Contracting with multiple suppliers: A multi‐item buyer's make versus buy decision

Abstract

We consider a multi‐item buyer's make versus buy decision. The buyer sells two substitutable products with uncertain demand. Each product requires a different item, which the buyer procures from two separate suppliers (one for each item) who offer quantity‐payment contracts under asymmetric information. The buyer can make one of the items but not both. We compare the buyer's expected profit between buying both the items and making one item (and buying the other), when it can make the item at the same cost as the supplier, as well as gain reactive production capability by making. We find that the buyer's profit when buying both items is higher than for making one item if product substitutability is above a threshold (but not extremely high). This is because the interaction between the suppliers' contracts increases the buyer's informational rents more when buying from both suppliers, as compared to one supplier, as product substitutability increases. On the contrary, in the complete information scenario (and under wholesale price contracts), we find that the buyer will never prefer to buy both the items as compared to making one item, as long as product substitutability is not extremely high. These are novel findings on how inter‐linked contracts under incomplete information and product substitutability can influence a buyer's make versus buy decision in a counter‐intuitive manner. Managerially, these results imply that a multi‐product buyer may prefer to keep buying from its supplier even when it can produce the item at the same cost as the supplier.

INTRODUCTION

Buyers (in retail or manufacturing) selling different (and substitutable) products have to sometimes purchase the different items corresponding to the products from separate suppliers. If the buyer is a retailer, then the items it purchases are the products themselves, but if the buyer is a manufacturer, then the items is procures are typically the components/materials that are used to make the products. For example, a retailer may sell two different (and substitutable) products in similar category (e.g., television) which it buys from two separate suppliers (brands like Vizio and Samsung). A processed food manufacturer selling two different (but substitutable) types of bottled olives may buy the ingredients (black olives and green olives) from different suppliers/growers. In other situations, the buyer (retailer or manufacturer) can make one of the item (product/component) while it continues to buy the other item from a supplier, for example, a retailer may develop its own private label which it sells along with an existing brand or the manufacturer may start its own production of an input while it continues to buy the other input from a supplier. This paper studies the buyer's make versus buy decision in such situations.

Examples that motivate the problem addressed in this paper come from both the retailing and manufacturing businesses. Best Buy (see Bustillo & Lawton, 2009) replaced some brands, for example, Vizio Inc's flat‐panel TVs, to bring its own private label Dynex televisions; all the while, it continued to sell Samsung's flat‐panel TV. Introduction of private label often results in the retailer changing the assortment of similar (substitutable) products (national brands) in its store. Analyzing data from US retailers Ma and Siebert (2021) find that retailers often remove some branded products (national brands) when introducing private label in a given category. Private labels are controlled by the retailers who are either buying the products from the manufacturers at close to the marginal price or the manufacturer might even be vertically integrated with the retailer (see Connor & Peterson, 1992). In another example, a processed food manufacturer in Australasia sells bottled/canned olives in the regional market (and in fact was the first to introduce them in the region in the late 1980s). It sells different types of bottled olives, for example, from Greek Kalamata olives (black smoothy‐meaty texture) or Greek Halkidiki olives (green, Greek). The olives are purchased from different wholesalers in Europe who offer price schedules based on volume. ¹ Climatically, it is possible for the food manufacturer to grow some of these olive varieties in Australasia but it continues to buy them from Europe. In an example from the chemical industry, a regional chemical manufacturer in Asia produces and sells textile chemicals (like fabric finishing agents and softening agents) to the textile manufacturers. ² The manufacturer sells different types of textile finishing agents, which are primarily differentiated by the use of either natural oil‐based emulsions or silicone‐based emulsions. ³ The manufacturer is producing its own natural oil‐based emulsions but has to buy silicone emulsions from a supplier because silicone, which is the critical feedstock in silicone‐emulsions, is controlled by the supplier who is a much bigger global chemical manufacturer (like Wacker Chemie or Dow Chemicals). ⁴ The (high) transport cost to price ratio of these emulsions limits the manufacturer to purchase silicone‐emulsion from the only available regional supplier of silicone emulsion. Finally, in the semiconductor industry, manufacturers (mainly in emerging economies) selling poly and mono crystalline silicon photovoltaic (PV) panels/modules might have facilities that produce poly‐crystalline silicon wafer (which are required to make the cell that are then assembled into modules) but are dependent on big suppliers (e.g., LONGi) for buying mono‐crystalline silicon wafer, because the manufacturer may not have the know‐how or capability to undertake the more advanced manufacturing process required for producing mono‐crystalline silicon wafers (see Indian Express).

These examples illustrate some of the salient features of the problem that we study in this paper, namely, a buyer selling two (or more) substitutable products. The items (which can be the products themselves or can be components/material used in manufacturing the products) required for the two products can sometimes be purchased from different suppliers (e.g., retailer buying from different brands within a segment or a regional processed food manufacturer buying foreign products). Sometimes the buyer only buys one of the item from a supplier while it produces the other item itself (e.g., chemical manufacturer making oil‐based emulsion while buying silicone‐based emulsion or PV‐manufacturer producing poly‐crystalline wafers while buying mono‐crystalline wafers), and sometimes the buyer can even replace the item that it was buying from a supplier with its own in‐house produced item while it continues to buy the other item from the respective supplier (e.g., retailer replacing one of the brands in a segment with its private label). Thus, these examples indicate that a buyer can decide whether it continues to buy different items from separate suppliers or whether it makes one of the item itself and only buys the other item from the concerned supplier. Moreover, in some of these cases, the suppliers can determine the contract terms that they offer to the buyer for their items (e.g., in case of brand names selling to retailer, or in case of growers selling regional products to foreign food manufacturers, or chemical suppliers that have regional exclusivity to the material they produce). This paper studies the buyer's make versus buy decision in such situations.

Specifically, we answer the question on whether a buyer (retailer/manufacturer), who sells two substitutable products, should buy the required items from both of the respective suppliers or should it buy only one item from a supplier and make‐in‐house the other item (that it was buying from the other supplier) when the buyer can make the other item at the same make cost of the supplier? We analyze the buyer's decision when supplier(s) offer it optimal menu of (price‐quantity) contracts. We analyze this question under a contractual framework for two reasons: (1) most of the examples that motivate the question include supplier(s) who can determine the contract terms that they offer to the buyer, for example, brands typically determine the contracts that they offer to the retailer or suppliers who have geographic exclusivity determine contracts that they offer to a manufacturer. (2) Many papers in OM literature have studied supplier determined contracts in situations where supplier is providing the exclusive product (see Corbett & de Groote, 2000; Corbett et al., 2004; Ha, 2001; Kostamis & Duenyas, 2011). In particular, Corbett et al. (2004) apply supplier determined contracts in retailing while Kostamis and Duenyas (2011) apply these contracts in manufacturing.

The main trade‐off that we investigate in analyzing the buyer's decision is based on the surplus that two suppliers can extract from the buyer when it buys the two items from the two respective suppliers versus the surplus that a single supplier can extract from the buyer when it buys a single item from the respective supplier and makes the other item itself. When making the item, the buyer saves on the surplus it was giving away to the supplier of the item and thus may save on its input cost (if its make cost is not greater than the price offered by the supplier). However, the supplier (from whom the buyer continues to procure the other item from) may extract a greater surplus because it no longer has to compete for the contract with another supplier. Thus, in making an item and buying the other item, while the buyer benefits in saving surplus it was giving to the supplier (of the item that the buyer now makes), it may instead have to give a higher surplus to the supplier from which it continues to buy. Whereas, in buying both the items from the two respective suppliers, while the buyer can benefit from competition between the contracts offered by the suppliers, it is also giving away surplus to both suppliers. Thus, upfront, it is not clear whether the buyer will increase its profitability by making one item (that it was procuring from the supplier) and buying the other item, as compared to buying both the items from the two respective suppliers, even if it can make the item at the same make‐cost as the supplier.

