound. A supermarket typically carries soft drinks from both Coca‐Cola and PepsiCo, each supplying multiple products in a category (e.g., canned drinks). Global brands such as Cadbury, Ferrero Rocher, and Nestlé all sell different chocolate products in a grocery store or supermarket. An electronic appliance retailer often carries a range of different refrigerators from each of the major manufacturers such as LG, Samsung, and Whirlpool. 

The prevalence of multiproduct firms notwithstanding, existing studies in supply chain contracting have predominantly focused on single‐product problems in which a manufacturer offers just one product to a retailer. That is, there is limited research on multiproduct contract design, despite there being a plethora of research in this area over the past two decades (Lariviere, 2016). This paper aims to fill this gap by studying multiproduct contract design in a supply chain that involves manufacturer competition.

With multiple products available from each manufacturer, a key question faced by the contracting parties is whether a separate price should be quoted for each product or an aggregate price for a combination of products. While contracting schemes may vary substantially in practice, it is useful to consider two dichotomous schemes depending on the specified contract terms. First, the manufacturers may charge the retailer an individual wholesale price for each product. Since the pricing of one product is independent of that of another, we refer to this scheme as individual contracting. This arrangement is essentially equivalent to applying a wholesale price or price‐only contract to each product (Lariviere & Porteus, 2001). This linear pricing policy has been widely applied in practice largely for its simplicity, for example, to yogurt and peanut butter (Sudhir, 2001), refrigerated orange juice (Dubé & Gupta, 2008), and breakfast cereal (Richards & Hamilton, 2015). Second, prices are sometimes quoted for a combination or bundle of different products. Since the pricing depends on the order quantities for more than one product, we refer to this scheme as aggregate contracting. This scheme is more complex than individual contracting and requires the manufacturers to adopt a nonlinear pricing policy. In practice, it comes in different forms such as a rebate applied on sales of an entire product line (Acciaro, 2010), a total quantity discount applied on the aggregate sales of multiple products (Crama et al., 2004), or package bids in the context of a procurement combinatorial auction (Kim et al., 2014).

In this paper, we consider a supply chain in which an arbitrary number of manufacturers (e.g., Coca‐Cola and PepsiCo) compete to supply their products to a single retailer (e.g., a grocery store or supermarket). Each manufacturer produces multiple products and must set prices accordingly. While both individual and aggregate contracting schemes have seen their applications in practice, we are not aware of any existing research that systematically studies these schemes. This paper sets out to develop a better understanding of both schemes and inform supply chain members of when best to use each of the schemes. The specific research questions we aim to address are as follows:

1.

What are the optimal contracting decisions for the manufacturers and the optimal pricing decision for the retailer under each scheme?

2.

How does each player's equilibrium profit depend on the market structure and product characteristics under each scheme?

3.

Does one contracting scheme always dominate the other from any player's perspective?

To answer these questions, we develop a Stackelberg model to study the manufacturers' contract design decisions and the retailer's pricing decision. In our model, the manufacturers each offer a multiproduct supply contract to the retailer, then the retailer determines the retail prices (or order quantities) of the offered products and finally sells the products to consumers. This sequence of events fits a situation where manufacturers with established brands and often substantial bargaining power move first by quoting a price for their products. For each contracting scheme, we characterize the equilibrium for the game and conduct sensitivity analysis on each player's profit with respect to key model parameters. Then we compare these schemes to study how each one performs from the supply chain perspective. We also examine each player's preferences between the contracting schemes and determine how the preferences are influenced by changing market conditions.

We use the multinomial logit demand model (MNL) to describe the purchase behaviors of consumers when they are presented with multiple product alternatives. MNL is widely used to model the demand and market share of products within a category. Starting from the seminal paper by Guadagni and Little (1983), there have been numerous empirical studies that use MNL to estimate the choices of consumers. MNL is also a workhorse model in revenue management used to examine firms' pricing, assortment, and inventory decisions (see e.g., Gallego & Topaloglu, 2019; van Ryzin, 2012). It is appealing to both academics and practitioners. To quote the commentary by Guadagni and Little (2008), “we have personally witnessed and/or participated in many applications of the [MNL] model in consumer packaged goods (CPG) firms.”

Given the model setup, we find under individual contracting that the equal‐margin policy where each manufacturer sets an identical wholesale price margin for all his own products is optimal. It has been established that a monopolistic retailer adopts an equal‐margin policy when setting retail prices under MNL (Anderson et al., 1992). Our findings suggest that this equal‐margin policy propagates through the supply chain to upstream manufacturers as well. ¹ A direct implication of this is that the total margin—the sum of retail and wholesale margins—remains identical for all the products from the same manufacturer. From a technical perspective, this property allows us to convert the multiproduct best response problem into a single‐product problem because what matters is the aggregate attractiveness of the offered products for each manufacturer. Since the market share increases as more products are offered, each manufacturer should offer all the available products to the retailer. We then show that there is a Nash equilibrium characterized by a system of nonlinear equations that can be solved by using the standard bisection search approach. The supply chain profit is not maximized in the equilibrium for individual contracting because of the positive wholesale margins.

Under aggregate contracting, a cost‐plus contract in which the manufacturers set their wholesale prices to cost and charge a fixed markup is optimal. The cost‐plus contract is simple and intuitive as only one contract parameter needs to be set by each manufacturer, making it easier to implement in practice. It is an extension of the two‐part tariff to a multiproduct setting where there is a fixed payment as well as a variable payment that is contingent on the aggregate order quantity of products available from a manufacturer. With this contract, the retailer's average purchase price decreases as her order quantity increases. Thus, it can also be seen as a variant of the total quantity discount contract where the discount is based on a purchase volume aggregated over several products (Crama et al., 2004). In equilibrium, the total supply chain profit is maximized, each manufacturer makes a profit equal to his marginal supply chain profit contribution, and the retailer receives the remaining profit, a result seen as a fair allocation of profits among supply chain members. We further show that this equilibrium is unique if the manufacturers are limited to offering a cost‐plus contract, which is indeed an optimal contract form.

We also compare these two schemes from the supply chain and each player's perspective. Qualitatively, aggregate contracting always delivers a higher supply chain profit, and we find that the efficiency loss for individual contracting decreases as the number of products or manufacturers increases in a symmetric setting with identical manufacturers. Using numerical studies, we also compare each player's profit between the two contracting schemes. A high‐level message is that no contracting scheme is dominant from any player's perspective. We further show that aggregate contracting is more beneficial to the retailer when there are more products available from each manufacturer or when consumer preferences are more homogeneous, while the opposite is true for the manufacturers. In addition, aggregate contracting is more beneficial to both the retailer and the manufacturers when there are fewer manufacturers.

Finally, we study a hybrid setting in which one manufacturer uses aggregate contracting and the other uses individual contracting. We fully characterize the equilibrium for the game and examine the impact of aggregate contracting on each player's equilibrium profit. By comparing the manufacturers' equilibrium profits in this hybrid setting with those in our baseline models, we find that each manufacturer prefers aggregate contracting regardless of the contracting strategy adopted by the other manufacturer. Therefore, in an extended game where the manufacturers are allowed to choose between aggregate contracting and individual contracting, both manufacturers will choose aggregate contracting in equilibrium. This, however, may result in a prisoner's dilemma under some parameter settings where both manufacturers can be better off switching to individual contracting.

This paper makes three main contributions. First, despite the prevalence of multiproduct firms in various industries, no prior research has studied competitive multiproduct contracting in a supply chain. The present paper fills this gap and advances our understanding of optimal contract design in a multiproduct context. Under both contracting schemes we fully characterize the equilibrium for the manufacturers and conduct a range of sensitivity analyses. Second, existing supply chain contracting models have predominantly used linear demand for tractability purposes. However, there is much empirical literature documenting the advantage of using discrete choice models to study consumer behavior in retail settings. Although discrete choice models (e.g., MNL) have been widely used in the marketing and revenue management literature, our paper is among the first to apply the MNL model to multiproduct contracting problems. Third, we find some results that to the best of our knowledge have not been reported in the literature, and based on these results, we develop some insights into the firms' contracting decisions. For example, under individual contracting the equal‐margin policy applies not only to the downstream retailer but also to the upstream manufacturers: An equal wholesale margin is applied to all the products from a single manufacturer. For aggregate contracting, cost‐plus contracts turn out to be optimal in our multiproduct setting, with the supply chain profit maximized in equilibrium. Comparing the two schemes, we identify the conditions under which respective players prefer one scheme over the other. The applicability of our model is further demonstrated using a practical example in the soft drink industry. These results provide useful guidance on optimal contract design for multiproduct firms.

The rest of the paper is organized as follows. Section 2 reviews the relevant literature, and Section 3 sets up the model and presents some preliminary results. We analyze the two contracting schemes and compare them in Section 4. Section 5 extends our model to a hybrid setting in which one manufacturer uses aggregate contracting and the other uses individual contracting. Finally, we conclude and discuss future research in Section 6. The online appendices include some additional technical results, a practical example in the soft drinks industry, and all the proofs.

LITERATURE REVIEW

This paper studies competitive multiproduct contract design in a supply chain. There are mainly two streams of literature related to this paper: (a) manufacturer competition within a supply chain, and (b) price and assortment competition.

