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In the value-based approach to business strategy, a firm’s value creation with a buyer—i.e., its value gap—is an important measure. In many formal and informal results, firm profitability depends on whether the firm can identify buyer segments in which it has the largest value gap—namely, a value-gap advantage. These results typically assume, either explicitly or implicitly, that firms have constant marginal costs. In this paper, we show that if a firm’s value gap is defined as its marginal value creation with a buyer, value gaps provide a foundation for a firm’s profitability more generally. By allowing for nonconstant marginal costs, two important factors emerge. First, a firm’s value-gap advantage with respect to a buyer should be based on comparing the firm’s marginal value creation with the buyer’s best alternative for value capture, not value creation. Second, a firm’s guaranteed profitability depends critically on buyer competition. We show that in addition to excluded buyers, there are two other sources of buyer competition. A competitor’s buyer acts like an excluded buyer of a given firm if it can create more value with the given firm. We call such buyers envious buyers. Additionally, because of linkages in buyer preferences—which we call market-price effects—a firm may benefit from a competitor’s excluded or envious buyer. Consequently, a firm can benefit from demand for seemingly unrelated products, even when that demand includes buyers who have zero willingness-to-pay for the firm’s product. These additional sources of buyer competition explain how, in environments of excess supply, a firm can still be guaranteed profits due to competition.

1. Introduction

The value-based approach to business strategy models a business’ competitive context without ex ante assumptions about the price-setting power of players, particularly firms. Pricing power, when it exists, is solely the consequence of competition. This approach is often chosen for one of two reasons. First, in many contexts—particularly those with business-to-business interactions—a preliminary assumption of price-setting power may not be appropriate. Second, the strategy field is often interested in providing insights that are applicable in a broad range of contexts. Because a value-based analysis focuses on the economic structure of a context—e.g., buyers’ preferences and firms’ costs—rather than on specific moves and countermoves, the analysis often produces such insights. A primary example is the importance of a firm’s value creation with a buyer, often variously described as the firm’s value gap, value proposition, or value stick. Simple intuition suggests that being better than one’s competitors should be a predictable route to profitability, and the value-based literature shows that a proper measure of “better” is not having a better product or a lower cost. Rather, to be better, a firm should have a larger value gap—that is, a value-gap advantage.

The reasoning behind this intuition is straightforward. To successfully compete for a buyer, a firm must feasibly deliver more value capture to the buyer than any other competitor can. The firm that provides the buyer with the largest value gap will always be able to do this. As well as being intuitive, this reasoning provides a succinct characterization of a firm’s positioning decision: to be profitable, choose products that give the firm value-gap advantages in identifiable buyer segments.

When firms have constant marginal costs, value-gap reasoning is well-defined. A firm’s value creation with a given buyer is unambiguously its marginal value creation with the buyer, and the buyer’s best alternative for value capture with a competing firm is unambiguously the buyer’s best alternative for value creation. Determining value-gap advantages with respect to a buyer (or segment of buyers) is, then, a simple matter of comparing the marginal value creation that a buyer creates with the different firms, and these advantages completely determine a firm’s range of possible profits (see Stuart 2004, Lemma 3).

In this paper, we address the question of whether value-gap reasoning holds more generally. In the absence of constant marginal costs, is it still the case that a firm’s value creation with a buyer is the foundation for its profitability? The results in this paper suggest a positive answer, provided that a value-gap advantage is appropriately defined. We show that the foundation for understanding a firm’s profitability is, in fact, its marginal value creation with a buyer. But the notion of a value-gap advantage should be a comparison between the firm’s marginal value creation with the buyer—its value gap—and the buyer’s best alternative for value capture. Using this definition of a value-gap advantage, we show that a firm’s range of possible profits is based on its value-gap advantages with respect to both its buyers and its competitors’ buyers.

At a technical level, this paper relaxes the assumption that firms have constant marginal costs. This assumption has enabled meaningful analyses in the value-based literature, including, for example, Adner and Zemsky (2006) on the sustainability of profitability, Chatain and Zemsky (2007) on horizontal scope, and Jia (2013) on relationship-specific investments. (Papers discussing the conceptual foundations of the value-based approach include Brandenburger and Stuart 1996, 2007; Stuart 2001; MacDonald and Ryall 2004; and Ryall et al. 2009.) There are at least two reasons to relax this assumption. First, firms often do not have constant marginal costs. In particular, this assumption precludes scale effects. Second, and more importantly, the assumption of constant marginal costs implies that no firm is guaranteed any profits. Because of the absence of ex ante assumptions about price-setting power, a value-based analysis will typically provide a range of possible profits for a firm, rather than a single number. Following Edgeworth (1881), the minimum of the range describes the profits that the firm is guaranteed to capture due to the effects of competition. The difference between the minimum and the maximum represents a residual bargaining problem, the resolution of which depends on factors other than competition. With constant marginal costs, buyers do not have to compete for firms, and so a firm is not guaranteed any profit. A key feature of many markets—competition guaranteeing profits to firms—has been assumed away.

If firms have more general cost functions, capacity constraints become possible. With capacity constraints, firms can have buyers competing for them, and this buyer competition can guarantee the firm a price above marginal cost.1 For a complete understanding of a firm’s guaranteed profitability, then, one must understand the sources of buyer competition.

The simplest example of buyer competition is excess demand for a firm’s product (see, for example, Kaneko1976). Intuitively, the presence of an excluded buyer allows a firm to credibly demand a higher price from its customers. However, our results show two additional issues arise with buyer competition. First, a competitor’s buyer may act as a source of excess demand for a given firm. We call such buyers envious buyers. The intuition is that the competitor’s buyer would prefer to transact with a given firm, but because that firm is at capacity, it cannot. If a firm has an excluded or envious buyer, it will be guaranteed a price above its marginal cost.

The second issue that arises with buyer competition is that a given firm can benefit from a competitor’s excluded or envious buyer. In extreme cases, a firm can be guaranteed a profit due to excess demand for a totally unrelated product. For instance, if one segment of buyers is interested in, say, only firm A or firm B, another in only firm B or firm C, and a third in only firm C, then an excluded or envious buyer of firm C will benefit firm A. However, no buyer in this example would ever view firm A and firm C as substitutes. Thus, with the possibility of excluded or envious buyers due to nonconstant marginal costs, linkages in buyer preferences become important. We call the consequences of these linkages market-price effects.

The main results can be stated as follows. To have the potential for profits, a firm must have buyers with the following property: the firm’s value gap with the buyer must be greater than the buyer’s maximum possible value capture with any other firm. To be guaranteed a profit, either a firm must have a marginal cost that exceeds its average cost,2 or there must be an unserved buyer or a competitor’s buyer with the following property: the buyer’s maximum possible value capture must be less than the value it could create with the firm. Informally, the results show that a firm’s potential profits are based on value-gap advantages with its customers, and its guaranteed profits are based on value-gap advantages with noncustomers.

