Social Network-Based Discriminatory Pricing Strategy - Springer Link

Marketing Letters 14:4, 239–256, 2003  2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Social Network-Based Discriminatory Pricing Strategy MENGZE SHI [email protected] Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, ON M5S 3E6, Canada

Abstract In this paper we study such pricing practices like MCI’s Friends and Family Program that employ price discriminations on the basis of callers’ social ties. We characterize a consumer’s personal communication network by the number of strong and weak ties that the consumer has. We then derive a consumer’s demand for communication service from the structure of the consumer’s personal communication network. A monopoly firm’s social network-based discriminatory pricing strategy consists of a menu of price plans, each plan targeting at one type of social networks. Our paper provides useful guidelines for the design of optimal social network-based discriminatory pricing strategies. We show that a firm may offer price discounts to communications between “friends and family members” in order to extract a larger profit from communications between callers with weak ties. Keywords:

pricing research, price discrimination, social network

1. Introduction Providing price discounts to communications between “friends and family members” has become an increasingly popular practice in telecommunication industry. A prominent example was MCI’s Friends and Family Program, which offered price discounts to long distance communications between customers and their pre-selected friends and family members. Sprint offered a similar plan that applied discounts automatically to those telephone numbers connected most frequently with customers during a billing period. Both examples can be seen as price discrimination based on strength of callers’ social ties. Casual observations indicate that such pricing practice has become particularly common in wireless service industry. For example, Bell Mobility of Canada’s Real Time Family Plan allowed multiple phones to enroll together to earn price discounts. For a monthly fee of CAN$85, the plan offered 500 minutes to be shared by two cellular phones that enrolled in the plan (CAN$42.5 for each phone), and a variable fee of CAN$0.30 per minute for communications with other phones. At the same time, for a monthly fee of CAN$45, a regular Real Time plan provided 400 calling minutes (http://www.bellmobility.ca/). Compared with the Real Time plan, the Real Time Family plan charged a lower price for communications between the participating phone sets but a higher price for communications with the others. The Real Time Family Plan clearly targeted at those consumers whose communications were concentrated among a small number of people; therefore, this is an example of price discrimination on the basis of callers’ personal communication patterns. In this paper, we

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use the term social network-based discriminatory pricing strategy to refer to any pricing policies that employ price discriminations based on strength of callers’ social ties, and/or pattern of callers’ communication networks. Despite its prevalence in practice, we have not seen any published research efforts about social network-based discriminatory pricing strategy. Thus, this paper provides the first formal study of such pricing policy. A unique feature of communication service is that two or more callers need to consume the service together. As a result, a consumer’s value of communication depends on whom the consumer communicates with. A consumer typically communicates with the members of her personal social network, which is often formed based on kinship, friendship, or professional relationship. Following social network literature, we describe the structure of a consumer’s social network with two key characteristics: strength of a tie and relational density of a personal network. The strength of an interpersonal tie depends on frequency of social contacts, importance of social relation, and type of social relation such as acquaintance and kinship. Following the convention of network analysis (Granovetter, 1973), we assume that strength of a tie can be strong, weak, or absent. We define a consumer’s personal communication network as the set of consumers with whom the focal consumer has either a strong or a weak tie. The relational density of a personal network refers to distribution of tie strengths within a personal network. In this paper we consider a market with two types of personal social networks that have the identical size. We name the type of networks that have more strong ties but fewer weak ties as dense networks, the other type of networks that have fewer strong ties but more weak ties as loose networks. We study a monopoly firm’s optimal pricing strategy. In our model, the firm provides a menu of two price plans, one plan targeting at consumers with dense networks while another plan targeting at consumers with loose networks. Each price plan consists of a fixed fee and two variable fees, one variable fee for the communications with strong ties and another for the communications with weak ties. The pricing policy contains two levels of price discrimination: communication pattern-based discrimination (difference between two price plans), and tie strength-based discount (difference between two variable fees within each price plan). Our results provide many useful guidelines for the design of optimal price plans. For instance, in a market where consumers highly value their initial communications with weak ties, we show that, first, the variable fees for the consumers with loose networks should be equal to marginal cost. Second, the variable fee for the consumers with dense networks should depend on strength of ties; specifically, the variable fee should be above marginal cost for communications associated with weak ties but below marginal cost for communications associated with strong ties. Thus, consumers with dense networks receive tie strength-based price discount. Our analysis indicates that a firm may offer such a pricing policy to extract a larger surplus from consumers with more weak ties. Our paper contributes to several streams of research. First, we study optimal multi-part tariffs for communication service; thus, we contribute to literature on non-linear pricing. Both Tirole (1988) and Wilson (1993) provided very comprehensive reviews of nonlinear pricing strategies. In the literature on using two-part tariffs for second-degree price discrimination, Oi (1971) studied a profit-maximizing theme park setting prices for the admission to the park (fixed fees) and for individual rides (variable fees). Mitchell (1978)

