Pricing and Quality Provision in a Supply Relationship: A Model of Efficient Relational Contracts Cristina Nistor∗

Matthew Selove†

February 14, 2018

Abstract We model how quality concerns affect relationships between a firm and its supplier. A firm concerned about uncontractible quality for a customizable good has to pay higher prices to sustain a relationship with the supplier. If the customizable good has sufficiently volatile demand, then a contract that includes a price premium only for this good cannot be sustained. Instead, the downstream firm pays a premium both for the customizable good and also for a good with more stable demand that is correlated with demand for the customizable good. To generate the greatest possible surplus, firms agree to a relational contract that fully compensates the seller for the value of its effort in each period, while also ensuring that the cost of effort in a period never exceeds the ongoing value of the relationship.

∗ †

University of Florida, [email protected] University of Florida, [email protected]

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1

Introduction

“You would expect that the customer pays us high prices for the value-added services we provide. Well, that doesn’t happen. [...] We cross-sell our other products to these customers by offering them significant breaks on the value-add products in return for their commitment to buy the book-and-ship products exclusively from us. In a way, in these relationships the commodity products subsidize the specialty products.” -Stephen Kaufman, Arrow Electronics Inc. CEO1 Many business transactions involve complex agreements that would be difficult to enforce with a formal contract. In this context, relationships become crucial: the value of the future transactions gives each party in a relationship an incentive to perform at a level beyond that which would be possible to enforce in a formal contract. For example, imagine you occasionally hire a contractor to make improvements to your home. Writing down precise details, for example, of which materials he should use for each task, circumstances in which you can request changes to the planned renovations, and how the schedule will change if problems arise, would require a long complex contract which a court might find practically difficult to enforce. However, if you establish a relationship with a particular contractor, you might reach an informal agreement that each time he works for you he will use high quality materials and make a reasonable effort to finish on time and meet your additional requests, and in return you will pay him a premium over market rates. As long as the value of the relationship to each party exceeds the cost of the additional effort or payments he must make, the relationship is sustainable. This paper formulates a model of payment schemes that will sustain a relationship between a supplier and its downstream partner at efficient quality levels. It also provides an explanation for a common phenomenon in marketing: cross-subsidized pricing. Formally, 1

Quoted in HBS Case “Arrow Electronics, Inc.” (Narayandas 1998)

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we explore how wholesale prices in a multi-product channel are affected by the downstream party’s needs for uncontractible quality of some of the goods provided. The analysis is in the context of repeated interactions, which give firms the opportunity to sustain a level of effort from the supplier that would not be possible in a one-time interaction. We develop a model that shows how channel members can use “relational contracts” (Baker et al. 2002) to provide uncontractible services to the downstream players who value them. The problem we study is quite general: relational contracts can be used in any channel where suppliers and downstream firms face a problem of uncontractible effort, costly monitoring of product quality, or a classic problem of channel coordination without a way to write enforceable contracts. Furthermore, cross-subsidized pricing is a common feature of such relationships (e.g., Narayandas 1998). Our model is a repeated game in which a supplier sells multiple goods to a downstream firm. Both players have the same information about demand, costs, and characteristics of the downstream firm. In each period, the supplier can add a dimension of quality, such as product customization, to one of the goods. After it receives the goods, the downstream firm learns the customizable good’s quality, which reveals how much effort the supplier exerted in the period. Firms cannot implement formal contracts in which payment is contingent on product quality, for example, because it would be too difficult for firms to specify and for courts to enforce such contracts (Iyer and Villas-Boas 2003). The supplier’s desire to sustain the relationship provides an incentive to exert effort. Furthermore, the downstream player pays higher wholesale prices than offered on the outside market, in order to reward the supplier’s past effort and sustain the relationship. If some goods are customizable and some are not, we might expect that the most efficient relationship (that is, the relationship that generates the greatest possible surplus) would set a price premium on the customizable goods to reflect the greater value of effort in times of high demand for these goods. However, if demand for the customizable goods is very volatile, 3

this creates a problem. When demand is very high, the premium payment the retailer is required to make may be so high that it exceeds the value of the effort that the supplier is willing to exert, given the (future) value of the relationship. Firms face a dilemma between wanting to provide higher total premium payments to reflect the greater value of effort in times of high demand and, on the other hand, not wanting to require unsustainable premium payments or unsustainable effort levels during these periods. We show that in some cases, setting price premiums on both a customizable good and a non-customizable good can help resolve this dilemma. If demand for different types of goods is correlated, but demand for non-customizable goods is less volatile, then a price premium on non-customizable goods can serve the dual role of providing the supplier with higher rewards in times of high demand while still limiting the “spikes” in these payments to a range that is acceptable to the retailer. Thus, the model has surprising insights about which types of payments can sustain optimal effort in a relational contract. We show that including a price premium on a non-customized good is sometimes an efficient way to reward a supplier for effort on a customizable good. Alternative payment schemes would result in the relationship breaking down or in lower effort. We interviewed a sales manager for a distributor that supplies fish, rice, and other inputs to sushi restaurants. The manager told us that some restaurants require frequent customization of fish. He considers them good clients who will pay more overall as they give him more future business and buy other products from his company. The manager provided us with a list of restaurants’ requests, for example, for fish to be “Clean and White please,” or for a specific size and cut of fish (“half loin please front part” or “15LBS back loin head part please !!!!!”), or even for how frozen the items should be (“BE Super Frozen Tuna Saku -1Bag BIGG” or “2lb not super frozen”). See Table 1 for a list of all customizable products sold by the distributor during an eleven month period. Instead 4

of using formal contracts to ensure these requests are satisfied, the manager said he was responsible for maintaining informal personal relationships with restaurant owners, who are willing to pay premium prices if the supplier consistently provides customized ingredients. The sushi supply manager we interviewed confirmed that some restaurants that require frequent customization of fish actually compensate the supplier with a price premium on non-customized products such a rice or seaweed.2 This arrangement is consistent with our model, as products like fresh fish have volatile demands that are quickly affected by supply shocks or other non-demand related shocks, so any payment scheme linked to the actual product being customized might not be able to sustain the relational contract that provides the customization effort. On the other hand, dry products like rice or seaweed are relatively stable goods that are perfect for serving as a basis for the premium payment to sustain the relational contract. Thus, our model provides one possible explanation for why restaurants that require customization of their fish might pay market prices for fish but a premium price for rice. More generally, suppliers of high-end restaurants often customize food ingredients to meet the particular needs of each restaurant they serve, as noted by the International Foodservice Distributors Association (IFDA) (Caldwell 2017). Writing down precise details, for example, of the storage temperature or the cut of each ingredient would require a long complex contract which a court might find practically difficult to enforce. However, if a supplier has an ongoing relationship with a restaurant, the two firms can establish an informal agreement that the supplier will make reasonable effort to satisfy the restaurant’s requests, and in return, the restaurant will pay the supplier a price premium over market rates for its ingredients. As long as the value of the relationship to each party exceeds the cost of the additional effort or payments they make, the relationship is sustainable. 2

For an example of two products that exhibit this pattern please see Appendix A.