The possibility of envious buyers and market-price effects complicates the assessment of value-gap advantages, but it is important to note that the results do suggest the following robust advice on a firm’s positioning decision: choose products that give the firm the largest marginal value creation in identifiable buyer segments, and limit capacity such that at least one of the buyers in the identified segments is excluded from the firm. This advice will always guarantee a profit for a firm.

In §2, we provide examples illustrating the main results. The first example uses constant marginal costs to discuss firm profitability in the absence of capacity constraints. The second is a standard commodity example to review the role of an excluded buyer in guaranteeing profits to a firm. The third introduces envious-buyer and market-price effects.

Section 3 contains the model and results. The model can be viewed as a generalization of either the Cournot results in Kaneko (1976) or the spatial competition results in Stuart (2004).3 (Telser 1972, Section V uses a similar model in discussing collusion versus cooperation; Kaneko and Yamamoto 1986 provide an existence result with slightly more restrictive cost functions.) Demand is assumed to be unitary, but there are no other restrictions on buyer preferences. Cost functions are general, except that we impose some structure to ensure that competition leads to stable outcomes. The results in §3 follow the sequence of examples in §2. Proposition 1 addresses contexts without capacity constraints, as demonstrated in Example 1. Proposition 2 describes the effect of excluded and envious buyers on guaranteed profitability, as shown in Examples 2 and 3. Proposition 3 characterizes a firm’s potential profitability and refers back to Examples 2 and 3. There is a sense in which potential profitability is about being needed—in terms of value creation—by either a firm or the market as a whole. Proposition 4 makes this notion more precise.

One implication of Proposition 2 is that a firm’s guaranteed profitability does not depend on the preferences of its own buyers. Rather, it depends upon the preferences of excluded buyers and competitors’ buyers. In §4, we explore some of the implications of this fact, as well as some implications for empirical investigation. The paper concludes with a brief summary of the main results.

2. Examples

In the first example, each firm has excess supply and constant, marginal costs. Because firms have to compete for buyers and cannot make a profit at a price equal to marginal cost, no firm has a guaranteed profit. Consequently, the computation of a firm’s value-gap advantages does not have to account for the effects of competition.

Example 1.

Suppose that there are three firms, each with the capacity to provide 11 units of product. Firm 1 has a cost of $2 per unit; firm 2 has a cost of $3 per unit; and firm 3 has a cost of $6 per unit. There are three segments of 10 buyers each, named A, B, and C. Each buyer is interested in acquiring only one unit of product. In the A segment, each buyer has a willingness-to-pay of $14 for firm 1’s product, $11 for firm 2’s product, and $16 for firm 3’s product. In the B segment, each buyer has a willingness-to-pay of $10 for firm 1’s product, $12 for firm 2’s product, and $13 for firm 3’s product. In the C segment, each buyer has a willingness-to-pay of $10 for firm 1’s product, $12 for firm 2’s product, and $16 for firm 3’s product.

With different segments of buyers, it is important to note that buyers from the different segments will have different perspectives on the worth of the firms’ products. Figure 1 is designed to emphasize this point. Each panel describes how a different buyer segment views the three firms.4 Thus, the only differences in each panel are the buyers’ willingnesses-to-pay. In a given panel, each vertical line depicts each firm’s marginal value creation—its value gap—with a buyer in the segment. Note, also, that the three firms in this example could be making completely different products—it is not at all necessary that the firms’ products represent different positions on some spectrum of, say, taste or quality. All that matters is that the buyers view each firm’s product as a possible alternative and that the firms view each other as competitors. The buyers’ willingnesses-to-pay in this example have been chosen to emphasize this fact. (For a suggestive interpretation, suppose that firm 1 makes a beanbag chair; firm 2 makes a standard chair; and firm 3 makes a premium chair. The B segment comprises mainstream buyers and the C segment wealthier buyers. The A segment comprises wealthy, “retro” buyers with a nostalgic interest in beanbag chairs.)

Figure 1 Segment Views of the Three Firms

We start with an intuitive analysis. In a competitive environment, note that firm 2 will successfully compete for all 10 buyers in the B segment. If firm 2 offers its products at any price below $4, say even $3.99, the other two firms cannot compete. Consider firm 1. Because buyers view firm 2’s products as $2 better than firm 1’s products—a buyer’s willingness-to-pay for firm 2’s product is $2 higher than that for firm 1’s—firm 1 would have to offer a price at least $2 less than firm 2’s price—in other words, $1.99 or less. Because firm 1’s cost per unit is $2.00, it cannot profitably do that. A similar analysis holds for firm 3. Because buyers view firm 3’s product as only $1 better than firm 2’s product, firm 3 could charge only up to $1 more than firm 2—i.e., $4.99. Firm 3’s cost per unit is $6, so it cannot profitably compete with firm 2, either.

From a value-creation perspective, Figure 1 visually depicts the source of each firm’s potential profit (see the dotted lines in the figure). In the B segment, firm 2 creates more value with a buyer—specifically, $9—than any other firm. From a segment B buyer’s perspective, the “next-best” firm is firm 1, which can create $8 of value with a buyer. Thus, there must exist a price at which firm 2 can give a segment B buyer at least $8 of value capture and still be profitable. Loosely, because it has the largest value gap, firm 2 can “win” a segment B buyer. Moreover, firm 2’s value-gap advantage over its buyer’s next-best option—i.e., $1—is firm 2’s potential per-unit profit. A similar analysis applies to firm 1 in the A segment and firm 3 in the C segment. Note, however, that although each firm has a potential profit, there is nothing preventing their respective buyers from trying to get a price close to each firm’s marginal cost. Without an ex ante assumption of price-setting power, each firm’s guaranteed profit is zero. Because there is no buyer competition for any of the firms, no firm is guaranteed a price above marginal cost.

This example has two other features worth highlighting. First, the outcome, though intuitive, is not trivial to prove. If each firm had a capacity of 20 units instead of 11, the outcome would be the same, and the proof would be easy (see Stuart 2004). Each firm’s profit would simply be any amount between zero and its marginal contribution (sometimes called added value in the strategy literature). In this example, firm 2’s marginal contribution is $75,5 but its potential profit is only $10. An extreme example of this phenomenon occurs when a firm has a positive marginal contribution but can make no profit, as in Postlewaite and Rosenthal (1974). (Firm C in Figure 3 illustrates this extreme.) Thus, this example demonstrates the limitations to using added value as a proxy for a firm’s profitability.

