Sale or Lease? Durable-Goods Monopoly with Network Effects Cyrus C.Y. Chu∗
Hung-Ken Chien
November 1, 2007
∗ Chien: Postdoctoral Fellow, Department of Economics, University of Bonn, Lenn´estrasse 37, 53113 Bonn, Germany. E-mail:
[email protected]. TEL: +49-228-73-5045. Chu: Distinguished Research Fellow, Institute of Economics, Academia Sinica. E-mail:
[email protected]. We are indebted to Professors Jeremy Bulow, Hal Varian, three anonymous referees, and an Area Editor for their useful comments and suggestions on earlier drafts.
i
Sale or Lease? Durable-Goods Monopoly with Network Effects
Abstract This paper studies the pricing problem of a durable-goods monopolist. It finds that contrary to the existing literature, profits from selling durable goods may be higher than from leasing when the products exhibit network effects. Under the influence of network effects, there exist multiple self-fulfilling equilibria that would sustain different network sizes at the same price. By employing the assumption that consumers are cautious about network growth, we find that consumption externalities among heterogeneous groups of consumers generate a discontinuous demand function, which requires a lessor to offer a low price if she wants to reach the mass market. In contrast, a seller enjoys a relative advantage in that she can build a customer base by setting a lower initial price and raise the price later in the mass market. Our finding that selling can be more profitable than leasing holds when consumers are more cautious about the prospect of the product’s success, which may be the case if, for example, the technology or manufacturer is relatively unknown.
Keywords: selling and leasing, penetration pricing, network externality, introductory pricing, product life cycles.
ii
Sale or Lease? Durable-Goods Monopoly with Network Effects
1
Introduction
When the market for personal computers started to flourish in the early eighties, Microsoft aggressively promoted its operating system by reducing the license fees of MS-DOS by half for a brief period of time.1 Yet as soon as Microsoft prevailed in both operating system and application software for personal computers, it adopted a new pricing plan in which business users paid annual subscription fees.2 The path of Microsoft’s pricing strategies defied the conventional wisdom that dictates the relative advantage of leasing over selling for durable goods. This paper proposes that Microsoft’s pricing strategies may be driven by the network effects of its product. That is, in order to take advantage of network effects the firm wants to establish an installed user base in the early stage of a product’s life cycle and selling proves to be a more effective strategy than leasing; the situation shifts when the market gets mature and the concern for building the network starts to fade. Therefore at a later stage the business strategy would resemble that of a conventional firm, which opts for leasing rather than selling a durable product. The idea of network effects employed here draws upon the work of Rohlfs (1974), which holds that the more total buyers there are in the market, the higher each individual’s willingness to pay. This attribute accurately describes a distinctive market characteristic for many products in the digital economy. As will be shown in the paper, it predicts the use of preemptive strategies such as the deep discount in the aforementioned example of MS-DOS. The goal of this paper is to provide a model that describes the two-staged pricing path during which the firm would switch from selling to leasing when it has sufficiently penetrated the market of a network product. The impetus of our analysis comes from the observation that adopting an emerging technology takes time. Normally the better-informed professional group adopts it first; and then the informa1
See Manes and Andrews (1993). Cabral et al. (1999) discuss other examples in computer software. Evidently, this new pricing strategy was so in favor of Microsoft that its deployment provoked strong opposition from the customers. Microsoft eventually suspended the Office XP subscription plan for the US market due to the pressure. 2
1
tion spreads out gradually, leading to the adoption by less-informed, less professional groups, etc.3 Rohlfs (1974) and Katz and Shapiro (1986) have pointed out that market expansion coupled with network effects may result in multiple market equilibria. In other words, the potential buyers’ expectations could sustain different network sizes for the same price. Therefore, equilibrium selection plays a crucial role when determining the firm’s pricing strategy. By assuming that consumers are able to coordinate their actions at the most efficient outcome, the conventional analysis of network effects is to select the equilibrium that maximizes consumers’ total surplus.4 In this paper, we envision a situation where consumers are more conservative about their expectations of network growth. Accordingly, we select the stable equilibrium with a different network size. Our selection criterion results in a possibly discontinuous demand function— a small decrease in price at the