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Product name

Table 1: Customizable Items % of Total Orders

Frozen Scottish Whole Salmon Y/F Tuna Loin Fluke(Hirame) Farm Rock Fz. Escolar Block California Uni Fz. Smoked Salmon Chunk Frozen Hamachi Fillet (Frozen) Scottish Salmon Fillet for Sushi Big Eye Tuna Loin Fz Hamachi Loin Farm Japan Asi Beff Gyoza Mushidako Octopus Fresh Hamachi Fillet(Japan) Aji Fz. Albacore Tuna Loin Fz. Escolar Block Live Mirugai(Geoduck) Madai (Japan) Fresh Kanpachi Fillet Unagi Spanish Mackeral (USA) OO-Toro Southern Blufin Atlantic Whole Salmon (Farm Raised) Tuna Ground Apex Y/F Tuna Saku AAA Blue Fin O-Toro YF Tuna Loin Bluefin Tuna Loin BE Super Frozen Tuna Saku Y.F. Tuna Loin Chillian Sea Bass BF Frozen O-Toro(Saku) Awabi (Abalone) YF Super Frozen Tuna Saku Overall Customized

9.09 3.68 2.64 2.53 2.20 1.83 1.81 1.66 0.85 0.78 0.55 0.51 0.49 0.43 0.43 0.42 0.41 0.41 0.29 0.24 0.24 0.23 0.22 0.21 0.20 0.14 0.13 0.10 0.08 0.03 0.02 0.02 0.01 0.00 0.00 32.87

% of Customized Orders 27.64 11.20 8.03 7.69 6.70 5.56 5.51 5.06 2.60 2.38 1.69 1.55 1.48 1.30 1.29 1.28 1.25 1.23 0.87 0.73 0.72 0.69 0.68 0.62 0.61 0.43 0.40 0.29 0.25 0.10 0.07 0.06 0.03 0.01 0.01 100.00

Based on sales data and customization requests from a supplier of sushi restaurants in the Southeast US, from April 2010 to February 2011.

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Arrow Electronics is another example of a company that uses a pricing structure like the one modeled in this paper to sustain a relationship with its customers. The company is a distributor for two types of micro-chips: standard commodity chips and “value-added” chips they customize for each client. The latter type of products requires effort investment from Arrow. As mentioned by the company’s CEO, the company forms relationships with clients who need customization, and these clients do not pay a price premium for valueadded products that are customized but instead pay higher margins on the standard chips. If these customers go to the outside market for the commodity products, their relationship with Arrow typically ends (Narayandas 1998). Another example of a supplier that uses this type of pricing structure is a healthcare consultant who prepares patient satisfaction reports for hospitals in a large city in the Southwestern United States.3 The clients are hospitals who have ongoing contracts for five years that can be terminated with one month notice. The clients sometimes have requests that are not written in the initial contract because it is difficult to know ahead of time what will be needed. They ask for added services, such as individual reports, results by area of management, or reports broken out by any variable in the data. The consultant generally does not charge for these extra projects, which he usually accepts, even though they are effort intensive. Instead, the price of the standard contract is set higher to make up for these added requests. The practice of having a price premium on the standard contracts and then offering extra services for free or lower prices is meant to sustain a relationship with the clients and encourage them not to price shop for each service individually, but rather think of all transactions as part of a relationship with the consultancy. Standard theoretical models of relational contracts assume bonus payments are a general function of observed performance outcomes (e.g., Levin 2003). Such models do not allow meaningful analysis of cross-subsidized pricing. These models typically assume there is only 3

This example is based on a personal interview with the consultant.

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one task or product. Also, because they assume payments can be a general function of outcomes for a task that requires effort, the principal does not pay a per-unit price premium for the task but instead offers a general bonus. By contrast, our model includes two products and a more realistic payment mechanism with linear prices. In particular, the base version of our model uses constant per-unit prices for each good, which is commonly used in practice (Schmalensee 1989). We then extend our model to allow for costly adjustments to the total payment in a given period; in this case, firms would like to set per-unit prices that generate optimal payments during the most likely states of demand, and incur adjustment costs during less likely states. We also extend the model to allow for a fixed payment component in addition to per-unit prices, similar to two-part tariffs. For both of these model extensions, we show that firms generally would like to use a relational contract that includes a premium payment on both customizable and non-customizable goods. Section 2 reviews related literature. Section 3 presents the theoretical model. Section 4 concludes with implications for channel management and directions for future research. Appendix A contains an example from a supplier of sushi restaurants. Appendix B contains formal proofs of all results.

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Related Literature

There is a growing theoretical literature on relational contracts (Baker et al. 2002; Levin 2003; Gibbons 2005; Plambeck and Taylor 2006; Li and Matouschek 2013; Halac 2012, 2015). Experimental and empirical and research has also documented how relational contracts can ensure reliable product supply (Brown et al. 2004; Macchiavello and Morjaria 2015). A key finding of this literature is discretionary payments can never become be too large or the principal would violate the contract, and effort specified in the contract can never become too large or the agent would violate the contract. Two key differences between this existing literature and our model are that we allow the supplier to sell two goods instead of just one,

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and our model uses linear payments with potential for costly payment adjustments instead of a general bonus payment. Thus, unlike the existing literature on relational contracting, our model has richer product variety and a more realistic payment mechanism in supply channels that allows cross-subsidized pricing between goods. Our paper uses an infinitely repeated game in the style of Rotemberg and Saloner (1986); Abreu (1988); Lal (1990); Dechenaux and Kovenock (2007); Thomadsen and Bhardwaj (2011); Piccolo and Miklos-Thal (2012). Like these earlier papers, we derive conditions in which repeated interactions enable firms to cooperate in equilibrium. By applying this framework to the problem of pricing and quality provision in a channel with multiple products, we generate new insights about the optimal pricing tactics that make cooperation sustainable, including cross-subsidized pricing. Unlike papers on folk theorem results, we do not focus on cases in which the discount factors approach one, and therefore it is not generally the case in our model that any outcome better than each firm’s worst case payoff is sustainable. As in standard in the theory on relational contracts (e.g., Baker et al. 2002; Levin 2003) and most other theory papers on repeated interaction (e.g., Rotemberg and Saloner 1986), we derive the most efficient sustainable equilibrium for discount factors that do not approach one. Previous theory on bundling and tying has developed models in which a monopolist requires customers who wish to purchase one product also to buy another product from the firm, as an anti-competitive measure (Whinston 1990; Rey and Tirole 2007) or to extract more surplus from consumers (Bakos and Brynjolfsson 1999). Our model involves a different, and in some sense opposite, motivation for multi-product relationships compared with this earlier literature. Rather than the supplier leveraging power over a price-taking customer to the detriment of the customer, firms in our model agree to prices that help ensure a longstanding and mutually valuable relationship persists, while providing the strongest possible incentives for the supplier to exert effort that benefits the customer. Also, in our 9

model, the supplier would not need to implement a formal legal requirement for customers to buy multiple products, as the threat of reduced supplier effort provides an incentive for the downstream firm to buy multiple products from the supplier. Literature on product line pricing has studied optimal prices when products are either complements or substitutes (Reibstein and Gatignon 1984; Dobson and Kalish 1988; Belloni et al. 2008). Our paper focuses on a different aspect of product line pricing. The goods in our model are neither substitutes nor complements on the demand side. Rather, firms agree on prices for multiple products that help sustain a reliable supply relationship with the greatest possible effort. Optimal prices in our model depend on the degree of volatility of demand for each good over time. If demand for the customizable good is sufficiently volatile, the downstream firm pays a premium both on that good and also on a basic good with more stable demand. Bargaining as a mechanism of setting wholesale prices has been analyzed in Iyer and Villas-Boas (2003). Our paper studies a different mechanism for sustaining effort, by allowing infinitely repeated interactions that make relational contracts possible. Another related stream of literature studies how information asymmetries affect channel partners (Jeuland and Shugan 1983; Desiraju and Moorthy 1997; Shaffer and Zettelmeyer 2002; Corbett et al. 2004; Busse et al. 2006; Guo and Iyer 2010). Our paper assumes that all parties in the channel have the same information about demand, costs and each other’s types. The marketing literature also has a long tradition of studying relationships and their importance in business-to-business environments, studying characteristics of how such relationships influence transactions (Spekman et al. 1998; Jap 1999; Ghosh et al. 2006; Shervani et al. 2007; Tuli et al. 2007, 2010). The present paper contributes to this literature by formulating a theoretical model of repeated interactions that provides insights into the mechanism behind prices and quality in channel relationships.

