# Is Channel Coordination All It Is Cracked Up To Be?

A fundamental task for supply-chain managers is to determine wholesale-prices. Such determination is a core theme in the marketing science literature on distribution channels--which seems to have concluded that channel coordination--setting wholesale-prices to maximize total channel profit--represents the best of all possible objectives. This judgment is based on analyses of bilateral monopoly models, within which profit may be redistributed to the benefit of all channel members. However, many manufacturers deal with multiple, competing retailers. The pure logic of bilateral monopoly models holds in the presence of multiple retailers if and only if the manufacturer can price discriminate between retailers. Although mechanisms for price discrimination exist, in many situations they are infeasible, illegal, or both.When retailers compete there are two feasible and legally permissible methods of achieving channel coordination: an explicit quantity-discount schedule or a menu of two-part tariffs. (The latter is derived in detail in this paper.) A feasible and legally permissible alternative to channel coordination is for the manufacturer to utilize a sophisticated Stackelberg two-part tariff--itself a form of a quantity-discount schedule. Although such a tariff cannot coordinate the channel, it is the best of all possible two-part tariffs from the viewpoint of maximizing manufacturer profit.

Because manufacturers are ultimately interested in their own profitability, it follows that channel coordination is manufacturer-optimal only if it generates at least as great a level of profit for the manufacturer as does noncoordination. In this paper we determine the conditions under which a channel-coordinating wholesale-price strategy will manufacturer-profit dominate a sophisticated Stackelberg two-part tariff We show that the optimal policy is dependent on (1) the retailers, fixed costs, (2) the relative size of the retailers, and (3) the degree of inter-retailer competition. We conclude that, from the perspective of a manufacturer, channel coordination is often undesirable relative to utilizing a noncoordinating, sophisticated Stackelberg price-strategy. Therefore, channel coordination can no longer be regarded as the ultimate goal which supply-chain managers should uncritically pursue.

INTRODUCTION

One of the more basic duties of supply-chain management is determination of an intra-channel wholesale-price at which a product should be sold. The marketing science literature on distribution channels provides a major source of guidance on how to perform this task. An essential conclusion of these writings is that the wholesale-price charged by a manufacturer should be set to assure that the channel is "coordinated." To marketing scientists the term "coordination" means setting all marketing mix variables to maximize total channel profit. The fundamental logic of coordination seems intuitively appealing: if a previously uncoordinated channel can be coordinated, then the additional profit can be divided among channel members so that all of them are better off than they were prior to coordination. [1]

This conventional wisdom has been bolstered by analytical research that identified two commonly used price-mechanisms--quantity-discount schedules and two-part tariffs--that can be used to achieve channel coordination (Jeuland and Shugan, 1983; Moorthy, 1987). But the fact that coordination is Pareto-optimal in a bilateral monopoly model--a model of one manufacturer selling through one retailer--does not assure that coordination benefits a manufacturer that sells through multiple retailers.

Does the logic of channel coordination extend to a manufacturer serving multiple retailers? If a manufacturer can charge different wholesale-prices to different retailers--if it can price discriminate between them--then a multiple-retailer channel may be conceptualized as a set of bilateral monopolies within which channel coordination is optimal. But there are reasons why the multiple-retailer scenario should not be cast as a set of bilateral monopolies. First, survey evidence suggests that manufacturers regard retailer-specific pricing arrangements as infeasible due to administrative, bargaining, and contract development costs (Lafontaine, 1990). Hence they tend to employ a single wholesale-price schedule, common across retailers, rather than treating each retailer separately. Second, there are legal restraints on channel pricing, at least in the United States. Section 2(a) of the Robinson-Patman Act ".... prohibits sellers from charging different prices to different buyers for similar products where the eff ect might be to injure, destroy, or prevent competition, in either the buyers' or sellers' markets" (Monroe, 1990, p. 394). In short, price discrimination is infeasible and is generally impermissible. [2]

If the manufacturer cannot employ a unique price-scheme for each retailer, another question arises: Are there wholesale-price schedules, common to all retailers, that are feasible, legally permissible, and that induce coordination when there are multiple retailers? The answer is yes. Concerning feasibility, either a quantity-discount schedule [3] or a menu of two-part tariffs [4] can be structured to achieve coordination. (A two-part tariff, which consists of a per-unit fee and a fixed fee, is a nonlinear quantity-discount schedule. With such a tariff the retailer's average, per-unit acquisition cost declines with increases in quantity purchased. Two-part tariffs are used explicitly in franchising, whereas nonlinear quantity-discount schedules are common in nonfranchise retailing. With a menu the retailer is allowed to select between a high fixed fee paired with a low per-unit charge or a lower fixed fee paired with a higher per-unit price.) Concerning legality, the Robinson--Patman Act requires every retaile r to have the right freely to select from the wholesale-price schedule: "the seller must offer the discount structure to all buyers" (Monroe, 1990, p. 339). Because the retailer determines its own quantity, a quantity-discount schedule meets the legality criterion. Similarly, because any element from a menu may be selected, a menu meets the legality criterion. Therefore channel coordination is both feasible and legally permissible.

This brings us to our central question: Is it in a manufacturer's interest to establish a wholesale-price policy that will induce coordination in a multiple retailer channel? There is limited evidence that it is not. Ingene and Parry (1995b) showed that a variety of noncoordinating two-part tariffs might manufacturer-profit dominate a channel-coordinating quantity-discount schedule. (The terminology "manufacturer-profit dominate" means that a particular wholesale-price strategy generates manufacturer profit that is at least as great as could be generated by the strategy to which it is being compared.) Their results depended on the model's parameters, of which the most critical was the difference in retailers' fixed costs. Subsequently the same authors (1998) derived the manufacturer-optimal two-part tariff by simultaneously determining the per-unit price and the fixed fee. They demonstrated that this feasible and legally allowed "sophisticated Stackelberg" two-part tariff manufacturer-profit dominates all ot her two-part tariffs--including those that dominate the quantity-discount schedule. It follows that under some parametric values the manufacturer prefers sophisticated Stackelberg noncoordination to quantity-discount coordination. What we do not know is whether the manufacturer prefers coordination to non-coordination in a multiple-retailer channel.

Should a manager seek to coordinate a multiple-retailer channel? The answer to this question is not merely of academic interest. For several reasons it is of considerable practical importance as well. First, virtually all supply-chains for consumer goods distribute through multiple outlets. Second, supply-chains regularly establish wholesale-price schedules that are common across many of their retailers. Thus, realistic supply-chains generally are not examples of bilateral monopoly models. Third, many supply-chain decision-makers understandably base their wholesale-price policies on knowledge that is currently accepted as state-of-the-art. Fourth, such knowledge has been based on insights gained from bilateral monopoly models. It logically follows that if "state-of-the-art knowledge" is applicable only under limited circumstances, then managerial decisions predicated on this knowledge may be applied inappropriately. In extreme situations these decisions may even have a deleterious impact on profit. Therefore , it is of practical as well as academic value to ascertain if a manufacturer's best interest is served by coordinating a multiple-retailer supply-chain.

In this paper we investigate the optimality of channel coordination (1) when the manufacturer distributes through competing retailers and (2) when price discrimination between the retailers is infeasible and/or illegal. We address two specific questions. First, "under what set of parametric values (if any) does the sophisticated Stackelberg two-part tariff manufacturer-profit dominate a channel-coordinating menu of two-part tariffs?"

Second, "under what set of parametric values does the sophisticated Stackelberg two-part tariff manufacturer-profit dominate a channel-coordinating quantity-discount schedule?" These comparisons permit us to make a definitive statement about the optimality of channel coordination. As a subsidiary insight, we determine the conditions under which a channel-coordinating menu of two-part tariffs manufacturer-profit dominates a channel-coordinating quantity-discount schedule.

Answers to the questions raised in the preceding paragraph have not appeared in the literature previously; indeed, the questions themselves do not seem to have been voiced. We shall see that the answers pivot on three factors: (1) the magnitude of the difference in retailers' fixed costs, (2) the relative size of the retailers and (3) the degree of competition between them. By "fixed cost" we mean the retailer's opportunity cost in addition to its actual, quantity-independent dollar outlay. Opportunity cost is the profit the retailer could obtain by devoting its scarce resources (such as shelf space) to selling the merchandise of a different manufacturer. Covering fixed cost is essential for retailer's voluntary participation in the channel. By "relative size" we mean the relative unit sales of the retailers. In the case of two retailers this may range from nearly zero (the smaller retailer is tiny compared to the larger one) to one (they are equal sized). We use unit, rather than dollar, sales because the f ormer is common to the competitors-they sell an identical product--whereas the latter is convoluted by the fact that they may charge different prices. By the "degree of competition" we mean the willingness of consumers to switch between retailers on the basis of price. Each of these factors is formalized in the fourth Section. The bottom line is that the manufacturer-optimal wholesale-price schedule depends on retailers' costs, demands, and competitive interaction, all of which are factors over which the manufacturer has little or no control.

We organize the paper as follows. In the next Section we describe our model and our assumptions and we review and extend key results from the relevant literature. Then we derive in detail the channel-coordinating menu of two-part tariffs.[5] In the following Section we define the theoretical conditions under which the sophisticated Stackelberg two-part tariff, the channel-coordinating menu of two-part tariffs, or the channel-coordinating quantity-discount schedule maximizes manufacturer profit. We also connect our research to three classic articles from the marketing science literature. In the subsequent Section we offer a range of practical illustrations in order to give a flavor to the parametric conditions under which channel coordination is, or is not, manufacturer-optimal. In the final Section we discuss our results and suggest directions for future research. A lengthy Technical Appendix that contains the mathematical details of each proof is available from the authors on request. To presage our results , we find that over almost all parametric values channel coordination rarely benefits the manufacturer. And, when it does benefit the manufacturer, coordination out-performs non-coordination only by a narrow margin. If a supply-chain manager (a manufacturer) seeks a rule of thumb, it should be that maximization of manufacturer profit is generally superior to maximization of channel profit.

THE MODEL AND RELEVANT RESULTS FROM THE LITERATURE

In this Section we establish our model of a distribution channel and briefly overview some relevant results from the marketing science literature. In the first sub-Section we present the model while in the second sub-Section we evaluate a vertically integrated manufacturer in order to define channel-coordinating values for prices, quantities and profits. The final two sub-Sections detail the channel-coordinating quantity-discount schedule and the sophisticated Stackelberg analysis with competing retailers.

The Model

We model the distribution channel as a single manufacturer selling its product through two competing retailers. In conformity with previous research we make six key assumptions. (1) The channel has only two levels: A manufacturer and its retailers. (2) Channel breadth is fixed at two competing retailers. (3) All decision-makers are profit-maximizers. (4) All decision-makers have full information. (5) Retailers cannot resell merchandise to each other. (6) Each retailer has a linear, downward-sloping demand curve of the form:

[Q.sub.i] = ([A.sub.i] - b[p.sub.i] + [theta][p.sub.j]) s.t. 0 [less than or equal to] [theta] [less than] b, i, j [epsilon] (1,2) (1)

In equation (1), [p.sub.i] denotes the price charged by the [i.sup.th] retailer while the intercept [A.sub.i] measures the [i.sup.th] retailer's demand at prices of zero (its "base demand"). Parameters b and [theta] reflect the retailer's own-price and cross-price sensitivity of demand, respectively. For there to be competition between the retailers requires [theta] [greater than] 0; for aggregate demand to be downward sloping requires [theta] [less than] b.

Our retailers may be envisioned as intra-type or as inter-type competitors. The former case (metaphorically, Macy's versus May Company) is compatible with a high degree of competition ([theta] close to b) and roughly equal sized retail outlets. The latter case (metaphorically, department stores versus apparel boutiques) is compatible with a moderate degree of competition and retail outlets of substantially different size. Of course, inter-type competitors could be of roughly equal size. The point is that our logic is applicable to a wide variety of competitive supply-chain management situations.

Our second assumption--fixed channel breadth--merits additional commentary. We know that determination of optimal channel breadth [6] is inordinately complex; the following pages will demonstrate that the topic addressed herein is also quite complicated. Therefore, in this paper we seek the manufacturer-optimal wholesale-price policy at constant channel breadth, leaving the convolution of breadth and wholesale-price optimality as a topic for future research.