Second, this example demonstrates why thinking in terms of generic strategies can be misleading. Porter (1980) argues that a successful firm should be low-cost, differentiated, or focused—i.e., low-cost or differentiated within a target segment. Cronshaw et al. (1994), by providing many examples of profitable firms whose strategies would not be described as either low-cost or differentiated, show that these conditions are not necessary. Similarly, in this example, firm 2’s profitability in segment B cannot be described as either a differentiation strategy or a low-cost strategy.6 Its product is perceived to be better than firm 1’s but not firm 3’s, so it is not differentiated. Firm 2 is profitable because it has the largest marginal value creation with buyers in the segment—it has a value-gap advantage in the segment. Because differentiation and cost advantages are intuitive terms, it is illuminating to describe firm 2’s profitability using these terms: Firm 2 has a net willingness-to-pay advantage over firm 1 and a net cost advantage over firm 3. Because the reason for firm 2’s profitability—a net cost advantage in one case, and a net willingness-to-pay advantage in the other—changes as each competitor is considered, we see why thinking in terms of generic strategies will not, in general, provide the reason for a firm’s profitability.

The second example uses a commodity product to highlight the role of an excluded buyer in guaranteeing profitability.

Example 2.

There are three firms, each with the capacity to provide two units of product. Firm A has a cost of $1 per unit; firm B has a cost of $3 per unit; and firm C has a cost of $4 per unit. There are seven buyers, each interested in acquiring only one unit from one of the three firms. Three buyers have a willingness-to-pay of $12; three buyers have a willingness-to-pay of $11; and one buyer has a willingness-to-pay of $10.

In this supplyand-demand example, firms are now guaranteed a profit. Because one buyer will be excluded from transacting, it is a source of competition to the remaining buyers. As a result, each firm is guaranteed a price of at least $10 per unit, and, thus, each firm is guaranteed a profit.7Proposition 2 will show that some form of buyer exclusion is necessary for a firm to be guaranteed a price above its marginal cost, and the case of excess demand for a commodity product is the simplest example of this fact.

This example also shows that to compute a firm’s potential profit, value-gap advantages must compare a firm’s value gap to the buyer’s best alternative for value capture. Firms B and C do not have the largest value-gap in any of the buyer segments, but their value-gaps are larger than a buyer’s best alternative for value capture. The results in Proposition 3 will show that for the first segment—buyers with a willingness-to-pay of $12—this best alternative is $1, and for the second segment, it is zero. (Loosely, buyers in the first segment enable an extra dollar of value creation, and one can think of the firms as competing for this extra dollar.)

As an aside, this example also illustrates that a firm can be worse than average in an industry, yet profitable. In Figure 2, the average cost per unit is $2.67, which is lower than the cost per unit of both firms B and C. However, both are profitable. Even if the demand is only five units rather than seven (as in Figure 3), the average cost per unit becomes $2.40, which is lower than the cost per unit of firm B, but firm B would still be profitable.

Figure 2 Commodity Product
Figure 3 Commodity Product with Excess Supply
Example 3.

There are three firms: uptown, downtown, and park, each with just one apartment to rent. There are three buyers, each of which is interested in renting only one apartment. Buyer 1 has willingness-to-pay of $10, $0, and $8 for the uptown, downtown, and park apartments, respectively; buyer 2 has willingness-to-pay of $8, $10, and $0; and buyer 3 has willingness-to-pay of $0, $8, and $5. Figure 4 depicts this situation. (Strictly for simplicity, the economic costs of the apartments are assumed to be zero.)

Figure 4 Envious-Buyer and Market-Price Effects

In this example, buyer 1 will rent the uptown apartment, buyer 2 the downtown apartment, and buyer 3 the park apartment. Because supply equals demand, there is no excluded buyer to guarantee a profit to any of the three firms. In fact, both the uptown and downtown firms are guaranteed a profit. The downtown firm’s profit is guaranteed by the presence of an envious buyer, and due to linkages in preferences, the envious buyer indirectly guarantees a profit to the uptown firm as well. As stated in §1, we call the consequences of such linkages market-price effects.

To demonstrate these two phenomena, first note that the park firm is not guaranteed any profit, so buyer 3’s maximum possible value capture is the value it creates with the park firm, namely $5. But notice that buyer 3’s willingness-to-pay for the downtown apartment is $8. This means that at any price for the downtown apartment less than or equal to $3, buyer 3 would prefer to rent the downtown apartment. Thus, buyer 3 acts like an excluded buyer with a willingness-to-pay of $3 for the downtown apartment.

To preview the formal results, we introduce some notation. Let pU, pD, and pP respectively denote the minimum prices that the uptown, downtown, and park firms will be guaranteed to receive. Let w3(D) and w3(P) denote buyer 3’s willingness-to-pay for the downtown and park apartments, respectively, and let cP denote the park firm’s marginal cost. With this notation, the term w3(P) − cP denotes the park firm’s value creation with buyer 3—that is, the park firm’s value gap in the buyer 3 segment. (This example has three segments of one buyer each.) We have just argued that buyer 3 could act like an excluded buyer if w3(D) > w3(P) − cP. Because this inequality holds, we know that the downtown firm is guaranteed a price of $3:

For intuition, if the downtown firm were to receive a price less than $3 from buyer 2, it could credibly sell to buyer 3 at a price of $3.

We call buyer 3 an envious buyer because it is envious of the downtown firm in terms of value creation. In other words, a firm’s buyer is envious of a competing firm if its value gap is smaller than its value gap with the competing firm. This can be seen visually in the far-right panel of Figure 4. Buyer 3’s value gap with the firm that serves it—park—is smaller than its value gap with the downtown firm.

The uptown firm has neither an excluded nor an envious buyer, but it, too, is guaranteed a profit. Because the downtown firm is guaranteed a price of $3, buyer 2 will capture, at most, $7. If buyer 2’s willingness-to-pay for the uptown apartment exceeds $7, then it will be a source of competition for the uptown firm. Using notation, if w2(U) > w2(D) − pD, then pUw2(U) − (w2(D) − pD). Using the numbers in this example, the uptown firm is guaranteed a price of $8 − ($10 − $3) = $1. To highlight the reason for this guaranteed profit, we can express the uptown firm’s guaranteed price by

In Proposition 2, we show that a firm’s guaranteed price will always have the structure of the above expression. For each firm, there is a base price that will be equal to either its marginal cost or the willingness-to-pay of an excluded buyer, whichever is larger. A firm’s guaranteed price will be either its base price or some other firm’s base price adjusted for linkages in buyer preferences. In this example, each firm’s guaranteed price is based on the park firm’s base price, namely its marginal cost—cP. The downtown firm’s guaranteed price is the park firm’s base price plus the term w3(D) − w3(P)—this is the linkage in preference between the uptown and park firms. The uptown firm’s guaranteed price is the park firm’s base price plus both w3(D) − w3(P) and w2(U) − w2(D). The first term links the park firm to the downtown firm, and the second term links the downtown firm to the uptown firm.

The root source of the uptown firm’s guaranteed profit is buyer 3’s envy of the downtown firm. Notice that buyer 3 has zero willingness-to-pay for the uptown firm. Because of market-price effects, the uptown firm benefits from a seemingly irrelevant buyer.