Upstream competition within a supply chain or distribution channel has received wide attention in the operations and marketing literature. An early paper by Choi (1991) considers a channel structure with two competing manufacturers and a common retailer who sells both manufacturers' products. McGuire and Staelin (2008) consider two suppliers each selling a single product to their exclusive retailers. Parallel to this model, Choi (1996) examines a setting where each supplier can supply to both retailers. More relevant to contract design, Cachon and Kök (2010) consider a retail supply chain where two manufacturers compete to supply their products to a common retailer, with a focus on the effect of different contract forms on equilibrium outcomes. A notable feature of the above models is that they all consider linear demand. While linear demand facilitates the equilibrium analysis and often allows a close‐form solution, discrete choice models provide a more empirically sound means of studying consumer behaviors in retail contexts (Guadagni & Little, 1983, 2008). Using the MNL demand model, Heese and Martínez‐de Albéniz (2018) study a setting in which multiple manufacturers each sell a single product to a common retailer and offer a wholesale price contract to the retailer (i.e., the manufacturers use the individual contracting scheme). Nevertheless, their focus is on understanding the effect of the retailer's assortment breadth on manufacturer competition.

Notably, in the above literature, each manufacturer supplies only one product. We extend this literature by considering the case where each manufacturer supplies multiple products. This extension is not trivial since the existing (single‐product) literature is silent on the pricing of multiple products for a manufacturer. To address this question, we consider two contracting environments: individual contracting and aggregate contracting. We further compare the two schemes to develop insights into each player's contracting scheme preferences.

In addition, there is a vast literature on horizontal competition in terms of price, quantity, lead time, and service levels (Vives, 1999). This literature focuses on competition between firms in the same echelon engaging in horizontal competition. A number of papers in this literature have considered discrete choice models such as MNL and its variants (see, e.g., Besbes & Sauré, 2016; Gallego & Wang, 2014; Kök & Xu, 2011; Li & Huh, 2011). While this literature does consider multiproduct competition, the models essentially apply to a direct sales environment with no downstream player involvement. Thus, contract design is not a concern in this literature.

MODEL SETUP AND PREPARATORY RESULTS

We consider an arbitrary number of manufacturers (“he”) and a single retailer (“she”). Each manufacturer produces one or more products and may choose to supply some or all of his products to the retailer. All the products belong to the same category or family. For example, a supermarket carries different varieties of canned drinks from both Coca‐Cola and PepsiCo.

We denote the manufacturers by

$i=1,\text{…},n$

where

$n\ge 2$

. The products of manufacturer i are denoted by

$j=1,\text{…},n_i$

where

$n_i\ge 1$

indicates the total number of products available from manufacturer i. Let

$N=\{1,\text{…},n\}$

be the set of manufacturer indices and

$N_i=\{1,\text{…},n_i\}$

be the set of product indices for manufacturer i. For convenience, we refer to product j of manufacturer i as product

$ij$

. We collect all the products in the set

$\mathcal{N}=\{ij:i\in N,j\in N_i\}$

. If

$n_i=1$

for all

$i\in N$

, the model reduces to the single‐product problem in which every manufacturer produces only one product.

We model consumer choices under a random utility framework. The consumer utility derived from purchasing product

$ij$

is given by

$$U_i^j=u_i^j-r_i^j+{\xi }_i^j,$$

1where

$u_i^j$

is the expected utility,

$r_i^j$

is the retail price, and

${\xi }_i^j$

is the random utility following a certain distribution. Here, the expected utility of a product depends on the product functionality, size, brand, quality, etc. Since all the products belong to the same category, we use the MNL demand model to represent consumer purchase behavior, where

${\xi }_i^j$

follows a Gumbel distribution characterized by

$Pr({\xi }_i^j\le x)=e^{-e^{-x/\mu +\gamma }}$

with mean 0 and variance

${\mu }^2{\pi }^2/6$

, and

$\gamma \approx 0.5772$

is Euler's constant. Here,

${\xi }_i^j$

accounts for idiosyncratic taste differences of the consumer population (Anderson et al., 1992, p. 32). The parameter μ is a positive constant, with a higher value of μ indicating a larger variance of

${\xi }_i^j$

and hence a greater degree of heterogeneity in consumer preferences (van Ryzin & Mahajan, 1999). One may consider two limit cases according to the value of μ (Anderson et al., 1992, p. 42). First, when

$\mu \to 0$

, that is, the variance of

${\xi }_i^j$

tends to zero, all the information about preferences is contained in the deterministic utility

$u_i^j-r_i^j$

. Consequently, MNL reduces to the deterministic utility model in which the consumers simply compare the deterministic utilities of all products and choose the product with the largest utility. Second, when

$\mu \to \infty$

, that is, the variance of

${\xi }_i^j$

tends to infinity; the deterministic utility

$u_i^j-r_i^j$

contains no information about consumer preferences. As a result, all products are chosen with an equal probability regardless of the product prices. Finally, we remark that the zero‐mean assumption is without loss of generality, and the Gumbel distribution is mainly used for tractability since it leads to a closed‐form solution for product demand. In addition, the MNL model with a Gumbel distribution has been commonly used to capture consumer purchase behaviors in both empirical studies (see e.g., Abdallah & Vulcano, 2021; Guadagni & Little, 1983) and analytical studies (see e.g., Besbes & Sauré, 2016; Heese & Martínez‐de Albéniz, 2018; van Ryzin & Mahajan, 1999).

For convenience, we define

$$v_i^j=e^{\frac{u_i^j-r_i^j}{\mu }},$$

2which measures the net attractiveness of product

$ij$

after accounting for the product retail price. Additionally, let

$v_0>0$

be the attractiveness of a dummy product representing the no‐purchase option, which captures the possibility of a consumer purchasing none of the available products. Without loss of generality we set

$v_0=1$

.

Consumers maximize their expected utility by choosing one of the products on offer or the dummy product whichever gives the highest expected utility. This utility maximization leads to the purchase probability for each product, which can be written in closed form thanks to the Gumbel distribution for

${\xi }_i^j$

. Specifically, we collect the products offered to consumers in the set

$\mathcal{S}=\{ij:i\in S\subseteq N,j\in S_i\subseteq N_i\}$

. Then the purchase probability for product

$ij$

is given by

$$d_i^j=\frac{v_i^j}{1+{\sum }_{i\in S}{\sum }_{j\in S_i}{v}_i^j},$$

3where

$v_i^j$

is defined in Equation (2). This measure is also referred to as market share in marketing and revenue management. An implicit assumption of the MNL model, as can be seen from Equation (3), is that when a product is not available for sale, the demand for any other product will increase. Different substitution effects can exist between any two different pairs of products, and the overall substitution structure depends on the attractiveness values of all available products. In turn, these attractiveness values depend on product prices, utility values and parameter μ. Without loss of generality, we consider the consumer population to be one. The demand for each product is thus equal to the purchase probability for that product.

Different products may incur different production costs. Let

$c_i^j$

be the unit cost for product

$ij$

. Similar to

$v_i^j$

, we define

$${\overline{v}}_i^j=e^{\frac{u_i^j-c_i^j}{\mu }-1}$$

4which measures the gross attractiveness of product

$ij$

(before accounting for any margins imposed by the retailer or the manufacturer). Further, we denote by

${\overline{v}}_i={\sum }_{j\in N_i}{\overline{v}}_i^j$

the aggregate gross attractiveness of all the products from manufacturer i.

This paper is concerned about the manufacturers' competitive contract design decisions and the retailer's pricing decision. As in competitive contracting models (see e.g., Cachon & Kök, 2010) and for the purpose of tractability, we do not consider any one‐off product carrying costs for the retailer. This is an appropriate assumption for retailers who typically carry a large range of products, all sharing a one‐off administrative cost. Thus, the marginal administrative cost of carrying any additional product is negligible compared to its purchase cost.

Product demand depends on retail prices, which in turn are influenced by the supply contracts offered by the manufacturers. While contracting schemes may vary substantially in practice, we consider two dichotomous schemes in our model: (a) individual contracting, in which the manufacturers charge a wholesale price for each of their products, and then the retailer sets her retail price which determines the order quantity for the product; and (b) aggregate contracting, in which each manufacturer specifies a payment function for a combination of his products, and then the retailer determines how many of each product to purchase from the manufacturers. The second scheme provides more flexibility to the manufacturers, allowing them to aggregate different products and specify the prices for multiproduct bundles. The model we have described is a Stackelberg game in which the manufacturers are leaders and the retailer is a follower. The manufacturers play a simultaneous Nash game, with every player aiming to maximize his or her own profit.

A supply chain perspective

Prior to analyzing the decentralized supply chain, this subsection examines the centralized problem where the manufacturers and the retailer are integrated into a single entity. The results in this subsection provide a basis for subsequent equilibrium analyses. In what follows, we take a two‐step approach, first examining the optimal pricing problem for a fixed set of products and then solving for the optimal set of products by anticipating the optimal prices for any given product set.

