Strategic Supply Chain Contracting to Stimulate Downstream Process Innovation

Stephen M. Gilbert¤and Viswanath Cvsay June 29, 2000

Abstract In a supply chain, investments that a ¯rm makes in reducing its own variable costs provide an obvious bene¯t to its suppliers: All else being equal, lower marginal costs cause the ¯rm to increase its own output, hence increasing consumption of suppliers' outputs. Without pre-commitment to wholesale prices from its supplier(s), a ¯rm will tend to underinvest in cost reduction because of fear of being held-up. Clearly, a supplier to such a ¯rm can eliminate this hold-up problem by pre-committing, instead of remaining °exible with respect to wholesale price. However, by making an advance commitment to wholesale price, it gives up an important means of responding to demand uncertainty. In this paper we examine the trade-o® that is faced by a ¯rm when its downstream channel partner has opportunities to invest in making relationship speci¯c marginal cost reductions. Should it commit to a price in order to encourage downstream investment in cost reduction, or should it remain °exible with respect to wholesale price in order to respond to demand uncertainty. We discuss several simple wholesale pricing mechanisms with respect to this trade-o®. (Channel Coordination, Channels of Distribution, Industrial Organization, Cost Reducing R&D )

¤

(Corresponding Author)The University of Texas at Austin, Management Department, CBA 4.202, Austin, TX 78712, [email protected] y

Case Western Reserve University, Department of OR and OM, 10900 Euclid Ave., Cleveland, OH 44106 [email protected].

1

Introduction

When one ¯rm in a supply chain invests in process innovation aimed at reducing its own variable production costs, collateral bene¯ts often acrue to its supply chain partners. All else equal, lower marginal costs tend to result in lower prices and higher quantities of output from a ¯rm, bene¯tting its customers and suppliers respectively. However, because some of the bene¯ts from cost reduction cannot be fully captured by the ¯rm that makes the investment, it will tend to invest less than would be optimal from the perspective of the chain as a whole. Consider, for example, the relationship between Frito Lay and 7-Eleven. These two ¯rms are currently involved in a contract that calls for 7-Eleven to sell Frito Pie, a concoction involving Frito-brand corn chips and freshly made chili, in its stores. There are any number of ways in which 7-Eleven might invest in reducing its marginal cost of delivering Frito Pie to customers in its stores. For example, variable costs could be reduced by automating the production of chili, or by developing a dispenser that allows customers to serve themselves. Alternatively, they might devote more resources to designing the packaging of the product that reduces the amount of time that their employees have to spend serving it or cleaning-up. Anything that 7-Eleven does along these lines to reduce their variable costs would tend to result in lower prices and more unit sales of Frito-pie, bene¯tting both ¯rms. However, there are two main forces that cause ¯rms such as 7-Eleven to underinvest in cost reduction relative to what would be most bene¯cial to its supplier or to the supply chain as a whole. First, since a portion of the bene¯t from a reduction in 7-Eleven's variable cost would spillover to its supplier(s), its investment decision is a®ected by a bene¯t externality , and its investment decision will ignore the portion of bene¯ts that spillover. This e®ect is very similar to the well-known "double-marginalization" e®ect that arises in supply chain pricing decisions. As with the double-marginalization e®ect, this bene¯t externality cannot be eliminated with linear, i.e. single-part, pricing mechanisms which are extremely common in practice. The second force that causes a downstream ¯rm ( such as 7-Eleven) to underinvest is the fear of supplier opportunism. That is, once a ¯rm reduces its variable costs, its supplier(s) could have an incentive to increase prices. In other words, by investing in process improvement, the downstream ¯rm subjects itself to being held-up by its supplier(s). In the absense of market uncertainty, the upstream ¯rm could costlessly eliminate this latter reason for underinvestment by making a commitment to a wholesale price before the investment decision is made. However, when demand uncertainty is signi¯cant, such commitment is not without cost: Process innovation projects, e.g. design for manufacturability, process improvement, etc., often have long lead times. Decisions to undertake such projects often must be made long before demand uncertainty is resolved. Thus, if the upstream ¯rm is going to make a wholesale price commitment to encourage investment, it must do it before market un-

certainty is resolved. However, as has been noted by VanMieghem & Dada (1999), wholesale pricing °exibility can often serve as an important bu®er against market uncertainty. Thus, the upstream ¯rm faces the following trade-o®: By maintaining wholesale price °exibility, it bu®ers itself against uncertain demand, but may also discourage the downstream ¯rm from investing in cost reduction. In this paper, we study how a seller (e.g. Frito-Lay) can construct a wholesale pricing contract with a buyer (e.g. 7-Eleven) to encourage investment in variable cost reduction in the presence of market uncertainty. Our perspective is primarily that of the seller, though we are also interested in how the buyer and the overall chain are a®ected. Because of their prevalence in practice, we focus on linear, i.e. single-part, pricing mechanisms. In particular, we examine three variations of a linear contract: One in which the seller retains pricing °exibility until after any cost reduction has been made and demand uncertainty has been resolved; one in which the seller makes a full commitment to a wholesale price in advance of both cost reduction and resolution of demand; and one in which the seller guarantees that the wholesale price will be no higher than a speci¯ed wholesale price ceiling, but may be lower if the market response is poor. Such contracts are not uncommon in practice and provide an interesting blend of pricing commitment, which encourages investment, and pricing °exibility, which bu®ers against demand uncertainty. The key features of our paper are that it identi¯es this trade-o® between pricing °exibility and strategic commitment to induce downstream cost reduction, and that it provides insight into how simple contracts that are commonly used in practice can be used to manage it. The remainder of our paper is orgaized as follows. In section 2, we review the relevant literature that is concerned with supply chain contracting, investments in cost reduction, or inter-industry interactions that in°uence marginal costs. In Section 3 we develop and analyze investment in cost reduction for both fully °exible pricing and full commitment to wholesale price. We draw upon these results in Section 4 where we analyze a ceiling contract that allows partial °exibility to decrease the wholesale price. In Section 5, we provide a numerical example, and in Section 6 we summarize the managerial implications of our results.