Although the above trade‐offs are clear, there is no model that characterizes this trade‐off in a buyer's make versus buy decision, which is precisely the objective of this paper. Indeed, there are many other considerations involved in a buyer's make versus buy decision. For instance, the typical advantages of buying include lower input cost from accessing lower cost supplier (adjusted by supplier margins), accessing scale economies, accessing better technology, faster product development times, lower fixed cost, group purchasing (see Peng et al., 2022), and freedom to focus on core competencies. Typical disadvantages include becoming over‐reliant on a supplier and potential loss of control, losing efficiencies in design for manufacturability, possible contractual hold‐ups, and other transactional costs. In this paper, we focus on an as yet unexplored factor that plays an equally relevant role (as the factors mentioned above) in a buyer's make versus buy decision. Specifically, we capture the interaction between suppliers' contracts to explore and compare profits of a buyer selling two substitutable products for which it sources items from two different suppliers to its profits when it sources only one item from the respective supplier and can make the other item at the same make cost as the supplier who supplies it. Note that this comparison will be trivial if the buyer can make both the items at the same make cost as the respective suppliers supplying the items—in which case the buyer can save entirely on the surplus it gives to the suppliers and will therefore have greater profit in making as compared to buying. However, as described in the examples above, technological limitations, manufacturing capability, control of feedstock, patent protection, limitations on capital, etc., may restrict the buyer from making all the items, but it may have the limited but necessary resources and capability to make one item (even more efficiently than the supplier).

To make this comparison in the buyer's profits, we model a buyer who offers two substitutable products in the market and whose demand may or may not be uncertain. Each product requires a different item which it can either procure from two separate suppliers (one for each item) or it can procure only one item from the respective supplier and make the other item itself (at the same make cost as the other supplier). The buyer is privately informed about its other product related cost. For instance, if the buyer is a manufacturer then it is privately informed about its production cost (which is not known to the suppliers). If the buyer is a retailer then it is privately informed of its marketing and retailing (operational) costs which are not known to the suppliers. We first consider the case where the buyer buys both the items from the respective suppliers. Each of the two suppliers offers a quantity‐payment contract (for its respective items) to the buyer, which we characterize in equilibrium using the common agency principal‐agent framework in which the two suppliers are principals and the buyer is the agent. The two suppliers's contracts are inherently interlinked due to substitutability of the two products that the buyer is selling in its downstream market. The common agency framework, developed by Stole and Martimort (see Martimort, 1992; Stole, 1991), is ideal for capturing this interaction between suppliers' contracts under asymmetric information. Specifically, the common agency framework can be used in a setting where multiple principals have to decide the contracts that they offer to a common agent (in our case, the principals are suppliers and the common agent is the buyer). In economics, the common agency framework is often used to study issues related to vertical contracting or in studying regulation (e.g., regulation of public utility [agent] by distinct regulators [principals]) or in lobbying (e.g., several lobbying groups [principals] wanting to influence a decision maker [the agent]). Using this framework, we characterize the buyer's expected profit when it contracts with two suppliers for the respective components. We then consider a buyer who can make the component/material for one of the products in‐house while it buys the component/material for the other product from a supplier. Thus, in the latter (buy) scenario, the buyer's expected profit can be characterized using the existing single principal agency model. Finally, we compare buyer's expected profit in the multi‐principal (buy) scenario versus the single‐principal (make) scenario. Note that we analyze a menu of price‐quantity contracts for two reasons: (1) as mentioned before suppliers can determine contracts and menu of contracts are appropriate for situations involving asymmetric information. In fact, several papers in the literature (mentioned before) have analyzed menu of contracts; however, different from them, we analyze principals competitively determining the menu of contracts in equilibrium. (2) More importantly, we show later that the basic trade‐off in buyer's make versus buy decision exists only in the case of information asymmetry.

We find that for product substitutability higher than a threshold (but not extremely high), the buyer will prefer to procure the respective items from the two suppliers, rather than make one of the items (and procure the other from the second supplier), even when the cost of producing that item is the same for the buyer as for the supplier. Conversely, lower product substitutability (below the threshold) will result in greater profitability when making‐in‐house as compared to buying. Thus, the answer to our original question is that the buyer may find it more profitable to procure from the supplier even when it can produce the item at the same cost as the supplier. The result arises from the differences in buyer's profitability when two suppliers offer optimal contracts in equilibrium under asymmetric information versus only one supplier offering it an optimal contract under asymmetric information.

More specifically, the buyer can extract more informational rents from both the suppliers (cumulatively) due to the interaction between the two suppliers' contracts (when the buyer procures the respective items from the two suppliers) as compared to the informational rents it can extract from a single supplier (when making the other item in‐house), because of the greater competition between the two suppliers as product substitutability increases. We therefore find that beyond a certain threshold on product substitutability (but not extremely high substitutability), the buyer's profits are higher when it buys the two items from two suppliers rather than making one of the items, even when the buyer can make the item at the same cost as the supplier. On the other hand, for product substitutability below this threshold, the buyer will prefer making one item over buying both the items. Note that both information asymmetry (and the resulting differences in the informational rents) and product substitutability are the primary drivers of this result. In fact, we find that when suppliers offer wholesale price contract under complete information (i.e., when suppliers know the buyer's cost information), the buyer will never prefer buying both the items from the respective suppliers over making one item in‐house (and only buying the other item from the supplier), as long as product substitutability is not extremely high. Thus, both information asymmetry and product substitutability are critical for the trade‐off in the buyer's make versus buy decision. Also note that the comparison in profits in both the incomplete and complete information settings becomes less straightforward in the limiting case of extremely high product substitutability (i.e., when products approach perfect substitutability). Moreover, the numerical results indicate that not only the buyer but the entire supply chain's (i.e., the buyer and the supplier(s)) profits are higher beyond a threshold on product substitutability when the buyer buys from both the suppliers as compared to buying from only one supplier (and making the other item itself).

Methodologically, we contribute to the common agency problem by extending the model to consider the impact of demand uncertainty on the make versus buy decision. Including demand uncertainty in the common agency model is challenging since with the introduction of demand uncertainty, the nice structural properties (namely concavity) of the suppliers' objective function are no longer preserved. Our work therefore contributes methodologically to the common‐agency framework by investigating the impact of demand uncertainty in a price‐setting model. In the specific case of demand uncertainty, the longer lead times involved in procurement result in the buyer having to procure items before demand uncertainty is resolved. On the other hand, making‐in‐house provides greater production reactivity (by lowering lead times). As a result, the buyer can postpone production of the item until demand uncertainty is resolved (while it procures the other item from the other supplier before uncertainty is resolved). Thus, under demand uncertainty, the buyer can not only make at the same cost as the supplier, but also gains the benefits of more reactive production when making in‐house as compared to buying. We show that if product substitutability is not extremely high, then under certain sufficient conditions, including both low and high demand uncertainty, there exists a threshold (as in the case of no demand uncertainty) for product substitutability above which the buyer gains greater expected profits from buying as compared to making, and below which it makes greater expected profits from making‐in‐house as compared to buying. As with the certain demand case, the comparison in profits becomes less than straightforward for extremely high product substitutability (i.e, as products approach perfect substitutability).

From a managerial standpoint, our results indicate that a multi‐item buyer procuring different product‐specific items from separate suppliers, via quantity‐payment contract under incomplete information, may actually reduce its profits by making one of the items in‐house as compared to procuring all the items, even if its make cost is the same as the make cost of the supplier, and even when the buyer gains reactive production advantages (due to demand uncertainty) in making the item in‐house. On the contrary, under full information (with wholesale price contracts), the buyer will typically never prefer buying both the items over making one item, if its make cost is same as the make cost of the supplier. Thus, in deciding whether to make or buy an item, the buyer should also consider (besides other concerns) the potential loss of the benefits obtained from the interaction between the suppliers' quantity‐payment contracts (under incomplete information) when it moves from buying to making an item. Thus, this work uncovers new insights into how the interaction between multiple suppliers quantity‐payment contract under asymmetric information and product substitutability may effect the buyer's make versus buy decision by analyzing suppliers' equilibrium contracts using a common‐agency framework.