Suppose the products in the set

$\mathcal{S}\subseteq \mathcal{N}$

are on offer. The price optimization problem from the supply chain's perspective is formulated as follows:

$$\underset{\{r_i^j:i\in S,j\in S_i\}}{\max }\mathrm{\Pi }=\sum _{i\in S}\sum _{j\in S_i}\left(r_i^j-c_i^j\right)d_i^j,$$

5where

$d_i^j$

is the demand of product

$ij$

as given in Equation (3) and Π is the total profit aggregated over all the products in

$\mathcal{S}$

. Solving this multivariate optimization problem, we obtain the optimal retail prices in the following lemma. Lemma 1

Given the set of products

$\mathcal{S}$

, the optimal retail prices

$\{{\overline{r}}_i^j:ij\in \mathcal{S}\}$

are characterized by the following conditions:

$${\overline{r}}_i^j-c_i^j=\mu +\sum _{i\in S}\sum _{j\in S_i}\left({\overline{r}}_i^j-c_i^j\right){\overline{d}}_i^j\equiv \mu +\overline{\mathrm{\Pi }},\forall ij\in \mathcal{S},$$

6where

${\overline{d}}_i^j$

is the demand for product

$ij$

and

$\overline{\mathrm{\Pi }}$

is the supply chain profit, both evaluated at the optimal retail prices.

Lemma 1 shows that it is optimal to set the same retail price margin for all the products on offer because the margins

${\overline{r}}_i^j-c_i^j=\mu +\overline{\mathrm{\Pi }}$

are independent of

$ij$

. We remark that this equal‐margin policy is a well‐established result in the literature (Anderson et al., 1992), and we repeat it here for completeness. Substituting Equation (6) into Equation (5) and rearranging terms yields

$$\frac{\overline{\mathrm{\Pi }}}{\mu }{e}^{\frac{\overline{\mathrm{\Pi }}}{\mu }}=\sum _{i\in S}\sum _{j\in S_i}{e}^{\frac{u_i^j-c_i^j}{\mu }-1}\equiv \sum _{i\in S}\sum _{j\in S_i}{\overline{v}}_i^j.$$

7Thus, we can write the optimal supply chain profit by using the Lambert's W function (Corless et al., 1996):

$$\overline{\mathrm{\Pi }}=\mu W\left(\sum _{i\in S}\sum _{j\in S_i}{\overline{v}}_i^j\right).$$

8

The above optimal supply chain profit can be seen as a set function of either the products or the manufacturers. Since the Lambert's W function is concave and increasing, we can show that

$\overline{\mathrm{\Pi }}$

is submodular in the product set

$\mathcal{S}$

as well as in the manufacturer set S. Proposition 1

The optimal supply chain profit

$\overline{\mathrm{\Pi }}(\mathcal{S})$

, as a set function of the products, is submodular and monotonically increasing in

$\mathcal{S}$

with

$\mathcal{S}\subseteq \mathcal{N}$

. Additionally,

$\overline{\mathrm{\Pi }}(S)$

, as a set function of the manufacturers, is submodular and monotonically increasing in S with

$S\subseteq N$

.

Submodularity exhibits the property of diminishing returns to the extent that the marginal contribution or value of a product decreases when the existing set of products expands. Similarly, as the existing set of manufacturers increases, the additional supply chain profit contributed by the introduction of a new manufacturer diminishes.

We now examine the optimal product selection decision. This decision turns out to be quite straightforward. Since the optimal profit

$\overline{\mathrm{\Pi }}$

increases as the product set expands (see Proposition 1), it is optimal to include all the available products. It is noted in our model that there is no maximum limit on the number of products the retailer can carry. This is a less restrictive assumption given that a majority of retailers nowadays have an online presence and are not subject to shelf space constraints. It also allows us to analytically characterize the equilibrium for both contracting schemes.

ANALYSIS

In this section, we analyze the two contracting schemes for the decentralized supply chain. Following backward induction, we first examine the retailer's pricing optimization given the manufacturers' contracts and then analyze each manufacturer's best response for determining the optimal contract. After characterizing the equilibrium for the manufacturers, we conduct sensitivity analysis with respect to key parameters. We also compare the equilibrium outcomes between the two schemes.

Individual contracting

Under individual contracting, each manufacturer offers a wholesale price for each individual product. We denote by

$w_i^j$

the wholesale price of product

$ij$

so manufacturer i needs to determine a set of wholesale prices

$\{w_i^j:j\in N_i\}$

.

It is helpful to consider a general form of the retailer's optimization problem for any given subset of products

$\mathcal{T}:=\{T_i:i\in N\}$

; that is, each manufacturer i offers products in

$T_i\subseteq N_i$

to the retailer at finite wholesale prices. ² Given the wholesale prices

$\{w_i^j:ij\in \mathcal{T}\}$

, the retailer's price optimization problem is formulated as follows:

$$\underset{\{r_i^j:i\in N,j\in T_i\}}{\max }{\pi }_R=\sum _{i\in N}\sum _{j\in T_i}\left(r_i^j-w_i^j\right)d_i^j,$$

9where

$d_i^j$

is given in Equation (3) associated with product set

$\mathcal{T}$

. The above problem is analogous to the supply chain problem in Section 3.1 except that product costs

$c_i^j$

are replaced by wholesale prices

$w_i^j$

, so we obtain a similar result to Lemma 1:

Given the wholesale prices

$\{w_i^j:ij\in \mathcal{T}\}$

, the optimal retail price for product

$ij$

, denoted by

${\widehat{r}}_i^j$

, is characterized by the following conditions:

$${\widehat{r}}_i^j-w_i^j=\mu +{\widehat{\pi }}_R,\forall ij\in \mathcal{T},$$

10where

${\widehat{\pi }}_R$

is the retailer's profit corresponding to the optimal retail prices. Substituting Equation (10) into the retailer's profit function in Equation (9), we obtain that the retailer's optimal profit is the unique solution to the following equation:

$$\frac{{\widehat{\pi }}_R}{\mu }{e}^{\frac{{\widehat{\pi }}_R}{\mu }}=\sum _{i\in N}\sum _{j\in T_i}{e}^{\frac{u_i^j-w_i^j}{\mu }-1}.$$

11Once the optimal value of

${\widehat{\pi }}_R$

is obtained, we can work out the demand for each product

$ij$

given the optimal retail prices:

$${\widehat{d}}_i^j=\frac{e^{\frac{u_i^j-w_i^j}{\mu }-1}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},\forall ij\in \mathcal{T},$$

12and the total market share is given by

$$\widehat{d}=\sum _{i\in N}\sum _{j\in T_i}{\widehat{d}}_i^j=\frac{{\sum }_{i\in N}{\sum }_{j\in T_i}{e}^{\frac{u_i^j-w_i^j}{\mu }-1}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)}=\frac{\frac{{\widehat{\pi }}_R}{\mu }}{1+\frac{{\widehat{\pi }}_R}{\mu }},$$

13where the second equality follows from Equation (11).

Having examined the retailer's problem, we now consider each manufacturer i's best response in determining the wholesale prices of his products. As discussed earlier, the best response problem should account for those cases where the manufacturer is only offering a subset of his products to the retailer. While from a modeling standpoint, this can be achieved by setting

$w_i^j=\infty$

for any product

$ij$

that he wishes to withhold (so that the demand for that product is always zero), we take a more convenient, two‐step approach. We first examine the optimal wholesale prices for any given set of products manufacturer i wishes to offer, and then study the optimal set of products for that manufacturer.

Suppose manufacturer i offers products in

$T_i\subseteq N_i$

. Given the contracts offered by his competitors,

$\{w_k^j:k\ne i,j\in T_k\subseteq N_k\}$

, manufacturer i's problem is to maximize his profit by choosing the optimal wholesale prices:

$$\begin{array}{c}\hfill \underset{\{w_i^j:j\in T_i\}}{\max }{\pi }_i=\sum _{j\in T_i}\left(w_i^j-c_i^j\right){\widehat{d}}_i^j=\sum _{j\in T_i}\left(w_i^j-c_i^j\right)\frac{e^{\frac{u_i^j-w_i^j}{\mu }-1}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},\end{array}$$

14

where we have substituted Equation (12) to obtain the second equality and

${\widehat{\pi }}_R$

is the retailer's optimal profit as implicitly determined by Equation (11). Solving this problem could be challenging since the objective function is not jointly concave in the wholesale prices. However, our next result shows that it is optimal to charge an identical margin for all the products offered by manufacturer i. Proposition 2

Suppose manufacturer i offers the products in

$T_i\subseteq N_i$

. Then it is optimal for manufacturer i to set an equal wholesale price margin for all his offered products, that is,

$w_i^j-c_i^j=m_i$

for

$j\in T_i$

where

$m_i$

denotes the wholesale price margin.

Proposition 2 shows that an identical wholesale price margin should be applied to all the products offered by the manufacturer. This mirrors the result for the retailer's pricing optimization problem in which the same retail price margin is applied to all the products carried by the retailer, as originally shown by Anderson et al. (1992) and also given in Equation (10). Proposition 2 extends this result to our supply chain setting, suggesting that the equal‐margin property holds not only at the retailer level but also at the manufacturer level.

From a technical standpoint, this proposition allows us to transform the original multidimensional optimization problem in Equation (14) into a single‐dimensional problem in which every manufacturer sets only one single margin for all the products he offers. While this transformation simplifies the analysis, we still need to examine whether each manufacturer should offer the full set or just a subset of his products. Such a problem becomes trivial in the single‐product setting studied by Heese and Martínez‐de Albéniz (2018) because offering the product is a (weakly) dominant strategy for each manufacturer. That said, we obtain a similar result in our multiproduct setting, as shown in the following proposition. Proposition 3

It is optimal for each manufacturer to offer the full range of his products to the retailer.