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analyzed a similar problem for local telephone service. In marketing literature, Dolan (1987) overviewed managerial applications of quantity discounts. More recently, Skander et al. (2002) investigated a firm’s access pricing policy when facing capacity constraints. But none of these papers addressed social network-based discriminatory pricing strategy. Second, our paper develops a social network approach to model consumer demand. We allow a caller’s valuation and a firm’s price of communication service to be dependent on the caller’s tie with another caller. Moreover, we explicitly model consumers’ heterogeneous calling patterns and let the firm design price plans tailored to different communication networks. In the best of our knowledge, only Laffont et al. (1998) and Shi et al. (2003) discussed calling pattens when formulating market demand. However, both papers only considered uniform calling pattern, where each consumer has the same chance to communicate with all other consumers in the market. Finally, our paper is related to social network analysis in consumer behavior research (Iacobucci, 1996). This stream of research studied impact of social network structure on the diffusion of new product information through word of mouth (Granovetter, 1973; Brown and Reingen, 1987; Frenzen and Nakamoto, 1993; Goldenberg et al., 2001), and consumers’ purchasing behavior in embedded markets (Granovetter, 1985; Frenzen and Davis, 1990). Unlike these studies, our paper focuses on the linkage between social network structure and demand for communication service as well as the implications for firm’s optimal pricing strategies. We present our model and analysis in section 2. 2. Model and Analysis In this section we model market demand for communication service and analyze firm’s pricing strategy. We describe market characteristics from a social network perspective in Section 2.1. We define firm’s pricing policy in Section 2.2. We model consumers’ value and demand for communication service in Section 2.3. Finally, in Section 2.4 we formulate firm’s optimization problem and present our results. 2.1. Communication Market We consider a market that consists of N consumers who demand for communication service. Value of communications between a pair of consumers depends on strength of their social tie. We represent strength of tie between two consumers with a onedimensional index variable, denoted by t. Consistent with literature on social network analysis, we assume three levels of tie strength: strong ties (denoted by t = s), weak ties (t = w), and absence of tie (t = 0) (Granovetter, 1973; Wasserman and Faust, 1994; Goldenberg et al., 2001). A consumer derives positive utility from communications with another individual if and only if these two consumers have either a strong or a weak tie. We define the set of individuals with whom a consumer has either a strong or a weak tie as the consumer’s personal communication network. We suppose that market consists of two different types of personal communication networks: dense networks, denoted by d, and loose networks, denoted by l. Dense networks

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are the type of personal communication networks that have more strong ties, and loose networks are those with fewer strong ties. In the rest of this paper, we denote the type of networks by T , where T ∈ {d, l}. We let ρd and ρl denote percentage of personal networks that are dense and loose, respectively, where ρd + ρl = 1. In other words, market consists of Nρd number of dense networks and Nρl number of loose networks. Within a network of type T, we let ntT denote the number of type t ties, where t ∈ {s, w} and T ∈ {d, l}. Finally, we assume that all personal communication networks are of the same size, which s w is denoted by n.1 Naturally, nsd + nw d = nl + nl = n. Since a dense network has more w strong ties and less weak ties than a loose network, we have nsd nsl and nw d nl . 2.2. Social Network-Based Discriminatory Pricing Policy A monopoly firm provides communication service at a constant marginal cost c. The firm follows social network-based discriminatory pricing policy by offering a menu of two price plans, denoted by {(Fd , pds , pdw ); (Fl , pls , plw )}. Each plan consists of a fixed fee (FT ) and two variable fees, one for the communications between consumers with strong ties (pTs ) and another for the communications serving weak ties (pTw ). Amount of service consumption can be measured either in minutes or seconds. A social network-based discriminatory pricing policy has two features. First, the policy contains a menu of two price plans, one targeting each type of communication network. We refer to this feature as communication pattern-based discrimination. Second, the variable fees within a plan may depend on strength of ties. We call the difference between two variable fees tie strength-based discount. 2.3. Value and Demand for Communication Service A consumer’s utility from communications with another person depends on strength of their tie. For a pair of consumers with a tie of strength t, we let ut (q) denote each consumer’s utility from their communications of amount q and propose a quadratic utility function as follows:2 ut (q) = α1t q −

α2t 2 q , 2

where α1t > 0, α2t > 0, and t ∈ {s, w}.