In this example, supply equals demand, but it is important to note that this is not critical. Suppose that we add a fourth firm, say park-2, for which the buyers’ willingness-to-pay are the same as for the park firm. The uptown firm will still be guaranteed a price of $1, and the downtown firm will still be guaranteed a price of $3. With this modified example, we see that both envious-buyer and market-price effects can occur in a context with excess supply. Arguably, this modified example demonstrates an essential aspect of many business contexts: the ability of firms to be guaranteed a profit despite excess industry supply.

Example 3 demonstrates the two nonobvious issues in understanding a firm’s guaranteed profitability: the role of envious buyers and market-price effects due to linkages in buyer preferences. Collectively, the examples suggest four other points.

First, an understanding of buyer preferences—and of the segmentation implied by buyer preferences—may be essential to understanding profitability. Although this is not surprising in a marketing context, strategy discussions often treat buyer preferences in a coarser way. For instance, the strategist might talk about a differentiated product, but the marketing analyst would take the extra step of identifying which buyer segments would pay more for the differentiation and which would not. As these examples show, and the formal results will show, the marketer’s extra step is necessary for a complete understanding of a firm’s profitability.

Second, the examples show that a value-based analysis emphasizes a point sometimes overlooked in a strategic analysis: even if a firm targets one buyer segment, it may be a source of competition in another. All three examples include instances in which a firm that is serving a particular segment provides competitive pressure on a firm serving another segment.

Third, and related to the first point, the assessment of who is in the game may require more care than one might initially suspect. The examples show that relationships in buyer preferences can lead to situations in which the buyers and providers of a seemingly unrelated product significantly affect a firm’s profitability. Consequently, in analyzing a firm’s competitive context, ideally, one should analyze the preferences of targeted customers, their alternatives, the preferences of the buyers of the alternatives, etc. Relying on an external categorization such as industry carries the risk of omitting critically important players. To illustrate, we consider Example 3 again.

Because buyer 3 is not interested in the uptown apartment, suppose that the uptown firm believes that buyer 3 is not relevant to an analysis of its profitability. Then, it might view the game as the scenario depicted in the two left-hand panels of Figure 5. Note that if this were the case, the uptown firm would no longer be guaranteed a profit. If this were a situation in which the uptown firm was analyzing potential entry, rather then already being in the game, it might decide not to enter.

Figure 5 Uptown Firm Ignoring Buyer 3

It is important to realize that omitting a player is not necessarily a problem. The two right-hand panels of Figure 5 depict the proper way to omit buyer 3. Notice that this requires an understanding that the downtown firm’s economic cost of providing its apartment—given the players in the game—is $3 + pD, where pD is the price that the park firm receives. This prompts the question as to why this is so. The answer, of course, is the existence of buyer 3. One could leave buyer 3 out of the analysis, but one would still need to understand the effect of buyer 3 on the remaining players in the game. Moreover, notice that to correctly omit buyer 3 from the analysis, one must assess the price that buyer 3 will pay to the park firm. (For a conservative analysis of the uptown firm’s guaranteed profitability, one would take pD = 0)

Finally, it might appear that there is a sense in which competing firms help a firm’s profitability. For instance, the higher a competing firm’s guaranteed price, the more likely it is that a competing firm’s buyer will be a source of buyer competition. But this is, at best, a matter of perspective. In this paper’s model, firms are not complements, and a firm would always do better without a competitor. It is true that if a firm has to have a competitor, it would like the competitor to be guaranteed as high a price as possible, and, at a sufficiently high guaranteed price, the competitor could have no effect. But the absence of the competitor would always be (weakly) better. To see this, revisit Example 3. Without the downtown and park firms, the uptown firm is guaranteed a price of $8 from buyer 2 being excluded. Without just the park firm, the downtown firm is guaranteed a price of $8 from buyer 3 being excluded, and this guarantees the uptown firm a price of $6. Both of these scenarios are better for the uptown firm than the original scenario.

3. Model and Results

As is usual in a value-based analysis, the model in this paper will be a transferable utility (TU) cooperative game (N; V), where N denotes the set of players and v is a mapping v: 2N → ℜ (the characteristic function). For any SN, the term v(S) denotes the maximum economic value that the players in S can create among themselves. An outcome of a TU cooperative game is described by an allocation x|N|, where component xi denotes the value captured by player i. The core of a TU cooperative game (N; v) is the set of allocations satisfying iNxi=v(N) and for all SN, iSxiv(S).

In this paper, the core is used to model free-form competition. The condition that iSxiv(S) for any group of players S can be interpreted as a requirement that “no good deal goes undone.” For instance, if it were the case that iSxi<v(S) for some set of players S, then these players could do better by creating v(S) on their own. The requirement that players, collectively, cannot capture more value than can be created in total can be stated as a feasibility condition: iNxiv(N). This feasibility condition, combined with the competitive condition iNxiv(N), implies the efficiency condition of the core: iNxi=v(N). Thus, when using the core to model competition, efficiency may be viewed as a consequence of feasibility and competition, not as an a priori assumption.

A firm’s core allocation determines its range of possible profits. As noted in §1, the minimum of this range is the profit guaranteed the firm due to competition, whereas the maximum is the firm’s potential profit. The difference between the maximum and minimum is a residual bargaining problem between the firm and its buyers. When the potential profit equals the guaranteed profit for every firm, there is no residual bargaining problem, and profits are perfectly determined (in the sense of Edgeworth 1881) by competition.

In this paper’s model, the player set N is composed of a set of firms F, and a set of buyers B—i.e., N = FB, with FB = ∅. We assume that there is at least one firm and one buyer: F ≠ ∅ and B ≠ ∅. For each firm iF let Ci(n) denote the firm’s cost of producing n units of product. For all iF, Ci is nonnegative and (weakly) increasing, and Ci(0) = 0. For any n > 0, let ci(n) denote the marginal cost—i.e., ci(n) = Ci(n) ƒ− Ci(n ƒ−ƒ 1). For each jB, let wj(j) denote buyer j’s willingness-to-pay for firm i’s product. Demand is assumed to be unitary, partly for convenience and partly for substantive reasons. For convenience, unitary demand ensures that value creation can be described by associating each buyer with some firm. (This could also be achieved by assuming that a buyer’s total demand is always small relative to a firm’s capacity.) An example best illustrates the substantive issue that the unitary demand assumption avoids. Consider a situation in which there are two firms with a capacity of four units each and three buyers who each want three units. In this example, demand exceeds supply, and there is a core outcome in which the price equals the buyers’ willingness-to-pay. But there are also many other core outcomes. Because at least one of the buyers will have to transact with more than one firm, competition has little to say about what should happen in such a case; see, for example, Moulin (1995, p. 75) or Stuart (2005, Example 2).