The above proposition shows that every manufacturer should offer all his products to the retailers. The intuition of this result is that, since the wholesale margin is the same for all the offered products and offering more products increases the market share, each manufacturer is better off offering the full range of his products to maximize his profit. Based on Propositions 2 and 3, we can transform the original multiproduct problem into an essentially single‐product problem. Manufacturer i's best response problem can then be formulated as follows (after setting

$w_i^j-c_i^j=m_i$

for all

$j\in N_i$

):

$$\underset{m_i\ge 0}{\max }{\pi }_i=m_i\frac{{\overline{v}}_i{e}^{-\frac{m_i}{\mu }}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},$$

15where

${\overline{v}}_i={\sum }_{j\in N_i}{\overline{v}}_i^j$

as defined earlier represents the aggregate gross attractiveness of manufacturer i's products, and

${\widehat{\pi }}_R$

is characterized by

$$\frac{{\widehat{\pi }}_R}{\mu }{e}^{\frac{{\widehat{\pi }}_R}{\mu }}=\sum _{i\in N}{\overline{v}}_i{e}^{-\frac{m_i}{\mu }},$$

16where we have substituted

$w_i^j-c_i^j=m_i$

into the right hand side of Equation (11) for each

$j\in N_i$

and used the definition of

${\overline{v}}_i$

. We are now ready to characterize the equilibrium for the manufacturers. Proposition 4 Equilibrium under individual contracting

There is a Nash equilibrium in which each manufacturer

$i\in N$

charges the same wholesale margin for all his products, and the equilibrium margin

$m_i^{\ast }$

of manufacturer i satisfies

$$\frac{1}{{\overline{v}}_i}\left(1-\frac{\mu }{m_i^{\ast }}\right)e^{\frac{m_i^{\ast }}{\mu }}=\frac{2+\frac{{\widehat{\pi }}_R}{\mu }}{e^{\frac{{\widehat{\pi }}_R}{\mu }}{\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)}^2},\forall i\in N,$$

17where

${\widehat{\pi }}_R$

is given in Equation (16) with

$m_i$

replaced by

$m_i^{\ast }$

.

Proposition 4 shows that, for competitive multiproduct contract design under MNL, the manufacturers compete at the firm level rather than at the product level, since what matters is the aggregate attractiveness of their products. Another observation is that the wholesale margins

$m_i^{\ast }$

must be greater than μ for all

$i\in N$

. Thus the total margin for each product is strictly positive (i.e., the sum of wholesale margin and retail margin,

$m_i^{\ast }+\mu +{\pi }_R^I$

, is greater than zero). This is expected since the manufacturers use the wholesale price contract for their products, and the well‐known double marginalization effect also occurs in our multiproduct supply chain. The immediate implication is that the total supply chain profit is maximized in equilibrium, leading to a supply chain efficiency loss. In Section 4.3, we examine how this efficiency loss is influenced by the market structure (e.g., the number of manufacturers along with the number of products from each).

We can also comment on the equilibrium retail prices. Since the total margin is the same for all the products from a same manufacturer, a higher production cost implies a higher retail price. If production costs are identical for all the products from a manufacturer, then their retail prices should be identical as well. This may explain an often observed phenomenon that the retailer tends to set the same price for mildly differentiated products of the same brand (e.g., color, or flavor) that cost the same to produce. A case in point is Coca‐Cola that has multiple soda products, each with a different flavor sold at the same retail price. ³ It is more problematic however to compare retail prices for products from different manufacturers because neither their wholesale margins nor product costs may be the same. ⁴ As a final note, product pricing is a complex decision faced by a retailer. Our model only factors consumer preferences/utilities and wholesale prices into the retailer's pricing decision. There remain however many other factors that may affect the pricing decision (e.g., operational costs, promotion schedules, and competition from other retailers).

We now examine each manufacturer's equilibrium profit under individual contracting. Substituting Equation (17) into manufacturer i's objective function in Equation (15) results in

${\pi }_i^I=(m_i^{\ast }-\mu )(1-1/(2+{\pi }_R^I/\mu ))$

, where

$m_i^{\ast }$

is manufacturer i's equilibrium wholesale margin and

${\pi }_R^I$

is the retailer's equilibrium profit (where the superscript indicates “individual contracting”). We now obtain some structural results concerning the relative sizes of wholesale margins and profits between the manufacturers. Corollary 1

Suppose

${\overline{v}}_i>{\overline{v}}_k$

where

$i,k\in N$

and

$i\ne k$

. Then we obtain

$m_i^{\ast }>m_k^{\ast }$

and

${\pi }_i^I>{\pi }_k^I$

.

The corollary shows that, consistent with our intuition, a manufacturer whose products have a greater aggregate attractiveness sets a higher wholesale margin and also makes a larger profit. In addition, since the retail margin is the same across all the products from all the manufacturers, the total margin—the sum of retail and wholesale margins—is also higher for the manufacturer with the greater aggregate attractiveness.

Next we perform a comparative statics analysis of the players' equilibrium profits and wholesale margins. For tractability, we focus on a semi‐symmetric setting in which the products of all the manufacturers have the same aggregate attractiveness, that is,

${\overline{v}}_i=\overline{v}$

for all

$i\in N$

. ⁵ The following result examines how the wholesale margin and the players' profits change with

$\overline{v}$

and n. Proposition 5

Suppose

${\overline{v}}_i=\overline{v}$

for

$i\in N$

. Then we obtain: (a)

$\frac{d{m}^{\ast }}{d\overline{v}}\ge 0$

,

$\frac{d{\pi }_R^I}{d\overline{v}}\ge 0$

, and

$\frac{d{\pi }_i^I}{d\overline{v}}\ge 0$

; (b)

$\frac{d{m}^{\ast }}{dn}\le 0$

, and

$\frac{d{\pi }_R^I}{dn}\ge 0$

. ⁶

Part (a) of the proposition shows that, as the aggregate gross attractiveness of each manufacturer increases, both the wholesale margin and every player's profit increase. To understand this, we know that an increase in gross attractiveness implies a greater market share for the retailer. As a result, the marginal profit with respect to the retail price is higher. This motivates the retailer to set a higher retail margin and thus she is better off with a larger value of

$\overline{v}$

. In addition, Equation (11) shows that the total market share increases in the retailer's profit. Thus, the total market share is also increased and each manufacturer faces a higher demand. Correspondingly, the manufacturers also set higher wholesale margins, thereby benefiting from an increase in

$\overline{v}$

as well.

Part (b) of the proposition shows that, when there are more manufacturers in the supply chain, the wholesale margin is lower and the retailer's profit is higher. As the number of manufacturers increases, competition intensifies. As a consequence, the wholesale margin is reduced and the total market share increases, a benefit to the retailer. Since we cannot analytically show the effect of n on each manufacturer's profit, we will resort to numerical studies for this analysis in Section 4.3, where we find that each manufacturer's profit decreases in n.

Aggregate contracting

Under aggregate contracting, each manufacturer offers a supply contract that sets the price as a function of the order quantity for every product. We denote the contract of manufacturer i by

$P_i(x_i^1,\text{…},x_i^{n_i})$

where

$x_i^j$

is the order quantity of product

$ij$

. ⁷ Since our intention is to consider the most general form of a supply contract under aggregate contracting, we do not impose any restrictions on

$P_i$

. However, to ensure that an optimal solution always exists for the retailer's optimization problem, it is necessary to assume

$P_i(x_i^1,\text{…},x_i^{n_i})$

is lower semi‐continuous.

Following backward induction, we start by considering the retailer's problem. Given the contracts offered by the manufacturers

$\{P_i(·):i\in N\}$

, the retailer aims to maximize her profit by determining an optimal retail price for each product. The retailer's optimization problem is formulated as follows:

$$\begin{array}{c}\hfill \underset{\{r_i^j:i\in N,j\in N_i\}}{\max }{\pi }_R=\sum _{i\in N}\left[\left(\sum _{j\in N_i}{r}_i^j{d}_i^j\right)-P_i\left(d_i^1,\text{…},d_i^{n_i}\right)\right],\end{array}$$

18where the demand

$d_i^j$

is a function of retail prices as given in Equation (3), and

$P_i(d_i^1,\text{…},d_i^{n_i})$

is the retailer payment to manufacturer i for an order of

$d_i^j$

for product

$ij$

where

$j\in N_i$

. Since the contracts are in general form, it is not possible to pin down precisely the retailer's optimal prices. As will be shown later however, an optimal contract for the manufacturers has a special form. On that basis, we will revisit the retailer's pricing problem.

Next, we consider each manufacturer's best response problem in determining the optimal contract, both in response to the competitors' contracts and in anticipation of the retailer's optimal retail prices for any given contracts. Suppose the contracts offered by the competitors are

$\{P_k(·):k\ne i,k\in N\}$

. Then each manufacturer i's contract design problem is to maximize his profit by choosing an optimal contract

$P_i(·)$

:

$$\underset{P_i(·)}{\max }{\pi }_i=P_i\left(d_i^1,\text{…},d_i^{n_i}\right)-\sum _{j\in N_i}{c}_i^j{d}_i^j,$$

19where

$d_i^j$

is the resulting order quantity for product

$ij$

, evaluated at the retailer's optimal retail prices as implicitly determined by Equation (18).