(1)

Values of coefficients α1t and α2t depend on tie strength t. When two consumers have no ties, they have zero valuation for communications. In the rest of this paper, we will only include strong and weak ties in computing communication demand. The utility function in equation (1) is concave, implying a decreasing marginal utility when amount of communication increases. Such a utility function fits well with communication service because people tend to transmit more important information at the beginning of their conversations. We also assume the absence of income effect because consumers’ communication expense typically accounts for a small proportion of their budget. A consumer determines the optimal amount of communication with another person by solving following utility maximizing problem:


qt pTt = arg max ut (q) − pTt q,

243 (2)

q

where ut (q) is given by equation (1), t ∈ {s, w}. When the optimal quantity of communication is positive, solution to above problem is determined by the first-order condition, ut (q) − pTt = 0. Substituting in the utility function given by equation (1), we obtain following inverse demand function for communication service between a pair of consumers with tie strength t in a communication network of type T : (t = w, s; T = d, l). (3) pTt = α1t − α2t qt pTt Equation (3) indicates that, for two consumers that have a tie of strength t, marginal value of their communications is α1t for the first unit and then decreases by α2t for each additional unit. We now distinguish a strong tie from a weak tie with respect to demand for communication service. In the social contexts where people do not pay for their communications, an individual tends to communicate more with those members with whom the individual has stronger ties.3 This implies that, when the variable price is sufficiently low, there should be a positive correlation between demand for communications and strength of social ties. Consistent with this notion, we define that a strong tie is associated with a larger demand for communication when the variable price is equal or below a firm’s marginal production cost (c). Following the demand function at equation (3), we state above definition as follows. 1 s 1 α1 − p > w α1w − p for p c. (4) s α2 α2 Essentially, equation (4) states that consumers with weaker ties have lower valuation for the extra amount of communications. Next, we calculate consumers’ surpluses from communications. When a firm charges zero fixed fee, consumers always obtain non-negative surplus. For a consumer that has a personal communication network of type T , we let vt (pTt ) denote the consumer’s surplus from communication amount qt (pTt ) when the fixed fee FT is 0. We substitute equation (3) into the consumer’s utility function (2) to calculate vt (pTt ). 2 1 t α1 − pTt vt pTt = t 2α2

(t = s, w; T = d, l).

(5)

2.3.1. Illustrating service demand and service valuation for a pair of consumers We depict in Figure 1 tie strength-based demand (equation (3)) and net valuation (equation (5)) for communication service. There are four different scenarios classified according to the relative sizes of intercepts, slopes of demand functions, and value of communications at a variable price equal to marginal cost. In each graph, line HQw and IQs represent service demand between a pair of consumers with a weak tie and a strong tie respectively. Point E locates marginal cost c. When the variable fee is equal to marginal cost, demand associated with a strong tie is EG, and demand associated with a weak tie is EF. Marginal value for initial amount of communications is indicated by intercept H for a weaker tie, and I for

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Figure 1. Demand Functions for Strong and Weak Ties. In each Graph, a Dot Line is Demand Function for a Strong Tie and a Solid Line is Demand Function for a Weak Tie.

a strong tie. When the variable fees are equal to marginal cost and the fixed fee is equal to zero, consumer surplus from communications with a weak tie (vw (c)) is represented by area EFH, and surplus from a strong tie (vs (c)) represented by area EGI. We now further describe each of four scenarios in more detail: Scenario A: α1w < α1s and α2w > α2s . In this scenario, marginal value of communications for a strong tie (line IQs ) is always higher than that for a weak tie (line HQw ). If a firm charges the same variable fee for strong and weak ties, value of communications is always greater for a strong tie than for a weak tie (area EGI is larger than EFH). Since α2w > α2s , marginal value of communications declines faster with a weak tie. As a result, difference