2

Literature Review

Most of the literature in supply chain contracting has focused on how contracts a®ect decisions regarding either capacity or inventory. Recent reviews of this literature can be found in Lariviere (1998) and Tsay, Nahmias & Agrawal (1998). In nearly all of this literature, variable costs are assumed to be exogenous. However, in practice decisions relating to process improvement and cost reduction can have a signi¯cant impact on the performance of a supply chain. A number of researchers have studied investments in cost reduction at a single level of a

2

supply chain in oligopolistic settings. In much of this work, the number of ¯rms in the industry is exogenous and entry and exit are precluded. For example, Brander & Spencer (1983) model the ways in which opportunities to invest in reduction of marginal costs a®ect a symmetric duopoly in which the ¯rms engage in Cournot competition. They ¯nd that such opportunities tend to depress prices and increase output, bene¯tting consumers but hurting the duopolists. Bester & Petrakis (1993) investigate a duopoly in which only one ¯rm has an opportunity to invest in cost reduction. They focus on comparing Bertand (price) competition with Cournot (quantity) competition. They ¯nd that when substitutability is low, the ¯rm will be more prone to invest in a given level of cost reduction under quantity competition than under price competition, but when substitutability is high, this relationship is reversed. Levin & Reiss (1988) consider investments in both quality improvement and cost reduction, analyzing the conditions under which the two are strategic complements versus substitutes. Several authors have endogenized industry structure in an attempt to understand its relationship with innovation in cost reduction. This research generally tends either to demonstrate how asymmetric industrial structure can arise as a result of innovative activity, or to determine the relationship between innovative activity and the intensity and or type of competition. Flaherty (1980) is an excellent example of the former. She develops a multi-period model in which ¯rms compete by determining investment levels in cost reduction and output quantities in each period. She uses this model to demonstrate that the only stable steady-states are asymmetric. A similar result is obtained by Petrakis & Roy (1999), who demonstrate the possibility of heterogeneity emerging along the dynamic equilibrium path of an industry that is initially composed of identical ¯rms, even though there is no uncertainty and all ¯rms have perfect foresight. Models with endogenized industry structure have also been used to provide a richer understanding of the relationship between innovative cost reduction and the nature of competition. Dasgupta & Stiglitz (1980b) develop a model in which a distinction is made between competition in the current product market, and competition in R&D. Their main results are that competition in the current product market tends to discourage R&D, while competition in R&D tends to reduce barriers to entry and encourages R&D. This presents a challenge to the conventionally held view that a market economy tends to lead to underinvestment in R&D, which is based on models that do not allow for entry. In another paper, Dasgupta & Stiglitz (1980a) examine the the ways in which industry structure, barriers to entry, elacticity of demand interact to a®ect investment in R&D. All of the above models consider a single industry, i.e. level in a supply chain, and tend to assume that demand is deterministic. We are aware of only three papers that look speci¯cally at how a ¯rm can in°uence marginal costs of a ¯rm(s) in another industry. Maksimovic (1990) demonstrates how bank loan commitments can induce a prisoner's dilemna game among duopolistic manufacturers such that at equilibrium, both ¯rms commit to pay a ¯xed fee in

3

return for lower marginal cost ¯nancing. This e®ectively reduces their marginal cost of production, signalling a willingness to produce more output. Gupta & Loulou (1998) demonstrate that a franchising arrangement, which allows a manufacturer to capture retail level pro¯ts through a lump sum payment, can discourage the maufacturer from reducing its costs. A di®erent perspective is adopted by Harho® (1996) who analyzes the the way in which a ¯rm can in°uence competition in a downstream industry by providing R&D subsidies. He identi¯es and analyzes the following trade-o®: Although R&D subsidies tend to increase entry, thereby stimulating competition, a larger number of ¯rms tends to depress the within-industry R&D investment per ¯rm. Thus, the individual ¯rms do not invest as much in reducing their own costs. One of the key features of our paper is that it considers the trade-o® that the upstream ¯rm faces between pricing °exibility and strategic commitment. Similar trade-o®s between operational °exibility and strategic commitment at a single level of a supply chain have been identi¯ed and modeled in both Appelbaum & Lim (1985) and Spencer & Brander (1992). Both of these papers examine how market uncertainty a®ects a ¯rm's willingness to make a strategic quantity commitment in a contestable market. This contrasts with our paper in which we look at how a strategic commitment to price by one ¯rm a®ects actions taken by a downstream channel partner.

3

The Model

In this paper, we examine a trade-o® that is faced by an upstream ¯rm (the seller) when a downstream channel partner (the buyer) has opportunities to invest in reducing its marginal costs. The trade-o® that we have identi¯ed is as follows: In order to respond to demand uncertainty, the seller would like to remain °exible with respect to the wholesale price. However, because °exible wholesale pricing creates an opportunity for the seller to hold-up the buyer after investments have been made, °exibility can deter such investments from being made in the ¯rst place. Although we assume that the seller can observe the costs of the buyer, we do not consider contracts that are explicitly contingent upon these costs. In practice, a seller may be able to estimate a buyer's variable costs from observations of product or process designs, but formal measures of such costs are notoriously imprecise, and enforcing a contract that is explicitly contingent upon such costs would be extremely di±cult. As discussed previously, our interest is in understanding how the incentive structure is a®ected by simple contracts that are easily implemented and observed in practice. Thus, we restrict our attention to single-part pricing mechanisms, mainly due to their prevalence in practice. For simplicity, we model the interaction between the seller and the buyer as a bilateral monopoly in which both ¯rms have access to the same information. Throughout the paper, we adopt the convention of using the pronouns she 4