critical point generates a discrete jump in sales. Thus, when the market expands, the monopolist should exercise penetration pricing by “offering a low price to invade another market” (Shapiro and Varian (1999), p. 288).5 In effect, the demand discontinuity places a restriction on the monopolist’s choice of quantities, a unique phenomenon that we call penetration-pricing constraint. Under this constraint, the firm can either access the niche market (small quantity) with a high price, or reach the mass market (large quantity) with a low price. The effects will depend on whether the monopolist sells or leases the product. In the first possible scenario, the seller has the option to establish a larger customer base in the beginning, which changes the residual demand and enables the seller to raise the price later and take full advantage of the network growth.6 Such an option is not feasible to the lessor, as she cannot demand renters committing to continuous renting.7 Essentially, the seller’s optimal strategy is to sacrifice her profits 3
Although Waldman (1993) considers a similar setup with different groups of consumers joining the market sequentially, consumers’ willingness to pay in his model is assumed to be identical within each group. Therefore, pricing strategy is irrelevant in the monopolist’s decision. 4 See, for example, Economides and Himmelberg (1995). Note, however, that some authors have analyzed various processes of coordination by consumers (e.g. Farrell and Katz (2005)). 5 Lee and O’Connor (2003) also assert that a penetration-pricing strategy during product launch performs better than a skimming pricing strategy for network effects products. 6 A seller cannot always commit customers to continue using her product. For instance, an owner of e-book reader devices can abandon his current reader at any time and adopt a new device that supports different formats of e-books. Such switching effectively curtails network externalities for a particular format, and thereby limits the seller’s benefits from her commitment power. We thank a referee for pointing out the caveat and suggesting this example. 7 When there are switching costs, the monopolist could lock in some early renters and establish a customer base in rental market. In effect, the presence of switching costs changes the demand of rental units in period 2 by shifting inverse demand upward for all x < x1` , where x1` is the period-1 rental quantity. Thus, the producer still encounters
2
in the early stage in order to capitalize the gains later. In contrast, the lessor extracts maximal profits early on, but her pricing choice later is limited by the penetration-pricing constraint. We further show that a selling scenario may generate higher profits than leasing, depending on the market conditions. When the market is burgeoning, the monopolist is likely to prefer selling the product. However, when the market gets mature, resorting to a rental-based pricing model may be optimal. Thus, our model fully describes the life cycle of a network product, and it fits well with the evolution of pricing strategies illustrated by the example of Microsoft. In sum, the central conclusion of this paper establishes that the existence of network effects will lead a monopolist to favor selling her durable products over leasing under certain market conditions. This finding is in stark contrast to the conventional wisdom in the literature of durable-goods monopoly as well as that of network effects, many of which focus on the selling scenario while assuming that leasing is not feasible. In comparison, the decision of selling or leasing in our model is endogenous, and therefore the results are more robust. Our work draws upon two streams of research. The first studies the implications of network effects, while the second examines a durable-goods monopolist’s choice between selling and leasing. Economics literature has analyzed various aspects of network effects, such as the issues of standardization and compatibility and their welfare implications (Katz and Shapiro (1985, 1986); Farrell and Saloner (1985)). Meanwhile, marketing research on network effects has mainly focused on the strategic implications for individual firms, such as pricing, choices of product features, product upgrade, etc. For instance, Dhebar and Oren (1985) demonstrate that introductory pricing is optimal for a monopolist to market a product that exhibits network effects. Xie and Sirbu (1995) extend this finding to the scenario with a duopoly. Sun et al. (2004) explore the market conditions under which different product strategies such as technology licensing and product-line extension can be more profitable. Chen and Xie (2007) demonstrate that asymmetry in customer loyalty can turn a first-mover advantage into a disadvantage in the presence of cross-market network effects. Hauser et al. (2006) provide an excellent survey and identify challenges for research on network effects. The leasing-selling decision has been analyzed in both economics and marketing literature. a discontinuous demand; the disadvantage of leasing due to the penetration-pricing constraint does not vanish.