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in marginal value of communications between a strong and a weak tie increases as the variable fee decreases. Scenario B: α1w < α1s and α2w < α2s . As in scenario A, value of communications is always higher for a strong tie than for a weak tie. Unlike in scenario A, since α2w < α2s , in this scenario marginal value of communications declines faster with a strong tie. Consequently, difference in value of communications between a strong and a weak tie narrows down as the variable price decreases. Scenario C: α1w > α1s , α2w > α2s , and vs (c) vw (c). In this scenario, marginal value of communications for a weak tie starts high for the initial amount (H is above I ), but declines quickly as quantity increases. When the variable fee is at marginal cost, consumer surplus from communications is greater for a strong tie than that for a weak tie (area EGI is larger than EFH). Scenario D: α1w > α1s , α2w > α2s , and vs (c) < vw (c). As in scenario C, marginal value of communications for a weak tie is higher for the initial amount of communications, but lower when amount of communications is sufficiently large. Unlike in scenario C, when the variable price is at marginal cost, consumer surplus from communications for a weak tie is greater than that for a strong tie (area EFH is larger than EGI). In summary, in both scenarios A and B, qs (p) > qw (p) because communication demand between a strong tie is always higher than that between a weak tie. In these two scenarios, demand functions for a strong and a weak tie never cross. For the same variable price p, consumer surplus for a strong tie is higher than the surplus for a weak tie; that is, vs (p) > vw (p). However, in scenarios C and D, demand function for a strong tie, qs (p), crosses demand function for a weak tie, qw (p), at a price above marginal cost c. While a strong tie values extra amount of communication more than a weak tie, a strong tie values the initial amount of communications less than a weak tie. The distinction between scenario C and D is that, at marginal cost price, value of communication vs (c) vw (c) in scenario C (as in scenarios A and B), but vs (c) < vw (c) in scenario D. In other words, a strong tie values service more than a weak tie under scenario C, but less than a weak tie in scenario D either because value difference associated with initial units of communication is sufficiently large or the rate of decrease in marginal value is small with weak ties. To understand scenario D, one may imagine a pair of consumers who have a weak tie and communicate only on important matters. This type of relationship leads to a high valuation for a small quantity of communications, but a low marginal valuation for additional communications. In contrast, a pair of consumers with a strong tie may hold long conversations regularly. By definition, a strong tie always holds higher valuations for an extended amount of communications than a weak tie. But a strong tie can have lower marginal valuations for short conversation than a weak tie because the contents of long conversations may not be as important. 2.3.2. Discussing a consumer’s total service demand and valuation With the absence of income effect, a consumer’s total demand for communication service is the sum over the consumer’s entire personal communication network. We let DT denote total communication demand for a consumer with a personal network of type T . Then we have w DT pTs , pTw = nsT qs pTs + nw (6) T qw pT ,

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Figure 2. Aggregate Demand Functions for Strong and Weak Ties (Scenario D).

where qt (.) is given by equation (3), T = d or l. Similarly, we can compute a consumer’s net value of communications from the entire personal communication network when the fixed fee is zero, denoted by VT for a personal network of type T . VT pTs , pTw = nsT vs + nw T vw s 2 2 n nw = Ts α1s − pTs + Tw α1w − pTw , 2α2 2α2 t ∈ {s, w} and T ∈ {d, l}. (7) From equations (6) and (7), we can calculate the difference in the total value of communications between consumers with dense and loose networks, for the same variable fees. w Vd − Vl = nsd − nsl vs + nw (8) d − nl vw = n(vs − vw ), w where n = nsd − nsl = nw l − nd measures the discrepancy between a dense and a loose network. Recall that, when the variable fees are equal to marginal cost, a strong tie values communications more than a weak tie in scenarios A, B, and C, but lower than a weak tie in scenario D. We can infer from equation (8) that, when the variable fees are at marginal cost, Vd > Vl in scenarios A, B and C, but Vd < Vl in scenarios D. In the rest of this paper, we refer to the segment of market that has higher (lower) total value of

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communications as high (low)-valuation segment. Specifically, the segment of consumers with dense personal communication networks is the high-valuation segment in scenarios A, B, and C, but becomes the low-valuation segment in scenario D. In Figure 2 we plot a consumer’s aggregate demand associated with strong and weak ties. Specifically, aggregate demand between strong ties is depicted by line IQds if a consumer has a dense network (nsd qs ), and by IQls if a consumer has a loose network (nsl qs ). Line HQdw and HQlw are aggregate demand functions between weak ties for a dense netw work (nw d qw ), and for a loose network (nl qw ), respectively. When the variable fees are at marginal cost and the fixed fees are equal to zero, consumer surplus from communications with a dense network is the sum of area EGd I (strong ties) and EFd H (weak ties). Consumer surplus from communications with a lose network is the sum of area EGl I (strong ties) and EF l H (weak ties). In scenario D, since Vd < Vl , area (EGd I + EFd H ) is smaller than area (EGl I + EFl H ). 2.4. Firm’s Pricing Strategy We now formulate firm’s pricing problem, following the standard approach for seconddegree price discrimination (Tirole, 1988; Wilson, 1993). A new feature of our problem is that we solve a menu of three-part tariffs, instead of the standard menu of two-part tariffs. We assume that it is optimal for the firm to serve both consumer segments (dense networks and loose networks). In other words, the firm earns more profits by serving both segments than only serving one of the two segments due to similar segment sizes and/or close service valuations. w w Max s w π = Nρd nsd qs pds pds − c + nw pd − c + Fd d qw pd s w {Fd ,pd ,pd ,Fl ,pl ,pl }

w w + Nρl nsl qs pls pls − c + nw pl − c + Fl , l qw pl

where w 2 nsT s nw T s 2 α α1 − pTw , T ∈ {d, l}, − p + T 1 s w 2α2 2α2 s w 2 nd s n s 2 + dw α1w − pdw − Fd s α1 − pd 2α2 2α2 s 2 2 n nw ds α1s − pls + dw α1w − plw − Fl , 2α2 2α2 w 2 nsl s n s 2 + l w α1w − plw − Fl s α1 − pl 2α2 2α2 w 2 nsl s nw l s 2 α α1 − pdw − Fd . − p + d 1 s w 2α2 2α2