The Vertically-Integrated Manufacturer

A vertically integrated manufacturer obtains the totality of channel profit: therefore, it will seek to maximize channel profit. Such maximization leads to the Nash-equilibrium, channel-coordinating margins and quantities:

[[[micro].sup.*].sub.i] [equivalent] ([[P.sup.*].sub.i] - [C.sub.i] - C) = [[[bQ.sup.*].sub.i] + [theta][[Q.sup.*].sub.j]/[b.sup.2] - [[theta].sup.2]] (2)

[[Q.sup.*].sub.i] = [[A.sub.i] - b([c.sub.i] + C) + [theta](c.sub.j] + C)/2] (3)

In the preceding equations C, [c.sub.i], and [c.sub.j] are per-unit variable costs of the manufacturer, and the [i.sup.th] and [j.sup.th] retailers respectively. These prices and quantities coordinate the channel by internalizing the inter-retailer externality implicit in demand curve (1). The relationship between retailers' outputs is determined by:

([A.sub.i] - [A.sub.j]) [GREATER THAN OVER LESS THAN] (b + [theta]) ([c.sub.i] - [c.sub.j]) [right arrow] [[Q.sup.*].sub.i] [GREATER THAN OVER LESS THAN] [[Q.sup.*].sub.j] (4)

To ease our exposition we will henceforth assume [[Q.sup.*].sub.i] [greater than] [[Q.sup.*].sub.j]. This can occur due to differences in base demand ([A.sub.i] - [A.sub.j]), differences in weighted per-unit costs ((b + [theta])([c.sub.i] - [c.sub.j])), or some combination thereof. Although our ensuing analysis is completely general, for expositional simplicity we will treat the retailers as having equal per-unit costs. Hence the [i.sup.th] retailer will be discussed as having the greater base demand. Finally, it can be proven that if [[Q.sup.*].sub.i] [greater than] [[Q.sup.*].sub.j], then the [i.sup.th] retailer is larger under all of the wholesale-price policies investigated in this paper.

A decentralized method of achieving coordinated prices and quantities is with a transfer-pricing scheme in which the [i.sup.th] retailer pays a marginal per-unit wholesale-price of:

[[W.sup.*].sub.i] = [theta]/b [[theta][[Q.sup.*].sub.j] + b[[Q.sup.*].sub.j]/([b.sup.2] - [[theta].sup.2])] + C (5)

Whether coordination is determined by centrally dictated prices or through use of retailer-specific transfer-prices [7] (5), the behavioral results are identical. The vertically-integrated manufacturer, charging prices defined by Equation (2), generates channel profit of:

[[[pi].sup.*].sub.c] = ([b[([[Q.sup.*].sub.i]).sup.2] + 2[theta][[Q.sup.*].sub.i][[Q.sup.*].sub.j] + b[([[Q.sup.*].sub.j]).sup.2]]/([b.sup.2] - [[theta].sup.2])) - [f.sub.i] - [f.sub.j] - F [equivalent] [[R.sup.*].sub.c] - [f.sub.i] - [f.sub.j] - F (6)

The terms F, [f.sub.i], and [f.sub.j] represent fixed costs for the manufacturing arm and the [i.sup.th] and [j.sup.th] retail outlets respectively and [[R.sup.*].sub.c] is the channel's net revenue--total dollar sales minus all variable costs. We make three observations. First, no wholesale-price scheme with independent retailers can induce channel coordination unless it reproduces the prices and quantities given by equations (2) and (3). Second, these values can only be obtained by wholesale-price schedules that generate marginal wholesale-prices of [[W.sup.*].sub.i] and [[W.sup.*].sub.j]--given by Equation (5). Third, from (5) we see that the optimal per-unit marginal cost must be larger for the retail outlet with the smaller optimal output. Therefore, no single two-part tariff can generate channel coordination. Rather, a per-unit discount is required.

The Channel-Coordinating Quantity-Discount Schedule

For a quantity-discount schedule to be channel coordinating requires that it generate a marginal wholesale cost to the [i.sup.th] retailer precisely equal to the [[W.sup.*].sub.i] of Equation (5). The most mathematically tractable of such schedules [8] is linear in quantity; this leads to a total cost to the [i.sup.th] retailer of: ([W.sup.QD] - [w.sup.QD][Q.sub.i])[Q.sub.i] + [phi]. With this schedule each retailer pays an identical fixed fee ([phi]) but a quantity-dependent per-unit wholesale-price. Hence the per-unit marginal cost to the [i.sup.th] retailer is ([W.sup.QD] - [2w.sup.QD][Q.sub.i]). Channel-coordinating values for [W.sup.QD] and [w.sup.QD] are:

[W.sup.[QD.sup.*]] = [[theta]([[Q.sup.*].sub.i] + [[Q.sup.*].sub.j])/[b.sup.2] - [[theta].sup.2]] + C (7)

[w.sup.[QD.sup.*]] = [[theta]/2b(b + [theta])] (8)

Note that [w.sup.QD] is the per-unit quantity-discount; Wit were zero we would have a two-part tariff.

This quantity-discount schedule yields the same channel-coordinating retail prices and quantities as obtained under vertical integration, while generating profit for the [i.sup.th] retailer of:

[[[pi].sup.[QD.sup.*]].sub.i] = [(1 - [bw.sup.[QD.sup.*]])[([[Q.sup.*].sub.i]).sup.2]/b] - [f.sub.i] - [[phi].sup.QD] [equivalent] [[R.sup.[QD.sup.*]].sub.i] - [f.sub.i] - [[phi].sup.QD] (9)

In Equation (9) [[R.sup.[QD.sup.*]].sub.i] is the [i.sup.th] retailer's net revenue and [f.sub.i] is its fixed cost (including the minimal economic profit--its opportunity cost--required for it to participate in the channel).

To retain both retailers in the channel, the (common) fixed fee cannot leave either retailer with insufficient profit for it to be willing to distribute the manufacturer's product. That is, it cannot exceed the lesser of ([[R.sup.[QD.sup.*]].sub.i] - [f.sub.i]) and ([[R.sup.[QD.sup.*]].sub.j] - [f.sub.j]) if both retailers are to be retained as channel members. And, to maximize manufacturer profit, the fixed fee cannot be less than the maximum value that is compatible with retaining both retailers. Thus we have:

[[phi].sup.[QD.sup.*]] = min{([[R.sup.[QD.sup.*]].sub.i] - [f.sub.i]), ([[R.sup.[QD.sup.*]].sub.j] - [f.sub.j])} (10)

There are three possibilities. (1) The [i.sup.th] retailer is more profitable, so [[phi].sup.[QD.sup.*]] = ([[R.sup.[QD.sup.*]].sub.j] - [f.sub.j]) and [[[pi].sup.[QD.sup.*]].sub.i] [greater than] 0 = [[[pi].sup.[QD.sup.*]].sub.j].

(2) The [j.sup.th] retailer is more profitable, so [[phi].sup.[QD.sup.*]] = ([[R.sup.[QD.sup.*]].sub.i] - [f.sub.i]) and [[[pi].sup.[QD.sup.*]].sub.i] = 0 [less than] [[[pi].sup.[QD.sup.*]].sub.j].

(3) The retailers are equally profitable, so [[phi].sup.[QD.sup.*]] = ([[R.sup.[QD.sup.*]].sub.i] - [f.sub.i]) = ([[R.sup.[QD.sup.*]].sub.j] - [f.sub.j]) and [[[pi].sup.[QD.sup.*]].sub.i] = 0 = [[[pi].sup.[QD.sup.*]].sub.j].

Because the difference in retailer profitability is ([[[pi].sup.[QD.sup.*]].sub.i] - [[[pi].sup.[QD.sup.*]].sub.j]) = ([[R.sup.[QD.sup.*]].sub.i] - [[R.sup.[QD.sup.*]].sub.j]) - ([f.sub.i] - [f.sub.j]), the more profitable retailer is given by the relationship:

([[[pi].sup.[QD.sup.*]].sub.i] [GREATER THAN OVER LESS THAN] [[[pi].sup.[QD.sup.*]].sub.j]) [right arrow] [[delta].sup.[QD.sup.*]] [equivalent] ([[R.sup.[QD.sup.*]].sub.i] - [[R.sup.[QD.sup.*]].sub.j])

= ((2b + [theta])([[Q.sup.*].sub.i] + [[Q.sup.*].sub.j])([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])/2b(b + [theta])) [GREATER THAN OVER LESS THAN] ([f.sub.i] - [f.sub.j]) (11)

The term [[delta].sup.[QD.sup.*]] is the difference in retailer net revenues. By virtue of our assumption concerning relative retailer size (i.e., [[Q.sup.*].sub.i] [greater than] [[Q.sup.*].sub.j]) we have [[delta].sup.[QD.sup.*]] [greater than] 0. Hence, we may think in terms of three "Zones" in ([f.sub.i] - [f.sub.j]) space that are defined as:

Zone [[Z.sup.[QD.sup.*]].sub.j]: [[delta].sup.[QD.sup.*]] [greater than] ([f.sub.i] - [f.sub.j])

Zone [[Z.sup.[QD.sup.*]].sub.ij]: [[delta].sup.[QD.sup.*]] = ([f.sub.i] - [f.sub.j])

Zone [[Z.sup.[QD.sup.*]].sub.i]: [[delta].sup.[QD.sup.*]] [less than] ([f.sub.i] - [f.sub.j]) (12)

The subscript on each [Z.sup.[QD.sup.*]] defines which retailer nets zero economic profit after paying the fixed fee. In game-theoretic terms, the magnitude of ([f.sub.i] - [f.sub.j]) determines the retailer for whom the channel participation constraint is binding.

The impact of a change in [f.sub.i] on manufacturer profit varies by Zone:

[partial][[[pi].sup.[QD.sup.*]].sub.M]([f.sub.i], [f.sub.j])/[partial][f.sub.i] = {$0 [FOR ALL] ([f.sub.i] - [f.sub.j]) [less than] [[delta].sup.[QD.sup.*]] - $2 [FOR ALL] ([f.sub.i] - [f.sub.j]) [greater than] [[delta].sup.[QD.sup.*]] (13)

At ([f.sub.i] - [f.sub.j]) [less than] [[delta].sup.[QD.sup.*]] increases in [f.sub.i] lower the [i.sup.th] retailer's profit but have no effect on [[phi].sup.[QD.sup.*]] or on manufacturer profit--the [i.sup.th] participation constraint is not binding. At ([f.sub.i] - [f.sub.j]) [greater than] [[delta].sup.[QD.sup.*]], retention of the [i.sup.th] retailer requires that a $1 increase in [f.sub.i] be matched by a $1 decrease in [[phi].sup.[QD.sup.*]] for both retailers--a $2 cost to the manufacturer. Figure 1 illustrates this graphically.

Summarizing, the quantity-discount schedule [Equations (7) and (8)] coordinates the channel at all ([f.sub.i] - [f.sub.j]) values and generates total channel profit [[[pi].sup.*].sub.c] [Equation (6)]. The manufacturer obtains all of [[[pi].sup.*].sub.c] only along the line in ([f.sub.i] - [f.sub.j]) space that is defined by the kink ([f.sub.i] - [f.sub.j]) = [[delta].sup.[QD.sup.*]]. On this line no noncoordinating wholesale-price schedule can manufacturer-profit dominate the channel-coordinating quantity-discount schedule.

The "Sophisticated Stackelberg" Two-Part Tariff

A manufacturer using a "naive Stackelberg" tariff optimizes its per-unit fee but accepts the fixed fee as a residual that extracts all profit from the less-profitable retailer. A naive two-part tariff can never coordinate a channel, although it leads to manufacturer profit that is defined in terms of three "Zones" (the middle one defined at a kink (call it [[delta].sup.nS]--similar to the Zones derived above). In fact, every possible two-part tariff will describe the same general shape of manufacturer profit (horizontal for ([f.sub.i] - [f.sub.j]) [less than] [[delta].sup.nS], declining at a rate of $2 for ([f.sub.i] - [f.sub.j]) [greater than] [[delta].sup.nS]). Only the value of [[delta].sup.nS] and the manufacturer profit at the kink differ with the per-unit fee.