3
Bulow (1982) and Stokey (1981) demonstrate that leasing avoids the problem of time inconsistency and hence is more profitable than selling. Other authors have proposed various circumstances under which the assertion no longer holds. These include entry of new customers that leads to price cycles (Conlisk et al. (1984), Sobel (1991)), entry deterrence of new competitors (Bulow (1986), Bucovetsky and Chilton (1986)), increasing marginal costs (Kahn (1986)), depreciation of goods (Bond and Samuelson (1987), Desai and Purohit (1998)), discrete demand (Bagnoli et al. (1989)), competitiveness of the market and reliability of the product (Desai and Purohit (1999)). Our analysis adds to the literature by pointing out that network externalities have similar effects that induce a durable-goods monopolist to favor selling over leasing. The rest of this paper will be arranged as follows. In Section 2, we present the basic model that characterizes network effects. By employing this model, we provide a theoretical foundation for inverted-U shaped demands that are commonly seen in the network-effect literature. In Section 3, we provide a numerical example of dynamic pricing to illustrate our point. We then formulate and solve the general dynamic-pricing problems faced by a monopolist in Section 4. The final section discusses the results and concludes.
2 2.1
Network Effect: A Characterization Consumer preferences We consider a durable good that exhibits network externalities. The good is sold by a monopoly
and lasts two periods. For simplicity, we assume that the monopolist’s discount factor is zero. The monopolist’s goal is to maximize the aggregate revenues. There are two heterogeneous groups of consumers in the economy. The first group consists of early adopters. These consumers arrive at the market in period 1, and hence have the opportunity to enjoy the product early on. The second group of consumers are the majority, who join the market only in period 2. Within group g (g = 1, 2), there is a continuum of consumers; each of them is indexed by a parameter xg , which is uniformly distributed on [0, mg ]. Following Rohlfs (2001, p. 209), we assume that an individual’s value for the good in question is composed of two parts: one is the
4
generic value and the other is the magnification of the network size. To facilitate comparison with the literature, we shall consider the linear form of generic valuation. To simplify the algebra, we further assume that the network effects for both groups are represented by the expected network size. These assumptions yield a specific demand structure as follows. For an early adopter x1 , her consumption value in period 1 is (A1 − a1 x1 )n1,1 .8 The term in parenthesis represents the generic part of valuation, while n1,1 is the expected network size in period 1.9 For the same consumer, her consumption value in period 2 will be (A1 − a1 x1 )(n1,2 + n2 ). Likewise, a group 2 consumer x2 has a consumption value of (A2 − a2 x2 )(n1,2 + n2 ) in period 2. Following Bulow (1982), we assume that a perfect secondhand market exists.10 This eliminates the possibility of price discrimination. Assuming there is no income effect for any individual, we shall focus on the equilibrium analysis of the market in the presence of network effects. In Online Appendix, we discuss extension of our linear model to a more general demand specification.
2.2
Inverted U-shaped Demands and the Critical Mass In the conventional analysis, a demand function associates any given price with a desired quan-
tity of the commodity. Nonetheless, with network effects, a price might correspond to multiple quantities. To derive the demand correspondences in our model, note that if a consumer indexed xg is willing to buy the object, anyone in the same group with a lower index x < xg will also demand it. Consequently, the total demand in the economy is determined by the marginal consumers. In period 1 only early adopters appear in the market, and the inverse demand is simply
p1 (x) = (A1 − a1 x)x.
(1)
The diagram in Figure 1(i) illustrates the demand correspondence. For any p < A21 /4a1 , p associates 8
By construction, individuals in each group are ordered decreasingly in terms of their generic willingness to pay, represented by a decreasing function qg (xg ) = Ag − ag xg . This particular ordering enables us to interpret qg (xg ) as an inverse generic demand, and thus facilitates our analysis. 9 Specifically, n1,i = |X1,i | for i = 1, 2, where X1,i ⊂ [0, m1 ] is a subset of group 1 and consists of those who are expected to consume the product in period i. In addition, we implicitly assume that an individual with the highest index mg has zero value for consumption regardless of network sizes, which implies mg = Ag /ag . 10 For instance, internet marketplaces such as Half.com or Amazon.com allow people to trade used computer software.