We remark that the analysis of the above best response problem poses several challenges. First, the manufacturer's contract must be searched from a function space. As a result, the conventional approach to analyzing contract design problems, which typically deals with a finite number of contract variables (e.g., a single variable for a wholesale price contract, or two variables for a buyback contract), does not apply in our problem. Second, for the best response problem, each manufacturer must take into account the retailer's optimal pricing decision while responding to the other manufacturers' contracts. Further, as discussed earlier, we cannot pin down the retailer's optimal prices since there is no specific form imposed on the manufacturers' contracts. To address these challenges, we adopt a different approach in which we first establish an upper bound for the manufacturer's profit and then demonstrate that this upper bound can be achieved with a certain type of contract. The following proposition presents the best response for each manufacturer. Proposition 6

(a). Given

$\{P_k(·):k\ne i,k\in N\}$

, the upper bound of manufacturer i's profit is

${\widehat{\pi }}_{i,R}(N)-{\widehat{\pi }}_R(N_{-i})$

, where

20(b). This upper bound can be achieved with the following cost‐plus contract:

21

It is useful to first explain the notation in the proposition:

${\widehat{\pi }}_R(N_{-i})$

is the retailer's optimal profit when she purchases only from the competing manufacturers other than manufacturer i, while

${\widehat{\pi }}_{i,R}(N)$

can be considered the optimal combined profit of manufacturer i and the retailer (given the competing manufacturers' contracts) since it is obtained by maximizing the retailer's profit when manufacturer i sets his product prices to cost. Thus, the upper bound

${\widehat{\pi }}_{i,R}(N)-{\widehat{\pi }}_R(N_{-i})$

measures the additional profit that can be earned when manufacturer i is included. Part (a) of Proposition 6 states that this is the maximum profit available to manufacturer i. The intuition of this result is easy to understand: Manufacturer i cannot obtain more profit than his added value to the supply chain; otherwise, the retailer would order nothing from him.

Part (b) of Proposition 6 further shows that the upper bound profit can be achieved by using a cost‐plus contract, as prescribed in the proposition. In other words, it is optimal for each manufacturer to set his unit variable margin to zero and charge a fixed markup equal to the additional contribution made by manufacturer i. While the manufacturers are free to search for an optimal contract from a function space, it turns out that an optimal contract has a surprisingly simple form, with only the markup needing to be determined by each manufacturer. This cost‐plus contract is a lower semi‐continuous, nonlinear function in which a fixed markup is added to the cost functions independent of the purchase quantities as long as the total purchase amount (aggregated over all his products) is positive. This resembles the two‐part tariff structure that is observed in practice, but for multiple products. With this cost‐plus contract, the average unit price decreases as the purchase quantity increases, thus the cost‐plus contract is essentially a type of quantity discount contract.

One may wonder at this point what the retailer's optimal decision would be, given that manufacturer i offers a cost‐plus contract. Since this contract only adds a fixed markup to cost price for a positive aggregate purchase volume, the optimal retail prices are identical to the solution to the following optimization problem as given in Proposition 6:

$$\begin{array}{c}\hfill \underset{\{r_k^j:k\in N,j\in N_k\}}{\max }\left\{\sum _{k\ne i}\left[\left(\sum _{j\in N_k}{r}_k^j{d}_k^j\right)-P_k(d_k^1,\text{…},d_k^{n_k})\right]+\sum _{j\in N_i}(r_i^j-c_i^j)d_i^j\right\}.\end{array}$$

22

We now proceed to the analysis of equilibrium for the manufacturers. We begin with an observation that, considering that the strategy of the manufacturer is a function, the above cost‐plus contract is not the unique solution to the best response problem. Indeed one may construct alternative solutions by twisting the cost‐plus contract in a way that allows the manufacturer to reap the same profit while ensuring the retailer's optimal retail prices match those under the cost‐plus contract. Nevertheless, we can obtain a sharper result on uniqueness if the manufacturers are restricted to offering a cost‐plus contract, as will be shown in the next proposition.

Another observation is that for the best response problem, the manufacturer's profit is a discontinuous function of his markup. That is, the profit function involves a jump, the placement of which depends on the other manufacturers' markups. While it can be shown that the profit function is quasiconcave, the standard equilibrium existence result based on the fixed point theorem (Debreu, 1952; Vives, 1999) requires the profit functions to be continuous. Therefore, we apply a more native approach for equilibrium analysis in the sense that we first conjecture a set of strategies and then demonstrate that no manufacturer has any incentive to deviate from the proposed strategies.

Before presenting the equilibrium, we note an important consideration in the equilibrium analysis of aggregate contracting; that is, the retailer's optimal choice may involve a tie. ⁸ This is a general concern in the bilevel programming literature that the optimal solution to the lower level problem is not uniquely determined (Dempe, 2002). To address this issue, we follow the literature by adopting the optimistic approach with the interpretation that the leader may be able to persuade the follower to select the solution which is best for the leader (Dempe, 2002, p. 123). Similar assumptions have been used in other studies in operations management (see e.g., Anderson et al., 2017; Cachon & Kök, 2010). One may consider an alternative rule in which the retailer randomly selects an optimal solution to break a tie. The issue with this rule, however, is that a pure‐strategy equilibrium may not exist. The proposition below characterizes an equilibrium for the manufacturers under aggregate contracting. Proposition 7 Equilibrium under aggregate contracting

(a)

There is a Nash equilibrium in which each manufacturer offers a cost‐plus contract and the equilibrium markup of manufacturer

$i\in N$

is

$\overline{\mathrm{\Pi }}(N)-\overline{\mathrm{\Pi }}(N_{-i})$

. In this equilibrium, the supply chain profit is maximized, each manufacturer makes a profit equal to his marginal contribution, and the retailer receives the remaining profit.

(b)

This is the unique equilibrium if the manufacturers are restricted to offering cost‐plus contracts.

For the equilibrium given in part (a), each manufacturer's markup is equal to his marginal contribution to the supply chain (i.e.,

$\overline{\mathrm{\Pi }}(N)-\overline{\mathrm{\Pi }}(N_{-i})$

), which is also his equilibrium profit. Given the proposed contracts, the retailer sets the retail prices that maximize the supply chain profit. This is because every manufacturer uses a cost‐plus contract in which no unit margin but a fixed markup is added to cost. In other words, the equilibrium retail prices are the same as the supply chain optimal prices

$\{{\overline{r}}_i^j:ij\in \mathcal{S}\}$

as shown in Lemma 1. Thus, in equilibrium there is no efficiency loss from the supply chain's perspective. This result is driven by the fact that the manufacturers have sufficient flexibility in terms of contract choice as they are allowed to offer a general nonlinear contract which can be in any form.

Part (b) of Proposition 7 goes a step further by showing that the equilibrium we establish is unique, provided that the manufacturers are restricted to the cost‐plus form. Notice that the cost‐plus contract is indeed an optimal contract form, as can be seen in Proposition 6. In addition, this contract is simple to administer as it only requires the manufacturers to determine a fixed markup. Thus, it offers the manufacturers a natural nonlinear contract to consider. This uniqueness result suggests that the equilibrium may provide a reasonable prediction of manufacturer competition dynamics.

It is interesting to consider the connection between the equilibrium profit allocation in Proposition 7 and the core, a common solution concept of cooperative game theory. A cooperative game is characterized by a set of players M (referred to as the grand coalition) and a characteristic value function

$z(·)$

that assigns a real number to every subset of M. The core of a cooperative game is a set of payoff profiles for all players such that no subset of the players has an incentive to deviate from the grand coalition. Denote by

$a_i$

player i's payoff for

$i\in M$

. Formally, a payoff profile is in the core if and only if: (i)

${\sum }_{i\in M}{a}_i=z(M)$

, and (ii)

${\sum }_{i\in S}{a}_i\ge z(S)$

for any

$S\subset M$

.

In our setting, the grand coalition consists of all manufacturers in N and the retailer which we denote by R (i.e.,

$M=N\cup \{R\}$

). The characteristic value refers to the optimal supply chain profit given a fixed set of players in the supply chain, and we can write the characteristic value function as follows: for any

$S\subseteq M$

,

$z(S)=\overline{\mathrm{\Pi }}(S\setminus R)$

if

$R\in S$

and

$z(S)=0$

if

$R\notin S$

. In particular, for the grand coalition we have

$z(M)=\overline{\mathrm{\Pi }}(N)$

, and for any subset that does not include the retailer, the characteristic value function assigns zero to that subset because the optimal supply chain profit is zero. Having set up the cooperative game, we are now ready to establish a connection between the aggregate contracting equilibrium and the core. Proposition 8

The equilibrium profit allocation in Proposition 7 is in the core of the corresponding cooperative game.

The above proposition shows that the equilibrium outcome under aggregate contracting satisfies all the conditions for the core. If a profit allocation is in the core, the grand coalition is stable under that allocation. Therefore, this result suggests that aggregate contracting encourages every player to stay in the supply chain and, in particular, the retailer does not want to use only a subset of the manufacturers.