RELATED LITERATURE

Our work is related to the operations management literature on principal‐agent models that investigates suppliers offering contracts to manufacturer. Özer and Raz (2011) consider a (big) supplier's take it or leave it offer to a monopolistic manufacturer. Wang and Shin (2015) investigate (take‐it‐or‐leave‐it) wholesale price contracts, under complete information, offered by the suppliers of substitutable components to a monopolistic manufacturer when suppliers can determine their innovation level. Ha (2001) considers a supplier offering an optimal menu of quantity‐payment contracts to a manufacturer. Similarly, Kostamis and Duenyas (2011) also consider supplier offering an optimal menu of contract to an OEM who is privately informed of its demand forecast and production cost. Corbett et al. (2004) study a bilateral monopoly structure to investigate the value of information under differ contract types (wholesale, two‐part linear, two‐part non‐linear) and under different information scenarios (complete and incomplete). Corbett and de Groote (2000) characterize the optimal quantity‐discount contract offered by supplier to a buyer. Like these papers, we too consider supplier(s) determining the optimal menu of contract(s) and/or wholesale price contracts to the buyer where the buyer can set prices in its downstream market. However, we depart from this stream of literature by studying the problem of two suppliers (principals) determining their optimal menu of contracts that they offer to a single buyer (agent) under equilibrium. Unlike the optimal menu of contract between single principal and single agent or equilibrium wholesale price contracts between multiple suppliers and a single buyer, the problem of optimal menu of contracts offered by multiple suppliers to a single buyer cannot be investigated by using the standard principal‐agent models. The problem in our paper requires the use of multi‐principal (also referred to as common agency) framework that investigates equilibrium contract offered by multiple principals to a single agent. The main difference between single principal‐agent and common agency models is that the information rents extracted by the principals in a common agency model get moderated by the competition between the two principals, and therefore, the agent might be better off in dealing with multiple principals as compared to a single principal.

Bargaining research in supply chain context has studied interaction in parallel negotiations between competing manufacturers and suppliers—see Feng and Lu (2012, 2013a, 2013b). Similar to this literature, our work also focuses on studying interaction between competing suppliers and the manufacturer as the key driver of buyer's (the manufacturer) profitability in either making or buying the item. However, in contrast to this literature, we look at interaction between principal‐agent contracts rather than interaction in multi‐lateral negotiations. More specifically, we adopt the multi‐principal agent approach in contrast to cooperative game theory approach taken by the bargaining literature.

Our work is also related to the extensive literature on decentralized assembly systems—see Gerchak and Wang (2004), Zhang (2006), Fang et al. (2008), Granot and Yin (2008), Nagarajan and Sošić (2009), Jiang and Wang (2010), and Jiang (2015) for some notable examples. Besides a few exceptions like Kalkanci and Erhun (2012), Chaturvedi (2021), Hu and Qi (2018), and Fang et al. (2014), the literature on decentralized assembly systems does not consider issues regarding asymmetric information between the supplier and assembler. In contrast to this literature, we explicitly model information asymmetry between the suppliers and manufacturer. Hu and Qi (2018) and Fang et al. (2014) consider an assembler's contract design problem when suppliers are privately informed about their costs. Instead, we consider the suppliers' contract design problem (under a common agency setup) when the manufacturer is privately informed about its cost, that is, the suppliers determine the contracts (like Özer and Raz (2011) and Wang and Shin (2015)) rather than the assembler. Thus, unlike Hu and Qi (2018) and Fang et al. (2014), we use a common agency framework, which is more pertinent to our study of the interaction between the suppliers' contracts. Like us, Chaturvedi (2021) also considers a common agency setting with substitutable products, but unlike us, he investigates the impact of production yield uncertainty (and correlation) on suppliers' contracts. We, on the other hand, investigate manufacturer's make versus buy decision, and moreover, we consider demand uncertainty rather than supply uncertainty. Kalkanci and Erhun (2012) have also used the multi‐principal common agency model to derive the optimal menu of contracts for suppliers to offer to a price‐taking newsvendor assembler who assembles complementary components from the suppliers into a single product. In contrast, we look at a price‐setting buyer, under both certain as well as uncertain demand scenarios, who procures substitutable components from the suppliers for each of the substitutable product it is selling in the market.

Finally, our work is also related to economics literature, in particular seminal work on the multi‐principal common agency model done by Martimort (1992) and Stole (1991). We use this framework to determine the optimal quantity‐payment contracts the two suppliers (principals) should offer to the buyer (agent) in the presence of asymmetric information. Methodologically, we extend the boundaries of their work by (1) considering asymmetric principals (suppliers who have different production cost) and (2) more importantly, by including operationally relevant demand uncertainty, which results in formulations that do not have neat concavity properties. We can nevertheless characterize the incentive compatible mechanisms the two suppliers will offer in equilibrium under certain sufficient conditions for the uncertain demand case.

MODEL

A buyer (who can either be a manufacturer or a retailer) sells two substitutable products A and B in the market. Each unit of product A and B requires a unit of a critical product specific item/material (in case of buyer being the retailer, the critical items will be the product themselves), such that the item for product A cannot be used in product B and vice‐versa. We first consider the buyer buys the item for product A from supplier

$S_A$

and the item for product B from supplier

$S_B$

. Supplier

$S_A$

cannot produce the item needed for product B and vice‐versa. The buyer buys quantity

$q_i$

of the item for product i from supplier

$S_i$

for a total payment of

$x_i$

(made to the supplier). Later, we consider the buyer can make‐in‐house the item it procures from supplier

$S_B$

. ⁵ The buyer (if it is a manufacturer) makes all the other items and materials required for the two products as well as produces the two final products, for which it incurs a cost θ per‐unit of each product, that is, we assume that θ is the same for product A and B (if buyer is a retailer, then θ can be interpreted as retailer's cost of sales and operations). The substitutable products mainly differ in critical inputs (which are bought from suppliers

$S_A$

and

$S_B$

, respectively), and except for these critical inputs, the production of the two products requires very similar items and production processes. This basic model setup reflects the motivating examples mentioned in Section 1, where the buyer was selling substitutable products (natural oil‐ and silicone‐based finishing agents, mono‐crystalline and poly‐crystalline PV panels, green and black bottled olives, or different types of flat screen TVs). The two types of product required two different (and critical) inputs (in the form of different types of emulsions for chemicals or different type of silicon wafers for PV panels, or different types of olives; or different TVs). In some cases, the buyer was producing the input for one product (natural oil emulsion for chemical, poly‐crystalline wafer for the PV panels, private labeled TVs) and buying the input for the other product from a supplier, and in some cases, the buyer was buying both the inputs from the respective suppliers (retailer buying TVs from brands before introducing its private label or the food manufacturer buying different types of olives).

The buyer faces a downward demand curve for the two substitutable products characterized by:

$$p_i=a-\frac{b}{2}{q}_i+d{q}_j,\text{for}i,j\in \{A,B\}\text{and}j\ne i,$$

1

where

$p_i$

is the price of product i,

$q_i,q_j$

are, respectively, the quantities of product i, and j the buyer brings into the market. The two products are substitutable, that is,

$d<0$

and moreover

$b>0$

; that is, the price is decreasing in quantities of both the products. We also assume that

$b^2>4d^2$

, that is, the products are not perfectly substitutable (where

$b=-2d$

would imply that the products are perfectly substitutable). Note that the buyer can set prices in its market, either because of geographical differentiation or horizontal differentiation (due to differences in technology, specification). In fact, in the motivating cases discussed, the chemical manufacturer (buyer) had a reasonably big regional market share mainly because of the high transport cost to product price ratio of textile chemicals; similarly, the food manufacturer (buyer) had a big market share (around 80%) of processed foods sold in the geographic market. In the case of the retailer (buyer), it is reasonable to assume multi‐brand retailers setting retail prices (both operations and marketing literature often models retailers as setting retail price for brands and their private labels; see Corbett et al., 2004; Timoumi et al., 2021; Subramanian et al., 2010; and Shulman et al., 2010, for some examples). Moreover, in Section 5, we will consider “a” to be a random variable for capturing demand uncertainty, ⁶ but first, we will investigate the model with fixed demand to better illustrate the approach. The buyer's profit can then be characterized as:

2

We will denote

$$\begin{array}{c}\hfill v(q_A,q_B,\theta )\equiv (a-\theta )(q_A+q_B)-\frac{b}{2}\left(q_A^2+q_B^2\right)+2d{q}_A{q}_B,\end{array}$$

3

such that

${\mathrm{\Pi }}_{manf}(q_A,q_B,x_A,x_B,\theta )=v(q_A,q_B,\theta )-x_A-x_B$

.