In contrast to a naive tariff, a "sophisticated Stackelberg" two-part tariff simultaneously optimizes the per-unit fee and the fixed fee in light of the difference in retailers' fixed costs. The resulting per-unit fee and the fixed fee in the sophisticated tariff are a function of ([f.sub.i] - [f.sub.j]), varying as this difference varies. [9] The former is constant for all values of ([f.sub.i] - [f.sub.j]), less than a lower boundary [L.sup.[SS.sup.*]], it declines at a linear rate until an upper boundary condition ([f.sub.i] - [f.sub.j]) = [U.sup.[SS.sup.*]] is met, and then it is constant again. The lower boundary is determined by the profit conditions [[[pi].sup.[SS.sup.*]].sub.i] [greater than or equal to] 0 =

[[[pi].sup.[SS.sup.*]].sub.j] - whereas the upper boundary is determined by [[pi].sup.[SS.sup.*]].sub.i] = 0 [less than or equal to] [[pi].sup.[SS.sup.*]].sub.j].In short, the boundaries are the points at which both participation constraints are binding. These boundaries are defined as:

[L.sup.[SS.sup.*]] [equivalent] 2(([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j]) [3[theta] [[Q.sup.*].sub.i] + (4b - [theta]) [[Q.sup.*].sub.j]/[(2b + [theta]).sup.2]

[U.sup.[SS.sup.*]] [equivalent] 2([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])[(4b - [theta]) [[Q.sup.*].sub.i] + 3[theta][[Q.sup.*].sub.j]]/[(2b + [theta]).sup.2] (15)

By virtue of our assumption that [[Q.sup.*].sub.i] [greater than] [[Q.sup.*].sub.j], we have [U.sup.[SS.sup.*]] [greater than] [L.sup.[SS.sup.*]] [greater than] 0. The fixed fee is equal to the net revenue of the jth retailer for all ([f.sub.i] - [f.sub.j]) [less than] [L.sup.[SS.sup.*]] and is equal to the net revenue of the ith retailer for all ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[SS.sup.*]]. Between the two boundaries the fixed fee is equal to the net revenue of both retailers.

Manufacturer profit under the sophisticated Stackelberg two-part tariff can be described graphically by connecting together the kinked points of all possible two-part tariffs (see Figure 2). In game-theoretic terms, this defines the envelope of manufacturer profits from all possible two-part tariffs that retain both retailers as channel members. As can be seen, the manufacturer extracts all profit from both retailers provided ([f.sub.i] - [f.sub.j]) lies between [L.sup.[SS.sup.*]] and [U.sup.[SS.sup.*]]. Below [L.sup.[SS.sup.*]] and above [U.sup.[SS.sup.*]] one retailer is able to retain an excess economic profit. In short, there are three "Zones" in ([f.sub.i] - [f.sub.j]) within which wholesale price, total channel profit, and the distribution of profit between manufacturer and retailers differ. The Zones are: [10]

Zone [[Z.sup.[SS.sup.*]].sub.j]: [L.sup.[SS.sup.*]] [greater than] ([f.sub.i] - [f.sub.j]) (16)

Zone [[Z.sup.[SS.sup.*]].sub.ij]: [L.sup.[SS.sup.*]] [less than or equal to] ([f.sub.i] - [f.sub.j]) [less than or equal to] [U.sup.[SS.sup.*]]

Zone [[Z.sup.[SS.sup.*]].sub.i]: ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[SS.sup.*]]

In Zone [[Z.sup.[SS.sup.*]].sub.j] the [j.sup.th] (the less profitable) retailer obtains zero economic profit, while in Zone [[Z.sup.[SS.sup.*]].sub.j] the [i.sup.th] retailer earns zero economic profit. In Zone [[Z.sup.[SS.sup.*]].sub.ij] both retailers net zero economic profits. Within Zone [[Z.sup.[SS.sup.*]].sub.j] manufacturer profit is unaffected by changes in the [i.sup.th] retailer's fixed cost. In Zone [[Z.sup.[SS.sup.*]].sub.ij] manufacturer profit declines at an increasing rate with changes in retail fixed costs ($0 [less than] [partial][[[pi].sup.[SS.sup.*]].sub.M]/[partial][f.sub.i] [less than] - $2) and in Zone [[Z.sup.[SS.sup.*]].sub.i] manufacturer profit is a constant, negative function of the [i.sup.th] retailer's fixed cost ([partial][[[pi].sup.[SS.sup.*]].sub.M]/[partial][f.sub.i] = -$2).

The sophisticated Stackelberg manufacturer is more profitable than is the naive Stackelberg manufacturer--or any other manufacturer utilizing a single two-part tariff. If any noncoordinating two-part tariff can ever manufacturer-profit dominate a channel-coordinating wholesale-price policy, it will be the sophisticated Stackelberg policy. Nonetheless, with a sophisticated Stackelberg two-part tariff, channel profit (defined as [[[pi].sup.SS].sub.C]) is generally less than what is earned by a channel-coordinating, vertically-integrated monopolist: [[[pi].sup.[SS.sup.*]].sub.C] [less than] [[[pi].sup.*].sub.C]. [11]

THE CHANNEL-COORDINATING MENU OF TARIFFS [12]

We now turn to the determination of a channel-coordinating menu of two-part tariffs. Because our primary purpose in devising a menu is to achieve coordination of a competing retailer channel, the menu must induce both retailers (1) to participate in the channel and (2) to sell their channel-optimal outputs. We begin with five observations. First, the fixed fee must leave each retailer with a non-negative profit; that is, the participation constraint must be met for both of them. Second, the per-unit fee must equal the optimal per-unit wholesale-price [defined by Equation (2.5)]; if it does not the channel will not be coordinated. Third, because the optimal per-unit wholesale-price differs by retailer, there must be a unique two-part tariff intended for each retailer. Fourth, because retailers may select whichever tariff they want, the fixed fee must prevent "defection" to the tariff intended for the other retailer: each retailer must prefer its "own" tariff of the menu. Fifth, within these constraints, fixed fees may be set to maximize the manufacturer's profit.

The Model

We model the decision process as a three-stage game. In Stage 1 the manufacturer creates a menu of two-part tariffs and offers that menu to the retailers. In Stage 2 each retailer selects the tariff that gives it the higher profit. In Stage 3 retailers set prices to maximize their own profit. To conserve space we sketch our argument for the first two stages of the game (the third stage involves a standard set of maximizations). Details of the mathematical logic for all three stages, and the resulting equilibrium values of all variables are in a Technical Appendix that is available from the authors.

To simplify our presentation we denote the [m.sup.th] two-part tariff as [[tau].sub.m] [equivalent] {[[W.sup.*].sub.m], [[phi].sub.m]}, m = i or j. We express the menu itself as [tau] [equivalent] {[[tau].sub.i], [[tau].sub.j]}. To assure coordination [[W.sup.*].sub.m] must be defined by Equation (5); only the fixed fee is a choice variable for the manufacturer.

The Manufacturer's Ideal Fixed Fees

For the manufacturer the ideal result would be for the menu to allow the manufacturer to extract all profit from a coordinated channel. There are two steps that must occur for this outcome to obtain. First, the manufacturer must set the [m.sup.th] fixed fee precisely equal to the [m.sup.th] retailer's maximum possible net revenue minus its fixed cost (i.e., [[[phi].sup.*].sub.m] [equivalent] ([[R.sup.*].sub.m] - [f.sub.m]), m = i or j). If a lower fixed fee is set the manufacturer will not receive all channel profit. Second, the [i.sup.th] retailer must select the [i.sup.th] tariff and the [j.sup.th] retailer must select the [j.sup.th] tariff when the fixed fee components of the menu are [[[phi].sup.*].sub.i] and [[[phi].sup.*].sub.j]. If there is "defection"--if a retailer selects the wrong element of the menu--the channel will not be coordinated. The only menu of two-part tariffs that satisfies the first step--provided the second step is met--is:

[[tau].sup.*] [equivalent] {[[[tau].sup.*].sub.i]. [[[tau].sup.*].sub.j] [equivalent] {([[W.sup.*].sub.i], [[[phi].sup.*].sub.i]), ([[W.sup.*].sub.j], [[[phi].sup.*].sub.j])} (17)

Under what conditions will both retailers behave in the requisite fashion if (17) is the menu? There are two necessary and sufficient conditions. First, the participation constraint, which sets an upper limit on the fixed fee, must be met for both retailers:

[[phi].sup.k] [less than or equal to] [{[[([[Q.sup.*].sub.kmn]).sup.2]/b] - f}.sub.k] [equivalent] ([[R.sup.*].sub.kmn] - [f.sub.k]), k [epsilon] (i, j) (18)

In Equation (18) the term [[R.sup.*].sub.kmn] denotes the net revenue of the [k.sup.th] retailer given that it selects the [m.sup.th] element of the menu when its rival selects the [n.sup.th] element. Second, both retailers must prefer the tariff that is intended for them. Each will do so if and only if the intended tariff generates the higher profit; this is the self-selection constraint. In our context channel coordination in the presence of self-selection requires that (1) the [i.sup.th] retailer to pick [[[tau].sup.*].sub.i] (rather than [[[tau].sup.*].sub.j]) when the [j.sup.th] retailer selects [[[tau].sup.*].sub.j] and that (2) the [j.sup.th] retailer pick [[[tau].sup.*].sub.j] (rather than [[[tau].sup.*].sub.j]) when the [i.sup.th] retailer selects [[[tau].sup.*].sub.j]. These statements reduce to the comparative profit requirements:

([[R.sup.*].sub.iij] - [[phi].sub.i]) [greater than] ([[R.sup.*].sub.ijj] - [[phi].sub.j]) and ([[R.sup.*].sub.jji] - [[phi].sub.j]) [greater than] ([[R.sup.*].sub.jii] - [[phi].sub.i]) (19)

These channel-coordinating tariff selections are an equilibrium set of choices provided neither retailer can improve its revenue by choosing an alternative tariff. However, there are actually three [13] possible responses to menu (17)--or to any other menu of two-part tariffs:

Case I: both retailers choose tariff [[[tau].sup.*].sub.j],

Case II: the [i.sup.th] retailer chooses tariff [[[tau].sup.*].sub.i] and the [j.sup.th] retailer chooses tariff [[[tau].sup.*].sub.j], and

Case III: both retailers choose tariff [[[tau].sup.*].sub.i].

Only Case II satisfies condition (19); thus only Case II leads to a coordinated channel with the manufacturer extracting all channel profit. Neither the Case I nor the Case III equilibrium leads to channel coordination or full profit extraction. Thus, two questions arise. (1) What is required for the Case II equilibrium to obtain? (2) If a Case I or a Case III equilibrium does occur, can the manufacturer modify the fixed fee to preclude defection, thereby restoring channel coordination?

In response to the first question, the Case II equilibrium occurs if and only if:

[L.sup.[Menu.sup.*]] [less than or equal to] ([f.sub.i] - [f.sub.j]) [less than or equal to] [U.sup.[Menu.sup.*]] (20)

where:

[L.sup.[Menu.sup.*]] [equivalent] h[bh[[Q.sup.*].sub.i] + (h + 2[theta](2[b.sup.2] - [[theta].sup.2]))[[Q.sup.*].sub.j]]([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])/[(b + [theta]).sup.2][(4[b.sup.2] - [[theta].sup.2]).sup.2] [greater than] 0 (21)

[U.sup.[Menu.sup.*]] [equivalent] h[(h + 2[theta](2[b.sup.2] - [[theta].sup.2]))[[Q.sup.*].sub.i] + bh[[Q.sup.*].sub.j]]([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])/[(b + [theta]).sup.2][(4[b.sup.2] - [[theta].sup.2]).sup.2] [greater than] [L.sup.[Menu.sup.*]] [greater than] 0 (22)

and where h [equivalent] (4[b.sup.2] + 2b[theta] - [[theta].sup.2]) [greater than] 0. These boundary conditions were derived by substituting the fixed fee definitions [[[phi].sup.*].sub.m] = ([[R.sup.*].sub.m] - [f.sub.m]) into the comparative profit requirement (19).