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3

Theoretical Model

The model has two risk-neutral players: a manufacturer and a retailer. The upstream firm can sell two goods to the downstream firm. Good one is a customizable good, the value of which increases if the seller invests (non-contractible) effort into customizing it for the retailer. Good two is a basic good, which cannot be customized. The players trade repeatedly at dates t = 0, 1, 2.... Let good one, which has an uncontractible quality dimension, be denoted by subscript u, while the basic good, good two, is denoted by b. On-time delivery, consistent good customer service, personal help, and even customized improvements to the physical good are possible examples of quality dimensions that may be uncontractible or unknown, such as in Iyer and Villas-Boas (2003), at the time the bargaining process over prices takes place. The model allows for these uncontractible quality dimensions to be quite general: it suffices that a formal contract may be hard to enforce, or hard to specify or costly to monitor in order for the results to be valid. The firms find it impossible to write completely enforceable contracts because of the complexity of the transactions (for example), but their desire to continue to do business with each other in the future acts as an incentive to maintain a relational contract. Under certain conditions, this relationship is self-enforcing and leads to optimal quality for the buyer because both parties fear the loss of future benefits if they deviate from cooperating. As is standard in the relational contracting literature (e.g., Baker et al. 2002; Levin 2003), including models of supply relationships (Macchiavello and Morjaria 2015; Andrews and Barron 2016), we abstract away from modeling the downstream firm’s customer demand. Rather, we assume the retailer needs quantities qt,u and qt,b of goods one and two, respectively, in period t. These amounts vary randomly from period to period, and we allow for qt,u and qt,b to be correlated in each period, with joint distribution F (qt,u , qt,b ) that is independent and identical across time periods. For now, the only other restriction we place on these

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distributions is that they are bounded, so there is a maximum possible quantity of each product. We later analyze the model for specific quantity distributions that allow us to explicitly derive optimal contracts that include a price premium on both products. The downstream firm’s utility from buying from the seller at time t is given by

URt = qt,u (αu + βet − Pt,u ) + qt,b (αb − Pt,b )

(1)

with et the non-contractible effort per unit put in by the seller at time t, where et ∈ [0, 1]. Setting et = 1 implies the supplier fully meets all of the customization needs of the retailer for period t, whereas et < 1 implies these needs are not fully met. β is the per-unit benefit of customization effort to the retailer, Pt,u and Pt,b are the unit prices at time t, and αu and αb are constants. The seller’s utility from selling to the retailer at time t is given by

USt = qt,u (Pt,u − γet ) + qt,b Pt,b

(2)

with γ being the per-unit cost of effort, where β > γ > 0, so the benefit to the retailer of customization effort exceeds the cost to the supplier.4

Note that optimal effort for

maximizing total channel profits is given by et = 1 in each period. This is the optimal per unit effort, which implies that optimal total effort costs for the seller are higher when quantity qt,u is high. We have assumed a linear effort costs, similar to Bond and Gomes (2009) and Ludwig et al. (2011), so that we can explicitly derive the maximum sustainable effort and the efficient price levels for particular demand distributions. Both firms have the option to buy or sell the goods on the outside market at unit prices P¯u and P¯b , where αu > P¯u > 0 and αb > P¯b > 0. This outside option does not provide 4

Without loss of generality, we have set marginal production costs equal to zero, so the only costs in the model are effort costs.

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customization to the retailer and does not reward the supplier for the effort expended on customization. For example, the outside market may consist of low-end retailers who do not value customization and low-end suppliers who cannot serve customization requests in a cost-effective manner. Thus, for periods in which they trade on the outside market, the players utilities are given by:5

URt = qt,u (αu − P¯u ) + qt,b (αb − P¯b )

(3)

USt = qt,u P¯u + qt,b P¯b

(4)

The retailer wants to maximize

P∞

t=0

δ t URt while the supplier wants to maximize

P∞

t=0

δ t USt ,

where each firm’s discount factor is denoted by δ ∈ (0, 1). 3.1

Timing

The parties initially agree on a relational contract (Pu , Pb , e(qt,u )) with constant unit prices Pu ≥ P¯u and Pb ≥ P¯b for each good, and effort which can be a function of quantity.6 It is important to allow per-unit effort to vary with quantity because in some cases the optimal effort cannot be sustained if quantity is very large. The restriction to contracts with constant unit prices, which is commonly found in many industries, has been justified on the grounds of simplicity (Schmalensee 1989). One way to interpret the constant unit prices assumption is that there are often costs to adjusting unit prices (Levy et al. 1997; Zbaracki et al. 2004). The results in this paper show how firms can set prices of two different goods to achieve 5

The seller’s utility represents the profits generated from the units of the goods it otherwise would have sold to the retailer if they had traded with each other instead of on the outside market. 6 With minor modifications to the model, we could allow a contract with one or both prices below those on the outside market, but such prices would not be optimal under the conditions of any of our formal results. For simplicity of exposition, we have restricted the contract to prices weakly greater than those on the outside market.

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some flexibility in their pricing without the need to incur such price adjustment costs.7 We later present a model extension in which firms can agree on a different payment level for a given period than the one implied by their relational contract, if they incur an adjustment cost in that period. After the players agree to their relational contract, they have the option to trade at times t = 0, 1, 2... based on this contract. At each time t, the game has 3 stages: 1. Nature draws qt,u and qt,b and the buyer and the seller observe this demand for that particular period. 2. The buyer can either agree to pay the unit prices from the relational contract, Pu and Pb , or it can deviate and go to the outside market, which offers P¯u and P¯b . 3. The seller decides how much effort to put into providing the service, et . The buyer observes this choice. Note that the buyer commits to a payment before observing the seller’s effort level. Therefore, the seller has an opportunity to shirk on the contract by accepting payment for the goods while providing less effort than agreed by the relational contract.8 If effort levels were legally enforceable, then the parties could always achieve optimal effort with a formal contract, and a relational contract would not be necessary. 3.2

Results

A subgame perfect equilibrium consists of an initial relational contract and strategies for each player after each possible history. There is a bad equilibrium in which the supplier 7

Recall that, for simplicity, we assume outside market prices are constant over time. If we instead allowed outside market prices to vary over time, then results equivalent to ours would hold if we allowed contracts with a constant per-unit price premium for each product, that is, a constant price mark-up over market prices. 8 Most of the theoretical literature on relational contracting assumes the agent decides effort before the principal decides the bonus payment, in which case the principal can benefit from the effort but then shirk on the bonus (e.g., Levin 2003). Using this alternative timing assumption would not significantly affect our key results. We believe the timing assumption used in this paper, in which the payment is decided before effort, is more realistic in the context of supply chain relationships.

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always invests zero effort, and the retailer always goes to the outside market. We will focus on equilibria in which the firms revert to this bad equilibrium if either player ever deviates from its equilibrium strategy, which provides the strongest possible incentives for each player to follow its equilibrium strategy in all periods. Given that effort is perfectly observable in our model, both firms always know with certainty whether either player has deviated. We define an equilibrium as “efficient” if it generates weakly greater total surplus than any other equilibrium, that is, if it maximizes the equilibrium value of:9

E

X ∞



t

δ (URt + USt )

(5)

t=0

We will also say that one relational contract is “more efficient” than another if the former contract leads to an equilibrium with greater surplus than any equilibrium under the latter contract. The most efficient possible outcome occurs if the firms always trade with each other and the supplier sets et = 1 in every period. For such a relationship to be sustainable in all periods, there can never be a time when either firm is better off deviating than staying in the relationship. Therefore, we first derive conditions for both the supplier and the retailer to stay in a relationship in which they trade with each other during all periods.10 At any time t, the utility for the buyer if it stays in the relationship is given by its utility in the present period added to the discounted stream of utilities it gets in the future while 9

In principle, for any equilibrium that maximizes this surplus generated, firms could divide the surplus in any manner they choose with an initial lump sum transfer between the players. 10 We refer to any equilibrium in which firms at least occasionally trade with each other as a “relationship.” Relationships can exist in which firms trade on the outside market during some periods. Our formal results derive conditions in which firms trade with each other during all periods, but the proofs in the appendix also account for the possibility of other alternative equilibria, in which firms sometimes trade with each other and sometimes trade on the outside market on the equilibrium path.