We assume that θ is the buyer's private information and the suppliers know that θ is distributed uniformly in the range

$[\underline{\theta },\overline{\theta }]$

. As mentioned in Section 1, asymmetric information is a very common assumption in OM literature. More importantly, as we show later (by comparing buyer's make versus buy decision in the complete information setting), both information asymmetry and product substitutability are critical to the trade‐off in buyer's make versus buy decision (because we find that the buyer will typically never prefer to buy in the complete information case). The two suppliers (principals) competitively determine the quantity‐payment contracts,

$(x_i,q_i)$

for

$i\in A,B$

, they offer to the buyer (the agent). As mentioned in Section 1, the suppliers have the power to set contract terms either when the buyer is a retailer or when the buyer is a manufacturer.

A supplier i's profit is

$${\mathrm{\Pi }}_{\mathrm{supp},i}=x_i-c_i{q}_i,\text{for}i\in A,B,$$

4

where

$c_i$

is the constant marginal production cost for supplier

$S_i$

, that is, the two suppliers are asymmetric.

We will first characterize the optimal contracts that the two suppliers (principals) will offer to the buyer in equilibrium, under asymmetric information. For this, we will use the multi‐principal mechanism design approach. We will then consider the alternative scenario in which the buyer can make‐in‐house the item which it procures from supplier

$S_B$

. To avoid trivial solution, we assume that the buyer can make this item at the same marginal cost

$c_B$

(as the supplier) while it continues to buy the other item from supplier

$S_A$

. Thus, the latter scenario corresponds to the usual single principal‐agent setting under asymmetric information. We then compare buyer's profit when it is buying both the items from two suppliers versus its profit when it buys only one item from a supplier and makes the other item in‐house. We find that under certain conditions, specifically for sufficiently high substitutability (‐d), the buyer would prefer to buy from both suppliers rather than make‐in house item for product B. We also compare the buyer's make versus buy profits under complete information scenario (when suppliers offer wholesale price contracts) and find that the buyer will prefer to make in‐house the item for product B in the complete information case (even when it would have preferred to buy under the incomplete information case), thus showing that the primary driver of the above result is information asymmetry.

ANALYSIS WITHOUT DEMAND UNCERTAINTY

Although the revelation principle is a widely used tool for mechanism design, for multi‐principal models, it loses some of its bite. As Attar et al. (2008) point out, the main issue in using the revelation principle for multi‐principal settings is that at the contracting stage, the agent's (buyer) private information includes both its type (θ) as well as the contract simultaneously offered by the other principal (i.e., the other supplier). Thus, a principal could design a mechanism that is contingent on the other principal's offer, but this in turn gives rise to an infinite regress in which the other principal's contract depends on his mechanism (and so on). Of course, no such problem arises in a single principal single agent setting, wherein by using the revelation principle one can restrict the search to class of direct mechanism. Martimort and Stole (2002) showed that in the multi‐principal common agency setting, there is no loss of generality in looking for an equilibrium in non‐linear prices

$x_i(q_i)$

for

$i\in \{A,B\}$

as far as pure strategies are concerned—this is termed the “delegation principle.” In an earlier paper, Martimort (1992) showed equivalence between the non‐linear pricing mechanism and a direct mechanism approach when the agent's utility (buyer in our case) is linear in payments. Essentially, he showed that any out of equilibrium deviation in a direct mechanism would also result in an out of equilibrium deviation in a non‐linear pricing mechanism (for a price schedule suitably extended over out of equilibrium choices), which would indeed invalidate equilibrium in non‐linear pricing and would thus contradict the “delegation principle.” Therefore, without any loss of generality, one can use the direct mechanism approach à la Martimort (1992) for the multi‐principal contracting problem in this paper.

The basic idea of the direct mechanism approach in a multi‐principal setting is that a principal, say supplier

$S_A$

, designs a direct mechanism

$x_A(\theta ),q_A(\theta )$

(for a buyer of type θ) for a

${\theta }_B(\theta )$

, which the buyer (agent) reports to supplier

$S_B$

's direct mechanism, such that

${\theta }_B(\theta )=\theta$

in equilibrium.

Following this approach, we consider the direct mechanism design problem in which suppliers

$S_A$

and

$S_B$

, respectively, announce mechanism

$q_A({\theta }_A),x_A({\theta }_A)$

and

$q_B({\theta }_B),x_B({\theta }_B)$

for a

${\theta }_A,{\theta }_B$

, which the buyer communicates to

$S_A$

and

$S_B$

, respectively. We formulate the mechanism design problem from supplier

$S_A$

's perspective and then solve the equilibrium of the mechanism (i.e., the contracts) offered by both the suppliers. From

$S_A$

's perspective, its mechanism will be incentive compatible for a buyer of type θ and for a given (

$q_B,x_B$

) if:

5

Also, by using envelope theorem, we get from (5)

$$\frac{d{\mathrm{\Pi }}_{manf}^{\ast }(\theta )}{d\theta }=\frac{\partial v(q_A(\theta ),q_B({\theta }_B(\theta )),\theta )}{\partial \theta }.$$

6

Note that the partial derivative above is taken with respect to the last argument only. From

$S_A$

's perspective (whose mechanism is incentive compatible when (5) is satisfied), the buyer's profit function should also satisfy the necessary first order condition for optimality of

${\theta }_B(\theta )$

, that is,

$$\begin{array}{c}\hfill \frac{\partial v(q_A(\theta ),q_B({\theta }_B(\theta )),\theta )}{\partial q_B}·\frac{d{q}_B({\theta }_B(\theta ))}{d{\theta }_B}-\frac{d{x}_B({\theta }_B(\theta )}{d{\theta }_B}=0.\end{array}$$

7

Note that to establish global optimality of (

$\theta ,{\theta }_B$

), one should also verify joint concavity of the buyer's profit function

${\mathrm{\Pi }}_{manf}(q_A({\theta }_A),q_B({\theta }_B),x_A({\theta }_A),x_B({\theta }_B),\theta )$

in

${\theta }_A,{\theta }_B$

, in addition to (7) and (6), which only require the necessary first order conditions. We will verify concavity ex‐post to show that for the derived mechanism, the buyer would indeed report

${\theta }_A={\theta }_B=\theta$

to maximize its profit.

Moreover, before formulating the suppliers' mechanism design problem, we need to ensure the mechanism satisfies the individual rationality constraint, that is,

${\mathrm{\Pi }}_{manf}^{\ast }(\theta )\ge 0,\forall \theta \in [\underline{\theta },\overline{\theta }]$

. From Equation (6), we see that

${\mathrm{\Pi }}_{manf}^{\ast }(\theta )$

is decreasing in θ (for non‐negative quantities

$q_i$

), and therefore, one needs to ensure that

${\mathrm{\Pi }}_{manf}^{\ast }(\overline{\theta })\ge 0$

. Indeed, when designing the optimal mechanism, the supplier would limit

${\mathrm{\Pi }}_{manf}^{\ast }(\overline{\theta })=0$

. Finally, supplier

$S_B$

would formulate similar conditions for its optimal mechanism which we will jointly solve with

$S_A$

's optimal mechanism problem.

Two‐suppliers' contract design problem

To ensure that condition (5) is satisfied, we substitute

$x_A(\theta )=v(q_A(\theta ),q_B({\theta }_B(\theta )),\theta )-{\mathrm{\Pi }}_{manf}^{\ast }(\theta )-x_B({\theta }_B(\theta ))$

in supplier

$S_A$

's profit function

${\mathrm{\Pi }}_{supp,A}$

, along with (6) (which will appear as a constraint in

$S_A$

's problem). We can now characterize supplier

$S_A$

's problem as follows (supplier

$S_B$

's problem will be symmetric):

8a

such that

$${\mathrm{\Pi }}_{manf}^{\ast }(\overline{\theta })=0$$

8b

$$\frac{d{\mathrm{\Pi }}_{manf}^{\ast }(\theta )}{d\theta }=\frac{\partial v(q_A(\theta ),q_B({\theta }_B(\theta )),\theta )}{\partial \theta }$$

8c

and

$$\begin{array}{c}\hfill \frac{\partial v(q_A(\theta ),q_B({\theta }_B(\theta )),\theta )}{\partial q_B}·\frac{d{q}_B({\theta }_B(\theta ))}{d{\theta }_B}-\frac{d{x}_B({\theta }_B(\theta ))}{d{\theta }_B}=0.\end{array}$$

8d

The Hamiltonian of

$S_A$

's problem can then be expressed as

9

where λ and μ are the Lagrangian multipliers. In Lemma below, we will characterize the solution using first order conditions with respect to

$q_A$

and

${\theta }_B$

. Note that the f.o.cs only provide necessary condition for optimality. We will later establish sufficiency of f.o.cs in

$S_A$

's problem by showing that H is jointly concave in

$q_A,{\theta }_B$

for a given mechanism (allocation and payment) offered by

$S_B$

—given that

$S_A$

's contract depends on

$S_B$

's, we can only verify concavity ex‐post. All the proofs are presented in the online companion. Lemma 1

The allocation

$q_A(\theta )$

that solves supplier

$S_A$

's mechanism design problem in (8) can be characterized by the differential equation

10

under the following assumptions: H is concave;

${\theta }_B(\theta )=\theta$

at equilibrium and

$\frac{d{q}_B(\theta )}{d\theta }$

is non zero.