5
6 J J J q1 (x) J J J J s s p J s J p1 (x) J J J 0
xmin
6 J J J J J J s s J s s s J pˆ J XXX q˜(x) XXXJ q1 (x) q2 (x) XJXX J XXX
-
x
xmax
x ˆ1
(i)
x ˆ2
p p˜(x)
-
x
(ii)
Figure 1: Single- and Double-hump Demand Curves in the First and Second Periods with three possible equilibria: x = 0 or the roots of the quadratic function (A1 − a1 x)x = p.11 As is well known, the larger root xmax in the downward-sloping part of the parabola is a stable equilibrium (see the Appendix of Rohlfs (2001)), and so is x = 0. Meanwhile, the smaller root xmin on the upward slope is an unstable equilibrium. xmin usually represents a critical mass, which the seller has to overcome in order to reach the stable and more profitable equilibrium. In period 2, the product’s appeal extends to the second group. For any price p, let x1 and x2 be the marginal consumers for groups 1 and 2, respectively. x1 and x2 must satisfy the demand correspondences: p = (A1 − a1 x1 )(x1 + x2 ) and p = (A2 − a2 x2 )(x1 + x2 ). Alternatively, one can first derive the aggregate inverse generic demand as q˜(x) by summing q1 (x) = A1 − a1 x and q2 (x) = A2 − a2 x horizontally. The aggregate inverse demand is then given by p˜(x) = q˜(x)x. In Figure 1(ii), q˜(x) is the dashed line, while p˜(x) is the double-hump shaped curve. For x ≤ x ˆ1 , p˜(x) is equal to A1 x − a1 x2 ; otherwise a1 a2 2 a2 A1 + a1 A2 ˜ −a x− x , p˜(x) = Ax ˜x2 = a1 + a2 a1 + a2 where the critical value x ˆ1 ≡
A1 −A2 a1
(2)
is determined by p1 (ˆ x1 ) = p˜(ˆ x1 ). From Figure 1(ii), we observe
that p˜(x) has two humps if and only if
A1 2a1
˜ 12 A 2˜ a.
The double-hump shape of demand is
11
Since the network effect is multiplicative, the willingness to pay by any consumer would be zero if he expects no one else to buy the product. Therefore, x = 0 is an equilibrium. 12
To be precise, a hump is a local maximum of the p˜(·) curve. In Figure 1(ii), p˜(x) reaches local maxima
6
A2 1 4a1
and
similar to the one drawn in Rohlfs (2001, p. 219), except that we provide a theoretical justification for it here.13 According to Rohlfs (2001) and Shapiro and Varian (1999), the monopolist can penetrate the section of p˜(x) beyond x ˆ1 in the second period only if the price is set lower than or equal to pˆ. This is the penetration-pricing constraint referred to in the literature. Due to the double-hump shape of demand, for any price p between pˆ ≡ p˜(ˆ x1 ) and
˜2 A 4˜ a,
p associates
with five equilibria (see the bullets in Figure 1(ii)), two (the third and the fifth from the left) of which are stable with positive quantities. In the presence of multiple stable equilibria, the criterion for equilibrium selection proves to be critical in our analysis of a monopolist’s market strategy.