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it is in the relationship:

qt,u (αu +βet (qt,u )−Pu )+qt,b (αb −Pb )+

∞ X

h i δ (T −t) E qT,u (αu + βeT (qT,u ) − Pu ) + qT,b (αb − Pb ) (6)

T =t+1

Note that we allow effort et to be a function of quantity qt,u as specified in the relational contract mentioned previously. On the other hand, if the retailer deviates and goes to the outside market, the relationship will end. In this case, prices in this period and all future periods will be P¯u and P¯b , and effort will be zero in the current period and all future periods. Thus, the retailer’s utility if it deviates at any time t is given by:

qt,u (αu − P¯u ) + qt,b (αb − P¯b ) +

∞ X

δ

(T −t)

h i ¯ ¯ E qT,u (αu − Pu ) + qT,b (αb − Pb )

(7)

T =t+1

By comparing expressions (6) and (7) and rearranging terms, we see that the retailer will always want to stay in the relationship if at each time t:  (Pu − P¯u ) qt,u +

     δ δ δ E[qT,u ] + (Pb − P¯b ) qt,b + E[qT,b ] ≤ β qt,u et + E[qT,u eT ] (8) 1−δ 1−δ 1−δ

where et and eT are current and future effort levels based on the relational contract. Intuitively, the present value of all the premium payments the retailer makes must be less than the present value of the benefits it receives due to the supplier’s higher effort in order for the relationship to be sustainable. The utility for the supplier if it stays in the relationship at any time t is:

qt,u (Pu − γet (qt,u )) + qt,b Pb +

∞ X

δ

(T −t)

h i E qT,u (Pu − γeT (qT,u )) + qT,b Pb

(9)

T =t+1

On the other hand, if the supplier deviates and provides lower effort than specified in the

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relational contract, the relationship ends and the supplier only receives prices P¯u and P¯b in all future periods. Therefore, at any time t, the supplier’s utility if it deviates and provides zero effort is: qt,u Pu + qt,b Pb +

∞ X

δ (T −t) E[qT,u P¯u + qT,b P¯b ].

(10)

T =t+1

By comparing expressions (9) and (10) and rearranging terms, we see that the supplier will always want to stay in the relationship if at each time t:   δ δ δ ¯ ¯ E[q ] + (P − P ) E[q ] ≥ γ q e + E[q e ] (Pu − Pu ) T,u b b T,b t,u t T,u T 1−δ 1−δ 1−δ

(11)

where et and eT are current and future effort levels based on the relational contract. Intuitively, the present value of the premium payments the supplier receives in all future periods must be greater than the present value of its cost of effort in the current period and all future periods in order for the supplier to stay in the relationship. We first study the case in which optimal effort is always sustainable. In this case, we prove that it is possible to create an efficient relational contract that sets a price premium only on the good with uncontractible quality. However, if demand for the good with uncontractible quality is too volatile, it is not possible to sustain the optimal effort in all periods. In this case, we derive conditions in which a contract that places a premium on both goods leads to a more efficient outcome than a contract that places a premium only on the customizable product. A necessary condition for the relationship always to be sustainable is that the retailer’s incentive constraint to stay in the relationship, given by (8), must hold in all time periods. Furthermore, a necessary condition for (8) to hold in all periods is that the expected total

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premium payment must be less than the retailer’s expected benefit of effort:11

(Pu − P¯u )E[qT,u ] + (Pb − P¯b )E[qT,b ] ≤ βE[qT,u eT ]

(12)

Combining this condition with the supplier’s constraint (11) and rearranging terms implies the following:

γqt,u et ≤ (β − γ)

  δ E qT,u eT 1−δ

(13)

Intuitively, for the relationship to be sustainable, the cost to the supplier of providing the required effort at time t must be less than the total expected discounted value of maintaining the relationship. If we insert the expression for optimal effort, et = 1, for all time periods, then for condition (13) to hold in all time periods requires:

γ max(qt,u ) ≤ (β − γ)

  δ E qT,u 1−δ

(14)

This condition states that optimal effort can be sustained only if the maximum quantity of the customizable good is small enough relative to its expected quantity, and if firms place enough weight on the future. Intuitively, the expected value of the relationship must be large enough to compensate for the effort required in times of greatest demand. Proposition 1. If it is possible to sustain optimal effort, et = 1, for all levels of demand (that is, if inequality (14) holds), then there is an efficient relational contract that sets a price premium only on the good with uncontractible quality (Pu > P¯u , Pb = P¯b ). Intuitively, in cases where optimal effort is sustainable, a premium on the good with uncontractible quality can be used to compensate the supplier for the full value of effort to 11

If this condition did not hold, it would imply that there must be quantity levels that occur with positive probability for which (8) is violated.

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the retailer in each period. The supplier is then willing to make the required customization effort in each period if condition (14) holds. Under the conditions of this proposition, there could also be other efficient equilibria that include a price premium on both goods, or only on the basic good. In particular, if firms are sufficiently patient (δ is sufficiently close to 1), a variety of different payment structures can sustain optimal quality, as long as average premium payments fall in the range that compels both parties to stay in the relationship. Thus, this proposition guarantees the existence of an efficient equilibrium with a premium payment only on the good with uncontractible quality, but it does not rule out other efficient equilibria. On the other hand, in cases where optimal effort is not always sustainable, we will show that there is no sustainable relationship in which firms always trade with each other with a price premium only on the customizable good. Formally, if we allow Pu − P¯u > 0 but impose the constraint Pb − P¯b = 0, then the buyer’s condition to stay in the relationship, given by (8), becomes:

 (Pu − P¯u ) qt,u +

   δ δ E[qT,u ] ≤ β qt,u et + E[qT,u eT ] 1−δ 1−δ

(15)

The seller’s incentive condition to stay in the relationship, given by (11), becomes:  δ δ γ qt,u et + E[qT,u eT ] ≤ (Pu − P¯u ) E[qT,u ] 1−δ 1−δ 

Multiplying both sides of (15) by

γ Pu −P¯u

and both sides of (16) by

β , Pu −P¯u

(16) and combining

these two constraints, implies that the following must hold:

 γ qt,u +

 δ δ E[qT,u ] ≤ β E[qT,u ] 1−δ 1−δ

(17)

Rearranging terms shows that this condition is equivalent to (14) during periods with peak demand of the customizable good, that is, when qt,u = max(qt,u ). Therefore, if (14) does not 19

hold, then it is not possible for both (8) and (11) to hold during all periods for any relational contract that has a price premium on the customizable good but not on the basic good. Intuitively, when optimal effort cannot be sustained during periods with highest demand for the customizable good, the premium payment during such periods must be lowered to reflect the seller’s actual effort level during peak demand, in order to induce the buyer to make this payment. However, if we constrain the contract to include a price premium only on the customizable good (and not on the basic good), then reducing the payment during peak demand proportionally reduces the payment level during other periods, resulting in even weaker effort incentives, which requires an additional payment reduction during peak demand to reflect the lowered effort levels, and so on, until any potential contract unravels. Proposition 2. If it is not possible to sustain optimal effort, et = 1, for all levels of demand (that is, if inequality (14)does not hold), then it is not possible to sustain any relationship in which firms always trade with each other with a price premium only on the good with uncontractible quality. This proposition rules out certain equilibria in which firms always trade with each other. Under the conditions of this proposition, in some cases there exists an equilibrium, with a price premium only on the good with uncontractible quality, in which firms trade with each other during periods of low demand and trade on the outside market during high demand. However, such an equilibrium is less efficient than an alternative equilibrium, which we will now derive, in which firms trade with each other during all periods. The next step in the analysis is to identify an arrangement that gives both the seller and the buyer the appropriate incentives to sustain a relational contract during all periods, even though optimal effort (first best) is not always sustainable. We will now derive an efficient contract that includes premium payments on both goods instead of only on the customizable good. To illustrate how this can occur, and to allow us to explicitly derive the most efficient