FT

(9)

(10)

(11)

The firm’s objective is to maximize its total profits from two segments. The optimal pricing policy needs to satisfy two types of conditions. First, individual rationality (IR)

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conditions given by equation (9) ensure that both segments expect nonnegative surplus from the service plan targeting at them. Second, incentive compatibility (IC) conditions given by (10) and (11) require that each type of consumers prefer the price plans designed for them. After solving the firm’s problem, we find that our results are consistent with standard second-degree price discriminations using a menu of two-part tariffs (see Tirole, 1988; Wilson, 1993). In general, the optimal variable prices for the high-valuation segment are equal to marginal cost. For the low-valuation segment, fixed fee is equal to a consumer’s total valuation of communications so that IR condition (9) is binding. Fixed fee for the high-valuation segment is such that the associated IC condition is binding. In the equilibrium, the high-valuation segment receives positive surplus. Otherwise, the high-valuation segment may adversely select the price plan designed for the low-valuation segment. As noted earlier, the key departure of our problem from standard second-degree price discrimination is the use of three-part tariffs to study social network-based price discrimination. When a monopoly firm offers a menu of two-part tariffs, the optimal variable price for the low-valuation segment is typically above the firm’s marginal cost in order to reduce the high-valuation segment’s surplus from the price plan designed for the lowvaluation segment. Such a deviation of variable price from marginal cost reduces the firm’s profit from the low-valuation segment. But the deviation also allows the firm to charge the high-valuation segment a higher fixed price, and hence increases its total profits from two segments. The optimal variable price can be below marginal cost when a firm designs a two-part tariff for two consumers whose demand functions cross as in scenario D of Figure 1 (see Oi, 1971). We extend above intuitions to the optimal menu of three-part tariffs in the context of social network-based discriminatory pricing strategy. We find that the optimal variable fees for the high-valuation segment are equal to marginal cost. But the optimal variable fees for the low-valuation segment are different from marginal cost; specifically, two variable fees for the low valuation segment deviate from marginal cost in opposite directions, resulting to tie strength-based price discount. We next discuss results in more detail. We first discuss scenario D, which is of our main interest, followed by scenarios A, B, and C. In Appendix we only place detailed analysis for scenario D. Analysis for other scenarios is available upon request. 2.4.1. Scenario D In scenario D, the high-valuation consumers have loose networks. We obtain the optimal pricing policy as follows. ∗

∗

pls = plw = c, nρl (α1s − c) ∗ pds = c − s , nd ρd − nρl nρl (α1w − c) ∗ pdw = c + w . nd ρd + nρl

(12) (13) (14)

Equation (12) indicates that the optimal variable fees for the consumers with loose networks are equal to marginal cost. Equations (13) and (14) show that the variable fees for the consumers with dense networks is above marginal cost for weak ties, but be-