We can be more explicit in terms of the profit split in equilibrium. Manufacturer

$i\in N$

makes a profit of

$${\pi }_i^A=\mu W\left(\sum _{k\in N}\sum _{j\in N_k}{\overline{v}}_k^j\right)-\mu W\left(\sum _{k\ne i,k\in N}\sum _{j\in N_k}{\overline{v}}_k^j\right),$$

23where we have used the definition of

$\overline{\mathrm{\Pi }}$

from Equation (8). Since the total supply chain profit is maximized, the retailer receives the remaining profit:

24Here, superscript A indicates “aggregate contracting.” We now conduct a sensitivity analysis for each player's equilibrium profit. Proposition 9

• (a)

As

$u_i^j$

increases or

$c_i^j$

decreases, both manufacturer i‘s profit and the retailer's profit increase, while all the other manufacturers’ profits decrease.

(b)

If manufacturer i has one more product, then both manufacturer i and the retailer make a higher profit, while all the other manufacturers make a lower profit.

(c)

If there is an additional manufacturer, then the retailer makes a higher profit, while all the existing manufacturers make a lower profit.

The results in Proposition 9 are self‐evident. From the definition of

${\overline{v}}_i^j$

in Equation (4), the gross attractiveness of a product increases as

$u_i^j$

increases or

$c_i^j$

decreases. The result in part (a) shows that as the attractiveness of a product,

${\overline{v}}_i^j$

, increases, both the manufacturer who produces that product and the retailer make a higher profit, but all the other manufacturers make a lower profit. To understand part (b), we know that having one more product on offer allows the manufacturer to secure a larger market share and thus a greater profit. As a result, the other manufacturers' market share is reduced, causing a profit decrease for those manufacturers. The retailer benefits because her overall market share also increases. Part (c) of the proposition shows that the retailer benefits from a larger pool of manufacturers. The existing manufacturers all make a lower profit, however, because their competition is intensified by the presence of the additional manufacturer.

As for individual contracting, we next focus on a special case in which aggregate attractiveness is the same across all manufacturers. For this case, we obtain the following results. Proposition 10

Suppose

${\overline{v}}_i=\overline{v}$

for

$i\in N$

. Then we obtain: (a)

$\frac{d{\pi }_R^A}{d\overline{v}}\ge 0$

; (b)

$\frac{d{\pi }_R^A}{dn}\ge 0$

, and

$\frac{d{\pi }_i^A}{dn}\le 0$

.

The first part of the proposition shows that the retailer's profit increases when the aggregate gross attractiveness of a manufacturer improves. The effect of

$\overline{v}$

on each manufacturer's profit is not analytically available, so we will revisit this in our numerical study. Part (b) of the proposition shows that, as the number of manufacturers increases, the retailer's profit increases but each manufacturer's profit decreases. This result is again driven by the fact that competition is greater when there are more manufacturers in the supply chain.

Numerical studies

In this section, we perform a numerical analysis to complement our analytical results. The purpose of this analysis is twofold: (a) to examine how each player's equilibrium profit changes with key parameters under each contracting scheme; and (b) to compare the two schemes and study how the profit differences for each player and the supply chain change with key parameters.

We focus on cases in which the manufacturers have the same aggregate gross attractiveness. We set

$\mu \in \{0.5,1,\cdots ,5\}$

,

$n\in \{2,3,4,5,6\}$

, and

$q\in \{2,3,\text{…},8\}$

where q refers to the number of products from each manufacturer. Each manufacturer's products are sorted in ascending order by the net utility; that is, the difference between the expected utility and the unit cost,

$u^j-c^j$

, is greater for a higher value of j where

$j=1,\text{…},q$

. We also assume that the utility gap between two consecutive products, denoted by δ, is the same, and we set

$\delta \in \{0.1,0.2,\text{…},1\}$

. Thus, under this specification, the first product has a net utility of

$u^1-c^1\in \{0.1,0.2,\text{…},2\}$

, the second product has a net utility of

$u^2-c^2=u^1-c^1+\delta$

, the third product has the net utility of

$u^3-c^3=u^1-c^1+2\delta$

, and so on. In total we consider

$70,000(=10\times 5\times 7\times 10\times 20)$

numerical instances.

Efficiency loss from individual contracting

We denote by

${\mathrm{\Pi }}^A$

and

${\mathrm{\Pi }}^I$

the supply chain profits under aggregate contracting and individual contracting, respectively. Let us first formalize a qualitative result from the comparison of equilibrium supply chain profits between the two contracting schemes. Corollary 2

We obtain

${\mathrm{\Pi }}^I<{\mathrm{\Pi }}^A$

.

Corollary 2 shows that the total supply chain profit under individual contracting is lower than that under aggregate contracting. As shown in Proposition 7, there is no supply chain profit loss in equilibrium under aggregate contracting. In contrast, under individual contracting, we find from Equation (10) that the wholesale margin for manufacturer i's products is

$m_i^{\ast }>\mu$

, which is greater than zero. Consequently, the total market share is lower than the supply chain optimum, hence the supply chain profit is not maximized due to the positive wholesale margins. Therefore, aggregate contracting always outperforms individual contracting from the supply chain's perspective.

We are more interested in the magnitude of efficiency loss (which is also the supply chain profit gap from individual contracting over aggregate contracting) and how it changes with varying market conditions. We use the following measure to quantify the efficiency loss from the individual contracting scheme (in percentage terms):

$$\mathrm{\Delta }\%=\frac{{\mathrm{\Pi }}^A-{\mathrm{\Pi }}^I}{{\mathrm{\Pi }}^A}\times 100\%.$$

25Across the 70, 000 numerical instances, the efficiency loss under individual contracting ranges from 0.02% to 15.89% with an average of 4.46%. Table 1 below presents the values of

$\mathrm{\Delta }\%$

for different combinations of the number of products from each manufacturer (q) and the number of manufacturers (n).

TABLE 1

Average efficiency losses under individual contracting for different combinations of q and n

	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$
$q=2$	12.74%	8.65%	6.58%	5.34%	4.52%
$q=3$	10.48%	6.64%	4.88%	3.90%	3.27%
$q=4$	8.90%	5.36%	3.86%	3.06%	2.56%
$q=5$	7.71%	4.47%	3.18%	2.50%	2.09%
$q=6$	6.78%	3.81%	2.68%	2.11%	1.76%
$q=7$	6.03%	3.31%	2.31%	1.81%	1.51%
$q=8$	5.41%	2.90%	2.02%	1.58%	1.32%

Note that the data entries in Table 1 are average efficiency losses across all numerical instances for parameters

$\mu ,\delta$

, and

$u^1-c^1$

. An overall message from the table is that the supply chain efficiency loss under individual contracting decreases when the number of products from each manufacturer q or the number of manufacturers n increases. To understand the impact of q on supply chain efficiency loss, we identify both the direct and indirect effects of increasing the number of products. The direct effect of an increase in q is that the total market share increases (since more consumers will be likely to purchase one of the available products). On the other hand, an increase in q also increases the aggregate attractiveness of each manufacturer, thus raising the wholesale margin as shown in Proposition 5. While an increase in wholesale margin may reduce the total market share thus widening the efficiency loss, it turns out that the direct effect prevails and the total market share increases because of a higher q. Therefore, the efficiency loss is reduced when there are more products from each manufacturer.

Regarding the impact of n, we know that competition between manufacturers becomes more intense as the number of manufacturers increases. It has been shown in Proposition 5 that this reduces wholesale margins and increases the total market share. Therefore, the efficiency loss is reduced when there are more manufacturers competing in the supply chain.

Each player's profit

This subsection examines the equilibrium profits of each manufacturer and the retailer. We start by calculating the difference in profits between the two contracting schemes for each player, and then derive various statistical measures across the numerical instances, as summarized in Table 2.

TABLE 2

Profit difference for each player between aggregate and individual contracting

	${\pi }_R^A-{\pi }_R^I$	${\pi }_i^A-{\pi }_i^I$
Minimum	−0.4900	−0.2413
Quartile 1	−0.0392	0.0037
Quartile 2/Median	0.0371	0.0373
Quartile 3	0.0937	0.0948
Maximum	0.6136	0.5403
Mean	0.0255	0.0684

While aggregate contracting always outperforms individual contracting for the supply chain, it remains unclear whether each player also benefits from aggregate contracting. Table 2 shows that the difference in the retailer profit (i.e.,

${\pi }_R^A-{\pi }_R^I$

) ranges from −0.4900 to 0.6136 with a mean of 0.0255. This indicates that the retailer on average prefers aggregate contracting, but in some numerical instances she prefers individual contracting. It also shows that the difference in each manufacturer's profit (i.e.,

${\pi }_i^A-{\pi }_i^I$

) ranges from −0.2413 to 0.5403 with a mean of 0.0684. This suggests that each manufacturer on average also prefers aggregate contracting but in some instances could benefit from using individual contracting. Overall, the results suggest that no contracting scheme is dominant from either the retailer's or the manufacturers' perspective. It is known that aggregate contracting allows a manufacturer to extract more rents in a one‐to‐one supply chain. In the presence of manufacturer competition, however, the use of aggregate contracting rather than individual contracting may intensify the competition. This negative effect of aggregate contracting may outweigh the positive rent extraction effect, and hence under some conditions, the manufacturers may prefer individual contracting. For the same reason, the retailer too may sometimes prefer aggregate contracting.