Note that we will verify the assumptions of Lemma 1 ex‐post, that is, after fully characterizing the mechanism offered by both the suppliers (which we do in the next Lemma and Proposition). From the definition of

$v(q_A,q_B,\theta )$

in (3), we can simplify the above differential equation to

11

The symmetric differential equation from

$S_B$

's problem would be

12

Note that, consistent with typical mechanism design approach, we have reduced the problem to differential equations (11) and (12) in allocations

$q_A$

and

$q_B$

only. However, different from a single principal‐agent type mechanism design problem, we now have two differential equations. The payments

$x_B$

and

$x_A$

can then be obtained, respectively, from the first order conditions characterized in Equation (8d) and from the symmetric equation in

$S_B$

's problem. Next, we find solutions to

$q_A$

and

$q_B$

for the differential equations (11) and (12) and show that

$\frac{d{q}_i(\theta )}{d\theta }$

is not zero. Lemma 2

For

$i\in \{A,B\}$

$$q_i=q_i(\underline{\theta })+t(\theta -\underline{\theta }),$$

13

solves (11) and (12), where

$$q_i(\underline{\theta })=\frac{(a-\underline{\theta })(b+2d)-b{c}_i-2d{c}_j}{b^2-4d^2},$$

14

for

$j\in \{A,B\}$

and for

$j\ne i$

. Finally,

$$t=\frac{b-8d\pm \sqrt{b^2+32d^2}}{4d(b-2d)}<0.$$

15

Indeed

$\frac{d{q}_i(\theta )}{d\theta }<0$

(thus verifying one of the assumption of Lemma 1). The next proposition characterizes the incentive compatible mechanism in a unique equilibrium by: (a) showing that both suppliers would offer the allocations characterized in Lemma 2 with a unique

$t=\frac{b-8d-\sqrt{b^2+32d^2}}{4d(b-2d)}$

; (b) characterizing the payment

$x_A$

and

$x_B$

subject to constraints (8b) and (8d) for

${\theta }_A={\theta }_B=\theta$

; (c) verifying the incentive compatibility of the suppliers' mechanisms characterized by the allocations in step (a) and payments in step (b) above, that is, a profit maximizing buyer would indeed report

${\theta }_A={\theta }_B=\theta$

. Note that in showing incentive compatibility, we also verify that

${\mathrm{\Pi }}_{manf}$

is concave in

$({\theta }_A,{\theta }_B)$

for the given mechanism. Finally, (d) we verify that the f.o.cs of

$S_A$

's problem are indeed sufficient for optimality (by showing that the hamiltonian for

$S_A$

's problem is concave in

$q_A,{\theta }_B$

) for the given mechanism (allocation and payment) offered by supplier

$S_B$

. Note that in showing points (c) and (d), we will have also verified the remaining assumption of Lemma 1 and thus fully characterized both suppliers' mechanisms in equilibrium. Proposition 1

The equilibrium of the suppliers' incentive compatible mechanism is characterized by allocation q in (13) determined by a unique

$t=\frac{b-8d-\sqrt{b^2+32d^2}}{4d(b-2d)}$

. Furthermore, the payment for

$i\in \{A,B\}$

in such a mechanism is characterized as

16

where

$$x_A(\overline{\theta })+x_B(\overline{\theta })=v(q_A(\overline{\theta }),q_B(\overline{\theta }),\overline{\theta }).$$

17

In (16), note that

$-1-(b-2d)t>0$

and

$t<0$

. Lemma 2 and Proposition 1 fully characterize the optimal contracts (direct mechanisms) the suppliers will offer to the buyer in equilibrium. In particular, we find that for a θ reported by the buyer, the suppliers will offer quantity as characterized by Equation (13) and payment as characterized by Equation (16). Moreover, the buyer will reveal its true θ (for such an allocation and payment mechanism) in order to maximize its profit.

Single supplier

In this section, we consider that the buyer can make the item for one of its product in‐house. Without loss of generality, we consider that the buyer can make in‐house the item required for product B, that is, it no longer procures the item from supplier

$S_B$

. The buyer continues to procure the item required for product A from supplier

$S_A$

. We further assume that the buyer can make the item for product B at the same per‐unit cost

$c_B$

as the supplier

$S_B$

, that is, the buyer does not incur any additional cost relative to the supplier

$S_B$

(and is as efficient as the supplier) in making item

$S_B$

. Moreover, the buyer's per‐unit cost of making the other components and the cost of producing the products remain the same as before at θ. These assumptions allow us to investigate whether the buyer would find it profitable to continue procuring the item from supplier

$S_B$

(rather than making the item itself) even when it can make the item as efficiently as the supplier.

Effectively, the buyer now only procures the item for product A from

$S_A$

. We are thus in the classic single principle‐agent setting. As before, supplier

$S_A$

offers a direct mechanism

$q_{A,s}({\theta }_{A,s}),x_{A,s}({\theta }_{A,s})$

that determines the respective allocation and payment for the buyer reporting its type as

${\theta }_{A,s}$

. ⁷ Using the revelation principle, one can look for optimal mechanisms, without any loss of generality, in the domain of direct mechanisms. Given that we are dealing with standard mechanism design principles, we have therefore relegated the derivation of optimal mechanism for the single supplier case to Section A1 of the online Appendix. Here, we directly state the optimal mechanism that the single supplier will offer to the buyer.

The optimal

$q_{A,s}(·)$

can be characterized as

$$\frac{\partial v_S(q_{A,s}(\theta ),\theta )}{\partial q_{A,s}}+(\theta -\underline{\theta })·\frac{{\partial }^2{v}_S(q_{A,s}(\theta ),\theta )}{\partial q_{A,s}\partial \theta }=c_A.$$

18

The payment can be characterized as

$${\displaystyle x_{A,s}(\theta )=v_S(q_{A,s}(\theta ),\theta )+{\int }_{z=\theta }^{\overline{\theta }}\frac{\partial v_S(q_{A,s}(z),y)}{\partial y}{|}_{y=z}dz.}$$

19

Comparison

We first compare the allocations (quantity of products A and B) in the single supplier case with the allocations in the two supplier case. Proposition 2

For

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

, one gets

$q_{A,s}(\theta )\le q_A(\theta )$

and

$q_{B,s}^{\ast }(\theta )\ge q_B(\theta )$

.

The above Proposition states that the buyer will increase the quantity of the item if it produces it in‐house (i.e., in the single supplier case) relative to the quantity if procuring the item from supplier (note that without any loss of generality, we have assumed the buyer produces the item for product B in‐house). Moreover, the buyer will reduce the quantity of the item it procures from the supplier in the single‐supplier case relative to the quantity it was procuring from the same supplier in the two‐supplier case. Indeed, it will cost the buyer less to produce the item for product B as compared to sourcing it from supplier

$S_B$

(note the buyer can produce the item at the same marginal cost as supplier

$S_B$

). The buyer therefore produces more of the item for product B and reduces the quantity of the item procured for product A (when it produces the item for product B) since the two products are substitutes.