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low marginal cost for strong ties. We summarize these results in the following proposition. Proposition 1. Under demand scenario D, the optimal variable fees are equal to marginal cost for the consumers with loose networks. For the consumers with dense networks, the optimal variable fee is above marginal cost for weak ties but below marginal cost for strong ties. Proposition 1 summarizes the optimal social network-based discriminatory pricing policy for scenario D. The firm offers a menu of two distinctive pricing plans, each targeting one consumer segment. Our results suggest setting the variable fees at marginal cost for the high-valuation segment (loose networks), but deviating from marginal cost for the lowvaluation segment (dense networks). Such deviations decrease the firm’s profit from the consumers with dense networks. However, the deviations, by reducing the difference between two segments’ service valuations, allows the firm to extract more surpluses from those consumers with loose networks. The firm gains profit from the offering of such price menus because the increase of profit from the high-valuation segment dominates the loss of profit from the low-valuation segment. Proposition 1 is consistent with the examples presented in the Introduction section. The tie strength-based price discount can be implemented explicitly as in MCI and Sprint’s programs or through quantity discounts (Dolan, 1987). Bell Mobility Real Time Family Plan provided menus of pricing plans that charged lower variable fees for communications with strong ties, but relatively higher variable fees for communications with other cellular phones that were not included in the plans. We can illustrate proposition 1 with Figure 2. As a benchmark for discussion, suppose that all variable fees are equal to marginal cost. To ensure that both types of consumers subscribe to the service, the optimal fixed fee should be equal to service valuation of a dense network, Vd (c, c), which is represented by area (EGd I + EF d H ). A consumer with a loose network would receive surplus represented by area (EGl I +EFl H ) − area (EGd I + EFd H ) = area (HF d Fl − IGd Gl ). To increase its profit by extracting more surpluses of the high-valuation consumers, firm moves variable fees for the dense networks away from marginal cost. Consider a move of variable fee pdw up from marginal cost to point Y . Since a dense network has less weak ties, increasing variable fee for the weak ties to decrease communications reduces a dense network’s service valuation less than a loose network. Next consider a move of variable fee pds below marginal cost to R. Since a dense network has more strong ties, decreasing variable fee for strong ties to increase communications raises a dense network’s service valuations more than a loose network. Both the downward move of pds and upward move of pdw reduce the gap of service valuations between a dense network and a loose network, thus enabling the firm to increase its profit by charging a higher fixed fee to consumers with loose networks.4 2.4.2. Tie strength-based discount From equations (13) and (14), we can obtain the tie strength-based price discount for consumers with dense communication networks.

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∗

∗

pdw − pds =

α1w − nw ρd d w nw l −nd ρl

c +1

+

α1s − c nsd

nsd −nsl

ρd ρl

−1

.

(15)

From equation (15) we can identify a number of factors that affect the size of tie strengthbased price discount. First, the discount increases with (α1w − c) and (α1s − c). In Figure 2, the distance of HE is α1w − c, and the distance of IE is α1s − c. Given others the same, as H moves up, firm gains more profit from an increase of pdw above marginal cost, leading to a higher optimal pdw . Similarly, as I moves up, firm gains more profit from a decrease of pds below marginal cost, leading to a lower optimal pds . Both moves increase the tie w strength-based price discount. Second, the discount increases with nsd /nsl and nw l /nd . Both ratios measure the structural discrepancy between two types of consumer networks. An increase of the ratio implies a larger difference between the high-valuation and loww valuation segments. In Figure 2, moving Qlw towards right would increase nw l /nd , making w the increase of pd from E to Y more profitable. Similarly, moving Qds towards right would increase nsd /nsl , making a decrease of pds below marginal cost more profitable. Third, the discount increases with ρl and decreases with ρd because deviating the variable prices of dense networks from marginal cost reduces the firm’s profit from dense networks, but increases its profit from loose networks. We can also relate our results to bundling pricing theory. In our model, we have two segments of consumers with different personal communication networks: a dense network with more strong ties but fewer weak ties than a loose network. Compared with a consumer with a loose network, a consumer with a dense network has higher service valuations for communications associated with strong ties but lower service valuations for communications associated with weak ties. This feature is similar with negative correlation assumption given in the classical bundling theory (Adams and Yellen, 1976). Under negative correlation assumption, the provision of bundling offers can help a firm to successfully extract more consumer surplus from multiple products collectively. In our result, the firm can charge consumers with loose networks more than actually valued on their communications with strong ties. As shown in Figure 2, when the variable fee for communications with strong ties is at R, a consumer with a loose network values the communications with weak ties by area IKR. But the fixed fee charged by the firm includes area IJR, which is larger than area IKR. The deficit is compensated by the excessive surplus that this consumer enjoys from communications with weak ties. Interestingly, the tie strength-based discount improves the efficiency of bundling design by forcing consumers with loose networks to pay extra for communications with strong ties. In this respect, our result can also be seen as the optimal design of multiple bundles targeting different segments.5 In the Appendix we also analyse the amount of communications and fixed fees. Amount of communications in a dense network is affected by the tie strength-based discount. We find that in a dense network, not only the amount of communication between strong ties, but also a consumer’s total communications increase with the deviations of variable fees from marginal cost. Note that the amount of communications in a loose network is not affected because the variable fees are equal to marginal cost. We can then infer that the tie strength-based price discount given by equations (13)–(14) widens the gap in communica-


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tion usage between consumers with different types of communication networks. Specifically, the firm is expected to provide much more calling minutes to those consumers with more strong ties. The above results are consistent with casual observations from telecommunication industry where firms often provide consumers with dense networks with a large number of free minutes. 2.4.3. Scenarios A, B, and C In scenarios A, B, and C, consumers with dense networks are the high-valuation segment. In the equilibrium, firm should set the variable fees for dense networks at marginal cost, but the variable fees for loose networks different from marginal cost. The fixed fee for loose networks is such that no surplus is left to consumers in this segment. We provide analytical results as follows (detailed analysis is available upon request), ∗