and he to refer to the seller and the buyer respectively. To represent the idea that investments in cost reduction often have signi¯cant implementation lead times, we assume that the downstream ¯rm must decide how much to invest before demand uncertainty is resolved. Under this assumption, we examine the way in which the seller's ex ante commitment to wholesale price a®ects the buyer's investment and various measures of supply chain performance. To model the demand in the market that is served by the buyer, let us assume that the inverse demand function is linear in the quantity produced. In particular, let q be the quantity, and de¯ne the market clearing price as follows: p(q) = a ¡ q where the density of parameter a, f (a), has a mean of a, standard deviation of ¾, and positive support in the range (amin ; amax ). The seller's variable production cost is Cs , and the buyer's variable cost, prior to any investment is Cb . For the sake of simplicity, we assume amin > Cs + Cb . This assumption, which is similar to the ones made by Spencer & Brander (1992) and by Appelbaum & Lim (1985), implies that for every realization of demand, there exists a non-zero quantity of output for which the market clearing price is greater than the marginal cost of production. By allowing us to avoid consideration of situations in which the buyer optimally responds to low realizations of demand by selling nothing, this assumption simpli¯es the presentation of our results. However, it does not alter the basic trade-o®s that we identify. Let r · Cb be the maximum cost reduction that can be attained through investment. This modeling device allows us to consider situations where the buyer's cost consists of several components, not all of which are controllable. For example, although the buyer might be able to eliminate labor costs by developing a more e±cient operation, the costs of raw materials and purchased parts are not controllable. To represent diminishing returns, we assume that the elimination of fraction µ of the maximum potential cost reduction requires an investment of Iµ2 . Note that this model of investment in cost reduction implies that it is independent from the scale of operations. This is highly consistent with investments in soft technologies: such as design for manufacturability, process improvement, etc. For example, when a decision is made to automate production, there is a big ¯xed cost associated with the design of the product and the process that is independent of the production volume. As long as capacity can be added in approximately continuous increments, then its cost can be viewed as variable. To guarantee that the pro¯t functions be concave in the amount invested, we require that it not be too cheap to reduce costs. Speci¯cally we require that 3I > r 2 . However, it is worth noting that, as described by Gupta & Loulou (1998), this relationship is typically satis¯ed easily by practical investment opportunities. 5

In what follows, we investigate several levels of °exibility in the wholesale pricing between the seller and the buyer. Throughout our analysis, we assume that the buyer must invest in cost reduction prior to his determination of output quantity. This is based on the observation that commitments to production quantities typically cannot be made until after most of the questions about product and process design have been resolved. Thus, in most of our analysis, the buyer must make advance commitment to the level of cost reduction, but he can postpone his production quantity decision. We also assume that the seller's announcement of the wholesale price preceeds the buyer's quantity decision. First, we consider the case of complete wholesale pricing °exibility where the wholesale price is determined only after the buyer has performed whatever cost reduction is to be done and the demand uncertainty has been resolved. We then consider the case of complete wholesale price commitment, where the seller commits to a wholesale price prior to both the buyer's cost reduction and the realization of demand. To assess the e®ect of the buyer's operational °exibility, we consider as a sub-case the situation where he cannot postpone his quantity decision until after demand uncertainty is resolved. Finally, we analyze what we refer to as a ceiling wholesale price, in which the seller makes advance commitment to only an upper bound on the wholesale price, reserving the option to charge lower wholesale prices once demand uncertainty is resolved.

3.1

Fully Flexible Wholesale Price

In this section we consider what happens when the seller remains fully °exible by delaying the announcement of wholesale price until after demand information is revealed. Thus, the buyer must determine how much to invest in reducing his variable costs with no guarantee regarding the wholesale price that will be o®ered by the seller. This analysis implicitly assumes that lead times for process innovation are long relative to production lead times. Speci¯cally, the buyer must make advance commitment to the level of cost reduction, but he can postpone his production quantity decision. This situation can be modeled as the following leader-follower game: The buyer moves ¯rst by determining the fraction (µ) of available variable costs to eliminate, and investing accordingly. After demand is observed, the seller reacts to both demand and the buyer's cost reduction by announcing a wholesale price (w). Finally, the buyer responds by choosing a quantity (q). The above game can be analyzed using backward induction. At the ¯nal stage of this game, the buyer takes as given: the realization of demand parameter (a), the wholesale price (w), and his own cost reduction (µ), and determines an order quantity (q) that maximizes the following pro¯t function: ¼b (a; µ; w; q) = q(a ¡ q ¡ w ¡ Cb + µr) ¡ Iµ2 6

(1)

It is easy to see that the optimal order quantity satis¯es: q(a; µ; w) =

1 (a ¡ w ¡ Cb + µr) 2

(2)

Thus, when the seller reacts to the realization of demand and sets a wholesale price, she does so with anticipation that the buyer will react as shown above. Thus, the seller sets the wholesale price to maximize the following pro¯t function: 1 ¼s (a; µ; w) = (w ¡ Cs )q(a; µ; w) = (w ¡ Cs ) (a ¡ w ¡ Cb + µr) 2 It is easy to see that the seller's optimal wholesale price response to the realization of demand and the buyer's cost reduction is: 1 w(a; µ) = (a + Cs ¡ Cb + µr) 2

(3)

We have now established how the game will be resolved once demand information is revealed. At the time that the buyer makes his investment, he anticipates these responses for every realization of demand, and seeks to maximize the following expected pro¯t function with respect to his decision, µ, the fraction of total available costs to eliminate. ¼bF F (µ)

=

Z amax amin

(a ¡ q(a; µ; w(a; µ)) ¡ w(a; µ) ¡ Cb + µr) ¡ Iµ2

where the two superscipts on the buyer's pro¯t refer to the level of °exibility of the seller and buyer respectively. The ¯rst F refers to the seller's °exibility that allows her to postpone her wholesale pricing decision until after observing demand, and the second F refers to the buyers °exibility that allows him to postpone his quantity decision until after observing demand. By substituting (2) and (3) into the above expression, and using ¯rst order conditions, it can be seen that the optimal cost reduction for the buyer satis¯es: µF F =

r (a ¡ Cs ¡ Cb ) 16I ¡ r2

(4)

By substituting (4) into (2) and (3), we can see how the seller's and buyer's responses to demand depend on the original parameters of the problem. wF F (a) = qF F (a) =

1 r2 16I (a + a ¡ (Cs + Cb ) ) + Cs 2 2 16I ¡ r 16I ¡ r 2 1 r2 16I (a + a ¡ (Cs + Cb ) ) 4 16I ¡ r2 16I ¡ r 2 7