Value creation is modeled as the gains-from-trade from buyers transacting with firms. To introduce the notation, we consider the value created by coalitions consisting of just one firm and a set of buyers. For any iF and SB, we will have

In words, to determine the maximum value created by a such a coalition, one would start with the buyer that has the highest willingness-to-pay for the firm, add in buyers in order of decreasing willingness-to-pay, and stop when either there are no more buyers in the coalition or the marginal cost exceeds the willingness-to-pay of any remaining buyers. To emphasize this latter point, note that if the optimal T is a strict subset of S, it follows that for any jS\T, wj(i)ci(|T|+1). In words, each buyer in S\T has a willingness-to-pay (weakly) lower than the firm’s marginal cost.

We now define the characteristic function for all coalitions. For all SN, if SF = ∅ or SB = ∅, let v(S) = 0; otherwise, let {Ti}iSF denote a collection of disjoint subsets of SB such that iSFTiSB. Then,


To interpret Equation (1), first note that value is created by matching buyers to individual firms. Although this is a common assumption, it does preclude situations in which competing firms are complements. Next, note that a collection {Ti}iSF is either a partition of SB or a partition of a subset of SB. The maximization in Equation (1) finds the partition of buyers that matches buyers to firms in the most efficient way. Thus, the set Ti denotes the buyers that transact with firm i. In situations in which it is not optimal for a given firm, say i, to be involved in the value created by the coalition S, the set Ti will be empty. In the case of a buyer, if it is not optimal for a given buyer, say j, to be involved in the value created by the coalition S, then j(SB)\iSFTi, where iSFTi is a strict subset of SB.

For convenience, let {Bi}iF denote a partition of B (or a subset of B) such that

where bi = |Bi|. Because the collection {Bi}iF describes which buyers will transact with which firms, it implicitly describes an optimal matching of firms to buyers. Consequently, in a slight abuse of terminology, we will call such a collection an optimal matching of B. Because the set Bi contains the buyers who will transact with firm i, if the set Bi is empty, then firm i has no buyers. If iFBi is a strict subset of B, there is unfulfilled demand. Alternatively, if iFBi=B, all demand is met. For notational convenience, given an optimal matching of B, we let B0=B\(iFBi). (The subscript zero is a reminder that a buyer j in B0 will have a marginal contribution of zero—i.e., v(N)v(N\{j})=0.) When the set B0 is nonempty, a buyer jB0 will be called an excluded buyer. Additionally, given an optimal matching of B, let F+F denote those firms that have buyers—i.e., F+ = {iF: Bi = ∅}.

In this paper, we consider only situations in which competition leads to a stable outcome. As a result, we need to identify additional conditions under which the TU game described by Equation (1) has a nonempty core. The following two conditions suffice.8 In any optimal matching of B, for any firm iF+ for 0 ≤ n < bi,

and for n > bi

Condition (C1) implies that a firm’s marginal cost (weakly) exceeds its average cost in an efficient outcome. This is a natural economic condition for stability (see, for example, Mas-Colell et al. 1995, Section 10.F). (The value bi represents firm i’s production quantity in an efficient outcome.) Condition (C2) states that a firm’s marginal cost is (weakly) increasing at its production quantity in an efficient outcome. Note that if a firm has either constant or (weakly) increasing marginal costs, the two conditions are immediate. Moreover, if firms have “u-shaped” average cost curves—that is, a fixed cost of production with (weakly) increasing marginal costs—then the conditions imply that firms must be at or above minimum efficient scale.

With Conditions (C1) and (C2), we have the following lemma.9 Proofs of all results are online in the electronic companion (available as supplemental material at

Lemma 1.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). The game has a nonempty core. Fix an optimal matching ofB. The guaranteed profit of a firmiFis

ifiF+and zero otherwise. In the core allocation in which each firm receives its guaranteed profit, each buyer receives its marginal contributioni.e., v(N) − v(N\{j}).

Lemma 1 serves the technical purpose of showing that with restrictions on the cost functions, competition will result in a stable outcome. More importantly, it provides a conceptual description of a firm’s guaranteed profit. It is the value that a firm creates with its buyers—v({i} ∪ Bi)—minus the sum of its buyers’ marginal contributions— jBi[v(N)v(N\{j})]. When we consider a firm’s profitability with capacity constraints, this description will provide the intuition for the results. As an initial step, we now consider a firm’s profitability when capacity is not constrained.

3.1. Profitability Without Capacity Constraints

From the perspective of a firm’s profitability, it is useful to note that an increase in marginal cost can act like a capacity constraint. In the case of a firm operating at its physical capacity, the marginal cost of an extra unit is infinite, so it is easy to see how a dramatic increase in marginal cost can be a capacity constraint. At the other extreme, if a firm can produce one more unit at the same marginal cost, it clearly has extra capacity, so the absence of an increase in marginal cost implies the absence of a capacity constraint. Thus, in an optimal matching, if we have ci(bi + 1) = ci(bi) for every firm, we can say that each firm’s capacity is not constrained. Using this fact, we have a result for firm profitability in the absence of capacity constraints.

Proposition 1.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). Fix an optimal matching. If for eachiF+, ci(bi + 1) = ci(bi), then each firmiF+has a guaranteed profit of

and a potential profit of

To interpret this result, first note that a firm’s guaranteed profit is due to the firm being guaranteed a price of ci(bi)—its marginal cost. (In the firm’s guaranteed profit, bici(bi) is the revenue and Ci(bi) is the cost.) Thus, a firm’s guaranteed profit will be positive only if the firm’s marginal cost strictly exceeds its average cost

The term jBi([wj(i)ci(bi)]max{0,maxrF\{i}wj(r)cr(br+1)}) is the sum of its value-gap advantages. The term wj(i) − ci(bi) is firm i’s value gap with its buyer j. The term max{0,maxrF\{i}wj(r)cr(br+1)} is buyer j’s best alternative for value capture. Loosely, because a given firm, say k, is selling only bk units, firm k is indifferent to selling (or not selling) another unit at its marginal cost ck(bk + 1). If buyer j’s willingness-to-pay is greater than this marginal cost—i.e., if wj(k)>ck(bk+1)—then buyer j could capture up to wj(k)ck(bk+1). (Because cr(br + 1) = cr(br),, this potential value capture is identified by the dotted lines in Figure 1.) In the case of constant marginal costs, the term bici(bi) − Ci(bi) equals zero, and we have the context of Example 1. A firm is not guaranteed a profit, and its potential profit is based on its value-gap advantages.

To summarize Proposition 1, to be profitable in the absence of capacity constraints, a firm must identify buyer segments in which it has the largest value gap. Moreover, to be guaranteed a profit, a firm’s marginal cost must exceed its average cost. If this is not the case, profitability will depend upon the firm’s ability to capture some of its potential profits—namely, some of its value-gap advantages—where this ability is based on factors other than the force of competition—a factor such as bargaining ability, for example.