Intuitive as the result may sound, it is still conditional on

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

. In fact, we find (numerically) that

$q_{B,s}^{\ast }$

could be strictly less than

$q_B$

when this condition does not hold. Then the buyer will make less of the item for product B (when making it in‐house) relative to if it is procuring it from supplier

$S_B$

, even though it will cost the buyer less to produce the item (because the supplier will add its margin). Effectively, this condition puts an upper bound on product substitutability, that is,

$-2d$

should be lower than

$b·(\frac{a-2\theta +\underline{\theta }-\max (c_A,c_B)}{a-2\theta +\underline{\theta }-\min (c_A,c_B)})$

for the result to hold. ⁸ For the symmetric supplier case (i.e.,

$c_A=c_B=c$

), this condition resolves to a simpler condition on

$-2d<b$

(as already assumed) and

$a-2\theta +\underline{\theta }-c>0$

. Basically, this condition implies that the buyer (of type θ) will procure or make strictly positive quantities of items for products A and B. On the other hand, when this condition does not hold, the buyer only produces one product due to the very high substitutability between the two products. For the sake of exposition, assume that

$c_B<c_A$

, such that the buyer only produces product B (i.e.,

$q_A=q_{A,s}=0$

) when substitutability is very high. Then, in the single‐supplier case, the buyer would evidently produce a

$q_{B,s}^{\ast }$

, which is similar to the single item monopolist quantity. However, in the two‐supplier case, the supplier

$S_B$

would offer a contract that would cost the buyer marginally more at the monopolist quantity (i.e., at

$q_B=q_{B,s}^{\ast }$

) and cost the buyer marginally less at a higher

$q_B$

, since that would incentivize the buyer to make a greater payment to the supplier and also increase its own profit. As a result, the buyer would procure more of the item for product B in the two‐supplier case relative to the quantity of the item it would make in the single supplier case.

We now compare the buyer's profit in the single supplier case (i.e., when it makes an item) with the two‐supplier case (i.e., when it procures both the items). From (8c) (for

${\theta }_B=\theta$

) and for

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

, the buyer's profit with two suppliers can be characterized as:

20

Similarly, for

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

, the buyer's profit for single supplier case can be written from Section 4.2, and from the proof of Proposition 2 (see Equation (A.32) for explicit characterization of

$q_{A,s}$

) as:

21

The next theorem characterizes the conditions under which the buyer is better off, that is, makes a higher profit procuring both the items from the two suppliers as compared to making one item in‐house at the same cost as the supplier (who previously supplied it) and procuring the other item from the second supplier. Theorem 1

For

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

, one gets

${\mathrm{\Pi }}_{manf}^{\ast }(\theta )\ge {\mathrm{\Pi }}_{manf,S}^{\ast }(\theta )$

if and only if

$-d\ge \frac{b}{\frac{3+\sqrt{17}}{2}}$

.

Theorem 1 states that for higher substitutability (i.e, when

$-d\ge \frac{b}{\frac{3+\sqrt{17}}{2}}$

) but not extremely high substitutability (i.e,

$-2d<b·(\frac{a-2\theta +\underline{\theta }-\max (c_A,c_B)}{a-2\theta +\underline{\theta }-\min (c_A,c_B)})$

), the buyer's profit is higher when it procures the items separately from the two suppliers, as compared to making one item in‐house at the same cost as the supplier (who was supplying it) while it continues to procure the other item from the second supplier. It is noteworthy that the result of the theorem does not depend on which item (for product A or product B) the buyer makes in‐house instead of procuring it from the respective supplier. ⁹ Also note that the condition

$-d\ge \frac{b}{\frac{3+\sqrt{17}}{2}}$

in Theorem 1 is both necessary and sufficient (as long as substitutability is not too high), and hence for low substitutability (i.e., for

$-d$

below the threshold

$\frac{b}{\frac{3+\sqrt{17}}{2}}$

), the buyer's profit from making one item in‐house (and procuring the other item from the other supplier) is higher as compared to its profit when it procures the items separately from the two suppliers. Note that

$-2d(a-2\theta +\underline{\theta }-\min (c_A,c_B))<b(a-2\theta +\underline{\theta }-\max (c_A,c_B))$

is a sufficient condition for the result of Theorem 1 because the comparison between profits is not as straightforward as products approach perfect substitutability (i.e., as

$-2d\to b$

), where the allocation (quantities) of one of the items might be 0. If substitutability is even higher (beyond that mentioned in the condition), then comparison of profits will also depend on the cost of the items (and obviously upon which item the buyer makes in‐house). However, such a result would be too limited given the very narrow range of product substitutability we would be analyzing (between

$b>-2d>b·(\frac{a-2\theta +\underline{\theta }-\max (c_A,c_B)}{a-2\theta +\underline{\theta }-\min (c_A,c_B)})$

), and hence we do not consider that scenario here. Also, note that except for extremely high substitutability (almost approaching perfect substitutability), the buyer's comparative profits between making one item versus buying both items do not depend on the make cost of the suppliers, nor on which item the buyer produces.

The basic intuition behind the result of Theorem 1 is that higher product substitutability increases the competition between the two suppliers (in the two‐supplier case), which allows the buyer to extract greater informational rents from the suppliers (in the two‐supplier case) as compared to the rents that the buyer can extract from a single supplier (in the single‐supplier case). Note that both higher competition and the informational rents that the buyer can extract (due to information asymmetry) are driving this result. With higher substitutability alone (and without information asymmetry, i.e., complete information) the buyer will prefer to produce more of whichever product costs less, which will disincentivize the supplier(s) to increase price in either the two‐ or the single‐supplier setting, because the buyer will otherwise order (or self‐produce) more of the other product in the two‐supplier (or single‐supplier) setting. What differentiates the buyer's profit between the single‐ and two‐supplier setting are the informational rents that it can extract, that is, with greater competition (higher substitutability), the buyer can extract more informational rents cumulatively from the two suppliers as compared to a single‐supplier.

In fact, to shed more light on the role of information asymmetry, we analyze the situation in which suppliers have complete information on the buyer's production cost θ, and they offer wholesale‐price contracts to the buyer. Comparing the buyer's profit between the two‐supplier and the single‐supplier setting in the complete information case we find (as shown in the next proposition and by the subsequent numerical experiments), the buyer will never make higher profits in the two‐supplier setting as compared to the singe‐supplier setting, as long as product substitutability is not extremely high. Proposition 3

When suppliers offer wholesale price contracts under complete information, the buyer will have to make greater profits in the single‐supplier case as compared to the two‐supplier case if and only if the below condition is true for

$x>1,z>0$

and

$y\in (-0.5,0)$

22

The above condition holds true for

$z=1$

.

The variables

$z,x,y$

in condition (22) correspond to

$z=\frac{c_B}{c_A}$

,

$x=\frac{a-\theta }{c_A}$

, and

$y=\frac{d}{b}$

. Thus, a

$z=1$

corresponds to symmetric suppliers (i.e., with similar costs). Numerically, we find that condition (22) holds true if

$-2d(a-\theta -\min (c_A,c_B))<b(a-\theta -\max (c_A,c_B))$

, that is, if substitutability is not excessively high. This condition is similar to the sufficient condition used in Theorem 1 for the incomplete information case. For extremely high substitutability, the comparison will depend on the cost of respective suppliers. Thus, from Proposition 3 and the numerical results, we find that under complete information, the buyer will never make greater profit in the two‐supplier setting as compared to single‐supplier setting, as long as the product substitutability is not excessively high (approaching perfect substitution). Whereas, from Theorem 1, we find that under asymmetric information, it will make greater profit in the two‐supplier case as compared to single‐supplier if product substitutability is above a threshold, but not extremely high.

We thus find that both, high product substitutability and information asymmetry, can together result in the buyer making greater profit with two‐suppliers as compared to one supplier. This result is in line with Martimort (1992) who has shown previously that with symmetric principals, the agent's profit is higher when products are substitutable as compared to when products are complementary. We therefore extend this insights for asymmetric principals and show that not only substitutability, but higher substitutability is required for the buyer to benefit from two principals (suppliers) as opposed to just one. Moreover, extending Martimort (1992), we find that higher substitutability increases the buyer's informational rents much more in the two‐supplier setting as compared to the single supplier, thus increasing the buyer's profits in the two‐supplier as compared to the single supplier setting as product substitutability increases.