Integrating over all possible values of the demand parameter, we can see that the expected °exible wholesale price and quantity are as follows: wF F

=

qF F

=

1 16I + Cs (a ¡ Cs ¡ Cb ) 2 16I ¡ r 2 1 16I (a ¡ Cs ¡ Cb ) 4 16I ¡ r 2

Note that in the absence of the opportunity to reduce costs, the expected wholesale price and quantity would be w = (a + Cs ¡ Cb )=2 and q = (a ¡ Cs ¡ Cb )=4 respectively. Thus, the opportunity to invest results in a higher wholesale price but also a larger quantity of output. The higher wholesale price is the result of the fact that the seller's optimal ex-post wholesale price is decreasing in the buyer's marginal cost. Once the buyer reduces his costs, the seller has an incentive to increase the price. The larger quantity occurs in spite of the higher wholesale price because it does not increase enough to fully o®set the reduction that was obtained through investment. Finally, we can substitute back into the pro¯t functions for the seller and buyer and integrate over all possible realizations of demand to obtain the expected pro¯ts associated with fully °exible wholesale pricing. The expected pro¯ts of the seller, the buyer, and the channel, denoted by E[¼sF F ]; E[¼bF F ]; E[¼cF F ] respectively, are presented in the ¯rst column of Table 1, along with the buyer's cost reduction and expected wholesale price and quantity. In the ¯rst colum of Table 1, it can be observed that the expected pro¯ts for the seller, the buyer, and the supply chain as a whole are increasing in the variance of demand. This is consistent with the notion that operational °exibility can often be compared with ¯nancial options.

3.2

Full Commitment to Wholesale Price with Postponed Quantity

In this section, we investigate the situation in which the seller commits to a wholesale price in advance of the buyer's investment and the realization of demand. For now, we continue with the assumption that although the buyer invests prior to the realization of demand, he can postpone his quantity decision. This situation can be modeled as the following leader-follower game: While demand uncertainty remains unresolved, the seller acts as a leader by committing to a wholesale price that is not contingent upon either demand or the investment of the downstream ¯rm, and the buyer responds by determining the fraction µ of controllable variable costs to eliminate. Finally, after demand information is revealed, the buyer makes a quantity decision. As before, this game can be analyzed using backward induction. In the ¯nal stage, the buyer's pro¯t and optimal order quantity are as shown in (1) and (2). When the buyer makes 8

his investment decision, he now faces uncertainty about demand, but the seller's wholesale price is no longer contingent upon demand. Thus, when the buyer invests, he takes the wholesale price as given and seeks to maximize the following pro¯t function with respect to his investment decision, µ. E[¼bCF (a; w; µ)] = =

Z amax

amin Z amax amin

(q(a; µ; w)(a ¡ q(a; µ; w) ¡ w ¡ Cb + µr) ¡ Iµ2 )f (a)da (

(a ¡ w ¡ Cb + µr)2 ¡ Iµ2 )f (a)da 4

(5)

Recall that the two superscipts on the buyer's pro¯t refer to the level of °exibility of the seller and buyer respectively. The C refers to the seller's ex ante commitment to wholesale price, and the F refers to the buyers °exibility that allows him to postpone his quantity decision until after observing demand. From ¯rst order conditions, we can see how the buyer's optimal cost reduction depends upon the seller's commitment to wholesale price: r µCF (w) = (a ¡ w ¡ Cb ) (6) 4I ¡ r2 Note that the buyer's cost reduction is decreasing in the wholesale price. This makes good intuitive sense. A lower wholesale price means that for a given mark-up, the buyer will have higher volumes, and at higher volumes, a given reduction in variable costs puts more money in the buyer's pocket. It is also interesting to observe that the buyer's investment depends upon the demand distribution only through its ¯rst moment. We can now consider the seller's problem of determining the wholesale price to which to commit. When she commits to a wholesale price, she does so anticipating the buyer's ex ante investment in cost reduction and his ex post order quantity response to to any realization of demand. Thus, the seller attempts to maximize the following pro¯t function with respect to wholesale price w: E[¼sCF (w)] = =

Z amax amin Z amax

(w ¡ Cs )q(a; µCF (w); w)f (a)da

amin

1 r2 (w ¡ Cs )(a ¡ w ¡ Cb + (a ¡ w ¡ Cb ))f (a)da 2 4I ¡ r 2

(7)

From ¯rst order conditions, it can be seen that the optimal wholesale price to which the seller will commit is: a + Cs ¡ Cb w CF = 2 By substituting back into (6) and (2), we can see how the buyer's cost reduction and order quantity (for any given realization of demand) depends upon the original parameters. µ CF

=

r(a ¡ Cs ¡ Cb ) 2(4I ¡ r2 ) 9

(8)

q CF (a) =

a¡a I + (a ¡ Cs ¡ Cb ) 2 4I ¡ r2

(9)

By integrating over all possible values of the demand parameter a, we have that the expected order quantity when the seller commits to the wholesale price is: q CF =

I (a ¡ Cs ¡ Cb ) 4I ¡ r2

(10)

To determine the pro¯ts of the seller, the buyer, and the combined channel when the seller makes a full commitment to wholesale price, we can substitute into (5) and (7) to characterize the expected pro¯ts of the buyer, seller, and the combined channel. The expressions for these expected pro¯ts, as well as the buyer's cost reduction and expected wholesale price and quantity are summarized in the second column of Table 1. By comparing the ¯rst two columns of Table 1, there are several observations that can be made about the e®ects of wholesale pricing °exibility when the buyer can postpone his quantity decision. ² The cost reductions that result from di®erent types of wholesale pricing °exibility are presented in the ¯rst row of Table 1. By comparing cases FF and CF in this row, it can be seen that µ CF > µ F F , i.e. the seller's commitment to wholesale price results in a larger investment in cost reduction by the buyer. ² The expected wholesale prices are given in the second row of Table 1. (Note that under committed wholesale prices, we refer to the price to which the seller commits as the expected wholesale price.) It is interesting to observe that the price to which the seller would commit is lower than the expected price that she would o®er if she remained °exible. This is in spite of the facts that a committed wholesale price results in lower downstream costs and that wholesale prices are typically decreasing in downstream costs. ² The expected output quantities are given in the third row of Table 1. It can be seen that, as a result of the increased investment and lower expected wholesale price, E[q CF ] > E[qF F ], i.e. the expected amount sold increases when the seller commits to wholesale price. ² The expected pro¯ts for the seller are given in the fourth row of Table 1. It can be seen that when the seller commits in advance to wholesale price, she no longer bene¯ts from the variance in demand since she cannot react to it. In other words, by committing to a wholesale price, she has given up the option to react to demand. For a given mean demand (a), as variance increases, the seller is better o® remaining °exible with respect to wholesale price. On the other hand, note that the term in E[¼sF F ] that is independent