3.2. Guaranteed Profitability with Capacity Constraints

To understand how capacity constraints affect a firm’s profitability, first note that a firm’s guaranteed profit— v({i}Bi)jBi[v(N)v(N\{j})]—can be written as

In the following lemma, we show that the term wj(i)[v(N)v(N\{j})] is constant for all of a firm’s buyers, implying that a firm’s guaranteed profit can be characterized by a single price.

Lemma 2.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). Consider an optimal matching ofB. For anyiF+and anyj,jBi,

Thus, a firm’s guaranteed profit can be described by
for anyjBi.

We will use Lemma 2 to develop intuition for a firm’s guaranteed profit. To understand a given buyer’s marginal contribution, it is useful to consider the reduction in value creation due to the (hypothetical) absence of the buyer. There are three possible scenarios: (i) no buyer takes the buyer’s place, and the buyer’s firm produces one fewer unit; (ii) an excluded buyer takes the buyer’s place; or (iii) another firm’s buyer takes the buyer’s place.

In the first scenario, a buyer’s marginal contribution will be just its marginal value creation with the firm—wj(i) − ci(bi). From Lemma 2, this implies a price of pi=ci(bi)—the firm’s marginal cost. In this situation, there is no buyer competition. Because no buyer would take the place of the absent buyer, no buyer is competing for the firm in question. (Example 1 demonstrates scenario (i).) Note that without capacity constraints, this will always be the scenario that occurs, as shown in Proposition 1.

In the second scenario, if an excluded buyer, say some jB0, has a willingness-to-pay greater than the firm’s marginal cost, then this excluded buyer takes the buyer’s place. The marginal value creation with the excluded buyer—wj(i)ci(bi)—will offset the loss of the given buyer’s marginal value creation—wj(i) − ci(bi). Thus, the given buyer’s marginal contribution will be wj(i)wj(i), implying a guaranteed price of pi=wj(i). In words, a firm is guaranteed a price equal to the highest willingness-to-pay among excluded buyers. An excluded buyer provides buyer competition. In such a scenario, we can show that there must be a capacity constraint. Because buyer j is excluded, we know that wj(i)<ci(bi+1). Because this excluded buyer could take another buyer’s place, we also know that wj(i)>ci(bi), implying that

This is a situation in which a moderate increase in marginal cost acts like a capacity constraint. Because of the increase, an otherwise viable buyer is excluded.

The third scenario is similar to the second in that there is a buyer providing competitive pressure that favors the firm. But now, the pressure comes from another firm’s customer rather than from an excluded buyer. In the absence of the given buyer, some buyer from another firm, j′ ∈ Bk, would take the place of buyer j. In the notation of our model, this would occur only if


In words, buyer j′ could create more value with firm i than with its current firm. Following the discussion of Example 3, we say that buyer j′ is envious of firm i, where the notion of envy is based on marginal value creation. Note that by the definition of an optimal matching, it must always be the case that

(Otherwise, there would be an increase in value creation if buyer j′ transacted with firm i instead of firm k. This cannot happen in an optimal matching. See Lemma A.2 in the online companion.) Thus, for Equation (E) to be possible, we must have ci(bi+1)>ci(bi), meaning there can be envious buyers only if some firm has strictly increasing marginal costs. (Buyer 3 in Example 3 is envious of the downtown firm.)

From scenarios (ii) and (iii), we see that for a firm to be guaranteed a price above marginal cost, there must be buyer competition. And to have buyer competition, there must be strictly increasing marginal costs. We summarize this fact with the following corollary.

Corollary 1.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). Fix an optimal matching ofB. If a firmiF+is guaranteed a price above its marginal cost, thenci(bi+1)>ci(bi).

Corollary 1 shows that to benefit from competition, a firm has to have some credible constraint on its ability to produce products. This constraint does not have to be a physical capacity constraint, but it does have to involve an increase in marginal cost. It is important to note that this is a necessary condition to create buyer competition, but it is not sufficient. For instance, in scenario (i), it is entirely possible for each firm to be at capacity.

For a characterization of a firm’s guaranteed price, we consider the linkages in buyer preferences described in the discussion of Example 3. In our model, the value capture of a buyer jBi, namely xj, can be written as wj(i)pj(i), where pj(i) denotes the price that buyer j pays to firm i. (Note that there is no ex ante requirement that pj(i)=pj(i) when j,jBi. Lemma 2 shows that this equality holds for a firm’s guaranteed price. It does not hold, in general, when a firm captures more than its guaranteed profit.) Thus, a core allocation x implicitly defines a collection of prices {pj(i)}. Using this notation, we have Lemma 3, which was used implicitly in the discussion of Example 3.

Lemma 3.

Consider a TU game described by Equation (1). If the core is nonempty, then in any core allocationx, in any optimal matching ofB, for any two distinct firmsi,kF+, and for any buyersjBiandj′ ∈ Bk,


Lemma 3 shows how the value capture of buyers from different firms must be related. The price a buyer j′ will pay to its firm k, namely pj(k), will be at least as high as the price paid by a competing firm’s buyer, adjusted for differences in preferences. Note that by rearranging terms, Lemma 3 can also be viewed as a revealed-preference condition: wj(i)pj(i)wj(k)pj(k).

Returning to the specific case of the guaranteed prices of the firms, Lemma 3 implies that for any distinct firms i,kF+, and any buyer j′ ∈ Bk, we must have

This relationship is the essence of the market-price effect. In words, firm i, say, can always benefit from firm k’s guaranteed price, provided that it compensates one of firm k’s buyers, say j′, for the difference in preference—that is, wj(i)wj(k). We use this relationship to determine a firm’s guaranteed price. Following the discussion of Example 3, firm i’s base price pi0 is defined as the larger of its marginal cost and the largest willingness-to-pay among excluded buyers:
From Lemma 3, for any firms i, k with buyers, for any buyer of firm k, we know that piwj(i)wj(k)+pk0. Similarly, for a firm hF+\{i,k} and a buyer jBi we have phwj(h)wj(i)+pi, yielding

Expressions of the form [wj(h)wj(i)]+[wj(i)wj(k)] can be viewed as market-price adjustments that show how the guaranteed price of one firm, say h, must be related to the base price of another firm, say k. The characterization of a firm’s guaranteed price is simply a matter of accounting for all possible market-price adjustments.10 This is straightforward conceptually, but not notationally. Consider SF+\{i,k},ik. Let s=|S|; let Π denote the set of permutations of the elements of S; let π denote a typical element of Π; and let πl denote the lth component of π. Let J be a vector of buyers from B(π,k)=Bπ1××Bπs×Bk with typical element (j1,,js+1). Then,

where m corresponds to the πm firm; m = 0 corresponds to firm i; and m = s + 1 corresponds to firm k. The term Λi(k) describes how competition could affect the price guaranteed to firm i based on the price guaranteed to firm k. If Λi(k) is positive, then firm i is guaranteed a price higher than firm k’s price. (Recall that pD>pP in Example 3.) Similarly, if Λi(k) is negative, then firm i’s guaranteed price is lower than firm k’s guaranteed price. In this latter case, it is important to note that even though Λi(k) is negative, firm i could still benefit. (Recall that pU<pD in Example 3.)