In what follows, we examine the effects of key parameters on each player's preferences over the contracting schemes. It is worth noting that when evaluating the effect of each parameter, the profits are averaged across all the numerical instances with respect to the other parameters. Figure 1 depicts how each player's average profit changes with the number of products, q. We find that as q increases, both the retailer's and each manufacturer's profits increase under both contracting schemes. This is consistent with our analytical results in part (a) of Proposition 5 and Proposition 10 that every player's profit increases in aggregate attractiveness

$\overline{v}$

, because

$\overline{v}$

increases in q in the considered setting. Additionally, as q increases, while the difference in the retailer's profit between the two schemes increases, the difference in each manufacturer's profit decreases. Further, as shown in the figure, aggregate contracting is more beneficial to the retailer only when the number of products available from each manufacturer exceeds four (

$q>4$

). In contrast, aggregate contracting is more beneficial to the manufacturers when the number of products is no greater than six (

$q\le 6$

). Overall, Figure 1 suggests that as each manufacturer produces more products, the retailer has a stronger preference for aggregate contracting over individual contracting, while the reverse is true for each manufacturer.

FIGURE 1

Effect of q on each player's profit

Figure 2 illustrates how each player's average profit changes with the number of manufacturers, n. As n increases, the retailer's profit increases but each manufacturer's profit decreases under both contracting schemes. This is consistent with part (b) of Proposition 5 and Proposition 10. In addition, the average profit differences for both the retailer and each manufacturer decrease in the number of manufacturers, implying that the benefit to supply chain members of using complex contract forms diminishes as competition intensifies. Overall, Figure 2 suggests that, with an increase in the number of manufacturers, aggregate contracting becomes less attractive to each player.

FIGURE 2

Effect of n on each player's profit

We now examine how μ affects equilibrium profit allocations, as illustrated in Figure 3.

FIGURE 3

Effect of μ on each player's profit

Two observations can be made from this figure. First, each player's profit increases in consumer heterogeneity parameter μ, suggesting that both the manufacturers and the retailer benefit from having more heterogeneous consumer preferences. A high‐level intuition is that as consumer preferences become more heterogeneous, more manufacturers are able to capture a share of the market, thus dampening competition. As a result, each manufacturer benefits. We find that the overall demand also increases, thereby benefiting the retailer as well. Second, the retailer's profit is higher under aggregate contracting when μ is low, and each manufacturer's profit is higher under aggregate contracting when μ is high. The intuition of this result is not immediate, but it relates to the effect of consumer heterogeneity on the extent of competition induced by both contracting schemes. Overall, Figure 3 suggests that when consumer preferences are increasingly heterogeneous, aggregate contracting becomes less attractive to the retailer but more so to the manufacturers.

AN EXTENSION WITH HYBRID CONTRACTING STRATEGIES

Our main analysis has focused on a symmetric setting where all manufacturers adopt the same contracting strategy. This may be appropriate when there is an industry norm for supply contracting or when the retailer specifies a particular contract arrangement for the manufacturers. If such a norm or requirement does not exist, however, it is useful to consider a hybrid setting in which manufacturers may adopt different contracting strategies. In this section, we examine a supply chain with two manufacturers (i.e.,

$N=\{1,2\}$

), in which manufacturer 1 uses individual contracting and manufacturer 2 uses aggregate contracting. Specifically, suppose manufacturer 1 offers his products in set

$T_1\subseteq N_1$

with a wholesale price

$w_1^j$

for each product 1j where

$j\in T_1$

, and manufacturer 2's contract specifies a payment function

$P_2(x_2^1,\text{…},x_2^{n_2})$

where

$x_2^j$

denotes the order quantity of product 2j where

$j\in N_2$

. Given the two contracts, the retailer's problem is to maximize her profit by choosing an optimal retail price for each product offered:

26where the demand

$d_i^j$

is a function of the product retail prices as given in Equation (3). Similar to aggregate contracting in Section 4.2, it is not possible to determine the retailer's optimal solution due to the general form of manufacturer 2's contract.

To proceed, we follow the same approach as in Section 4.2 by first examining the best response for manufacturer 2 who adopts aggregate contracting. Since individual contracting is a special case of aggregate contracting, the payment function for manufacturer 1 can be written as

$P_1(x_1^1,\text{…},x_1^{n_1})={\sum }_{j\in N_1}{w}_1^j{x}_1^j$

. Therefore in this extension, the results of Proposition 6 also apply to manufacturer 2. This means that an optimal contract for manufacturer 2 has the cost‐plus form. In the subsequent analysis, we focus our attention on the case in which manufacturer 2 offers a cost‐plus contract with

$P_2(x_2^1,\text{…},x_2^{n_2})=F+{\sum }_{j\in N_2}{c}_2^j{x}_2^j$

for

${\sum }_{j\in N_2}{x}_2^j>0$

and

$P_2(0,\text{…},0)=0$

, where F is a constant to be determined by manufacturer 2. The objective of manufacturer 2 is to maximize F on the proviso that the retailer places a positive order with him.

We now revisit the retailer's pricing problem with manufacturer 1 offering a wholesale price contract and manufacturer 2 offering a cost‐plus contract. The retailer's pricing problem in Equation (26) is reformulated as follows:

27Since F is a constant, it does not affect the retailer's optimal retail prices. Therefore, the above problem is the same as the supply chain problem in Section 3.1 except that manufacturer 1's costs

$c_1^j$

are replaced by wholesale prices

$w_1^j$

. Thus, the optimal retail prices are characterized by the following conditions:

$$\begin{array}{c}\hfill {\widehat{r}}_1^j-w_1^j=\mu +{\widehat{\pi }}_R,\forall j\in T_1,\text{and}{\widehat{r}}_2^j-c_2^j=\mu +{\widehat{\pi }}_R,\forall j\in N_2,\end{array}$$

28where with slight abuse of notation we use

${\widehat{\pi }}_R$

to denote the retailer's optimal profit without accounting for F in the remainder of this section. Substituting the above conditions into the retailer's profit function, we obtain the retailer's optimal profit as the unique solution to the following equation:

$$\frac{{\widehat{\pi }}_R}{\mu }{e}^{\frac{{\widehat{\pi }}_R}{\mu }}=\sum _{j\in T_1}{e}^{\frac{u_1^j-w_1^j}{\mu }-1}+\sum _{j\in N_2}{e}^{\frac{u_2^j-c_2^j}{\mu }-1}.$$

29Once the optimal value of

${\widehat{\pi }}_R$

is obtained and knowing the optimal retail prices, we can work out the demand for each product

$ij$

:

$$\begin{array}{c}\hfill {\widehat{d}}_1^j=\frac{e^{\frac{u_1^j-w_1^j}{\mu }-1}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},\forall j\in T_1,\text{and}{\widehat{d}}_2^j=\frac{e^{\frac{u_2^j-c_2^j}{\mu }-1}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},\forall j\in N_2.\end{array}$$

30

We now consider manufacturer 1's best response in determining the wholesale prices of his products. Similar to the analysis of individual contracting in Section 4.1, we take a two‐step approach: we first examine the optimal wholesale prices for any given set of products offered by manufacturer 1, and then study the optimal set of products for the manufacturer.

Suppose manufacturer 1 offers products in

$T_1\subseteq N_1$

. Given the cost‐plus contract offered by manufacturer 2, manufacturer 1's problem is to maximize his profit by choosing the optimal wholesale prices:

31where

${\widehat{\pi }}_R$

is the retailer's optimal profit as implicitly determined by Equation (29). The above problem is the same as Equation (15) except that we have two manufacturers and manufacturer 2's wholesale prices are equal to costs. The results in Proposition 2 then apply to manufacturer 1, and it is optimal to set an identical margin for all the products offered by that manufacturer, that is,

$w_1^j-c_1^j=m_1$

for

$j\in T_1$

where m ₁ denotes the wholesale price margin.

Using the above result, we can transform the multidimensional optimization problem in Equation (31) into a single‐dimensional problem where manufacturer 1 determines a single margin for all the products he offers. Following the same approach as in Section 4.1, Proposition 3 also applies, and it is optimal for manufacturer 1 to offer his full range of products to the retailer. Therefore, manufacturer 1's best response problem can then be reformulated as follows (after setting

$w_1^j-c_1^j=m_1$

for all

$j\in N_1$

):

$$\underset{m_1\ge 0}{\max }{\pi }_1=m_1\frac{{\overline{v}}_i{e}^{-\frac{m_1}{\mu }}}{e^{\frac{{\widehat{\pi }}_R}{\mu }}\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)},$$

32where

${\overline{v}}_1={\sum }_{j\in N_1}{\overline{v}}_1^j$

as defined earlier represents the aggregate gross attractiveness of manufacturer 1's products, and

${\widehat{\pi }}_R$

is characterized by

$$\frac{{\widehat{\pi }}_R}{\mu }{e}^{\frac{{\widehat{\pi }}_R}{\mu }}={\overline{v}}_1{e}^{-\frac{m_1}{\mu }}+{\overline{v}}_2.$$

33

We are now ready to present the equilibrium for this extension with hybrid contracting strategies. Proposition 11 Equilibrium in hybrid setting

There is a Nash equilibrium in which manufacturer 1 sets the same wholesale margin

$m_1^{\ast }$

for all his products, which satisfies

$$\frac{1}{{\overline{v}}_1}\left(1-\frac{\mu }{m_1^{\ast }}\right)e^{\frac{m_1^{\ast }}{\mu }}=\frac{2+\frac{{\widehat{\pi }}_R}{\mu }}{e^{\frac{{\widehat{\pi }}_R}{\mu }}{\left(1+\frac{{\widehat{\pi }}_R}{\mu }\right)}^2},$$

34where

${\widehat{\pi }}_R$

is given in Equation (33) with m ₁ replaced by

$m_1^{\ast }$

, and manufacturer 2 uses a cost‐plus contract with the fixed markup

$F^{\ast }={\widehat{\pi }}_R-{\widehat{\pi }}_R^{-2}$

, where

$${\widehat{\pi }}_R^{-2}=\underset{\{r_1^j,j\in N_1\}}{\max }\sum _{j\in N_1}\left(r_1^j-c_1^j-m_1^{\ast }\right)d_1^j,$$

35with

$d_1^j$

given in Equation (12) when

$\mathcal{S}=\{11,12,\text{…},1n_1\}$

.