2.3
The Penetration-Pricing Constraint and Equilibrium Selection It has long been recognized that a market price can associate with multiple network sizes in
equilibrium (Oren and Smith (1981), Oren et al. (1982), Katz and Shapiro (1986)). Following Katz and Shapiro (1986), the conventional approach is to select the equilibrium that is Pareto-preferred by consumers.14 As seen from Figure 1(ii), this selection criterion implies a different demand ˜
˜
A A curve faced by the monopolist lessor so that she can choose to lease any quantity from [ 2˜ a, a ˜ ] in
period 2. In this paper, however, we assume that consumers are more conservative in terms of their expectations of network growth. Consequently, we focus on the stable equilibrium with the smallest network size for any given price. The resulting demand corresponding with the price range ˜
˜2
0 ˆ ] instead of [ A , x 0 [ˆ p, A 1 4˜ a ] is [x1 , x 2˜ a ˆ2 ], where x1 is the larger root of p1 (x) =
˜2 A 4˜ a
and x ˆ2 ≡
A2 a ˜
satisfies
p˜(ˆ x2 ) = p˜(ˆ x1 ). One way to justify our criterion for equilibrium selection is that it validates the dichotomy of pricing strategies when marketing a new product, as proposed by Dean (1976). The first segment A1 of demand for x ∈ [ 2a ,x ˆ1 ] represents the strategy of skimming pricing with which the monopolist 1 ˜2 A 4˜ a
at x =
A1 2a1
˜ A1 A , respectively. However, if x ˆ1 ≤ 2a , p˜(x) is increasing 2˜ a 1 ˜ ˜ A A . The argument for the case with x ˆ1 ≥ 2˜ is analogous. 2˜ a a β
and
∀x ∈ [0, x ˆ1 ], and thus has only one local ˜
A1 A1 A2 A maximum at x = Note that 2a < 2˜ = 2a + 2a . a 1 1 2 13 Rohlfs (2001, p. 215) assumed q(x) = cx and f (n) = kn. However, this will not generate the inverted-U figure he suggested. For under his specification, p(x) = q(x)f (x) = ckxβ+1 , which (depending on the size of β) is either increasing or decreasing in x. Thus, the willingness to pay by the marginal consumer would be increasing or decreasing in the whole range of x. 14 Other authors such as Oren et al. (1982) simply assumed a single-peaked willingness-to-pay function, which eliminates the possibility of multiple equilibria.
7
˜
targets the elite group of consumers; while the second segment of demand for x ∈ (ˆ x2 , Aa˜ ] corresponds to the strategy of penetration pricing that directs at the general public. In this paper, we focus on the situation under which penetration pricing dominates skimming pricing.
3
Dynamic Pricing: A Numerical Example
In this section, we consider a specific set of parameters and demonstrate that selling is more profitable than leasing under this particular demand specification. Let xi` be the quantity produced by the monopolist lessor up to the ith period, and xiS the quantity produced by the monopolist seller up to the ith period.15 Consider the following example. Example: Suppose A1 = 4, a1 = 2, A2 = 1, and a2 = 41 . One obtains A˜ = zero production costs, the optimal leasing quantities are given by x1` = optimal sales are x1S =
3 2
4 3
4 3
and a ˜ = 29 . Assuming
and x2` = 29 , while the
and x2S − x1S ≈ 2.8. The pricing strategies yield a profit below 9.12 for
leasing the product, which is lower than 9.23 by selling it. We shall explain why this is so below. If the monopolist wants to lease the product, she maximizes π` = p1 (x1` ) · x1` + p˜(x2` ) · x2` by choosing x1` and x2` . It is readily verified that the first-best prices are p1 = and p2 =
16 9
(x1` =
4 3)
16 9
(x2` = 4). Nonetheless, as we have discussed in Section 2.3, the penetration-pricing constraint limits the range of feasible x2` ’s. In particular, (x2` , p2 ) = 4, 16 is not an achievable 9 strategy, because the second-group demand cannot be “penetrated” by the price p2 =
16 9 .
In the presence of the penetration-pricing constraint, the lessor can only select x2` from [1, 32 ] and ( 92 , 6]. With this discontinuous demand curve, the optimal strategy under leasing is to set p1 =
16 9
and p2 slightly below
3 2
so that x1` =
4 3
and x2` slightly above
9 2.
achievable two-period profits under leasing are bounded from above by π` =
As such, the highest 16 9
·
4 3
+
3 2
·
9 2
≈ 9.12.
Now consider the selling strategy. The monopolist’s second-period strategy is still restricted by the need to penetrate the market. However, being able to build a customer base in the first period alleviates the constraint for the seller. In our example, if x1S < 32 , the corresponding range 15
We adopt the notations so that x2` or x2S represents the total number of units on the period-2 market. Period-2 production is therefore x2j − x1j for j = ` or S.