10

of demand variance (¾ 2 ) is less than the corresponding term in E[¼sCF ] since: 32I 2 32I 2 8I 2 I < = < 2 2 2 2 2 2 (16I ¡ r ) (16I ¡ r )(16I ¡ 4r ) (16I ¡ r )(4I ¡ r ) 2(4I ¡ r 2 )

(11)

Therefore, when demand uncertainty is low, the seller earns more by committing fully to a wholesale price than by remaining °exible. ² The expected pro¯ts for the buyer are presented in the ¯fth row of Table 1. It can be shown that E[¼bCF ] > E[¼bF F ], i.e. the buyer earns a larger expected pro¯t when the seller makes full commitment to wholesale price, regardless of the amount of variance. Together with the previous observation, this implies that for su±ciently small variance, i.e. predictable demand, both the buyer and the seller are better o® when the seller commits in advance to wholesale price than when she remains °exible. ² Indeed, by comparing the combined pro¯ts, which are given in the last row of Table 1, it can be seen that the rate at which supply chain pro¯ts increase with demand variance is larger when the seller remains °exible than when she commits to wholesale price. Therefore, for su±ciently large variance, supply chain pro¯ts will be larger when the seller retains wholesale price °exibility. On the other hand, we note that the term in E[¼cF F ] that is independent of demand variance (¾ 2 ) is less than the corresponding term in E[¼cCF ] since: I(48I ¡ r2 ) 3I I(48I ¡ r2 ) < < (16I ¡ r2 )2 (16I ¡ r2 )4(4I ¡ r2 ) 4(4I ¡ r2 ) Thus, as variance goes to zero, the supply chain earns larger pro¯ts when the seller commits ex ante to a wholesale price.

3.3

Full Commitment to Wholesale Price with Early Quantity Response

In some supply chains, the buyer lacks the operational °exibility to delay his quantity decision until more information is available about demand. This in°exibility may be the result of long lead-times, from either the seller that we have considered endogenously, or from other suppliers that are not explicitly represented in our model. In this section, we analyze how the buyer's in°exibilty a®ects the performance of the chain when the buyer commits in advance to a wholesale price. Note that because we know of no situations in which a buyer commits to a quantity before the seller announces the wholesale price, we do not consider °exible wholesale pricing when the buyer must make advance commitment to quantity. As before, this situation can be modeled as a leader follower game, but in this case, all of the decisions are made (and all of the costs are incurred) prior to the realization of demand. We assume that the seller makes the ¯rst move by o®ering a wholesale price, and that the buyer 11

reacts by simultaneously investing in cost reduction and determining his quantity. Thus, for a given wholesale price, the buyer maximizes the following pro¯t function with respect to both µ and q: E[¼bCC (w; µ; q)]

=

Z amax amin

(q(a ¡ q ¡ w ¡ Cb + µr))f (a)da ¡ Iµ2

where the superscripts C and C refer to the seller's commitment to wholesale price and the buyer's commitment to quantity in advance of observing demand. It is easy to show that the buyer's optimal cost reduction and quantity are the following functions of the wholesale price: r 4I ¡ r 2 2I q CC (w) = (a ¡ w ¡ Cb ) 4I ¡ r 2

µCC (w) = (a ¡ w ¡ Cb )

(12) (13)

In anticipation of this response, the seller sets the wholesale price w to maxmimize his own expected pro¯t, which can be expressed as: E[¼sCC (w)] = =

Z amax amin

(w ¡ Cs )qCC (w)f (a)da

amin

(w ¡ Cs )(a ¡ w ¡ Cb )

Z amax

= (w ¡ Cs )(a ¡ w ¡ Cb )

2I f (a)da 4I ¡ r 2

2I 4I ¡ r 2

It is easy to see that the seller's optimal wholesale price is: wCC =

a + Cs ¡ Cb 2

By substituting back into (12) and (13) it can be seen that when the buyer must both invest and incur production costs prior to observing demand, the equilibrium cost reduction and quantity are: µCC q CC

r 2(4I ¡ r 2 ) I = (a ¡ Cb ¡ Cs ) 4I ¡ r 2 = (a ¡ w ¡ Cb )

By substituting these results into the appropriate pro¯t functions above, we can develop expressions for the expected pro¯ts of the seller, buyer, and the chain as a whole. These expressions are presented in the third column of Table 1. It is interesting to compare how the behavior of the supply chain depends upon whether the buyer has quantity °exibility when the seller commits to wholesale price. The following observations can be made from comparing the second and third columns in Table 1: 12

² By comparing the ¯rst row of these two columns, it can be seen that, as long as the seller commits to wholesale price in advance of observing demand, the buyer's cost reduction is the same, regardless of whether he has quantity °exibility. ² The seller's optimal committed wholesale price does not depend upon whether the buyer has to make an advance commitment to quantity. ² Because the buyer's °exible quantity is linear in the realized demand parameter, his expected quantity when he has quantity °exibility is identical to the quantity to which he would commit if he lacked °exibility. ² By comparing the buyer's pro¯ts, we see that E[¼bCF ] ¸ E[¼bCC ]. The seller loses the ability to bene¯t from demand variablity when the buyer has to make an advance commitment to quantity. ² By comparing the seller's pro¯ts we see that E[¼sCF ] = E[¼sCC ]. The expected pro¯t of the seller is the same regardless of whether the buyer can postpone his quantity decision. ² As a result of how their individual pro¯ts are a®ected, the absense of quantity °exibility for the buyer results in reduced combined pro¯ts, i.e. E[¼cCF ] ¸ E[¼cCC ].