20

contract, we assume the quantities traded for each product have the following distribution:

P rob(qt,u = L, qt,b = L) = 1 − ω

(18)

P rob(qt,u = H, qt,b = M ) = ω

(19)

where 0 < L < M < H and ω ∈ (0, 1). Note we are assuming demand for the goods is perfectly correlated, and there are only two possible states of demand in each period: one state in which both goods have low demand, and another state in which both goods have relatively high demand. In the next section of the paper, we present an extension that allows demand for the two goods to be uncorrelated or imperfectly correlated.12 We focus on the case in which the quantity distributions satisfy these conditions:13  γM < (β − γ)

 γH > (β − γ)

δ 1−δ

h i (1 − ω)L + ωM

(20)

δ 1−δ

h i (1 − ω)L + ωH

(21)

Note that (21) implies that (14) is violated, and thus the demand spikes for the good with uncontractible quality are so large that first-best effort (et = 1) cannot be sustained during high demand. We will now derive the optimal second-best contract which is sustainable during all periods. This second-best contract involves setting et = 1 during low demand, and setting the maximum sustainable effort, denoted e∗H , during high demand. We will show that the optimal high-demand effort level can be found by solving the following equation: 12 For simplicity, we have assumed demand for the two goods is the same (qt,u = qt,b = L) in the low condition. However, this assumption could be relaxed, and the prices for the good that does not have uncontractible quality could be adjusted accordingly to produce equivalent results.   13

It is possible to choose values of M and H that satisfy both of these conditions if γ < (β −γ)

in which case (20) holds if M sufficiently close to L, and (21) holds if H is sufficiently large.

21

δ 1−δ

<

γ ω,

γHe∗H = (β − γ)

i δ h (1 − ω)L + ωHe∗H 1−δ

(22)

When this equation holds, effort costs during high demand equal the expected discounted value of the relationship, and so this effort level is sustainable. Note that each side of (22) is a linear function of e∗H . Furthermore, (20) implies the left side of (22) is less than the right side if we set e∗H =

M , H

whereas (21) implies the left side of (22) is greater than the right side

, 1) that satisfies equation if we set e∗H = 1. Therefore, there is a unique value of e∗H ∈ ( M H (22). By solving equation (22), we find that: # " δ (1 − ω) (β − γ) L 1−δ e∗H = δ H γ − (β − γ) 1−δ ω

(23)

Equation (23) gives the maximum sustainable per-unit effort level during high demand. Note that e∗H is increasing in the marginal benefit of effort (β) and the weight firms place on the future (represented by δ), but decreasing in the marginal cost of effort (γ) and the relative magnitude of high-demand quantity of the customizable good (H). Given two prices and two possible demand states, the firms can design a contract with the precise total payment they would like to set in each demand state.14 We show that the relational contract can sustain effort et = 1 during low demand and et = e∗H during high demand by setting the following prices:

Pu − P¯u =



 He∗H − M β H −M

Pb − P¯b = β − (Pu − P¯u )

(24)

(25)

As the maximum sustainable effort during high demand (e∗H ) falls, the contract shifts the 14

In the model extension that follows, there are four demand states and two prices, and so the firms sometimes need to incur an adjustment cost to achieve their desired payment level.

22

price premium toward the basic good, that is, Pu − P¯u decreases and Pb − P¯b increases by the same amount. As e∗H →

M , H

the contract converges to placing the entire premium payment

on the basic good. On the other hand, as e∗H → 1, the contract converges to placing the entire premium payment on the good with uncontractible quality. Proposition 3. If the quantity distributions satisfy conditions (20) and (21), then optimal effort, et = 1, cannot be sustained during high demand, and the most efficient relational contract includes premium payments on both goods, with the high-demand effort level given by (23) and prices given by (24) and (25). Intuitively, including a price premium on the more stable, non-customized good provides larger total premium payments to the supplier in times of high demand without letting the total premium payment grow so large that the retailer would want to exit the relationship. This same contract is not sustainable if the same premium payment is placed only on the good with uncontractible quality because the premiums become more than the retailer is willing to pay during the large spikes in quantity traded of this good. Thus, if demand for the good with uncontractible quality is sufficiently volatile, but demand for the basic good is not too volatile, an efficient relational contract includes a premium payment on both of the goods. As is standard in the theoretical contracting literature, we have focused on the most efficient contract in terms of total surplus generated, without analyzing how that surplus is shared between the players (see Weitzman 1980; Bolton et al. 2005). In fact, in our model, all of the surplus goes to the supplier, in order to provide the strongest possible incentives for supplier effort during high demand. In principle, firms could split the surplus differently by using an initial lump sum transfer. For example, the supplier could offer the retailer a one-time “new customer discount” on the first payment as they begin their relationship. Because this initial payment occurs before any effort by the supplier, such a discount would not affect the supplier’s effort incentives. Therefore, this slightly modified version of the 23

relationship we have derived, with an initial lump sum transfer to the buyer, would also be sustainable. 3.3

Numerical Example

We now present a numerical example in which optimal effort is not sustainable during high demand. To illustrate the benefit of including a price premium on both products, we compare the optimal relational contract derived in the previous section (which includes a premium payment on both goods) with a relational contract that includes only a fixed bonus payment in each period.15 We use the following parameter values: Table 2. Parameter values used in the numerical example L=5

Quantity of each good during low demand

M = 10

Quantity of basic good during high demand

H = 25

Quantity of customizable good during high demand

ω = 0.25

Probability of high demand in each period

γ = 1.0

Per-unit cost of effort for the supplier

β = 1.5

Per-unit benefit of effort for the buyer

δ = 0.8

Each firm’s discount factor

These parameter values imply much larger demand spikes for the customizable good than for the basic good (H is larger than M ). In particular, these parameters satisfy conditions (20) and (21), so Proposition 3 applies. Based on the derivations in the previous section, the optimal contract given these parameter values specifies effort levels e(L) = 1 and e(H) = 0.6, and prices Pu − P¯u = 0.5 and Pb − P¯b = 1.0. Thus, the most efficient sustainable contract involves 60% of the first-best effort level during high demand, and this contract places more of a premium on the basic good than on the customizable good. The expected discounted 15

This comparison involves a slight modification of our model, as the analysis in the previous sections did not allow for fixed bonus payments.

24

  1 value generated by this relationship is (β − γ) 1−δ E qT,u eT = 18.8.16 For comparison, we now compute the best possible contract based on a fixed bonus payment in each period. Formally, we allow the total payment the buyer makes to the seller in each period to take the form qt,u P¯u + qt,b P¯b + y, where the bonus payment is y ≥ 0. All other aspects of the model set-up remain the same. Given this alternative payment arrangement, the analysis from the previous section no longer allows us to compute the optimal contract directly. Therefore, we search for a sustainable contract with the following numerical procedure: (i) Initialize by setting effort levels e(L) = e(H) = 1. (ii) Compute the maximum bonus the buyer would pay based on the current effort levels and a constraint analogous to (8). (iii) Compute the maximum effort levels the seller would exert given the current bonus level and a constraint analogous to (11). (iv) Repeat steps (ii) and (iii) until the algorithm converges. Based on this procedure, we find that it is possible to sustain effort levels e(L) = 1 and e(H) = 0.45, with a per-period bonus of y = 9.38. The total expected value generated by   1 E qT,u eT = 16.4. Thus, compared with a fixed bonus payment, this contract is (β − γ) 1−δ the optimal contract results in 33% higher effort during high demand (0.6 vs. 0.45) and 14.3% greater expected value of the contract (18.8 vs. 16.4). The optimal contract, which includes a price premium on both goods, provides the strongest possible incentives for effort by fully compensating the supplier for the value of its effort in each period. Any contract with different payments than the optimal contract either 16

This expression includes the expected value of effort in the current period (in addition to future periods), 1 δ which is why it includes the term 1−δ rather than 1−δ .