Answering the basic make versus buy question we posed (in Section 1), the result of Theorem 1 implies that under information asymmetry and above a certain product substitutability, the buyer may find it more profitable to continue buying from a supplier even when it can produce the item in‐house at the same cost as the supplier. More specifically, for a multi‐product buyer who buys different product‐specific items from different suppliers via quantity‐payment contracts (under information asymmetry), the benefits gained from the interaction between the suppliers' contracts when procuring all the items will offset the savings in supplier margins if the buyer makes one of the items in‐house at the same make cost as that of the supplier. In the next section, we will show a similar result for uncertain product demand, that is, when the buyer makes an additional gain on production reactivity from making in‐house.

In fact the benefits of procuring from both suppliers (in the two‐supplier setting) as compared to sourcing from a single supplier are not just limited to the buyer but also extend to overall supply chain profitability (i.e., the sum of buyer and the supplier(s) profits) as product substitutability increases. In Figure 1, we show the results from numerical experiments comparing supply chain profits in the single‐ versus the two‐supplier setting. ¹⁰ Note that the marginal production costs in either setting do not change because the buyer can produce the item for product B at the same marginal cost

$c_B$

as supplier

$S_B$

. However, as seen in Proposition 2, the buyer increases the quantity of product B and decreases quantity of product A in the single supplier setting, as compared to the two‐supplier setting. Greater product substitutability further increases the differences in the production level of the two products between the single‐ and two‐supplier setting, because the buyer will increasingly prefer to produce more of the item for product B (in the single supplier setting)—since it costs less than procuring it from the supplier in the two‐supplier setting. However, this results in more unbalanced production of the two products in the single supplier setting as compared to the two‐supplier setting, which in turn negatively impacts the total supply chain profitability as product substitutability increases. Thus, interestingly, we find that not only the buyer but also the overall supply chain is also better off if both the product specific items are supplied by two different suppliers rather than the buyer making one of the items (even if the buyer can make the item at the same cost as the supplier).

FIGURE 1

Difference in supply chain profits between the single‐supplier and the two‐supplier settings. Here

$a=100,b=0.1,c_A=0.5,c_B=0.1,\theta =0.8,\underline{\theta }=0.1,\overline{\theta }=1.1$

UNCERTAIN DEMAND

In the previous section, we analyzed a buyer's make‐versus‐buy decision for a setting where there is no demand uncertainty. However, in most real‐life settings, the buyer is faced with demand uncertainty, especially since decisions like make versus buy are made well in advance of the product's launch in the market. In this section, we investigate the implications of uncertain demand on the make‐versus‐buy decision of the buyer. Moreover, the analysis in this section serves as a robustness check on whether the insights of the previous section (on a buyer's profitability from buying being higher than making as product substitutability increases, under asymmetric information) carry over to the uncertain demand scenario. Finally, incorporating demand uncertainty in the common agency model for a price setting buyer is technically challenging since the principals' (suppliers') mechanism design problem, in equilibrium, no longer preserves the neat properties that we had leveraged in the previous section to obtain our results. Thus, this section also contributes methodologically to the common agency literature in deriving the equilibrium contracts (under sufficient conditions) when a price‐setting buyer faces uncertain demand.

We now model the downward demand curve for the two substitutable products by:

$$p_i=\mathbf{a}-\frac{b}{2}{q}_i+d{q}_j,\text{for}i,j\in \{A,B\}\text{and}j\ne i,$$

23

where a is a random variable. Given that make‐versus‐buy decision is made well in advance (due to the long lead times involved in procurement), we model demand uncertainty with a two‐point distribution, such that a takes a value of

$a_h$

with probability δ and takes a value of

$a_l$

with probability

$1-\delta$

. We consider

$a_h>a_l$

such that

$a_h$

represents high demand whereas

$a_l$

represents low demand. This model of demand uncertainty is appropriate for the setting since at the time of taking the make‐versus‐buy decision, the main uncertainty on product demand faced by the buyer would be whether the product will be hit (corresponding to high demand) or a miss (low demand). Indeed, the extent of the uncertainty will depend on the exact parameter values, that is, on

$a_h,a_l$

, and δ.

Due to longer lead times in procurement as compared to producing and assembling in‐house, we assume that the buyer determines procurement quantities before demand uncertainty is realized, but produces all other components and assembles the products after demand uncertainty is resolved. More specifically, in the case of the buyer procuring both the specific items (required for each product) from two separate suppliers,

$S_A$

and

$S_B$

respectively, the buyer determines procurement volume

$q_i$

for both the items before demand uncertainty is resolved, but determines the quantity

$q_{i,f}(\mathbf{a})\le q_i$

of the respective product assembled (along with the other components produced in‐house) and brought to the market only after demand uncertainty is resolved. Thus, the buyer's expected profit for a given quantity

$q_A$

and

$q_B$

of the items it procures for product A and B, respectively, and for a payment of

$x_A$

and

$x_B$

can be characterized as:

24

On the other hand, in the case of buyer producing in‐house the item required for product B while procuring the item for product A from the supplier

$S_A$

, the buyer determines production quantity,

$q_{B,s}$

, of product B (and the items/components required for it) only after demand uncertainty is resolved, whereas it determines the procurement volume

$q_{A,s}$

of the item required for product A before demand uncertainty is resolved and decides on the quantity of other components and the quantity of product A only after uncertainty is resolved. Thus, in this case, the buyer's expected profit for a quantity,

$q_{A,s}$

, of the item for product A procured from supplier

$S_A$

for a payment of

$x_{A,s}$

can be characterized as:

25

Effectively, in the make scenario (i.e., when buyer decides to produce the item for product B in house), the buyer can determine the production quantity of the item for product B after demand uncertainty is resolved whereas it has to commit to the procurement quantity of that item from supplier

$S_B$

in the buy scenario. Thus, the buyer can leverage production reactivity better in the make scenario (due to lower production lead time) as compared to the buy scenario. As in the previous section, we assume the buyer can produce the item for product B in‐house at the same marginal cost,

$c_B$

, as the supplier

$S_B$

. In the previous section, we saw that without demand uncertainty the buyer will prefer to buy the item for product B (despite the same production cost) when substitutability between two products is high. In this section, we will like to investigate whether (and under what conditions) the buyer might prefer to buy even when demand uncertainty favors making in‐house. Finally, for tractability, we assume that suppliers face the same marginal cost of production, that is,

$c_A=c_B=c$

.

Similar to the two‐suppliers and the single‐supplier case (when demand is certain) which we analyzed in Sections 4.1 and 4.2, respectively, we now analyze the two‐suppliers and the single‐supplier case, when demand is uncertain. Specifically, with uncertain demand: (1) We first characterize the equilibrium contracts offered by the two suppliers (for the items for product A and B respectively). (2) We then characterize the contract offered by supplier

$S_A$

when the buyer only buys the item for product A from the supplier and produces the item for product B in‐house. (3) Finally, we compare the buyer's profits between the two‐suppliers case and the single‐supplier case when demand is uncertain. The specific details of the two‐suppliers and single‐supplier case for uncertain demand, along with the results on suppliers optimal contract have been presented in the online companion (in Sections A2 and A3 for the two‐suppliers and the single‐supplier case, respectively). Here, we proceed directly toward comparing the profits between the two‐suppliers and the single‐supplier case when demand is uncertain.