Case I equilibrium occurs if [L.sup.[Menu.sup.*]] [greater than] ([f.sub.i] - [f.sub.j]) and Case III equilibrium occurs if ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[Menu.sup.*]]. In short, just as with the previous wholesale-price policies, there are three "Zones" in ([f.sub.i] - [f.sub.j]) space. These Zones are:

Zone [[Z.sup.[Menu.sup.*]].sub.j]: [L.sup.[Menu.sup.*]] [greater than] ([f.sub.i] - [f.sub.j]) Right arrow] Case I

Zone [[Z.sup.[Menu.sup.*]].sub.ij]: [L.sup.[Menu.sup.*]] [less than or equal to] ([f.sub.i] - [f.sub.j]) [less than or equal to] [U.sup.[Menu.sup.*]] Right arrow] Case II

Zone [[Z.sup.[Menu.sup.*]].sub.i]: ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[Menu.sup.*]] Right arrow] Case III (23)

Adjusting the Fixed Fees to Assure Channel-Coordination

How can the fixed fees be modified to assure channel coordination when either a Case I or a Case III defection occurs? Because the problem is symmetric, it is sufficient for us to focus on Case I as illustrative of both Cases. In Case I both retailers select the [j.sup.th] tariff when offered the menu {[[[tau].sup.*].sub.i], [[[tau].sup.*].sub.j]}. If the [i.sup.th] retailer were not to defect it would obtain zero economic profit since its profit before paying the fixed fee ([[R.sup.*].sub.i] - [f.sub.i]) would be equal to the fixed fee [[[phi].sup.*].sub.j] = [[[R.sup.*].sub.i] - [f.sub.i]]. By defecting the [i.sup.th] retailer is able to pay a lower fixed fee [[[phi].sup.*].sub.j] = [[[R.sup.*].sub.j] - [f.sub.j]] [less than] [[[phi].sup.*].sub.i]. But defection also entails a higher per-unit fee; thus the [i.sup.th] retailer net revenue is reduced from [[R.sup.*].sub.i] to {[[R.sup.*].sub.i] - [B.sub.j]}. The net revenue reduction is:

[B.sub.j] [equivalent] [theta](2[b.sup.2] - [[theta].sup.2])([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])/b[(b + [theta]).sup.2][(4[b.sup.2] - [[theta].sup.2]).sup.2] [(8[b.sup.3] + 6[b.sup.2][theta] - 2b[[theta].sup.2] - [[theta].sup.3])[[Q.sup.*].sub.i] + [theta] (2[b.sup.2] - [[theta].sup.2])[[Q.sup.*].sub.j]] [greater than] 0 (24)

The important point is that in a Case I equilibrium the [i.sup.th] retailer defects to the [j.sup.th] tariff because its sacrificed profit from a reduction in net revenue is less than its saving from a lower the fixed fee. As a result, the [i.sup.th] retailer's profit rises. Formally we have:

[[pi].sub.ijj] [equivalent] ({[[R.sup.*].sub.i] - [B.sub.j]} - [f.sub.i]) - [[[R.sup.*].sub.j] - [f.sub.j]] [greater than] ([[R.sup.*].sub.i] - [f.sub.i]) - [[[R.sup.*].sub.i] - [f.sub.i]] [equivalent] [[pi].sub.iij] = 0 (25)

Expression (25) simply states that profit due to defection is greater than the zero profit that is earned without defecting. On both sides of the inequality the [bracketed] term is the relevant fixed fee and the (parenthetical) term is retailer profit prior to paying the fixed fee. The term in {braces} on the LHS is the [i.sup.th] retailer's net revenue when it defects.

To assure coordination the manufacturer must eliminate the [i.sup.th] retailer's incentive to defect. How can it achieve this? It cannot adjust the per-unit fees because doing so would make coordination impossible. Thus, it must alter a fixed fee. If the manufacturer were to raise [[[phi].sup.*].sub.j] it would encourage the [j.sup.th] retailer to defect; thus the manufacturer has only one option: [[[phi].sup.*].sub.i] must be lowered. But how far? The answer is implicit in expression (25). We know that when it defects the [i.sup.th] retailer pays a fixed fee of [[[phi].sup.*].sub.j] and sacrifices net revenue of [B.sub.j]. Thus, the maximum fixed fee that the manufacturer can extract from the [i.sup.th] retailer without causing defection is ([[[phi].sup.*].sub.j] + [B.sub.j]). As a result, instead of using [[[tau].sup.*].sub.i] [equivalent] {[[W.sup.*].sub.i], [[[phi].sup.*].sub.i] = ([[R.sup.*].sub.i] - [f.sub.i])} as the [i.sup.th] element of the menu, the manufacturer must instead offer [[[tau].sup.L].sub .i] [equivalent] {[[W.sup.*].sub.i], [[[phi].sup.L].sub.i] = (([[R.sup.*].sub.j] - [f.sub.j]) + [B.sub.j])} as the [i.sup.th] element. With this modification the tariff intended for [i.sup.th] retailer ([[[tau].sup.L].sub.i]) and the tariff intended for the [j.sup.th] retailer ([[[tau].sup.*].sub.j]) generate equal profit for the [i.sup.th] retailer. We break such "ties" in favor of the channel-coordinating solution. In summary, in a Case I equilibrium the manufacturer-optimal menu of two-part tariffs is {[[[tau].sup.L].sub.i], [[[tau].sup.*].sub.j]}. This menu leads to the [i.sup.th] retailer earning a positive economic profit while the [j.sup.th] retailer "breaks even."

Similarly, in a Case III equilibrium, which occurs when ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[Menu.sup.*]], both retailers select tariff [[[tau].sup.*].sub.i]. Defection by the [j.sup.th] retailer can be prevented if the [j.sup.th] element of the menu is set to [[[tau].sup.U].sub.j] [equivalent] {[[W.sup.*].sub.j], [[[phi].sup.U].sub.j] = (([[R.sup.*].sub.i] - [f.sub.i]) + [B.sub.i])}, where [B.sub.i] is defined as:

[B.sub.i] [equivalent] [theta](2[b.sup.2] - [[theta].sup.2])([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j])/b[(b + [[theta]).sup.2][(4[b.sup.2] - [[theta].sup.2]).sup.2] [[theta](2[b.sup.2] - [[theta].sup.2])[[Q.sup.*].sub.i] + (8[b.sup.3] + 6[b.sup.2][theta] - 2b[[theta].sup.2] - [[theta].sup.3])[[Q.sup.*].sub.j]] [greater than] 0 (26)

Now both the tariff intended for the [i.sup th] retailer ([[[tau].sup.*].sub.i]) and the (modified) tariff intended for the [j.sup.th] retailer ([[[tau].sub.j].sup.U]) yield the same positive profit to the [j.sup.th] retailer ([[R.sup.*].sub.j] - [f.sub.j]) - [[[phi].sub.j].sup.U],

[[pi].sub.jii] [equivalent] ({[[R.sup.*].sub.j] - [B.sub.i]} - [f.sub.j]) - [[[R.sup.*].sub.i] - [f.sub.i]] = ([[R.sup.*].sub.i] - [f.sub.i]) + [B.sub.i]] [equivalent] [[pi].sub.jji] [greater than] 0 (27)

In expression (27) the LHS is the [j.sup.th] retailer's profit from defecting when the menu is ([[[tau].sup.*].sub.i], [[[tau].sup.*].sub.j]} and the RHS is the profit from not defecting when the menu is {[[[tau].sup.*].sub.i], [[[tau.sup.U].sub.j]}. On both sides the [bracketed] term is the relevant fixed fee, the (parenthetical) term is profit before paying the fixed fee, and the term in {braces} on the LHS is the [j.sup.th] retailer's net revenue when it defects. The term [B.sub.i] is the net revenue reduction to due defection. [14]

To summarize, in Case I or Case III equilibrium the manufacturer-optimal, channel-coordinating menu of two-part tariffs varies across Zones. The respective per-unit fees ([[W.sup.*].sub.i] and [[w.sup.*].sub.j]) are the same in every Zone--as is necessary to assure coordination--but the manufacturer must vary the fixed fees to ensure that both retailers select the proper element of the menu. The manufacturer-optimal fixed fees are:

Zone Zonal Definitions [[Z.sup.[Menu.sup.*]].sub.j] [L.sup.[Menu.sup.*]] [greater than]([F.sub.i] [F.sub.j]) [[Z.sup.[Menu.sup.*]].sub.ij] [L.sup.[Menu.sup.*]] [less than or equal to] ([f.sub.i] - [f.sub.j]) [less than or equal to] [U.sup.[Menu.sup.*]] [[Z.sup.[Menu.sup.*]].sub.i] ([f.sub.i] - [f.sub.j])[greater than] [U.sup.[Menu.sup.*]] Fixed Fees Zone [j.sup.th] Retailer [[Z.sup.[Menu.sup.*]].sub.j] [([[R.sup.*].sub.j] - [f.sub.j]) + [B.sub.j]] [[Z.sup.[Menu.sup.*]].sub.ij] [[R.sup.*].sub.i] - [f.sub.i] [[Z.sup.[Menu.sup.*]].sub.i] [[R.sup.*].sub.j] - [f.sub.i] Zone [j.sup.th] Retailer [[Z.sup.[Menu.sup.*]].sub.j] [[R.sup.*].sub.j] - [f.sub.j] [[Z.sup.[Menu.sup.*]].sub.ij] [[R.sup.*].sub.j] - [f.sub.j] [[Z.sup.[Menu.sup.*]].sub.i] [([[R.sup.*].sub.i] - [f.sub.i]) + [b.sub.i]]

In Zone [[Z.sup.[Menu.sup.*]].sub.j] the [j.sup.th] (the less profitable) retailer obtains zero net economic profit, whereas in Zone [[Z.sup.[Menu.sup.*]].sub.i] the [i.sup.th] retailer is the less profitable--and it obtains zero net economic profit. In Zone [[Z.sup.[Menu.sup.*]].sub.ij] both retailers net zero economic profits after paying their fixed fees. We conclude that an appropriately specified menu of two-part tariffs can coordinate the channel regardless of the actual difference in retailers' fixed costs. The "cost" of coordination is lower manufacturer profit under some distributions of retailers' fixed costs; this is detailed graphically in Figure 3.

Note that in Zone [[Z.sup.[Menu.sup.*]].sub.j] manufacturer profit is invariant with the [i.sup.th] retailer's fixed cost because an increase in [f.sup.i] lowers the profit of the [i.sup.th] (the more profitable) retailer. In zone

[Z.sup.[Menu.sup.*]] manufacturer profit declines at a constant rate ([partial][[[pi].sup.[Menu.sup.*]].sub.M]/[partial][f.sub.1] = -$1) because a $1 increase in [f.sub.i] lowers the fixed fee in tariff [[[tau].sup.*].sub.i] by $1. In Zone [[Z.sup.[Menu.sup.*]].sub.i] an increase in [f.sub.i] decreases the profit of the [i.sup.th] (now the less profitable) retailer--requiring a fixed fee reduction for both retailers in order to preclude (1) the [i.sup.th] retailer from not participating in the channel and (2) the [j.sup.th] retailer from defecting; accordingly [partial][[[pi].sup.[Menu.sup.*]].sub.M]/[partial][f.sub.i] = 1$2. The effect of a change in [f.sub.j] is precisely symmetric.

COORDINATION VERSUS NON-COORDINATION: THEORETICAL BASES

The preceding Sections described three pricing mechanisms: (1) a channel-coordinating quantity-discount schedule, (2) a sophisticated Stackelberg two-part tariff that cannot coordinate the channel but that manufacturer-profit dominates any single two-part tariff, and (3) a channel-coordinating menu of two-part tariffs. Will the manufacturer ever prefer the non-coordinating two-part tariff? If channel coordination is preferable, which method of coordination will yield greater manufacturer profit--a quantity-discount schedule or a menu of two-part tariffs? This Section answers these questions theoretically. Additionally, in the final sub-Section we link our results to three classic articles from the marketing science literature; Jeuland and Shugan (1983), McGuire and Staelin (1983), and Moorthy (1987).

Dimensionality: a Basis for Comparison

We re-parameterize manufacturer profit for each of our wholesale-price policies in terms of two dimensions. They are (1) the degree of competition between the retailers and (2) the relative size of the retailers. We formalize these dimensions in the following manner. First, the degree of competition between the retailers is expressed as the ratio of cross-price to own-price sensitivity ([chi] [equivalent] ([theta]/b)). The ratio of [theta] and b is a pure number. Further, because [theta] [greater than] 0 is required for the retailers to be in competition, and because [theta] [less than] b is necessary for the aggregate demand curve to be negatively sloped, [chi] lies in the unit interval. Second, the relative size of the retailers is expressed as their output ratio in the baseline, vertically integrated state ([Q.sup.*] [equivalent] ([[Q.sup.*].sub.j]/[[Q.sup.*].sub.i]. Because we have assumed [[Q.sup.*].sub.i] [greater than] [[Q.sup.*].sub.j], [Q.sup.*] is a pure number that lies in the unit interval.

The parameters [phi] and [Q.sup.*] form a two-dimensional unit-square within which we may map manufacturer profit for each wholesale-price policy (see Table 1). Although the profit levels are presented in terms of b, [theta], [[Q.sup.*].sub.i], and [[Q.sup.*].sub.j], they are actually combinations of the primitive values [A.sub.i], [A.sub.j], [c.sub.i], [c.sub.j], C, b, and [theta] as can be seen by writing out [Q.sup.*]:

[Q.sup.*] [equivalent] [[Q.sup.*].sub.j]/[[Q.sup.*].sub.i] [equivalent] ([A.sub.j] - b([c.sub.j], + C) + [theta] (c.sub.i] + C)/[A.sub.i] - b([c.sub.i] + C) + [theta] ([c.sub.j] + C). (28)

Thus, manufacturer profit may be expressed in the abbreviated notation of [Q.sup.*] and [chi].