By defining Λi(i) to be 0, the guaranteed profitability result can be stated as follows.

Proposition 2.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). The price guaranteed to a firmiF+is

Further, firmiis guaranteed a price above its marginal cost if, and only if, either there exists an excluded buyerjB0such that
or there exists akF+\{i}and aj′ ∈ Bksuch that

Two implications of Proposition 2 should be noted. The first is that the expression for a firm’s guaranteed price does not contain a willingnesses-to-pay of any of its buyers. In other words, given a firm iF+, pi will never contain a term wj(i) where jBi. Thus, only the willingnesses-to-pay of “noncustomers” affect a firm’s guaranteed price. In §4, we explore some of the consequences of this fact.

The second implication is that guaranteed profitability, like potential profitability, has a value-gap advantage interpretation. In other words, a firm is guaranteed a price above its marginal cost only if it has a value gap with a noncustomer—an excluded buyer or a competitor’s buyer—that is larger than the noncustomer’s value capture. In Proposition 2, these two scenarios are represented by wj(i)ci(bi)>0 and wj(i)ci(bi)>wj(k)pk. Thus, Proposition 2 can be summarized as follows: for a firm to be guaranteed a profit above its marginal cost, it must identify noncustomers with whom it has a value-gap advantage.

We close this section by discussing why we use the term market-price effects for the effects of linkages in preferences. With a commodity product, buyer preferences will satisfy wj(i)=wj(k)=wj for all firms i and k. This implies that Λi(k)=0, which, in turn, implies that pi = pk for all i,kF+. In other words, a firm’s guaranteed price will be a market price. Thus, if commodity products are viewed as a limiting case of buyer preferences, then, in the limit, the linkages generate a traditional market price. (For instance, in Example 2, all three firms are guaranteed the same price—$10 per unit. This is the minimum value in the intersection of the supply and demand curves in Figure 2.)

3.3. Potential Profitability with Capacity Constraints

If the core is nonempty, it is straightforward to show that a firm’s potential profit can be determined by establishing the guaranteed profit—namely, guaranteed value capture—for each of its buyers. (See Lemma A.1 in the online companion.) Conceptually, a buyer’s guaranteed value capture is similar to a firm’s guaranteed price. There is a base value capture, and then there is the possibility that a buyer is guaranteed more than this base because of market-price effects. Recalling the discussion following Proposition 1, a buyer can always capture value based on using another firm’s extra unit of capacity. Thus, we can define the lower bound for a buyer’s guaranteed value capture as follows. Consider an optimal matching of B, and define, for any iF+ and jBi,

To incorporate market-price effects, we can use Lemma 3 to show that for any buyer jBi, iF+, kF+\{i}, and jBk, we must have
where mj and mj denote the guaranteed value capture of buyers j and j′, respectively. This fact suggests that a buyer’s value capture can be defined iteratively. In the proof of Proposition 3, we show that the following procedure produces a buyer’s guaranteed value capture. For any buyer jBi, iF+, define
mjn=max{mjn1,maxkF+\{i}maxjBk[wj(k)wj(k)+mjn1]}and mj=mjn,
where n is the smallest n such that mjn=mjn for all n>n. For any jB0, set mj = 0 The term mj is buyer j’s guaranteed value capture. In the discussion of Proposition 3, it will be useful to interpret mj as buyer j’s best alternative for value capture with a competing firm.

Having defined a buyer’s guaranteed value capture, we have the following result.

Proposition 3.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). Fix an optimal matching ofB. In the core, each firmiF+has a potential profit of


In this expression, the left-hand term is the sum of its value-gap advantages—the firm’s marginal value creation with the buyer minus the buyer’s best alternative for value capture—and the right-hand side is the profit from marginal cost exceeding average cost. (For a demonstration of Proposition 3, consider Example 2 again. The buyers with a willingness-to-pay of $12 are guaranteed to capture $1; all other buyers are guaranteed zero. Therefore, firms A, B, and C have a potential price of $11 per unit—the maximum value in the intersection of the supply and demand curves in Figure 2.)

Proposition 3 states that a firm’s potential profit is determined by the guaranteed value capture of its buyers. In particular, a firm iF+ has the potential for prices above marginal cost if, and only if, there exists a buyer jBi such that mj<wj(i)ci(bi). But this prompts the question of when, or under what conditions, this inequality will hold—i.e., when will a firm have a buyer (or buyers) for whom the guaranteed value capture is strictly less than the buyer’s marginal value creation? As with a firm’s guaranteed profit, the answer to this question is complicated by the possibility of market-price effects. There is, however, an intuitive sufficiency result.

Corollary 2.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). Fix an optimal matching. Each firmiF+has a potential profit of at least


Corollary 2 provides the foundation for the robust positioning advice provided in §1. As long as a firm has buyer segments in which its marginal value creation with a buyer is greater than any other firm’s value creation with a buyer, it will have the potential for profits. And if the firm undersupplies such a segment, Proposition 2 implies that a firm iF+ will be guaranteed a price


We close this section by examining the intuition that a firm should have the potential for profitability if it is needed for value creation. Proposition 4 shows that for a firm to have the potential for prices above its marginal cost, it must be needed by either specific buyers or the market as a whole.

Proposition 4.

Consider a TU game described by Equation (1) and satisfying Conditions (C1) and (C2). A firmiF+will have the potential for prices above marginal cost if, and only if, either

(i) there exists a buyerjsuch that in every optimal matching{Br}rF, jBi; or

(ii) ci(bi)<ci(bi+1)in an optimal matching that minimizes the size ofBi.

Condition (i) corresponds to situations in which a firm is needed by a specific buyer. To maximize the value that can be created in the game, there is only one firm that the buyer can transact with. When the left-hand side of the lower bound in Corollary 2 is positive, this condition is easily met, because if wj(i)ci(bi)>maxrF\{i}{wj(r)cr(br)}, then buyer j clearly needs firm i from a value-creation perspective. If a firm does not have any buyers who specifically need it, then there is no sense in which the firm can obtain potential profits by being better than the competition. But a firm may still have the potential for profit because it is needed to meet market demand. In this latter scenario, a subtlety arises due to market-price effects: a firm can be the source of its own competition. This subtlety is one of the main insights of Postlewaite and Rosenthal (1974), and Firm C in Figure 3 provides an example. Condition (ii) of the proposition prevents such cases. The requirement that the firm have some sort of capacity limitation—i.e., that ci(bi)<ci(bi+1)—prevents a firm from being a perfect competitor with itself.