4

Ceiling Wholesale Price with Quantity Flexibility

In the previous section, we demonstrated the role that a seller's commitment to a wholesale price can play in encouraging investments in reducing variable costs by the buyer. It eliminates the buyer's fear of being held-up by the seller. However, in environments that are characterized by large amounts of demand uncertainty, we showed that the channel pro¯ts are higher if the seller retains wholesale pricing °exibility than if she sacri¯ces it to provide a stronger incentive for the buyer to invest in cutting variable costs. Much of the reason for this is that, without wholesale pricing °exibility, the seller cannot respond to poor market conditions by dropping her price; both she and the buyer are burdened with the commitment that she has made. In this section, we attempt to analyze a ceiling contract on wholesale price that places an upper limit on the wholesale price that the seller will o®er. Under such a contract, the seller makes the ¯rst move by setting a wholesale price ceiling, which we denote by wc. We refer to this as partial commitment since it commits the seller only to o®ering wholesale prices no larger than wc, rather than to a speci¯c wholesale price. The buyer responds to this price ceiling by investing in cost reduction. Let µP F (wc) be the buyer's cost reduction response to wholesale price ceiling wc. Note that the ¯rst superscript P refers to the seller's partial commitment to wholesale price, while the second superscript F refers to the buyer's °exibility that allows him to

13

postpone his quantity decision until after observing demand. (Since it is reasonable to assume that the wholesale pricing decision must preceed the quantity decision, wholesale price °exibility is only possible when the buyer has quantity °exibility.) After the seller observes the cost reduction (µ) and demand realization (a), she announces a wholesale price, z(a; wc; µ) that is no larger than the ceiling speci¯ed in the contract. Finally, the buyer responds to a given wholesale price (z) with a quantity decision, q(a; z; µ) that also depends on the amount (µ) by which the buyer has reduced his variable costs and the realization (a) of demand. Because the actual wholesale price (z(a; wc; µP F (wc))) at which goods are transferred is the minimum of a demand-realization-speci¯c pro¯t maximizing wholesale price and the ceiling wc, it is di±cult to obtain closed form results for this model. However, by making comparisons to the previous models, it is possible to develop insight into the performance of the wholesale price ceiling. Proposition (i) There always exists a wholesale price ceiling wc such that both the seller's and the buyer's expected pro¯ts exceed those earned under the seller's optimal fully committed wholesale price. Formally, there exists a wholesale price ceiling (wc) such that: E[¼sP F (wc)] ¸ E[¼sCF ]

E[¼bP F (wc)] ¸ E[¼bCF ]

(14) (15)

(ii) The seller's expected pro¯ts under an optimal price ceiling contract are always at least as large as those under either an optimal full commitment (CF) contract, or those under a full °exibility (FF)contract. Formally: E[¼sP F ] ¸ E[¼sCF ]

(16)

¸

(17)

E[¼sP F ]

E[¼sF F ]

Proof:(i) Let wc = wCF = (a + Cs ¡ Cb )=2. When the buyer determines his cost reduction, µ, he seeks to maximize his expected pro¯ts, which can be expressed as the following function of the wholesale price ceiling wc, and his cost reduction: E[¼bP F (wCF )] = =

M ax E[¼bP F (wCF ; µ)] ¸ E[¼bP F (w CF ; µCF )] µ¸0

Z amax

(a ¡ q(a; z(a; w CF ; µCF ); µ CF ) ¡ z(a; wCF ; µ CF ) ¡ Cb + µCF r)

Z amax

(a ¡ q(a; w CF ; µCF ) ¡ w CF ¡ Cb + µCF r)q(a; w CF ; µCF )f (a)da

amin

q(a; z(a; w CF ; µCF ); µCF )f (a)da ¡ I(µ CF )2 ¸

amin CF 2

¡I(µ

) = E[¼bCF ] 14

where the latter inequality follows from the fact that, for every realization of demand parameter (a), the buyer's pro¯ts are non-increasing in the wholesale price, and by de¯nition: z(a; wc; µ) · wc = w CF . To show (14), we will ¯rst show that the buyer's investment and subsequent reduction in cost is larger when the seller commits to a wholesale price ceiling of wc = w CF = (a+Cs ¡Cb )=2 than when she makes full commitment to a wholesale price w = wCF . Consider the ¯rst derivative of the buyer's pro¯t (under the ceiling contract) with respect to his cost reduction (µ), evaluated at µ = µCF : dE[¼bP F (w CF ; µCF )] dµ

= ¸ =

Z amax r

(a ¡ z(a; wCF ; µCF ) ¡ Cb + µCF r)f (a)da ¡ 2IµCF 2 r (a ¡ w CF ¡ Cb + µCF r)f (a)da ¡ 2IµCF amin 2 dE[¼bCF (wCF ; µCF )] =0 dµ

a Z min amax

Note that the inequality follows from the de¯nition of a wholesale price ceiling that implies that z(a; wCF ; µ CF ) · w CF . The equating of the the right-hand-side to zero follows from the de¯nition of µCF as the buyer's pro¯t maximizing cost reduction in response to the seller's full commitment to wholesale price w CF . It follows from the above that µP F (wCF ) ¸ µCF (w CF ) = µCF . Thus, if the seller announces a wholesale price ceiling of wc = w CF , her expected pro¯ts are: Z amax

(z(a; w CF ; µP F (wCF )) ¡ Cs )

¸

1 2

Z amax

(w CF ¡ Cs )(a ¡ w CF ¡ Cb + rµ P F (w CF ))f (a)da

¸

1 2

Z amax

(w CF ¡ Cs )(a ¡ w CF ¡ Cb + rµ CF (wCF ))f (a)da

E[¼sP F (w CF )] =

1 2

amin

(a ¡ z(a; w CF ; µP F (wCF )) ¡ Cb + rµP F (w CF ))f (a)da

amin

amin

= E[¼sCF ]