∗

pds = pdw = c, (ns − nsl )ρd (α1s − c) ∗ pls = c + sd , nl ρl + (nsd − nsl )ρd s (nw − nw ∗ l )ρd (α1 − c) plw = c + wd . w nl ρl + (nd − nw l )ρd

(16) (17) (18)

Equation (16) indicates that variable fees for dense networks (high-valuation segment) are equal to marginal cost. Equations (17) and (18) show that variable fees for loose networks depend on strength of ties. Specifically, the variable fee for strong ties is above marginal cost, but below marginal cost for weak ties, implying an increasingly high unit price for larger volumes of consumption.6 We summarize the above discussions in proposition 2. Proposition 2. In scenarios A, B, C, the optimal variable fees are equal to marginal cost for dense networks. For loose networks, the variable fee for weak ties is below marginal cost but above marginal cost for strong ties. Proposition 2 differs from proposition 1 because the consumers with dense networks are the high-valuation segment in scenarios A, B, and C. In the equilibrium, variable fees for dense networks are at marginal cost. However, for consumers with loose networks, both variable fees for strong and weak ties are distorted in the directions that reduce the difference between two segments’ overall service valuations. Specifically, the variable price for strong ties is above marginal cost, while the variable price for weak ties is below marginal cost. In other words, the variable fee for loose networks is very low for an initial amount of communications, but becomes much higher after a specified quota.

3. Conclusions In this paper we have investigated a monopoly firm’s optimal social network-based discriminatory pricing policy. Our analysis shows that the optimal pricing policy depends

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on structure of social networks and tie strength-based service valuations. In a scenario where consumers highly value the initial amount of communications with their weak ties, consumers with fewer strong ties (loose networks) obtain higher service valuation. In this scenario, the optimal social network-based discriminatory pricing policy charges the consumers with fewer strong ties (loose networks) variable fees always at marginal cost and a large fixed fee. Meantime, the optimal pricing policy offers consumers with more strong ties (dense networks) tie strength-based discount, i.e., a lower variable fee for the communications between strong ties. The firm provides tie strength-based price discount to these consumers in order to extract more surpluses (profits) from the consumers with loose networks. These results offer an explanation for the offering of price discounts to communications with “friends and family members” as well as social network-based price menus observed in the market. Our results underscore the importance of developing market demand for communications from a social network perspective. Our paper has taken an initial step towards this direction. Future research may model network structures more realistically. For instance, our paper only considers two characteristics of social network structure, namely, tie strength and network density. In reality, a network can have more than two levels of tie strength. Moreover, relationship can be directional, and hence communication demand between two consumers can be asymmetric. In order for marketers to reach target consumers and implement network-based discriminatory pricing strategies, it is also useful to understand the composition variables that characterize consumers and their relationship. Future research is also needed to extend the paper to a competitive setting. A competing firm may use social network-based pricing strategy to attract and retain networks of consumers. For example, AOL provides free messaging service to their subscribers. Offering free messaging service is essentially same as charging a below-marginal cost variable fee. (Fixed fee is included in total access fee for the bundle that includes both Internet access and messaging service.). In this paper we study below-marginal cost variable fee as a result of a monopoly firm’s second-degree price discrimination. In a competitive setting, as users provide lists of people that are on their “buddy list”, AOL may use such a pricing strategy to acquire new customers through their current subscribers. Moreover, having a network of consumers within the same firm can help prevent customers from defecting to rival services (Angwin, 2001).

Acknowledgements Mengze Shi is an Assistant Professor of Marketing at the Rotman School of Management, University of Toronto. This paper has benefited substantially from the suggestions of the Editor, the anonymous reviewers, and comments from Robert Wyer.

Appendix. Analysis of Scenario D Firm’s pricing problem can be reformulated as follows with equilibrium conditions,

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Max

{Fd ,pds ,pdw ,Fl ,pls ,plw }

w w π = Nρd nsd qs pds pds − c + nw pd − c + Fd d qw pd w w + Nρl nsl qs pls pls − c + nw pl − c + Fl , l qw pl

where 2 2 nsd s nw α1 − pds + dw α1w − pdw , s 2α2 2α2 s w 2 2 2 nl s n ns α1 − pls + l w α1w − plw − Fl = l s α1s − pds s 2α2 2α2 2α2 2 nw + l w α1w − pdw − Fd . 2α2

Fd =

(A.1)

(A.2)