It is important to recognize that although [Q.sup.*] represents the ratio of optimal retailer outputs for a profit-maximizing, vertically-integrated firm, the parametric value [Q.sup.*] is exogenous from the perspective of the wholesale-price policy chosen by the decentralized manufacturer. It is exogenous because a decentralized price strategy has no effect on the values of either [[Q.sup.*].sub.i] or [[Q.sup.*].sub.j]. Thus, the manufacturer's profit from a wholesale-price policy is endogenous whereas the basic parametric values [Q.sup.*] and [chi] are exogenous.

Manufacturer profit is also affected by fixed costs at retail and at manufacturer. The latter value (F) affects each price-policy equally, shifting manufacturer profit uniformly upward or downward in response to changes in F; hence we ignore F in what follows. In contrast, the difference in retailers' fixed costs, which may be of any sign, has a complex impact on profitability because it affects what Zone is relevant for each wholesale-price policy.

The Manufacturer-Optimality Conditions

In this sub-Section we investigate how the manufacturer's choice of a wholesale-price policy varies with the parameters that are encapsulated within the dimensions [Q.sup.*] and [chi]. For ease of exposition we begin with Figure 4, drawn for the special case of equal retailers' fixed costs; this provides a starting point for our analysis. Also, because the marketing science literature on distribution channels largely ignores fixed costs, this case provides the basis for a comparison with extant knowledge. [15,16]

Within the unit-square of Figure 4 we identify four Regions, each of which has its own distinctive, manufacturer-optimal wholesale-price strategy. These Regions are defined in terms of the following three Conditions, which we phrase as questions:

Condition 1: Does coordination with a channel-coordinating menu of two-part tariffs always manufacturer-profit dominate a sophisticated Stackelberg tariff?

Condition 2: Does sophistricated Stackelberg maximization always manufacturer- profit dominate a channel-coordinating menu when differences in retailers' fixed costs are "large"? (By "large" we mean ([f.sub.i] - [f.sup.j]) [greater than] [U.sup.[SS.sup.*]].)

Condition 3: Does coordination with a quantity-discount schedule ever manufacturer-profit-dominate coordination with a menu of two-part tariffs?

The following Table illustrates that the answer to each question is "yes" in only a single Region:

Condition 1: Is [[[pi].sup.[Menu.sup.*]].sub.M][greater than or equal to] [[[pi].sup.[SS.sup.*].sub.M] for all ([f.sub.i] - [f.sub.j])? Region 1 Yes Region 2 No Region 3 No Region 4 No Condtion 2: Is [[[pi].sup.[Menu.sup.*]].sub.M] [less than [[[pi].sup.[SS.sup.*]].sub.M] for all [U.sup.[SS.sup.*]] [less than] ([f.sub.i] - [f.sub.j])? Region 1 No Region 2 Yes Region 3 No Region 4 No Condition 3: Is [[[pi].sup.[Menu.sup.*]].sub.M][less than or equal to] [[[pi].sup.[QD.sup.*]].sub.M] for some [[delta].sup.[QD.sup.*]] [less than]([f.sub.i] - [f.sub.j]) [less than][L.sup.[Menu.sup.*]]? Region 1 No Region 2 No Region 3 No Region 4 Yes

Condition 1 holds only in Region 1; Condition 2 holds only in Region 2; none of the Conditions holds in Region 3; and Condition 3 holds only in Region 4. We now describe the optimal wholesale pricing by Regions. (The intuition and formal proofs for our results are sketched in the Technical Appendix, available from the authors.)

The Manufacturer-Optimal Wholesale-Price Strategy by Region

In this sub-Section we identify the optimal pricing strategy within each Region defined in the preceding Table. In each case we present a decision rule that summarizes the manufacturer's optimal wholesale-price policy.

Region 1

Region 1 occupies a small, parabolic space in the upper left-hand corner of Figure 4. From the Table immediately above we see that in Region 1 the manufacturer always prefers the channel-coordinating menu (1) to the sophisticated Stackelberg tariff (Condition 1 is met) and (2) to the channel-coordinating quantity-discount schedule (Condition 3 is not met). Thus the manufacturer-optimal wholesale-price strategy in Region 1 is:

* Employ the channel-coordinating menu of tariffs.

Notice that this Region exists only for a [Q.sup.*] ratio close to zero and a [chi] ratio close to one. [17] To understand what this means consider the special case of equal costs across retailers ([c.sub.i] = [c.sub.j]). Under this assumption inter-retailer differences arise solely from variations in base levels of demand ([A.sub.i] and [A.sub.j]). Numerical analysis makes clear that the parametric values necessary to be in Region 1 yield a peculiar result: the [i.sup.th] retailer has a slightly higher equilibrium price but sells a massively larger quantity-even though the two retailers are near perfect competitors! We find it difficult to conceive of a real-world example that is compatible with these characteristics so we are inclined to regard Region 1 as a rare occurrence. This is important because Region 1 is the only Region of the unit-square within which channel coordination is optimal for all values of ([f.sub.i] - [f.sub.j]).

Region 2

Region 2 occupies the bottom of Figure 4. As in Region 1, the channel-coordinating menu always dominates the channel-coordinating quantity-discount schedule (Condition 3 is not met). The manufacturer's optimal wholesale-price strategy is a choice between the channel-coordinating menu and the sophisticated Stackelberg tariff. But Region 2 is defined by the condition that the sophisticated Stackelberg two-part tariff is manufacturer-preferred to the channel-coordinating menu for any ([f.sub.i] - [f.sub.j]) [greater than] [U.sup.[SS.sup.*]]. Two other conditions of Region 2 influence the optimal manufacturer price-strategy. First, it can be shown that the sophisticated Stackelberg two-part tariff is manufacturer-preferred to the channel-coordinating menu when ([f.sub.i] - [f.sub.j]) [less than] [L.sup.[SS.sup.*]]. Second, we have already established that, because the channel-coordinating menu extracts all profits from both retailers, the menu dominates the sophisticated Stackelberg tariff whenever [L.sup.[Menu.sup.*]] [less than or equal to]([f.sub.i] - [f.sub.j]) [less than or equal to] [U.sup.[Menu.sup.*]].

Taken together, these three conditions imply the existence of two distinct two ([f.sub.i] - [f.sub.j]) values at which the sophisticated Stackelberg and the menu policies are manufacturer-profit-equivalent. Let [V.sub.1] denote the critical value in the interval [L.sup.[SS.sup.*]] [less than] [V.sub.1] [less than] [L.sup.[Menu.sup.*]] that satisfies the condition:

([f.sub.i] - [f.sub.j]) = [V.sub.1] Right arrow] [[[pi].sup.[Menu.sup.*]].sub.M] = [[[pi].sup.[SS.sup.*]].sub.M] (29)

Similarly, let [V.sub.2] denote the critical value located in the interval [U.sup.[Menu.sup.*]] [less than] [V.sub.2] [less than] [U.sup.[SS.sup.*]] that satisfies:

([f.sub.i] - [f.sub.j]) = [V.sub.2] Right arrow] [[[pi].sup.[Menu.sup.*]].sub.M] = [[[pi].sup.[SS.sup.*]].sub.M] (30)

These critical values define the manufacturer-optimal wholesale-price strategy in Region 2:

* If ([f.sub.i] - [f.sub.j]) [less than] [V.sub.1] employ the sophisticated Stackelberg tariff;

* If [V.sub.1] [less than or equal to] ([f.sub.i] - [f.sub.j]) [V.sub.2] use the channel-coordinating menu of tariffs; and

* If ([f.sub.i] - [f.sub.j]) [greater than] [V.sub.2] employ the sophisticated Stackelberg tariff.

We numerically illustrate this rule in the next Section.

Region 3

Region 3 occupies the middle of Figure 4. This Region is defined by the fact that none of the three Conditions hold. This is consistent with using the sophisticated Stackelberg tariff at "small" values of ([f.sub.i] - [f.sub.j]) and the channel-coordinating menu at higher values. (Because Condition 3 is not met the quantity-discount schedule is not manufacturer-preferred anywhere in this Region.)

We can show that this strategy is indeed manufacturer-optimal by use of the following results. (We stress that these results hold in Region 3, not throughout the unit-square.) First, for ([f.sub.i] - [f.sub.j]) [less than] [L.sup.[SS.sup.*]] the manufacturer prefers the sophisticated Stackelberg two-part tariff to the channel-coordinating menu. (This statement also held in Region 2.) Second, for any ([f.sub.i] - [f.sub.j]) [greater than or equal to] [L.sup.[Menu.sup.*]] the manufacturer prefers the channel-coordinating menu of tariffs. Taken together, these two conditions imply the existence of a ([f.sub.i] - [f.sub.j]) value at which the sophisticated Stackelberg tariff and the menu are manufacturer profit-equivalent. This critical value, which we denote as [V.sub.3], is located in the interval [L.sup.[SS.sup.*]] [less than] [V.sub.3] [less than] [L.sup.[Menu.sup.*]]. It satisfies the condition:

([f.sub.i] - [f.sub.j]) = [V.sub.3] Right arrow] [[[pi].sup.[Menu.sup.*]].sub.M] = [[[pi].sup.[SS.sup.*]].sub.M] (31)

Because this Region is characterized by a single critical value, we summarize the manufacturer-optimal wholesale-price strategy in Region 3 as:

* If ([f.sub.i] - [f.sub.j]) [less than] [V.sub.3] employ the sophisticated Stackelberg tariff; and

* If ([f.sub.i] - [f.sub.j]) [greater than or equal to] [V.sub.3] utilize the channel-coordinating menu of tariffs.

Region 4

Region 4 occupies the upper right-hand portion of Figure 4. This Region is defined by the fact that, for some values of ([f.sub.i] - [f.sub.j]), the manufacturer prefers the channel-coordinating quantity-discount schedule to the channel-coordinating menu of tariffs. It can be shown that this occurs only when [[delta].sup.[QD.sup.*]] [less than] [L.sup.[Menu.sup.*]].

Three results are needed to describe fully this Region's optimal pricing strategy. First, because the manufacturer obtains all channel profit at ([f.sub.i] - [f.sub.j]) = [[delta].sup.[QD.sup.*]], the channel-coordinated quantity-discount schedule is manufacturer-preferred at this point (and in its vicinity). Second, for any ([f.sub.i] - [f.sub.j]) [less than] [L.sup.[SS.sup.*]] the manufacturer prefers the sophisticated Stackelberg two-part tariff to the channel-coordinating quantity-discount schedule. Third, for any ([f.sub.i] - [f.sub.j]) [greater than] [L.sup.[Menu.sup.*]] the manufacturer prefers the channel-coordinating menu of tariffs.

Taken together, these three conditions imply the existence of two distinct ([f.sub.i] - [f.sub.j]) values at which the sophisticated Stackelberg and the menu policies are manufacturer-profit-equivalent. In particular, let [V.sub.4] denote the critical value that is located in the interval [L.sup.[SS.sup.*]] [less than] [V.sub.4] [less than] [[delta].sup.[QD.sup.*]] and that satisfies the following condition:

([f.sub.i] - [f.sub.j]) = [V.sub.4] Right arrow] [[[pi].sup.[QD.sup.*]].sub.M] = [[[pi].sup.[SS.sup.*]].sub.M] (32)

Similarly, let [V.sub.5], located in [[delta].sup.[QD.sup.*]] [less than] [V.sub.5] [less than] [L.sup.[Menu.sup.*]], denote the critical value satisfying the condition:

([f.sub.i] - [f.sub.j]) = [V.sub.5] Right arrow] [[[pi].sup.[Menu.sup.*]].sub.M] = [[[pi].sup.[QD.sup.*]].sub.M] (33)

Then the manufacturer-optimal wholesale-price strategy in Region 4 may be defined as:

* If ([f.sub.i] - [f.sub.j]) [less than] [V.sub.4] employ the sophisticated Stackelberg tariff;

* If [V.sub.4] [less than or equal to] ([f.sub.i] - [f.sub.j]) [less than or equal to] [V.sub.5] use the channel-coordinating quantity-discount schedule; and

* If ([f.sub.i] - [f.sub.j]) [greater than] [V.sub.5] apply the channel-coordinating menu of tariffs.

Regional Summary

We opened this Section by asking (in effect) "Is it more profitable for the manufacturer to coordinate or not to coordinate the channel?" The following Table summarizes our observations. In brief, the answer to our question is dependent on the difference in retailers' fixed costs ([f.sub.i] - [f.sub.j]) and on both the relative size of the retailers ([Q.sup.*]) and the degree of competition between them ([chi]). The pair ([Q.sup.*], [chi]) determines which Region is relevant and the absolute difference in retailers' fixed costs determines which wholesale-price policy is manufacturer profit-preferred.