25

is not sustainable or results in weaker effort incentives. The fixed bonus contract derived above results in weaker effort incentives because it does not take advantage of the buyer’s willingness to pay a higher bonus during high demand. 3.4

Model Extension: Payment Adjustment Costs

In the previous sections, firms agreed to a relational contract with constant unit prices. Ideally, firms would like to use such a contract, or other simple contracts, to generate the optimal premium payment for every possible state of demand. However, if the number of possible states of demand is too large, it may be too complicated to construct a contract that specifies the optimal payment for every possible state of demand. In such cases, firms could agree to a contract that generates optimal payments for the most likely states of demand, while also agreeing to negotiate a different premium payment than the one specified by the contract during periods when relatively unlikely states occur. In this section, we extend our model to allow firms to negotiate a different premium payment than the one implied by the unit prices in their contract, if they incur an adjustment cost for the period. For example, suppose firms agree to a contract that includes a premium payment on both a basic good and a customizable good. This contract may yield optimal payments when the quantity of both goods is low and when the quantity of both goods is relatively high. However, the firms may want to negotiate a lower total payment than the one implied by these prices during periods when demand for the basic good is high but demand for the customizable good is low, in order to reflect the relatively low value of effort during such periods. Similarly, they may want to negotiate a higher total payment than the one implied by the contract’s prices during periods when demand for the basic good is low but demand for the customizable good is high. More generally, if there are many possible states of demand, and it is too complicated to construct a contract that specifies the optimal payment for every possible demand state,

26

firms could set prices that generate the optimal payment for the most likely demand states, while also reaching an understanding of the types of situations in which they will deviate from these prices. Formally, the relationship derived in this model extension represents an incomplete contract (Hart and Moore 1999). We assume firms initially agree to a relational contract that specifies values (Pu , Pb , e(qt,u )), representing unit prices for each good and the seller’s effort as a function of quantity of the customizable good. However, in each period, we allow firms to agree to a different payment than implied by these prices, similar to the model of procurement contract renegotiation by Bajari and Tadelis (2001).17 The timing in each period is the following: 1. Nature draws qt,u and qt,b and the buyer and the seller observe this demand for that particular period. 2. The seller has the option, if it incurs an adjustment cost d, of proposing an alternative bonus payment Bt (in excess of what the buyer would pay on the outside market) for the current period, which can be different than the bonus implied by the prices in the relational contract. 3. If the seller has proposed such an alternative payment, the buyer can either agree to make this proposed payment, or it can go to the outside market. If the seller has not proposed an alternative payment in the current period, the buyer can either agree to pay the unit prices specified in the relational contract, or it can go to the outside market. 4. The seller decides how much effort to put into providing the service, et . The buyer observes this choice. 17 In principle, we could allow firms to negotiate new prices that continue to hold after the adjustment. However, it is more efficient to negotiate a one time payment adjustment and then revert to the original contract prices, if the original prices were chosen optimally for the given demand distributions.

27

When the seller proposes and the buyer agrees to an alternative bonus Bt , the buyer’s utility for the period is:

URt = qt,u (αu + βet − P¯u ) + qt,b (αb − P¯b ) − Bt

(26)

And the seller’s utility for the period is:

USt = qt,u (P¯u − γet ) + qt,b P¯b + Bt − d

(27)

where the adjustment cost for the period is represented by d > 0. For this modified version of the model, we characterize subgame perfect equilibria based on an initial relational contract (Pu , Pb , e(qt,u )), a set of demand states D for which the seller will propose an alternative bonus, and the alternative bonus payment B(qt,u , qt,b ) for each demand state (qt,u , qt,b ) ∈ D. As before, there is a bad equilibrium in which the buyer always goes to the outside market and the seller always invests zero effort, and firms revert to this bad equilibrium if either party ever deviates from the proposed equilibrium. We now generalize the demand distribution used in the final proposition of the previous section, allowing for imperfect correlation in product quantities. In particular, we assume the quantities traded for each product have the following distribution:

P rob(qt,u = L, qt,b = L) = (1 − ρ)(1 − ω)2 + ρ(1 − ω)

(28)

P rob(qt,u = L, qt,b = M ) = (1 − ρ)ω(1 − ω)

(29)

P rob(qt,u = H, qt,b = L) = (1 − ρ)ω(1 − ω)

(30)

P rob(qt,u = H, qt,b = M ) = (1 − ρ)ω 2 + ρω

(31)

where 0 < L < M < H, ω ∈ (0, 1), and ρ ∈ [0, 1]. For these distributions, each product has probability (1 − ω) of having a low quantity and probability ω of having a relatively 28

high quantity. The parameter ρ reflects their strength of correlation, with ρ = 0 implying no correlation and ρ = 1 implying perfect correlation. If the correlation in product demands is sufficiently strong, then the two most likely states are that both products have low demand and that both products have relatively high demand. In particular, states (L, L) and (H, M ) are each more likely than state (H, L) and more likely than state (L, M ) if the following holds:  ρ > max

−1 + 2ω 1 − 2ω , 2ω 2 − 2ω

 (32)

If the correlation in product quantities is strong enough that this conditions holds, an efficient contract uses unit prices to generate optimal payments at states (L, L) and (H, M ), and firms make price adjustments at states (H, L) and (L, M ). Furthermore, we show that such a contract involves premium payments on both the customizable and the basic good. The derivations of this contract are similar to those for the contract described in Proposition 3. We now present a brief version of these modified derivations. We focus on cases in which quantity distributions satisfy these conditions.  γM <

 γH >

δ 1−δ

h i (β − γ)[(1 − ω)L + ωM ] − 2d(1 − ρ)ω(1 − ω)

(33)

δ 1−δ

h i (β − γ)[(1 − ω)L + ωH] − 2d(1 − ρ)ω(1 − ω)

(34)

Note these are modified version of conditions (20) and (21). The term 2d(1 − ρ)ω(1 − ω) reflects expected adjustment costs in each period, that is, the adjustment cost, d, times the probability of state (H, L) or (L, M ) occurring. Condition (34) implies first-best effort cannot be sustained during high demand. Given these conditions, if the adjustment cost d is sufficiently small, then the optimal contract involves setting et = 1 during low demand and et = e∗H during high demand, where the 29

maximum sustainable effort during high demand is found by solving:

γHe∗H

 =

δ 1−δ

h i (β − γ)[(1 − ω)L + ωHe∗H ] − 2d(1 − ρ)ω(1 − ω)

(35)

Conditions (33) and (34) ensure there is a value e∗H ∈ ( M , 1) that solves this equation. H Solving for e∗H , we find: 1 e∗H = H

"

δ 1−δ

 # (β − γ)(1 − ω)L − 2d(1 − ρ)ω(1 − ω) γ−

δ (β 1−δ

− γ)ω

(36)

Note that the maximum sustainable effort during high demand is decreasing in the adjustment cost d and increasing the the strength of product quantity correlation ρ. As in Proposition 3, optimal prices as a function of e∗H are given by the following: Pu − P¯u =



 He∗H − M β H −M

Pb − P¯b = β − (Pu − P¯u )

(37)

(38)

Firms make payment adjustments at states (H, L) and (L, M ), and the associated bonus payments are: B(H, L) = βHe∗H

(39)

B(L, M ) = βL

(40)