Propositions 2 and 4 can be summarized with loose, but suggestive, terms. To be guaranteed a price above marginal cost, a firm should be an object of competition. And to have the potential for profitability, a firm should be needed, but it shouldn’t be a competitor to itself.

4. Strategic and Empirical Implications

The results of this paper show that a firm’s positioning decision relies critically on the value that the firm creates with buyers, both those that it would like to serve and those that it would like to have as a source of competition. One consequence of these results is that the pattern of buyer preferences and the technological nature of cost functions are not, in isolation, informative. Thus, it is generally not possible to assess a strategic action based on, say, the product variety of an industry or the extent of scale effects in the industry. Rather, one needs knowledge of how preferences and costs interact in the specific context. Perhaps ironically, the generality of the results from a value-based analysis suggest a complementary need for context-specific knowledge.

Consider, for example, a context with both differentiated products and economies of scale. The scale effects could be so strong that in the value-gap calculations, the differentiation doesn’t matter. Profitable positions would be based on cost capabilities. Or the differentiation effects could outweigh the scale effects. Profitable positions would be based on segmentation. More importantly, there could easily be a combination of these two extremes. For instance, there could be a profitable position where a large firm with a significant cost advantage serves multiple buyer segments, but there could also be other positions based on other buyer segments with strong enough preferences for some firms with relative cost disadvantages. To understand what a firm’s position should be in this context requires detailed knowledge of the context.

Although the results of this paper do not provide simple positioning advice, they do show how to think about a firm’s positioning decision, as the prior example shows. Similarly, the results show how to better understand the rationales—and limitations—of strategic rules-of-thumb. For example, the advice to lower costs seems trivial, and if a firm has buyer competition, lowering costs will generally improve both guaranteed and potential profitability. But if the reduction in costs results in the firm winning more buyers and reducing buyer competition, guaranteed profitability could actually decrease (see Proposition 2). Again, it takes context-specific knowledge to assess the true impact.

A similar story can be told about increasing the willingness-to-pay for a firm’s product. For instance, the advice to “focus on the customer” might seem robust, but if a firm cares only about its guaranteed profit, investing in a product improvement that increases customers’ willingness-to-pay is not worthwhile unless it also increases noncustomers’ willingness-to-pay. From Proposition 2, conditional on having buyers, improving only the willingness-to-pay of customers has no effect on guaranteed profitability.

For a final example, consider the rule-of-thumb that “commoditization” of an industry is undesirable. Consider a given firm with a shortage of demand and a (differentiated) competitor with excess demand. If this excess demand comprises buyers with no interest in the given firm, then the given firm does not benefit. But if products become commoditized, linkages in preferences could arise, and the given firm could then benefit from the excess demand. (In fact, commoditization is not necessary. Creating linkages in buyer preferences would have the same effect.) Whether this would actually happen is an empirical question.

The results in this paper show that answering empirical questions about actual buyer preferences and firm costs—the context-specific knowledge previously mentioned—is essential. Furthermore, any value-based analysis will prompt broader empirical questions. There is the fundamental question of whether a value-based analysis is appropriate. If not, then the context would most likely be better modeled with a noncooperative game. If a value-based analysis is appropriate, there is the question of whether firms act to maximize guaranteed profit, potential profit, or an amount in between. Most of the informal examples in this section consider firms that act according to guaranteed profit. But if firms act to maximize potential profit, their behavior changes. A firm will generally make decisions that increase overall value creation. In particular, serving more buyers and increasing the firm’s value gap with buyers will increase potential profitability. And if firms do expect to capture more than their guaranteed profit, there are the questions of how much more they capture and whether there is a systematic reason for why they capture more. These last questions have received the most attention in the literature (see, for example, Bennett 2013, Chatain 2011, Grennan 2013), but the first two questions are also essential for strategy research. Ideally, a firm would like to know what contexts are appropriately characterized by the severe competition implied by a value-based analysis. And among those contexts, it would like to know when the profits guaranteed by competition are the only route to profitability.

5. Conclusion

The value that a firm can create with a customer would seem to be an important measure in a competitive environment. In a value-based analysis this is, in fact, the case. To have the potential for profitability, a firm must identify segments of buyers with whom it can create more value than the buyers could capture otherwise. To be guaranteed a price above its marginal cost, a firm must limit its capacity sufficiently to create buyer competition. If the firm is in a context with excess demand, then it is guaranteed to have buyer competition. But a firm can also have buyer competition due to envious buyers and because of market-price effects, due to a competitor’s excluded or envious buyer. Consequently, a firm can benefit from buyer competition even when there is excess supply.


The author thanks Adam Brandenburger, Luis Cabral, Ken Corts, Nan Jia, Dan Levinthal, Patrick Sileo, and seminar participants at Carnegie Mellon—Qatar, NYU Stern, and Wharton. The author also thanks the senior editor and both referees for advice that significantly improved this paper. Financial support from Columbia Business School is gratefully acknowledged.


1 MacDonald and Ryall (2006) show that for a player to be guaranteed positive value capture in the core of any transferable utility (TU) game, there must be at least two groups of players competing for it. The need for buyer competition to guarantee profits above marginal cost can be viewed as an example of their result.

2 In a price-setting model, this result would be trivial, as a firm would not price below its marginal cost. But because a value-based model has no ex ante assumptions of pricing power, this fact becomes a result.

3 For the Cournot results, see, also, Moulin (1995). For an application of the model to monopoly, see Muto et al. (1988) or Stuart (2005).

4 For teaching purposes, a picture like Figure 1 can be called a Rashomon diagram, after the Japanese film classic of that name. In that movie, all the witnesses to a murder recount different perspectives on the same crime. In much the same way, each buyer segment sees the value of the three firms’ products differently.

5 Without firm 2, one segment B buyer would transact with firm 1; one would transact with firm 3; and the others would no longer transact.

6 Firm 2 and segment B are based on Brandenburger and Stuart (2007, Example 5.3).

7 For this type of example, Kaneko (1976) shows that firm profitability is consistent with standard supply and demand reasoning. A uniform price will emerge, and the difference between a firm’s guaranteed and potential profit is due only to the discreteness of the demand curve.

8 Stuart (2015) provides necessary and sufficient conditions for a nonempty core in the game of Equation (1).

9 Telser (1978) and Kaneko and Yamamoto (1986) identify similar conditions for core stability.

10 In Example 3, the uptown firm was linked to the downtown firm via [w2(U)w2(D)]+[w3(D)w3(P)]. Note that this was more favorable than the direct linkage w3(U)w3(P). Lemma 3 implies that the most favorable linkage must be used in determining a firm’s guaranteed price.


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Harborne W. Stuart, Jr. is an adjunct professor in the Decision, Risk and Operations Division at Columbia Business School. He received his PhD from Harvard University. His research focuses on the development of business theory using game-theoretic approaches, including the use of cooperative game theory to study businesses as the central players in economic value creation.

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