Relationship to the Literature

In this sub-Section we frame our results relative to three classic articles in the analytical channels literature. We shall show that two of them are special cases of our model whereas the third may be interpreted within the context of Figure 4.

First, Jeuland and Shugan (1983) examined a bilateral monopoly channel. The parallel within our context is given by assuming that the optimal output ratio [Q.sup.*] is zero. Of course, this effectively means that the degree of substitutability is also zero since there is only one retailer! In our language, the Jeuland and Shugan bilateral monopoly model is an analysis of the manufacturer-optimal wholesale-price strategy under the assumptions [Q.sup.*] = 0 and [chi] = 0. (Graphically this is a solitary point--the lower left-hand corner--of Figure 4.) They showed that a properly specified quantity-discount schedule permits the manufacturer to maximize channel profit, all to its own benefit. [18] Second, Moorthy (1987) used the same bilateral monopoly model to demonstrate that the manufacturer can obtain all channel profit with a channel-coordinating two-part tariff.

In the context of our model it can be proven that, under the restricted parametric values [Q.sup.*] = 0 and [chi] = 0, the sophisticated Stackelberg tariff also maximizes channel profit and permits the manufacturer to extract all profit from the retailer. This is a critical feature of the sophisticated Stackelberg tariff. Unlike the naive Stackelberg tariff that can never coordinate any channel, the sophisticated version of the Stackelberg approach does coordinate a bilateral monopoly channel. Thus, our menu, our quantity-discount schedule and our sophisticated Stackelberg tariff yield the same behavioral results in the special case of a bilateral monopoly. However, our approach of adding a second, competing retailer expands our analytical capability from a single corner point to the entire unit-square.

Third, McGuire and Staelin (1983) examined a market containing two channels with identical parametric values--hence the channels have equal output. [19] This is equivalent to assuming that the optimal output ratio is unity ([Q.sup.*] = 1); however, the degree of competition may range from zero to one. It is easy to show that a properly specified quantity-discount schedule permits the manufacturer to maximize channel profit and extract all profit from both retailers--provided fixed costs at retail are identically zero [as is assumed by McGuire and Staelin (1983)]. Similarly, a channel-coordinating menu has the same properties, but due to the fact that the retailers are identical the menu has a single element--it is a degenerate menu. Finally, under this set of parametric values the sophisticated Stackelberg tariff also maximizes channel profit and extracts all profit from both retailers--again duplicating the menu and quantity-discount schedules. Thus, by permitting our retailers to have different parametric values we are able to extend the analysis of competing retailers from the special case of equal market shares--a single vertical line in Figure 4--to all possible market shares as depicted by the unit square. Nonetheless, there is a fascinating parallelism between our work and their work. McGuire and Staelin (1983) were concerned with how the degree of competition affects a manufacturer's decision whether to vertically integrate or to utilize an independent retailer. Our approach deals with how the degree of competition affects a manufacturer's decision to coordinate or not to coordinate the channel. In the next Section we illustrate this phenomenon.

We conclude by stating that for the models analyzed in this sub-Section neither a channel-coordinating menu nor a channel-coordinating quantity-discount schedule has any inherent advantage for a manufacturer relative to a channel non-coordinating sophisticated Stackelberg. The sophisticated tariff is a powerful tool for a manufacturer even for the special cases of selling through a single retailer or serving a pair of competing, but identical, retailers.

COORDINATION VERSUS NON-COORDINATION: PRACTICAL ILLUSTRATIONS

We have seen in Figure 4 that the manufacturer's choice of a wholesale-price policy is dependent upon relative retailer size and the degree of inter-retailer competition when the retailers have equal fixed cost. In this Section we numerically illustrate the implications for the manufacturer-optimal wholesale-price policy when the retailers' fixed costs are unequal. In effect, by analyzing different fixed cost values we are converting our unit-square to a "unit-cube" with the third dimension being the difference in retailer fixed costs. [20]

As a practical matter it is clear that while we can write out the conditions under which coordination or non-coordination is manufacturer-profit-optimal, managers have a more difficult time making the same calculations. At the heart of this differential capability is the fact that as academics we may assume full information about parametric values, but managers must actually work with imperfect knowledge of the three fundamental parametric values: relative retailer size ([Q.sup.*]), the difference in fixed cost ([f.sub.i] - [f.sub.j]), and the degree of competition ([chi]). In general we expect managers to have an excellent grasp of [Q.sup.*], but to have less accurate information about the other two dimensions. Therefore, to illustrate the impact of these two parameters we hold [chi] fixed in the first sub-Section while varying ([f.sub.i] - [f.sub.j]). In the second sub-Section we hold ([f.sub.i] - [f.sub.j]) fixed and vary [chi].

Illustration of the Effects of Changes in ([f.sub.i] - [f.sub.j])

To keep our presentation manageable we present an example based on a single magnitude of competition. We work with the parametric values [A.sub.i] = 150, [A.sub.j] = 100, [c.sub.i] = [c.sub.j] C = $10, and (b - [theta]) = 0.5. Hence [[Q.sup.*].sub.i] = 70, [[Q.sup.*].sub.j] = 45 and relative retailer size is [Q.sup.*] = ([[Q.sup.*].sub.j]/[[Q.sup.*].sub.i]) = 0.643. Note that by focusing upon a single magnitude of competition--a vertical slice of Figure 4--we can clearly delineate the effects of the other two factors: [chi] and ([f.sub.i] - [f.sub.j]). This particular slice is compatible with Regions 2 through 4. Other possible "slices" would generate similar observations (except at very low magnitudes of competition where Region 1 would appear at sufficiently high [chi] values). To further simplify matters we set F = $1,000 and [f.sub.j] = $0 so that ([f.sub.i] - [f.sub.j]) = [f.sub.i].

These three examples represent progressively higher degrees of competition. Variations in the degree of competition may be interpreted in terms of geographic space or customer service. In the spatial realm, close geographic proximity of the retailers implies a high degree of competition. For example, two retailers located in the same shopping mall will be stronger competitors than they would be if they were in different malls. In the service realm, comparable customer service across the two retailers implies stronger competition than differential service would imply. Other marketing mix factors could be considered in the same vein.

A Low Degree of Competition

We start with our lowest degree of competition: [theta]/b = 0.286. Details are presented in Table 2. The first profitability column is based on an arbitrary value ([f.sub.i] - [f.sub.j]) = $0. As can be inferred from the Table, the sophisticated Stackelberg strategy is manufacturer-profit dominant for all 0 [less than or equal to] [f.sub.i] [less than] $3,567.61. Thus the critical value [V.sub.I] is $3,567.61. The menu is strictly preferred for $3,567.62 [less than] [f.sub.i] [less than] $3,792.90. Finally, at [f.sub.i] [greater than] $3,792.91 (the critical value [V.sub.2]) the sophisticated Stackelberg strategy is again preferred. Profit columns 2 and 4 provide information on the [f.sub.i] level at which the menu and the sophisticated Stackelberg strategies generate manufacturer profit-equivalence, whereas profit column 5 (at which [f.sub.i] = [U.sup.[SS.sup.*]] demonstrates that at a sufficiently high fixed cost level non-coordination is again manufacturer-preferred. Note that the quantity-discount strateg y can never be preferred for this set of parametric values because the chosen values of b and [theta] are incompatible with Condition 3. For completeness we note that the sum of the [i.sup.th] retailer's fixed cost and fully coordinated channel profit (denoted as [[[pi].sup.*].sub.C]) is a constant in this and successive Tables.

A Moderate Degree of Competition

At a slightly higher degree of competition ([chi] = .444) the sophisticated Stackelberg strategy is manufacturer-profit-preferred only at "low" levels of the fixed cost difference; it does not again become manufacturer-preferred at high values. For the example at hand we have the sophisticated Stackelberg tariff being manufacturer-profit-optimal for 0 [less than or equal to] [f.sub.i] [less than] $2,633.31 (the critical value [V.sub.3]) and the menu being preferred at all higher ([f.sub.i] - [f.sub.j]). To conserve space we do not provide a Table to demonstrate this result.

Our terminology "low difference in fixed costs" embraces the very real case of a negative difference (i.e., [f.sub.i] [less than] [f.sub.j]). The notion that the smaller retailer may have a higher fixed cost is plausible once we recognize that fixed cost incorporates opportunity cost, land rent, and other factors. If the smaller quantity retailer is located in a mall, while the larger is not, the former may pay higher rent than does the latter. Similarly, if the smaller retailer is an upscale boutique, while the larger retailer is a mid-market purveyor, then the former may have a greater opportunity cost than the latter. What is important from this brief argument is that a "low" (or even a negative) value of ([f.sub.i] - [f.sub.j]) is eminently plausible.

A High Degree of Competition

Finally, the impact of a very high degree of competition ([chi] = .750) is characteristic of Region 4. In particular, the sophisticated Stackelberg tariff is manufacturer-preferred at low levels of ([f.sub.i] - [f.sub.j]) the quantity-discount schedule is preferred at "moderate" levels of ([f.sub.i] - [f.sub.j]), and the menu is manufacturer-optimal at even higher fixed cost differences. The precise critical values are [V.sub.4] = $1,094.47 and [V.sub.5] = $1,131.69.

Channel-Coordinating Wholesale-Price Policies

Contrasting these results demonstrates that there is an inverse relationship between the degree of competition and the ([f.sub.i] - [f.sub.j]) value at which the sophisticated Stackelberg and a channel-coordinating policy attain manufacturer profit-equality. The result with a high degree of competition illustrates the benefit of a quantity-discount schedule to the manufacturer: it allows channel-coordination without the potential defection cost associated with offering a menu of tariffs. A menu enables the manufacturer to maximize total channel profit over a wide range of ([f.sub.i] - [f.sub.j]) values, but coordination comes at the price of sacrificing potential manufacturer profit to prevent defection. Defection is more likely when the elements of the menu are similar--which is the case when there is a high degree of competition between the retailers. Thus it should not be surprising that the quantity-discount strategy is a viable supplement to the channel-coordinating menu only at high degrees of competi tion when the retailers are of roughly comparable size.

Illustration of the Effects of Changes in the Degree of Competition

In this sub-Section we utilize a graphical presentation--based on numerical analysis--to illustrate the effect on manufacturer's profit of changes in the degree of competition ([chi]) under each of our wholesale-price policies. We evaluate the impact of [chi] over four combinations of [Q.sup.*] and ([f.sub.i] - [f.sub.j]):

[Q.sup.*] = .67 [Q.sup.*] = .11 [f.sub.i] = 0 = [f.sub.j] Figure 5 Figure 7 [f.sub.i] = .65 = [f.sub.j] Figure 6 Figure 8

These four examples represent sharply different relative retailer size and fixed costs. In Figures 5 and 6 we examine the case of the smaller retailer having a 40% market share ([Q.sup.*] = .67 = 40%/60%). This relative retailer size roughly reprises our parametric value from Tables 2 through 4 (where we had [Q.sup.*] = .643). More importantly, it is representative of competition between rivals who are of approximately equal size--metaphorically Macy's versus May Company. We shall refer to this as "competition between equals." In Figures 7 and 8 we examine the case of the smaller retailer having a 10% market share ([Q.sup.*] = .11 = 10%/90%). This is representative of any "David versus Goliath" competition (e.g., a small apparel boutique versus a department store or a convenience food store versus a supermarket). We shall refer to this as "competition between unequals." In both cases we show the manufacturer-profit relationship between each of our wholesale-price policies at every possible degree of competit ion. In Figures 5 and 7 we focus on an absence of fixed costs (again reflecting general usage in the literature) whereas in Figures 6 and 8 we set fixed cost at a substantial level, 65% of a retailer's net revenue in a coordinated channel.