Comparison

We can now compare buyer's expected profit for uncertain demand when it buys items for both the products from the two respective suppliers with its expected profit when it buys only one item from the supplier and makes the other item in‐house at the same cost as the supplier from whom it procured the other item (i.e., we compare the buyer's expected profit

${\mathrm{\Pi }}_{manf}^{\ast }$

of Section A2 with its expected profit

${\mathrm{\Pi }}_{manf,S}^{\ast }$

of Section A3 under the optimal mechanisms offered in the respective scenarios). This comparison can be made for both the high and the low demand states as characterized in Proposition A1 and Proposition A2. Do note from Sections A2 and A3 that the mean of a is

$a_l+\delta (a_h-a_l)$

, and accordingly, we refer to a lower value of

$\delta (a_h-a_l)$

as the low demand state because it implies the mean of a will be closer to the lower draw

$a_l$

. Conversely, we refer to a high value of

$\delta (a_h-a_l)$

as the high demand state because it implies the mean of a will be closer to the higher draw

$a_h$

. Moreover, extreme values of δ (i.e., either when δ is close to 0 or 1—in other words, one knows with reasonable certainty whether demand will be high or low) or a low

$a_h-a_l$

imply that demand uncertainty is low. Conversely, a high

$a_h-a_l$

and/or a δ close to 0.5 (i.e., demand can be high or low with equal likelihood) implies higher demand uncertainty. Thus, we see that both low or high demand uncertainty are represented within the low or the high demand state—because low demand uncertainty will typically imply either low demand state (when both δ and

$a_h-a_l$

are low) or high demand state (when δ is high) and high demand uncertainty will typically imply high demand state (i.e., when δ is close to 0.5 and

$a_h-a_l$

is high). In the next Theorem, we show that buyer's make versus buy decision are invariant in either the low or the high demand state, that is, the buyer's make versus buy decision will not typically depend on whether demand uncertainty is high or low. Theorem 2

For either demand state, that is,

$\delta (a_h-a_l)\le c$

and

$\max (2\theta ,\frac{bc}{b+2d})<a_l-c+\underline{\theta }$

or

$\delta (a_h-a_l)\ge \frac{(b-2d)c}{b}-\frac{{(b-2d)}^2}{b}t(\overline{\theta }-\underline{\theta })$

and

$-2d<b(\frac{\delta (a_l-\theta )-c}{\delta (a_l-\theta )})$

one gets

${\mathrm{\Pi }}_{manf}^{\ast }(\theta )\ge {\mathrm{\Pi }}_{manf,S}^{\ast }(\theta )$

if and only if

$\frac{3+\sqrt{17}}{4}\ge -\frac{b}{2d}$

.

Thus, for either low demand (i.e., low

$\delta (a_h-a_l)$

) or high demand state (i.e., high

$\delta (a_h-a_l)$

), we find that higher substitutability, that is,

$-d$

greater than a threshold is both necessary and sufficient for

${\mathrm{\Pi }}_{manf}^{\ast }(\theta )\ge {\mathrm{\Pi }}_{manf,S}^{\ast }(\theta )$

as along as the underlying allocations are strictly positive (which requires conditions

$\max (2\theta ,\frac{bc}{b+2d})<a_l-c+\underline{\theta }$

and

$-2d<b(\frac{\delta (a_l-\theta )-c}{\delta (a_l-\theta )})$

in the low and high demand states, respectively). We found similar conditions in Theorem 1 of Section 4, that is, buyer profits are higher in the two‐supplier scenario as compared to the single‐supplier scenario if substitutability is higher than a threshold but not extremely high, that is, when substitutability is higher than a threshold but products do not approach perfect substitutability. Thus, irrespective of whether demand uncertainty is high or low (or even inexistent), we find that as long as product substitutability is higher than a given threshold, the buyer will make higher expected profits when buying as compared to making‐in‐house. Conversely, a lower product substitutability (below the threshold) will result in higher expected profits for making‐in‐house as compared to buying. Managerially, this result indicates that even under the significant advantages of production reactivity (due to high demand uncertainty) when making‐in‐house versus buying, higher rents due to supplier competition from high product substitutability result in buying offsetting those make‐in‐house gains, thus leading to higher expected profits in buying as compared to making.

CONCLUSION

In this paper, we have shown that when suppliers offer optimal quantity‐payment contracts under information asymmetry, the buyer may (under high product substitutability) make higher profit by procuring two items, one for each of its products, from two separate suppliers as compared to procuring only one item from a supplier and making the other item in‐house at the same cost as the other supplier. We have further shown that this result is independent of whether the buyer faces uncertain demand or not. This implies that the buyer may find it more profitable to continue buying from a supplier even when it can produce the item in‐house at the same cost as the supplier and even when it can benefit from production reactivity (while making at similar cost as the supplier, like before) as compared to buying. On the contrary, we find that the buyer will typically never buy the item over making the item (while it buys the other item) when suppliers have complete information and offer wholesale contracts. Thus, indicating that the buyer's make versus buy decision is primarily driven by the benefits that a buyer gains from the interaction between optimal contracts offered by multiple suppliers under asymmetric information. Using the multi‐principal mechanism design approach allows us to capture these interactions and characterize the conditions under which buying or making is more profitable for the buyer under asymmetric information. In particular, we show that if product substitutability is higher than a threshold (but not extremely high), then buying is more profitable, while below that threshold making in‐house is more profitable for the buyer, irrespective of whether demand uncertainty is low (or even inexistent) or high. Essentially, higher product substitutability increases buyer's informational rents more when buying both the items from respective suppliers as compared to making one of the items (and buying the other from a supplier) due to increasing competition between the suppliers in offering the contracts. In contrast, when making one of the items in‐house, the sole supplier can extract a greater surplus and moderates the higher informational rents that the buyer gets (as substitutability increases) since the supplier is the only principal and does not have to compete with the other supplier in offering the contract. Thus, it is the primarily the interaction between the contracts, under asymmetric information, offered by competing suppliers that drives the make‐versus‐buy profitability of the buyer.

These results imply that a multi‐product buyer procuring separate product‐specific items from different suppliers (via quantity‐payment contracts) can actually reduce its profits by making one of the items in‐house (when product substitutability is high), even when its make cost is same as that of the supplier and even when it can gain on production reactivity advantages (due to demand uncertainty) from making it. Of course, other considerations can determine the make versus buy decision, such as the risk of buyer's loss of control (favoring make) or focusing on core competencies or the risk of controlling feedstock for production (favoring buy). What we have shown is that under asymmetric information, the buyer can benefit from the interaction between the suppliers' contracts (when contracting with multiple suppliers) and it gives up these benefits when it decides to make rather than buy some components. Thus, these contractual interactions can also have a relevant impact on buyer's make versus buy decisions.

Conversely, the results also imply that a buyer producing one of the items in‐house as efficiently as any other supplier may want to stop production of the item and instead buy the item from a supplier. In doing so, the buyer will also willingly give up the benefits of reactive production and instead expose itself to demand uncertainty by procuring from a supplier who may have longer lead‐times as compared to making in‐house. In fact, it is not uncommon to see manufacturers spinning‐off production divisions that are as competitive (in production efficiency) as the market. The most common explanation for spinning‐off an efficient (and profitable) division is that the manufacturer can focus assets on its core competencies (i.e., not spreading the assets too thinly). However, we show that spinning‐off an efficient production division can be interesting (in terms of increasing profitability) for a manufacturer even if one ignores the benefits of increasing the return on assets.

We are considering that suppliers have all the bargaining power, that is, the suppliers offer buyer optimal contracts. One can also consider more nuanced scenarios where the buyer and suppliers' bargaining power are more distributed. To understand the impact of more distributed power, consider the opposite extreme in which the buyer has all the bargaining power. In that situation, the buyer will offer take it or leave it contracts to the suppliers. ¹¹ Since the buyer is the only principal, it will be in a position to extract all the surplus from the supply chain where it is fully informed about the suppliers, irrespective of whether the buyer is buying from one supplier or two suppliers. In such a scenario, the buyer would indeed prefer to make one of the item/product itself if it can do so at a cost that is less than one of the supplier's cost. Thus, we see that on one extreme (with buyer having all the power), the buyer will always prefer to make whereas on another extreme (suppliers having all the bargaining power) the buyer might prefer to buy. One can conjecture that as suppliers' power decreases and buyer's increases, the buyer would increasingly prefer to buy. Whether the increase is monotonic or not remains to be seen. A more complete analysis of distributed bargaining power can be done by investigating multi‐party bargaining solution under incomplete information, which would have to be a new study in itself. Finally, we have not endogenized buyer's make option in the contract terms that the suppliers offer. The suppliers contracts will not change (even after endogenizing buyer's make option) for conditions where the buyer prefers to buy under our model. The reason being that suppliers know that buyer will buy so why change contract terms which are optimal for the suppliers in equilibrium. The contract terms would indeed change (when endogenizing for make option) in scenarios where buyer prefers to make as compared to buy, because then the suppliers will offer additional surplus to the buyer in order to incentivize it to buy rather than make. However, our result makes a stronger statement in showing that even without such an outside option, the buyer would still extract sufficient rents from the suppliers to prefer buying over making (even when its make cost is same as that of the supplier).