(18)

where the latter inequality results from the fact that µ P F (w CF ) ¸ µ CF (wCF ), as shown above, and the former inequality results from the de¯nition of z(a; w CF ; µ) which implies that for each realization of demand parameter a and any cost reduction µ: (z(a; w CF ; µ) ¡ Cs )(a ¡ z(a; w CF ; µ) ¡ Cb + rµ(w CF ))

¸ (w CF ¡ Cs )(a ¡ w CF ¡ Cb + rµ(w CF )) } 15

(ii) To see that (16) is true, it su±ces to observe that: E[¼sP F ] =

M ax E[¼sP F (wc)] ¸ E[¼sP F (w CF )] ¸ E[¼sCF ] wc

(19)

where the ¯nal inequality results from (18). Note that (17) is trivially true since, by setting the wholesale price ceiling arbitrarily high, i.e. wc ¸ amax , the fully °exible wholesale price contract is a special case of a ceiling contract. } For general demand distributions that have positive support only in the range [amin ; amax ] we have established that there exists a wholesale ceiling price contract that improves the pro¯ts of both buyer and seller, and that when the seller moves ¯rst to set the ceiling, then she is always better o® under the ceiling contract than under either a fully committed or a fully °exible wholesale price.

5

Numerical Study

In this section we perform a numerical study to gain further insight into how the three levels of wholesale pricing °exibility a®ect the downstream investment in cost reduction, and subsequently the level of output and pro¯tability of the supply chain. In particular we ¯rst compare and contrast the performance of the supply chain when the seller retains full °exibility in pricing to the case where the seller commits to a wholesale price before the uncertainty in demand is revealed. Our intention in this analysis is to determine when the seller prefers wholesale °exibility to commitment and vice versa. We then extend the numerical analysis to better understand how a ceiling wholesale price contract in°uences the buyer's investment in cost reduction. Although we developed our results in the earlier sections for a general distribution, for the purposes of the numerical study we use a bernoulli distribution (see ¯gure 1) to model the demand uncertainty. In the ¯gure, Time 0 corresponds to the time prior to the resolution of demand, while Time 1 corresponds to the time after demand uncertainty has been resolved. There are several interesting features in this distribution which make it convenient for a numerical study. Note that the mean of the distribution is ¯xed at a, but as ® increases to 1, the upper realization increases rapidly and the lower realization reaches a constant a¡¢. Since the variance ¢2 , we can increase the variance of the distribution as of this distribution is given by ¾ 2 = ®(1¡®) large as we desire by increasing ® towards 1 while still preserving the mean at a. This gives us the desired °exibility to conduct a thorough numerical study. For the purposes of comparing °exible and committed wholesale pricing, we assume that the buyer has the ability to postpone his quantity and pricing decisions. Let us arbitrarily assume 16

a + ∆/(1-α) 1-α

α

a - ∆/α

Time 0

Time 1

Figure 1: Demand distribution at time 1 as viewed from time 0. that the parameters of the problem are as follows: a = 50; ¢ = 10; I = 20; r = 5; Cb = 10; Cs = 0. We can now explore the way in which the seller's preference between wholesale pricing °exibility and commitment are a®ected by the demand uncertainty when the mean demand is held constant. In ¯gure 2 we plot the seller pro¯ts as a function of ®. (Recall that demand variance is increasing in ® and becomes in¯nite as ® approaches unity.) The ¯gure clearly demonstrates that, for committed wholesale pricing (case CF), the seller's pro¯ts do not depend upon the variance of demand, while for °exible pricing (case FF) her pro¯ts are increasing. As we can see, when the demand variance is small (® close to 0.5), the seller prefers commitment, while for su±ciently large variance in this case that threshold value is 0.73), she prefers °exibility. If the seller must choose between complete pricing °exibility and complete commitment, then Figure 3 shows the buyer's and the supply chain pro¯ts under the seller's preferred pricing policy. Note that given either one of the two policies, the pro¯ts of the buyer and the total supply chain pro¯ts are monotonic and convex in ®. But as shown in ¯gure 2 the seller will switch from full commitment in wholesale price to full °exibility when the uncertainty (®) reaches a certain threshold. This switchover in the seller's policies reintroduces the ine±ciency in the supply chain abruptly and causes discontinuous drops in the buyer's cost reduction and pro¯ts. This discontinuity carries through to the combined pro¯ts of the chain as well. Thus, when the seller can choose between commitment and °exible wholesale pricing, pro¯ts of the buyer and of the supply chain are neither monotonic nor convex in the variability of demand. Under the ceiling contract the decision problem for the seller is to set the wholesale price ceiling prior to both the cost reduction investment and the realization of demand. Assuming all

17

Figure 2: Seller's pro¯ts as function of ®.

Figure 3: Total supply chain and Buyer's pro¯ts as function of ®. the subsequent decisions are made optimally, ¯gure 4 shows the buyer's investment decision (µ) as a function of the wholesale price ceiling. The shape of this graph can be explained as follows: For su±ciently low wholesale price ceilings (wc), the buyer's cost reduction is large enough to eliminate the seller's incentive to exercise his °exibility by reducing his wholesale price even for the low realization of demand. Thus, the wholesale price that is o®ered will be the ceiling price, regardless of the outcome of demand. 18

Figure 4: Buyer's investment decision (µ)as a function of sellers wholesale price (w) As the ceiling price (wc) increases, we see that the investment (µ) decreases and ¯nally reaches a discontinuity and has a steep drop. This drop occurs at the point where the buyer's cost reduction is no longer large enough to eliminate the seller's incentive to exercise her pricing °exibility for the low realization of demand. For price ceilings above this threshold, the seller retains and exercises the °exibility to change the price in case of a lower realization of demand. The sudden appearance of °exibility in the seller's pricing serves to reduce the buyer's marginal bene¯t from cost reduction. At the margin, an increase in cost reduction now serves to increase the wholesale price that will be charged under the low realization of demand. Thus the ceiling contract reintroduces the hold-up problem that exists when the seller retains full °exibility in the wholesale price. On the other hand, as we showed in section 4, the seller can always do better with an optimal ceiling contract than with either full °exibility or full commitment to wholesale price.