Proposition 4. If d is sufficiently small and the quantity distributions satisfy conditions (32), (33), and (34), then optimal effort, et = 1, cannot be sustained during high demand, and the most efficient relational contract includes premium payments on both goods, with the high-demand effort level given by (36), unit prices given by (37) and (38), and payment adjustments given by (39) and (40). Thus, if there is a sufficiently strong correlation in demand for a customizable good 30

with very volatile demand and a basic good with less volatile demand, including a premium payment on both goods can generate an optimal payment when demand for both goods is low and when demand for both goods is relatively high. The firms can then make costly payment adjustments when other less likely states occur.18 The stylized example in this section illustrates the more general point that, when the number of possible demand states is too large for the contract to specify a payment in every possible state, firms can use unit prices on multiple goods to achieve some pricing flexibility with a relatively simply contract. Finally, note that our model has assumed, in each given period, the buyer can buy either from the seller or from the outside market, but not from both. If we allowed the buyer to restrict its quantity demanded from the seller by splitting its order for high-demand periods between the seller and the outside market, doing so would represent an alternative mechanism for limiting premium payments during high demand. However, as long as the transaction costs of sourcing from multiple suppliers exceed the cost of price adjustment, it would never be optimal for the buyer to split its order between two suppliers for a period. Rather, it would be more efficient to adjust prices when necessary, using the relational contract derived in this section. Furthermore, going to the outside market would not allow firms to increase the premium payment during periods of low demand for the basic good but high demand for the customizable good, which is another reason payment adjustments are potentially more efficient than sourcing from multiple suppliers. 3.5

Model Extension: Two-part Tariffs

We now extend the model to allow contracts with a fixed payment component and per-unit prices, similar to two-part tariffs. Although including a fixed payment can improve contract efficiency, we show that fixed payments do not generally eliminate the benefit of setting a 18

On the other hand, if the correlation in product demands is weak enough that (32) is violated, there is an efficient contract that places a price premium only on the customizable good. For example, if (L, L) and (L, M ) are the two most likely states, there is an efficient contract that places zero price premium on the basic good, sets a price premium on the customizable good to generate optimal payments when demand for this good is low, and involves payment adjustments when demand for the customizable good is high.

31

per-unit price premium on both the basic good and the customizable good. Firms can still use per-unit price premiums on multiple goods, in order to provide additional degrees of freedom in determining the premium payment in each period. Formally, we allow firms to agree to a relational contract that specifies values (y, Pu , Pb , e(qt,u )), where y is the fixed payment component, which can be either positive or negative (or zero). We also return to the original timing assumptions of the model. In principle, one could allow payment adjustment, as in the previous model extension, and also allow a fixed payment, but for simplicity, we do not allow payment adjustment in this section. When the buyer agrees to the payment specified in the relational contract, the buyer’s utility for the period is:

URt = qt,u (αu + βet − Pu ) + qt,b (αb − Pb ) − y

(41)

And the seller’s utility for the period is:

USt = qt,u (Pu − γet ) + qt,b Pb + y

(42)

Using a fixed payment and per-unit prices on two products allows three degrees of freedom, which generally allows firms to generate the exact premium payment they would like in three different demand states. In order to derive an efficient contract for this case, we extend the demand distribution used in Proposition 3 in order to include three possible demand states. In particular, we assume the quantities traded for each product have the

32

following distribution:

P rob(qt,u = L, qt,b = X) = 1 − ωM − ωH

(43)

P rob(qt,u = M, qt,b = X + M − L) = ωM

(44)

P rob(qt,u = H, qt,b = X + M − L) = ωH

(45)

where 0 < L < M < H, X > 0, ωM > 0, ωH > 0, and ωM + ωH < 1. This distribution implies both products exhibit an increase in quantity of M − L as they move from the low demand state to the medium state; however, the customizable good has a further increase in quantity of H − M whereas the basic good has no further increase in quantity as they move from the medium demand state to the high state. Other distributions with three states could be used to produce similar results, with appropriate adjustments to the fixed payment and per-unit prices, as long as each product’s quantity is not a perfectly linear function of the other product’s quantity. We focuses on cases in which this distribution satisfies the following conditions:  γM < (β − γ)

 γH > (β − γ)

δ 1−δ

δ 1−δ

h

(1 − ωM − ωH )L + (ωM + ωH )M

i

h i (1 − ωM − ωH )L + ωM M + ωH H

(46)

(47)

Note these are modified versions of conditions (20) and (21). Condition (47) implies firstbest effort cannot be sustained during high demand. The optimal contract involves setting et = 1 during low and medium demand, and setting et = e∗H during high demand, where the maximum sustainable effort during high demand is found by solving:

γHe∗H

i δ h ∗ = (β − γ) (1 − ωM − ωH )L + ωM M + ωH HeH 1−δ 33

(48)

Conditions (46) and (47) ensure there is a value e∗H ∈ ( M , 1) that solves this equation. H Solving for e∗H , we find: # " δ (β − γ) [(1 − ω − ω )L + ω M ] 1 M H M 1−δ e∗H = δ H ωH γ − (β − γ) 1−δ

(49)

The optimal per-unit prices and fixed payment that sustain these effort levels are as follows:

Pu − P¯u =



 He∗H − M β H −M

(50)

Pb − P¯b = β − (Pu − P¯u )

(51)

y = βL − (Pu − P¯u )L − (Pb − P¯b )X

(52)

Proposition 5. If the quantity distributions satisfy conditions (46) and (47), then optimal effort, et = 1, cannot be sustained during high demand, and the most efficient relational contract includes a fixed payment and per-unit premium payments on both goods, with the high-demand effort level given by (49), unit prices given by (50) and (51), and the fixed payment given by (52). Thus, even when the relational contract includes a fixed price component, it is generally optimal to include a per-unit premium payment on both goods to achieve additional pricing flexibility.

4

Conclusion

We develop a model in which a downstream firm pays premium prices on the goods it purchases, in order to compensate its supplier for customization services that are not enforceable through a formal contract. We show that the payments that can sustain such a relationship are partially linked to goods with stable demand, even if these goods are not customizable. Moreover, this relational contract ensures efficient provision of uncontractible services for

34

the customizable goods. Our results provide one explanation for why firms often use cross-subsidized pricing. In some cases, it is efficient for a firm to pay a price premium on one good to compensate a supplier for effort expended on another good. We show that this pricing strategy can help channel members form a simple relational contract that provides price flexibility to compensate suppliers for greater effort during high demand, without making the total premium payment too large to sustain. We do not explicitly model customer demand and retail prices, which could lead to double marginalization. Further research could explore relational contracts as a tool to solve the problem of double marginalization. In addition, it would be interesting to explore what happens if the retailer also can incur uncontractible effort that increases sales and benefits the channel, so that the contract needs to provide effort incentives for both firms. Future research could also model how firms initially bargain over the terms of a relational contract. In our model, formulating an efficient contract requires knowing the supplier’s cost and the retailer’s benefit of effort, and the distribution of the retailer’s quantity needs. Firms would need to share such information in order to construct a contract that sustains efficient effort while minimizing payment adjustment costs. If each firm is concerned that sharing such information weakens its bargaining position, it would be interesting to study conditions in which channel partners can overcome this challenge to formulate an efficient contract (see Wernerfelt 2012). An important implication of our research is that partners should cooperate to formulate a relationship that maximizes joint surplus, while avoiding situations in which one party has an incentive to violate the terms of the contract or in which the parties need to make costly adjustments to those terms.

35

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Appendix A Example: Salmon and Seaweed Price Premiums

Example is based on sales data and customization requests from a supplier of sushi restaurants in the Southeast US, from April 2010 to February 2011. Restaurants were divided into volatile and stable demand using a median split of sales variation. Low End restaurants are those with menu prices under $10 on Yelp.com or UrbanSpoon.com while High End restaurants are those with prices above that. 94% of all customization requests come from the high-end restaurants.