Competition Between Equals

When the rival retailers are of roughly comparable size we find that the sophisticated Stackelberg tariff manufacturer-profit dominates both of the channel-coordinating whole-sale-price policies at all degrees of competition. This dominance is relatively small at high values of [chi], but is distinctly noticeable at lower levels of competition, especially when fixed costs are substantial. The general inference, that the sophisticated Stackelberg tariff is a powerful managerial tool, holds as long as fixed costs are less than two-thirds the channel coordinated net revenue. At larger fixed cost levels the menu and (sometimes) the quantity-discount schedule are manufacturer-profit-optimal at high levels of [chi] However, even when this is the case the sophisticated Stackelberg tariff generates a minimum of 96% of the manufacturer-profit obtained with the optimal coordinating policy. Interestingly, at lower levels of [chi] the sophisticated Stackelberg tariff outperforms the channel-coordinating policies by well in excess of 10%. We conclude that, when retail competition is between "equals" who have roughly equal fixed costs, the sophisticated Stackelberg wholesale-price policy--which does not coordinate the channel--is apt to be the single best price-policy for the manufacturer to adopt when the precise degree of competition is unknown. It is the actual optimal policy over most of parametric space, and when it is not-optimal it is an excellent alternative choice because it does not fall much short of the optimum. In contrast, when the coordinating schedules are not optimal they are often poor performers relative to the sophisticated Stackelberg tariff. [21]

Competition Between Unequals

When the rival retailers are of dramatically different size we again find that the sophisticated Stackelberg tariff manufacturer-profit dominates both channel-coordinating wholesale-price policies at all degrees of competition. The basic message from Figures 7 and 8 is the same as above; the sophisticated Stackelberg two-part tariff is manufacturer-profitoptimal; channel-coordination is not in the manufacturer's best interest. What is surprising is that with a substantial disparity in competitor size the sophisticated Stackelberg tariff massively out-performs the two channel-coordinating wholesale-price policies at low degrees of competition. Of course, as [chi] goes to one the manufacturer's advantage from non-coordination shrinks. Our comments in the previous paragraph (on the relative merits of the price policies when fixed costs exceed two-thirds of the net revenue under coordination) continue to hold. At a substantial disparity in retailer size (i.e., at all [Q.sup.*] [less than] .07), combined with a very high deg ree of competition, Region 1 becomes relevant (see footnote 17 for details). Within this Region the menu is manufacturer-optimal. However, our numerical analysis shows that the menu only generates about 1% higher profit for the manufacturer than does the sophisticated Stackelberg tariff. (At degrees of competition less than "very high" the sophisticated Stackelberg policy is optimal.)

The Practicality of Coordination versus Non-Coordination

We have seen that any of the three wholesale-price policies may be manufacturer-profit-optimal, depending on the specific parametric values of (1) the degree of competition ([chi]), (2) the relative size of the competitors ([Q.sup.*]), and (3) the absolute difference in the retailers' fixed costs ([f.sub.i] - [f.sub.j]). However, extensive numerical analysis strongly suggests that the noncoordinating sophisticated Stackelberg strategy is the manufacturer-preferred wholesale-price policy unless competition is very unequal or the level of fixed costs is very high. Even then, competition must be very intense for coordination to be manufacturer-optimal. Further, when a coordinating wholesale-price policy does outperform the sophisticated Stackelberg policy it does so only by a small margin. The reverse is not true. When non-coordination is manufacturer-preferred the sophisticated Stackelberg policy substantially outperforms either coordinating policy over a wide-range of parametric values. From this we conclude that unless a supply-chain manager is quite certain of the specific parametric values that its retailers face, and unless those values fall in narrowly circumscribed ranges, use of a noncoordinating sophisticated Stackelberg wholesale-price strategy is the manufacturer's best option. In short, the standard prescription from the literature to coordinate the channel rarely generates a substantial improvement in manufacturer-profit over what could be obtained without seeking coordination and often yields sharply lower profit. Stated simply, channel coordination is not all that it is cracked up to be.

DISCUSSION AND CONCLUSION

Should a manufacturer that sells through competing retailers seek to coordinate every one of its retail relationships? The marketing science literature has long argued "Yes, because coordination maximizes total channel profit, profit that may be re-distributed between channel members to the benefit of all." This argument is clearly correct for a single channel dyad, but its extension to multiple dyads is logically predicated upon the manufacturer "cutting a separate deal" with each of its retailers. However, common practicalities such as administrative, bargaining and contract development costs (Lafontaine, 1990) and legal restrictions such as the Robinson-Patman Act (Monroe, 1990) suggest that manufacturers do not regularly cut such deals with their retailers. Rather, the evidence is that manufacturers typically employ a wholesale-price policy that is common to many retailers.

Recent analytical work by Ingene and Parry (1995b) raised the possibility that channel-coordination might not be in the manufacturer's interest when competing retailers are treated comparably. Their results were based on a limited number of specific wholesale-price policies. Thus it has not been clear if they glimpsed a broadly generalizable principle (that channel-coordination is not all that it has been cracked up to be) or if they found a set of special circumstances under which channel-coordination is non-optimal for the manufacturer.

In this paper we have clarified the issue of generalizable principle versus special circumstances by analyzing a basic channel situation: a single manufacturer selling through a pair of differentiated, competing retailers facing their own unique demand and cost curves. We constrained the manufacturer to treat its retailers comparably and we assumed a fixed cost [denoted as [f.sub.k], k [epsilon] (i, j)] for each retailer. The legal and practical issues cited above justify our equal treatment constraint; the nontrivial opportunity cost of channel participation justifies our assumption of fixed costs.

Our analytical results make two critical contributions the theory of distribution channel structure. First, we now know that a manufacturer's choice of an optimal wholesale-price policy is dependent upon three factors: the difference in retailers' fixed costs ([f.sub.i] - [f.sub.j]), the relative size of the retailers (i.e., the magnitude of competition), and the degree of inter-retailer competition. Second, we proved that over virtually all possible combinations of degrees and magnitudes of inter-retailer competition there exist plausible ([f.sub.i] - [f.sub.j]) values for which channel-coordination is not in the manufacturer's best interest. Given that our analysis compared a sophisticated Stackelberg tariff (the best of all single two-part tariffs) with a channel-coordinating menu of two-part tariffs, our conclusion is definitive over the range of all possible two-part tariffs. [22] (A two-part tariff, which consists of a per-unit fee and a fixed fee, is common in franchising. More importantly, it is a no nlinear quantity-discount schedule: the retailer's average, per-unit acquisition cost declines with increases in quantity purchased.) The sophisticated Stackelberg tariff was also shown to manufacturer profit-dominate a linear, channel-coordinating quantity-discount schedule. Therefore, we may safely conclude that channel coordination is not all that it has been cracked up to be.

We believe that a well-intentioned "managerial implication" from the literature ("seek to coordinate your channel") may have inadvertently misled managers. Such advice encourages pursuit of a strategy that is manufacturer profit-reducing over a wide range of parametric values. Our analysis clearly demonstrates that there is no reason to expect it to be in the best interests of different manufacturers to offer their retailers the same wholesale-price strategy; just as each dyad within a channel faces different demand and cost conditions, so does every manufacturer face diverse parametric values. It follows that identical advice ("seek channel coordination") cannot be appropriate for all channels at all times. Finally, if one insists on looking for a single rule-of-thumb to guide wholesale-price decisions, the appropriate advice to supply-chain managers is to "forget channel coordination and use a sophisticated Stackelberg tariff."

The larger and (we believe) more important message is that commonly held modeling assumptions are not always innocuous. Within the context explored here such key assumptions from the extant literature are (1) fixed costs do not affect behavior within a channel and (2) conclusions drawn from bilateral monopoly models can be generalized to non-bilateral situations. We have proven that the difference in retailers' fixed costs matters a great deal, for it affects which wholesale-price policy the manufacturer will offer and how much economic profit each retailer will earn. Further, we have shown that conclusions valid in a single dyadic framework are generally inaccurate for multiple dyads. This indicates that other channel management recommendations may be similarly affected by assumptions that seem innocuous.

One candidate for future assessment involves channel breadth. In this paper we have followed the conventional modeling assumption of a fixed channel breadth. We did not pursue the question of manufacturer-optimal channel breadth because doing so would have made an already intricate problem even more complicated. We speculate that, if one retailer's profit contribution is small relative to the other retailer's, then the manufacturer may find it more profitable to deal solely with the larger retailer. The reason is that by serving a single retailer, thus sacrificing market coverage, the manufacturer can extract all channel profit through a properly specified two part-tariff. We believe our intuition is reasonable, but it should be confirmed in future research. Similarly, we believe that assumptions in other realms of the broad field of supply-chain management should be rigorously investigated to ascertain their robustness. This conviction follows directly from the results presented in this paper, which provide dramatic evidence that seemingly innocuous assumptions can dramatically affect the conclusions reached regarding optimal marketing decisions.

Acknowledgment: Appreciation is expressed to the guest editors and anonymous reviewers for helpful comments on earlier drafts.

Charles A. Ingene, University of Washington, School of Business. Box 353200, Seattle, WA 98195-3200 (e-mail: caingene@u.washington.edu). Mark E. Parry, University of Virginia, Colgate Darden Graduate School of Business Administration, Box 6550, Charlottesville, VA 22906-6550 (e-mail: PARRYM@Darden.virginia.edu).

NOTES

(1.) Channel coordination is often linked to the concept of Pareto-optimality. In the context of channels, a situation is Pareto-optimal if and only if no channel member can be made better off without harming another channel member.

(2.) An exception is wholly owned subsidiaries--such as Sherwin-Williams--where different prices across outlets are allowed.

(3.) Jeuland and Shugan (1983) demonstrated that coordination could be achieved in a bilateral monopoly context through the judicious application of an appropriately valued quantity-discount schedule. Ingene and Parry (1995b) extended this argument to the case of competing retailers. They mentioned the possibility of a channel-coordinating menu of two-part tariffs.

(4.) In this paper we prove the existence of a feasible, channel-coordinating menu of two-part tariffs. (5.) Ingene and Parry (1995b) sketched a channel-coordinating menu without providing much detail.

(6.) Ingene and Parry (1995a) examined optimal channel breadth for a manufacturer selling to any number of non-competing retailers. They used a general demand curve rather than the linear demand curve of Equation (1). A general demand curve precludes obtaining closed-form solutions. The assumption of no competition between retailers dramatically simplifies the problem, although a glance at that paper should make clear that the solution is still very complex.

(7.) Recall that retailer-specific transfer-prices are legally permissible for wholly owned subsidiaries.

(8.) Although this sub-Section follows Ingene and Parry (1995b), we develop several points that did not appear in that paper. In particular, only Equations (7)--(1O) appeared (1995b) and did so only for the special case of b = 1. The Zones that are developed in expressions (12) are original with this paper.

(9.) A Technical Appendix, available from the authors, provides details on the wholesale margin and fixed fee by Zones.

(10.) Expressions (14)--(16) appeared in Ingene and Parry (1998) with substantially different notation. Our notational change facilitates the ensuing comparison across wholesale-price regimes.

(11.) It is typical to attribute an absence of "channel coordination" ([[[pi].sup.[SS.sup.*]].sub.C] [less than] [[[pi].sup.[SS.sup.*]].sub.C] to the "double marginalization" that occurs with Stackelberg pricing--that is, to the fact that manufacturer and retailers obtain a positive margin (Gerstner and Hess 1995). Equivalence between lack of channel coordination and double marginalization is strictly accurate only within a bilateral monopoly model. With competing retailers the channel-optimal wholesale-price entails a positive per-unit manufacturer margin, just as the optimal retail-price entails a positive margin for both retailers; with competing retailers "double marginalization" is required for channel coordination.

(12.) This Section substantially expands a two-page Section of Ingene and Parry (1995b). Those pages included only two of the subsequent Equations-(18) and (19)-and then only for the special case of b = 1. The Zones discussed below were not developed, nor was the issue of possible "defection" accorded attention.

(13.) A fourth case appears to be theoretically possible: the [i.sup.th] retailer chooses tariff [[[tau].sup.*].sub.j] and the [j.sup.th] retailer chooses tariff [[[tau].sup.*].sub.i]. However, it can be shown that this possibility can never be an equilibrium outcome.

(14.) Incidentally, a comparison of the reduction in net revenues due to defection shows that [B.sub.j] [greater than] [B.sub.i] [greater than] 0.

(15.) Notice that fixed costs that are equal to each other is more general than fixed costs that are equal to zero.

(16.) Notice also that Figure 5 is drawn for [f.sub.i] = 0 = [f.sub.j]. A fundamentally similar Figure is obtained for all [f.sub.i] = [f.sub.j] [less than] .66. At higher fixed costs values Region 1 becomes more prominent.

(17.) Specifically, the parabola can be characterized as .99 [greater than or equal to] [chi] [greater than or equal to] .85 and [Q.sup.*] [less than or equal to] .07.

(18.) Jeuland and Shugan used a general demand curve, while we assume a specific demand curve. Our assumption is necessary to obtain analytical results when [Q.sup.*] [neq] 0 and [chi] [neq] 0. They also stressed that channel profit could be split between channel members by negotiation. Our textual comment is that a greedy manufacturer can obtain all profit.

(19.) We are indebted to an anonymous reviewer for suggesting this comparison. We stress that the McGuire-Staelin analysis differs from the analysis presented in this paper in two key respects. (1) Each of their retailers is supplied by a different manufacturer. (2) They analyze an optimal one-part tariff-they do not consider a nonzero fixed fee; hence their results cannot entail channel coordination except when a channel is vertically integrated.