6

Discussion

In this paper, we have explored the way in which commonly implemented variations on linear wholesale pricing policies a®ect downstream investments in innovation. This is a dimension of supply chain behavior that has not been adequately addressed in the operations literature. Note that although we focused our analysis on investments that reduce variable costs, most of our results would also hold if the downstream investments were instead assumed to be demand enhancing. 19

The key trade-o® that we have identi¯ed is the one faced by a seller when she must balance the need for °exibility to respond to demand uncertainty against the need to insure the buyer against exploitation by committing in advance to a wholesale price. We have shown that from the buyer's perspective, commitment to wholesale price is always preferred. From the perspective of the seller and the supply chain as a whole, full commitment dominates full °exibility for low amounts of uncertainty, while the opposite is true above a certain threshold amount of uncertainty. In addition, we have demonstrated that, for the seller, a ceiling contract always dominates both full °exibility and full commitment to wholesale price. To highlight the trade-o® that the seller faces between his own pricing °exibility and the need to encourage downstream process innovation, we have intentionally restricted ourselves to a simple bilateral monopoly. Although we believe that this basic trade-o® would also exist if the downstream industry were instead an oligopoly, it is certainly of interest to investigate how its dynamics would be a®ected by more complicated competitive settings. We have also restricted ourselves to assuming that investments in cost reduction are separate from decisions regarding the scale of production. As we discussed, this represents investments in soft technologies relatively well. However, in many industries where economies of scale are signi¯cant, a single decision determines both scale and variable cost. For example, our model would not adequately represent a situation in which the downstream ¯rm cannot add capacity in approximately continuous increments. In such a situation, the investment decision may simultaneously determine both variable costs and capacity level. We hope that our paper can serve as a building block for further research into how supply chain interactions a®ect operational decisions other than the price and quantity decisions that have received much of the attention until now.

20

References Appelbaum, E. & Lim, C. (1985), `Contestable markets under uncertainty', Rand Journal of Economics 16(1), 28{40. Bester, H. & Petrakis, E. (1993), `The incentives for cost reduction in a di®erentiated industry', International Journal of Industrial Organization 11, 519{534. Brander, J. A. & Spencer, B. J. (1983), `Strategic commitment with R&D: The symmetric case', Bell Journal of Economics 14, 225{235. Dasgupta, P. & Stiglitz, J. (1980a), `Industrial structure and the nature of innovative activity', The Economic Journal 90, 266{293. Dasgupta, P. & Stiglitz, J. (1980b), `Uncertainty, industrial structure, and the speed of R&D', Bell Journal of Economics 11, 1{28. Flaherty, M. T. (1980), `Industry structure and cost-reducing investment', Econometrica 48, 1187{1209. Gupta, S. & Loulou, R. (1998), `Process innovation, production di®erentiation, and channel structure: Strategic incentives in a duopoly', Marketing Science 17(4), 301{316. Harho®, D. (1996), `Strategic spllovers and incentives for research and development', Managment Science 42, 907{925. Lariviere, M. A. (1998), Supply chain contracting and coordination with stochastic demand, in S. Tayur, M. Magazine & R. Ganeshan, eds, `Quantitative Models for Supply Chain Management', Kluwer Academic Publishers. Levin, R. C. & Reiss, P. C. (1988), `Cost reducing and demand creating R&D with spillovers', Rand Journal of Economics 19, 538{556. Maksimovic, V. (1990), `Product market imperfections and loan commitments', Journal of Finance 45, 1641{1653. Petrakis, E. & Roy, S. (1999), `Cost reducing investment, competition, and industry dynamics', International Economic Reveiw 40, 381{401. Spencer, B. J. & Brander, J. A. (1992), `Pre-commitment and °exibility; applications to oligopoly theory', European Economic Review 36, 1601{1626. Tsay, A., Nahmias, S. & Agrawal, N. (1998), Modeling supply chain contracts: A review, in S. Tayur, M. Magazine & R. Ganeshan, eds, `Quantitative Models for Supply Chain Management', Kluwer Academic Publishers. 21

VanMieghem, J. A. & Dada, M. (1999), Price versus production postponement: Capacity and competition, to appear in Management Science.

22

Table 1: Comparison of Wholesale Pricing Flexibility

µ¤

Case FF Flexible Wholesale Price Flexible Quantity r (a ¡ Cs ¡ Cb ) 16I¡r2

E[w¤ ]

8I 16I¡r2 (a

¡ Cs ¡ Cb )

E[q ¤]

4I 16I¡r2 (a

¡ Cs ¡ Cb )

E[¼s ]

32I 2 (16I¡r 2 )2 (a

E[¼b ]

I 16I¡r 2 (a

E[¼c ]

48I¡r2 (16I¡r 2 )2 I(a

Case CC Committed Wholesale Price Committed Quantity

r a¡Cs ¡Cb 4I¡r 2 2

r a¡Cs ¡Cb 4I¡r 2 2

a¡Cs ¡Cb 2

a¡Cs ¡Cb 2

I 4I¡r2 (a

¡ Cs ¡ Cb)2 +

¡ Cs ¡ Cb)2 +

Case CF Committed Wholesale Price Flexible Quantity

¾2 8

I 2(4I¡r2 ) (a

¾2 16 2

¡ Cs ¡ Cb )2 + 3¾ 16

I 4I¡r 2 (a

¡ Cs ¡ Cb )

¡ Cs ¡ Cb )

I (a¡Cs ¡Cb )2 4I¡r2 2

¡ Cs ¡ Cb)2

I 4(4I¡r2 ) (a

¡ Cs ¡ Cb )2 +

¾2 4

I 4(4I¡r 2 ) (a

¡ Cs ¡ Cb )2

3I 4(4I¡r2 ) (a

¡ Cs ¡ Cb )2 +

¾2 4

3I 4(4I¡r 2 ) (a

¡ Cs ¡ Cb )2