Constraint (A.1) ensures that consumers with dense networks receive zero surplus. Constraint (A.2) is the IC condition that consumers with loose networks are indifferent between plans (Fl , pls , plw ) and (Fd , pds , pdw ). Substitute the fixed fees solved by (A.1)–(A.2) into the firm’s objective function, we derive the first-order conditions with respect to the variable fees. From the first-order conditions, we can obtain following solution. ∗

∗

pls = plw = c, nρl (α1s − c) ∗ pds = c − s , nd ρd − nρl nρl (α1w − c) ∗ pdw = c + w . nd ρd + nρl Substituting (A.3)–(A.5) into equations (A.1)–(A.2), we obtain the fixed fees. 2 2 w 2 nsd ρd nw nsd (α1s − c)2 nw d (α1 − c) d ρd Fd = + , 2α2s nsd ρd − nρl 2α2w nw d ρd + nρl 2 nsd ρd (α1s − c)2 s nl + n s Fl = 2α2s nd ρd − nρl 2 nw ρd (α w − c)2 w nl − n w d . + 1 w 2α2 nd ρd + nρl

(A.3) (A.4) (A.5)

(A.6)

(A.7)

Increased Fixed Fee for Loose Networks In the benchmark case, fixed fee F0 is equal to Vd (c, c) given by equation (7). Comparing equation (A.7) with Vd (c, c), we have the following difference. 2 (α s − c)2 nsd ρd − 1 Fl − Vd (c, c) = 1 s n 2α2 nsd ρd − nρl 2 (α w − c)2 nw d ρd + 1 w n 1 − . (A.8) 2α2 nw d ρd + nρl

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Since both components of equation (A.8) are positive, we have Fl − Vd (c, c) > 0. Equation (A.8) shows that the increase of fixed fee is positively related to value of communications between strong ties (vs ) and weak ties (vw ), the structural discrepancy between two types of personal communication networks (nsd /nsl if measured in number of w strong ties and nw l /nd if measured in number of weak ties), and number of loose networks (ρl ).

Usage In the benchmark case, total usage of communication service by a consumer with network type T is as follows: w DT (c, c) = nsT qs pTs + nw T qw pT nw ns = Ts α1s − c + Tw α1w − c (T = d, l). (A.9) α2 α2 From (A.3), we infer that the variable fees and hence the total usage of communication service remains the same for a consumer with a loose network. From (A.4)–(A.5), we compute the total demand for communication service by consumers with dense networks as: nsd nρl (α1s − c) nρl (α1w − c) nw d ∗ s w Dd = s α1 − c + s + w α1 − c − w α2 nd ρd − nρl α2 nd ρd + nρl s α −c 1 = Dd (c, c) + nρl 1 s α2 ρd − ρl n/nsd αw − c 1 . (A.10) − 1 w α2 ρd + ρl n/nw d From (4), qs (c) =

1 s 1 α1 − c > w α1w − c = qw (c). s α2 α2

Since 1 1 1 > > , ρd − ρl n/nsd ρd ρd + ρl n/nw d we have Dd∗ > Dd (c, c) > Dl (c, c). Therefore the difference in total usage between a dense network and a loose network becomes wider.

Notes 1. Analysis of the case where some personal communication networks are larger than the others is available from the authors. We find that the outcome follows the same intuitions that are presented in this paper.


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2. A consumer’s service usage increases with the frequency and length of communications. We do not explicitly model the frequency or length because the firm’s profit is determined by the total amount of communications only. 3. Research on the structure of social networks has found heterogeneous communication patterns both within personal networks (tie strength) and between personal networks (network density). For example, in an experimental study on electronic communications among 50 social network researchers (Freeman, 1986), individuals sent a larger number of messages to receivers with whom they had stronger acquaintanceship. Moreover, total number of messages that each individual sent during the period of experiment varied significantly. Dimmick et al. (1996) found that frequency of phone calls between randomly selected pairs of subjects increased with strength of their ties, which were described by proximate and affective relations. 4. Our logic behind charging a below-marginal-cost variable fee for strong ties is similar with Oi (1971). However, we extend the result to a model where a consumer’s service valuation varies by the strength of social tie with the other caller. Skander et al. (2002) showed that a below-marginal-cost price could also be optimal when competing firms face capacity constraints. 5. While each consumer communicates with (n − 1) people, the firm offers service plans that bundle all (n − 1) lines of communications and charge one fixed fee. Theoretically speaking, the firm could un-bundle these (n − 1) lines of communications and provide a contract separately for each line of communications. However, in practice the additional transaction costs involved in tracking, contracting, and billing would be prohibitively high to the firm. 6. These results depend on the assumed constant size of networks. If a high-valuation consumer has more strong ties and more weak ties, both variable fees for the low-valuation consumer will be above marginal cost.

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