Several points are worth mentioning about Proposition 11. First, the conditions that characterize the equilibrium wholesale margin of manufacturer 1 share a similar structure with those of Proposition 4, with the difference that Equation (33) depends only on wholesale margin m ₁ (as manufacturer 2 offers a cost‐plus contract). Further, this wholesale margin is independent of manufacturer 2's fixed markup. Second, we note that

${\widehat{\pi }}_R^{-2}$

gives the retailer's optimal profit when the retailer orders from manufacturer 1 alone. Thus, the equilibrium markup of manufacturer 2 is set in such a way that the retailer is indifferent between purchasing nothing from him (and not paying the fixed markup) and purchasing amounts

${\widehat{d}}_2^j$

for

$j\in N_2$

(and paying the fixed markup). This is understandable because manufacturer 2 can always improve his profit as long as his markup is lower than

$F^{\ast }$

, but the retailer will order nothing from manufacturer 2 if his markup is higher than

$F^{\ast }$

. Third, using the results in Section 3.1, we write down manufacturer 2's profit more explicitly:

${\widehat{\pi }}_R=\mu W({\sum }_{j\in N_1}{\overline{v}}_1^j{e}^{-\frac{m_1^{\ast }}{\mu }}+{\sum }_{j\in N_2}{\overline{v}}_2^j)$

, and

${\widehat{\pi }}_R^{-2}=\mu W({\sum }_{j\in N_1}{\overline{v}}_1^j{e}^{-\frac{m_1^{\ast }}{\mu }})$

. Thus,

$$\begin{array}{c}\hfill F^{\ast }=\mu W\left(\sum _{j\in N_1}{\overline{v}}_1^j{e}^{-\frac{m_1^{\ast }}{\mu }}+\sum _{j\in N_2}{\overline{v}}_2^j\right)-\mu W\left(\sum _{j\in N_1}{\overline{v}}_1^j{e}^{-\frac{m_1^{\ast }}{\mu }}\right).\end{array}$$

36This allows us to compare manufacturer 2's profit (

$F^{\ast }$

) in the hybrid setting with his profit under aggregate contracting as studied in Section 4.2. Corollary 3

We obtain that

$F^{\ast }\ge {\pi }_2^A$

.

Corollary 3 shows that manufacturer 2 makes a greater profit when manufacturer 1 uses individual contracting rather than aggregate contracting, suggesting that manufacturer 2 prefers the other manufacturer to use individual contracting. Another question is whether one can find a relationship between

$F^{\ast }$

and manufacturer 2's profit under individual contracting (i.e.,

${\pi }_2^I$

). As the analytical analysis is not amenable, we use numerical studies to answer this question. We consider the same setup as in Section 4.3 except that there are two manufacturers in this extension. The numerical results show that

$F^{\ast }>{\pi }_2^I$

indicating that manufacturer 2 is always better off switching from individual contracting to aggregate contracting when manufacturer 1 uses individual contracting. Similarly, we find that

${\pi }^A>{\pi }_1^H$

, where

${\pi }_1^H$

is manufacturer 1's equilibrium profit in the hybrid setting. This suggests that manufacturer 1 is also better off switching from individual contracting to aggregate contracting when manufacturer 2 uses aggregate contracting. Collectively, these numerical results suggest that it is a dominant strategy to use aggregate contracting. We further show that both

$F^{\ast }-{\pi }_2^I$

and

${\pi }^A-{\pi }_1^H$

increase in each of μ and the number of products from each manufacturer q. In other words, aggregate contracting is more attractive to the manufacturers when consumer preferences are more heterogeneous, or when there are more products on offer from each manufacturer.

The above results have some implications for an extended three‐stage game where each manufacturer first determines his contracting strategy, then accordingly determines the specific contract to offer the retailer, and finally, the retailer determines the retail prices (or the order quantities). This three‐stage game is appropriate when no regulatory or managerial requirements are imposed on the contracting strategy, so that the manufacturers can freely choose between aggregate contracting and individual contracting. As shown in our numerical studies, the dominant strategy is to use aggregate contracting, so in equilibrium both manufacturers will choose aggregate contracting. However, Section 4.3 demonstrates that the manufacturers make a higher profit under individual contracting when each offers a sufficiently large number of products (a high q), or there is a low level of consumer heterogeneity (a low μ). In these circumstances, therefore, the manufacturers may find themselves in a prisoner's dilemma where both manufacturers are worse off using aggregate contracting rather than individual contracting. We can illustrate this by using an example where

$\mu =1$

,

$u^1-c^1=1$

,

$\delta =1$

, and

$q=10$

. It can be shown that the manufacturers' equilibrium profits are

${\pi }_1^A={\pi }_2^A=0.614$

under aggregate contracting, and

${\pi }_1^I={\pi }_2^I=0.848$

under individual contracting, thus creating a prisoner's dilemma.

CONCLUSIONS AND FUTURE RESEARCH

Existing models in supply chain contracting focus primarily on single‐product problems. In this paper, we bridge a gap in the literature by studying competitive multiproduct contracting in a supply chain consisting of multiple manufacturers and a single retailer. With multiple products on offer from each manufacturer, a critical decision around optimal contract design is whether to quote a separate price for each individual product or an aggregate price for a combination of products. This paper considers two dichotomous contracting schemes: individual contracting where the manufacturers charge a wholesale price for every product separately, and aggregate contracting where each manufacturer designs a general nonlinear contract for a combination of his products. We fully characterize the equilibrium for the players under each scheme and obtain the following results. Under individual contracting, each manufacturer offers his full range of products to the retailer and charges an equal margin for each of his products. This result may be noteworthy in its own right as it extends the well‐known equal‐margin result obtained at the retailer to upstream manufacturer level. For aggregate contracting, the conventional approach to equilibrium analysis does not work in our model. Instead, we adopt a more native approach in which we first conjecture a set of strategies and then demonstrate that the conjectured strategies form an equilibrium. We show that a simple and intuitive cost‐plus contract is optimal for the manufacturers. The merit of this contract is its simplicity, with the manufacturers only needing to determine what fixed markup to add to their cost functions. Further, if restricted to cost‐plus contracts, we are led to a unique Nash equilibrium where the retailer sets retail prices in a way that maximizes the total supply chain profit. In terms of the equilibrium profit split, each manufacturer earns a profit equal to his marginal contribution to the supply chain, and the retailer earns the remaining profit.

We also examine the effect of changes in key parameters on equilibrium outcomes under each contracting scheme as well as the conditions for each player under which one scheme dominates the other. We derive analytical results whenever possible, and in cases where analytical analysis is not amenable, we resort to a numerical study. Overall, individual contracting results in efficiency losses while aggregate contracting does not. The efficiency loss from individual contracting decreases as the number of products available from each manufacturer, or the number of manufacturers, increases. Furthermore, no contracting scheme is always dominant from any player's perspective. We further examine how each player's contracting scheme preferences change along with varying market conditions. Specifically, we find that aggregate contracting benefits the retailer when each manufacturer offers more products, when there are fewer manufacturers in the supply chain, or when consumer preferences are less heterogeneous. For the manufacturers, aggregate contracting is more beneficial when there are fewer manufacturers or products on offer, or when consumer preferences are more heterogeneous. These sensitivity analysis results have important implications for optimal contract design. Indeed, managers should take into account the number of products, the concentration of the supply market, and consumer preferences when choosing an appropriate contracting scheme. Finally, as an extension, we study a hybrid setting where two manufacturers use different contracting strategies and demonstrate how a prisoner's dilemma may occur in a three‐stage game where manufacturers can choose between the two contracting strategies in the first stage.

This paper opens up several research opportunities in multiproduct contracting. First, we have chosen the standard MNL model, as suitable for the considered setting where all the products fit into the same category. Other discrete choice models may, however, be more appropriate for products from different categories. A natural extension of our model is to consider variants of MNL such as nested logit and hierarchical choice models (Gallego & Topaloglu, 2019; Li & Huh, 2011). Second, the supply chain considered in this paper involves multiple manufacturers and a common retailer. One may also study multiproduct contract design in settings with competing retailers. Third, in purely focusing on the firms' pricing and contracting decisions, we have refrained from considering asymmetric information. Incorporating information asymmetries in multiproduct contracting represents another avenue for future research. Finally, multiproduct contracting is an important problem in other retail contexts as well (e.g., agency selling and online retail platforms). Extending the current work along these lines also presents interesting opportunities for research.