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Appendix B Proof of Proposition 1 Under the conditions of the proposition, we show that a contract that sets Pu − P¯u = β and Pb − P¯b = 0 can sustain effort e(qt,u ) = 1 in every period. Substituting these values into the buyer’s constraint (8), we have:    δ δ β qt,u + E[qT,u ] ≤ β qt,u + E[qT,u ] 1−δ 1−δ 

(53)

Because this constraint always holds with equality, the buyer never has an incentive to deviate and go to the outside market. Substituting the proposed prices and effort level into the supplier’s constraint (11), we have:   δ δ E[qT,u ] ≥ γ qt,u + E[qT,u ] β 1−δ 1−δ

(54)

Rearranging terms, this condition is equivalent to:

γqt,u ≤ (β − γ)

δ E[qT,u ] 1−δ

(55)

Condition (14) ensures that this constraint is always satisfied, so the supplier never has an incentive to deviate from setting e(qt,u ) = 1. Therefore, the proposed relationship is sustainable with optimal effort in all periods. QED Proof of Proposition 2 We show that a relationship in which firms always trade with each other with a price premium only on the customizable good cannot be sustained if condition (14) does not hold. The derivations in the body of the paper show that, given Pu − P¯u > 0 and Pb − P¯b = 0, if we combine the buyer’s constraint, given by (8), with the seller’s constraint, given by (11), 41

together these constraints imply:  γ qt,u +

 δ δ E[qT,u ] ≤ β E[qT,u ] 1−δ 1−δ

(56)

Note the derivations of this condition do not depend on the particular effort levels specified by the contract. Therefore, this condition must hold for such a relationship (in which firms always trade with each other with a price premium only on the customizable good) to be sustainable for any effort levels. Rearranging terms, this condition is equivalent to:

γqt,u ≤ (β − γ)

  δ E qT,u 1−δ

(57)

If (14) does not hold, then the above condition does not hold when qt,u = max(qt,u ), which implies at least one firm must have an incentive to deviate during periods of peak demand for the customizable good, and such a relationship cannot be sustained. QED Proof of Proposition 3 Given the conditions of the proposition, we show that a relationship that sets e(L) = 1, e(H) = e∗H from equation (23), and prices according to (24) and (25) is sustainable and generates strictly greater surplus than any other sustainable relationship. The total premium payment during low demand is (Pu − P¯u )L + (Pb − P¯b )L, which, given the proposed prices, equals βL. The total premium payment during high demand is (Pu − P¯u )H + (Pb − P¯b )M , which, given the proposed prices, equals βHe∗H . For these payment values and the proposed effort levels, the buyer’s constraint (8) during low demand becomes

βL +

i i h h δ δ β (1 − ω)L + ωHe∗H ≤ βL + β (1 − ω)L + ωHe∗H 1−δ 1−δ

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(58)

During high demand, the buyer’s constraint becomes

βHe∗H +

i i h h δ δ β (1 − ω)L + ωHe∗H ≤ βHe∗H + β (1 − ω)L + ωHe∗H 1−δ 1−δ

(59)

Thus, the buyer’s constraint always holds with equality, and so the buyer never has an incentive to deviate. Inserting the proposed payment and effort levels into the seller’s constraint, given by (11), during low demand, we have:

β

i  i δ h δ h (1 − ω)L + ωHe∗H ≥ γ L + (1 − ω)L + ωHe∗H 1−δ 1−δ

(60)

During high demand, the seller’s constraint becomes

β

i  i δ h δ h (1 − ω)L + ωHe∗H ≥ γ He∗H + (1 − ω)L + ωHe∗H 1−δ 1−δ

(61)

Recall that e∗H was chosen to solve equation (22), which implies (61) holds with equality. Furthermore, conditions (20) and (21) imply that e∗H must lie in the set ( M , 1) to solve (22). H Therefore, He∗H is greater than M , which is greater than L. Thus, if (61) holds with equality, then (60) hold strictly. Because the constraints for the buyer and seller to stay in the relationship are always satisfied, the proposed relationship is sustainable. Any relationship with lower effort, or in which firms trade on the outside market, during either high or low demand would generate lower surplus than the proposed relationship. Furthermore, as shown in the discussion following (22), any effort level higher than e∗H during high demand would imply that effort costs for the period exceed the ongoing value of the surplus generated by the relationship; therefore, condition (13) would be violated during high demand, which implies the relationship could not be sustainable. Thus, the proposed contract is the most efficient relational contract that 43

can be sustained. Finally, we show that no other pair of prices can sustain these efficient effort levels. In order for the seller’s constraint (11) to be satisfied during high demand, any alternative contract with the same effort levels would have to provide at least the same expected total premium payment as the proposed contract. In particular, the expected discounted value of h i δ (1 − ω)L + ωHe∗H . future premium payments would have to be at least β 1−δ Any alternative prices that generate the same expected premium payment as the proposed contract would imply either higher total payments during high demand (and lower total payments during low demand) or higher total payments during low demand (and lower total payments during high demand). However, because the buyer’s constraint (8) always holds with equality in the proposed equilibrium, any such alternative contract that resulted in a total premium payment greater than βL during low demand would result in (8) being violated during low demand, and similarly, any such contract that included a total premium payment higher than βHe∗H during high demand would result in (8) being violated during high demand. Thus, only the proposed prices can sustain efficient effort levels. QED Proof of Proposition 4 Given the conditions of the proposition, we show that the proposed equilibrium is sustainable and generates strictly greater surplus than any other sustainable equilibrium. Most steps of this proof are similar to the proof of Proposition 3. The proposed prices and payment adjustments imply the total premium payment is βL whenever demand for the customizable good is low and βHe∗H whenever demand for the customizable good is high. Therefore, as in the proof of the previous proposition, the buyer’s constraint during low demand for the customizable good is given by (58), the buyer’s constraint during high demand of this good is given by (59), and these constraints are always satisfied with equality.

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The left side of (35) gives the seller’s effort cost during high demand, and the right side of (35) gives the expected discounted value of future premium payments minus future effort costs and future adjustment costs, given the proposed relational contract. The effort level e∗H was chosen so that the two sides of this equation are equal, which implies the seller’s constraint to provide the required effort is satisfied with equality during high demand for the customizable good, and its constraint is satisfied strictly during low demand for this good. For any period in which the seller needs to incur an adjustment cost and propose a total bonus payment lower than the one implied by the contract prices, the difference between the equilibrium bonus payment and effort cost for that particular period provide an incentive for the seller to incur this adjustment cost and to propose the equilibrium bonus. If the seller deviated from the equilibrium path and proposed any other bonus or stayed with the contract prices, the relationship would end, so the seller would not receive any bonus for that period. Therefore, as long as the adjustment cost is small enough, the seller would not deviate. Because neither player ever has an incentive to deviate, the proposed relationship is sustainable. Lower effort levels or trading on the outside market would result in lower surplus. Furthermore, conditions (33) and (34) imply that any effort level higher than e∗H during high demand leads to effort costs during those periods that exceeds the ongoing value of the relationship, so such effort could not be sustained. Any alternative total premium payments imply either that the seller’s effort incentives are weaker (because the expected premium payment is smaller), or that the buyer’s constraint to stay in the relationship is violated during periods when the total premium payment is higher than in the proposed contract.

If d is sufficiently small, it is more efficient

to generate an optimal payment in each period rather than to incur these consequences of suboptimal payments. Given that (H, L) and (L, M ) are the two least likely demand states, the proposed equilibrium involves the lowest possible expected adjustment costs that 45

generates the flexibility to set an optimal payment in each period. QED Proof of Proposition 5 The proof is similar to the proof of Proposition 3. The proposed equilibrium implies the total premium payment is βL demand is low, βM when demand is medium, and βHe∗H when demand is high, and that the benefit to the buyer of effort in each period is always equal to these premium payments. The left side of (48) gives the seller’s effort cost during high demand, and the right side of (48) gives the expected discounted value of future premium payments minus future effort costs, given the proposed relational contract. The effort level e∗H was chosen so that the two sides of this equation are equal. Any alternative payments imply either that the buyer’s constraint to stay in the relationship is sometimes violated or that the seller’s effort incentives are weaker. QED

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