(20.) Strictly speaking the unit-cube requires that each retailer's fixed cost be normalized by its net revenue under coordination. This standardized fixed cost lies in the unit interval.

(21.) Inspiration for these Figures comes from McGuire and Staelin (1983); their Figure 2 showed the manufacturer's profit-at various levels of competition-for various channel structures.

(22.) If a noncoordinating menu of two-part tariffs were preferred to the tariffs explored here it would reinforce our point: channel coordination is often manufacturer non-optimal. (We stress that we do not know if such a menu exists or if it could be calculated analytically.)

REFERENCES

Gerstner, E. and J. Hess. (1995). "Pull Promotions and Channel Coordination," Marketing Science, 14 (Winter) 43-60.

Ingene, C. and M. Parry. (1995a). "Coordination and Manufacturer Profit Maximization: The Multiple Retailer Channel," Journal of Retailing, 71 (Summer) 129-151.

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_____ and _____. (1998). "Manufacturer-Optimal Wholesale Pricing," Marketing Letters, 9 (1) 65-77.

Jeuland, A. and S. Shugan. (1983). "Managing Channel Profits," Marketing Science, 2 (Summer) 239-272.

Lafontaine, F. (1990). "An Empirical Look at Franchise Contracts as Signaling Devices," Pittsburgh: Graduate School of Industrial Administration, Carnegie-Mellon University.

McGuire, T. and R. Staelin. (1983). "An Industry Equilibrium Analysis of Downstream Vertical Integration," Marketing Science, 2 (Spring), 161-192.

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The Channel-Coordinating Menu of Two-Part Tariffs: Channel Members' Profits by zone

Channel Member's Profits in Zone [[Z.sup.[Menu.sup.*]].sub.j]

The [j.sup.th] retailer: [[[pi].sup.[Menu.sup.*]].sub.j] = 0

The [j.sup.th] retailer; [[[pi].sup.[Menu.sup.*]].sub.j] = 0

Manufacturer: [[[pi].sup.[Menu.sup.*]].sub.M] = [[omega].sup.*] + [2[[R.sup.*].sub.j] - 2[f.sub.j] + [B.sub.j]] - F

Margin on units sold: [[omega].sup.*] [equivalent] [[theta].sup.2] [([[Q.sup.*].sub.i]).sup.2] + 2b[theta][[Q.sup.*].sub.i][[Q.sup.*].sub.j] + [[theta].sup.2] [([[Q.sup.*].sub.j]).sup.2]/b([b.sup.2] - [[theta].sup.2])

Channel: [[[pi].sup.[Menu.sup.*]].sub.C] = [b[([[Q.sup.*].sub.i]).sup.2] + 2[theta][[Q.sup.*].sub.i][[Q.sup.*].sub.j] + b[([[Q.sup.*].sub.j]).sup.2]]/([b.sup.2] - [[theta].sup.2]) - [f.sub.i] - [f.sub.j] - F.

Channel Member's Profits in Zone [[Z.sup.[Menu.sup.*]].sub.ij]

The [j.sup.th] & [j.sup.th] retailers: [[[pi].sup.[Menu.sup.*]].sub.i] = 0 = [[[pi].sup.[Menu.sup.*]].sub.j]

Manufacturer: [[[pi].sup.[Menu.sup.*]].sub.M] = [b[([[Q.sup.*].sub.i]).sup.2] + 2[theta][[Q.sup.*].sub.i][[Q.sup.*].sub.j] + b[([[Q.sup.*].sub.j]).sup.2]]/([b.sup.2] - [[theta].sup.2]) - [f.sub.i] - [f.sub.j] - F.

Channel: [[[pi].sup.[Menu.sup.*]].sub.C] = [[[pi].sup.[Menu.sup.*]].sub.M]

Channel: Member's Profits in Zone [[Z.sup.[Menu.sup.*]].sub.i]

Reversing i's and j's in the profit expressions for Zone [[Z.sup.[Menu.sup.*]].sub.j] yields the results for Zone [[Z.sup.[Menu.sup.*]].sub.i].

The Sophisticated Stackelberg Two-Part Tariff: Channel Members' Profits by Zone

Channel Member's Profits in Zone [[Z.sup.[SS.sup.*]].sub.j]

The [j.sup.th] retailer: [[[pi].sup.[SS.sup.*]].sub.j](j) = 0

The [j.sup.th] retailer: [[[pi].sup.[SS.sup.*]].sub.j](j) = 2[3[theta][([[Q.sup.*].sub.i]).sup.2] + 4(b - [theta])[[Q.sup.*].sub.i][[Q.sup.*].sub.j] - (4b - 3[theta])[([[Q.sup.*].sub.j]).sup.2]/[(2b + [theta]).sup.2]] - ([f.sub.i] - [f.sub.j])

Manufacturer: [[[pi].sup.[SS.sup.*]].sub.M](j) = [(4[b.sup.2] - 4b[theta] + 9[[theta].sup.2])[([[Q.sup.*].sub.i]).sup.2] - 2(4[b.sup.2] - 16b[theta] + 3[[theta].sup.2])[[Q.sup.*].sub.i][[Q.sup.*].sub.j] + (20[b.sup.2] - 12b[theta] + [[theta].sup.2])[([[Q.sup.*].sub.j]).sup.2]/2(b - [theta])[(2b - [theta]).sup.2]] - 2[f.sub.i] - F

Channel: [[[pi].sup.[SS.sup.*]].sub.C](j) = [(4[b.sup.2] + 8b[theta] + 3[[theta].sup.2])[[([[Q.sup.*].sub.i]).sup.2] + [([[Q.sup.*].sub.j]).sup.2]] + 2(4[b.sup.2] 5[[theta].sup.2])[[Q.sup.*].sub.i][[Q.sup.*].sub.j]/2(b - [theta])[(2b - [theta]).sup.2]] - [f.sub.i] - [f.sub.j] - F

Channel Member's Profits in Zone [[Z.sup.[SS.sup.*]].sub.ij]

The [i.sup.th] & [j.sup.th] retailers: [[[pi].sup.[SS.sup.*]].sub.i]([alpha]) = 0 = [[[pi].sup.[SS.sup.*]].sub.j]([alpha])

Manufacturer: [[[pi].sup.[SS.sup.*]].sub.M]([alpha]) = {[[[pi].sup.[SS.sup.*]].sub.C](j) + 8[alpha](1 - [alpha])(b - [theta])[([[Q.sup.*].sub.i] - [[Q.sup.*].sub.j]).sup.2]/[(2b + [[theta]).sup.2]}

Channel: [[[pi].sup.[SS.sup.*]].sub.C]([alpha]) = [[[pi].sup.[SS.sup.*]].sub.C]([alpha])

Value of a: 1 [greater than or equal to] [equivalent] (([f.sub.i] - [f.sub.j]) - [L.sup.[SS.sup.*]])/([U.sup.[SS.sup.*]] - [L.sup.[SS.sup.*]]) [greater than or equal to] 0

Channel Member's Profits in Zone [[Z.sup.[SS.sup.*]].sub.i]

Reversing i's and j's in the profit expressions for Zone [[Z.sup.[SS.sup.*]].sub.j] yields the results for Zone [[Z.sup.[SS.sup.*]].sub.i].

Manufacturer-Optimal Wholesale-Price Strategies ([f.sub.i - [f.sub.j]) [less than or equal to] [L.sup.[SS.sup.*]] Region 1 Menu Region 2 SS Region 3 SS Region 4 SS [L.sup.[SS.sup.*]] [less than] ([f.sub.i]-[f.sub.j]) [less than] max ([U.sup.[SS.sup.*]], [U.sup.[Menu.sup.*]]) Region 1 Menu Region 2 SS, Menu, SS Region 3 SS, Menu Region 4 SS, QD, Menu max ([U.sup.[SS.sup.*]], [U.sup.[Menu.sup.*]]) [less than or equal to] ([f.sub.i]-[f.sub.j]) Region 1 Menu Region 2 SS Region 3 Menu Region 4 Menu Legend: Menu = Channel-Coordinating Menu of Two-Part Tariffs SS = Channel Non-Coordinating Sophisticated Stackelberg Two-Part Tariff QD = Channel-Coordinating Quantity-Discount Schedule We examine the following three sets of (b, [theta]) parametric values: b [theta] [chi] [equivalent] [theta]/b Region 0.7 0.2 0.286 2 0.9 0.4 0.444 3 2.0 1.5 0.750 4 Region 2: The Sophisticated Stackelberg Tariff and The Channel-Coordinating Menu Manufacturer Profit Rankings [[[pi].sup.[SS.sup.*]].sub.M] [greater than] [[[pi].sup.[Menu.sup.*]].sub.M] [f.sub.i] $ 0.00 [[[pi].sup.*].sub.C] $12,572.22 [[[pi].sup.[SS.sup.*]].sub.M] $ 9,217.19 [[[pi].sup.[Menu.sup.*]].sub.M] $ 8,998.49 [[[pi].sup.[QD.sup.*]].sub.M] $ 8,921.43 [[[pi].sup.[SS.sup.*]].sub.M] = [[[pi].sup.[Menu.sup.*]].sub.M] [f.sub.i] $3,567.61 [[[pi].sup.*].sub.C] $9,004.61 [[[pi].sup.[SS.sup.*]].sub.M] $8,998.49 [[[pi].sup.[Menu.sup.*]].sub.M] $8,998.49 [[[pi].sup.[QD.sup.*]].sub.M] $8,921.43 [[[pi].sup.[Menu.sup.*]].sub.M] [greater than] [[[pi].sup.[SS.sup.*]].sub.M] [f.sub.i] $3,680.26 [[[pi].sup.*].sub.C] $8,891.96 [[[pi].sup.[SS.sup.*]].sub.M] $8,878.87 [[[pi].sup.[Menu.sup.*]].sub.M] $8,891.96 [[[pi].sup.[QD.sup.*]].sub.M] $8,862.50 [[[pi].sup.[Menu.sup.*]].sub.M] = [[[pi].sup.[SS.sup.*]].sub.M] [f.sub.i] $3,792.91 [[[pi].sup.*].sub.C] $8,779.31 [[[pi].sup.[SS.sup.*]].sub.M] $8,733.27 [[[pi].sup.[Menu.sup.*]].sub.M] $8,733.27 [[[pi].sup.[QD.sup.*]].sub.M] $8,637.20 [[[pi].sup.[SS.sup.*]].sub.M] [greater than] [[[pi].sup.[Menu.sup.*]].sub.M] [f.sub.i] $4,082.03 [[[pi].sup.*].sub.C] $8,490.19 [[[pi].sup.[SS.sup.*]].sub.M] $8,240.63 [[[pi].sup.[Menu.sup.*]].sub.M] $8,155.03 [[[pi].sup.[QD.sup.*]].sub.M] $8,058.96

Legend: [[[pi].sup.[X.sup.*]].sub.M] [equivalent] manufacturer profit with pricing policy "X." [[[pi].sup.[X.sup.*]].sub.M] [equivalent] Vertically-Integrated Channel; [[[pi].sup.[SS.sup.*]].sub.M] [equivalend] Sophisticated Stackelberg;[[[pi].sup.[Menu.sup.*]].sub.M] [equivalent] Channel-Coordinating Menu of two-part tariffs; [[[pi].sup.[QD.sup.*]].sub.M][equivalent] Channel-Coordinating Quantity-Discount Schedule; [f.sub.j] [equivalent] fixed cost of the [i.sup.th] retailer.

Note: We set the following parametric values [A.sub.i] = 150, [A.sub.j] 100, [c.sub.i] = [c.sub.j] = C = $10, [f.sub.j] = $0, and F = $1,000. We hold (b - [theta]) = 0.5 and specify b = 0.7 and [theta] = .2; [U.sup.[SS.sup.*]] = $4,082.03, [L.sup.[SS.sup.*]] = $3,105.47, [U.sup.[Menu.sup.*]] = $3,746.87, [L.sup.[Menu.sup.*]] = $3,573.73 and [[delta].sup.[QD.sup.*]] = $3,650.79. It follows that [[Q.sup.*].sub.i] 70, [[Q.sup.*].sub.j] = 45, [[[rho].sup.*].sub.i] $148.89 and [[[rho].sup.*].sub.j] = $121.11.

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Comment: | Is Channel Coordination All It Is Cracked Up To Be? |
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Author: | INGENE, CHARLES A.; PARRY, MARK E. |

Publication: | Journal of Retailing |

Article Type: | Brief Article |

Geographic Code: | 1USA |

Date: | Dec 22, 2000 |

Words: | 16113 |

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