8
for feasible x2S ’s is (x1S , 23 ] ∪ ( 29 , 6], and the penetration-pricing constraint is similar to that in the leasing scenario. On the contrary, if x1S ≥ 32 , the seller can select any x2S from [3, 6]. The total profits under the selling regime are πS = (p1 (x1S ) + p˜(x2S )) · x1S + p˜(x2S )(x2S − x1S ). Let xp2 (x1S ) denote the solution to the last optimization problem subject to the penetration-pricing constraint. It is straightforward to verify that the constraint is binding when x1S < 32 , and nonbinding otherwise. Therefore, the seller’s problem in the example is given by max (p1 (x1S ) + p˜(xp2 (x1S ))) · x1S + p˜(x2S )(x2S − x1S ),
x1S ,x2S
s.t.
9 3 3 x2S ∈ (x1S , ] ∪ ( , 6] for x1S < , 2 2 2
The optimal sales that solve the above problem are x1S = highest profits available to the seller are πS ≈
3 2
·
3 2
3 x2S ∈ [3, 6] for x1S ≥ . 2
or
3 2
and x2S ≈ 4.3. It follows that the
+ 1.62 · 4.3 ≈ 9.23, which are greater than the
leasing profits.
4
The General Analysis
The previous section illustrates the possibility that leasing may be less profitable than selling. In this section, we shall identify the general conditions under which the comparison holds true. To highlight the implications from imposing the penetration-pricing constraint, we start with the benchmark case where the monopolist is not bounded by the constraint.
4.1
The Case without Penetration-Pricing Constraints An important characteristic that sets apart our model is the assumption of a growing market.
In this subsection we show that this property alone does not lead to our findings. Without the penetration-pricing constraint, the analysis and conclusion are similar to those in Bulow (1982), even though the latter assumes an identical market in both periods. The monopolist lessor’s aggregate profits are π` = (p1 (x1` ) − c) · x1` + p˜(x2` ) · x2` − c · (x2` − x1` ), where c is the marginal cost of production.16 Let M R1 (x1 ) ≡ p1 (x1 ) + p01 (x1 ) · x1 and M R2 (x2 ) ≡ 16
It is possible that x2` < x1` , in which case the last term shall be zero. Nonetheless, one can show that x2` > x1`
9
p˜(x2 ) + p˜0 (x2 ) · x2 be the marginal revenues for period-1 and aggregate production, respectively. The first order conditions can then be written as M R1 (x1` ) = 0 and M R2 (x2` ) = c. Solving these equations yields the interior solution as follows.
x∗1`
2 A1 = , 3 a1
x∗2`
A˜ 1 = + 3˜ a 3
s A˜ 2 a ˜
−
3c . a ˜
(3)
For (3) to solve the monopolist lessor’s problem, we need the following regularity assumptions. Assumption 1. c <
˜2 A . 4˜ a
Assumption 2. π` (x∗1` , x∗2` ) > maxx (2p1 (x) − c) · x. Assumption 1 is equivalent to requiring either x∗2` >
˜ A 2˜ a
or p˜(x∗2` ) > c. Therefore, the assumption
guarantees that x∗2` lies on the downward sloped part of p˜(x) and yields positive profits. Throughout the paper, we assume that Assumption 1 always holds. Note that the monopolist has a simple strategy where she ignores group-2 consumers in period 2, and focuses on leasing to group 1 in both periods. Assumption 2 ensures that this alternative strategy is less profitable than (x∗1` , x∗2` ).17 Proposition 1. Suppose the growth of market is significant for the lessor in the sense of Assumption 2. Then, (x∗1` , x∗2` ) in (3) maximizes the monopoly lessor’s profits in the absence of penetration-pricing constraints. We now consider the seller’s problem. Let x∗2 (x1S ) ≡ arg maxx2S (˜ p(x2S ) − c)(x2S − x1S ) be the solution to the second-stage optimization problem. Realizing her own choice in period 1 restricts her action in period 2, the seller’s problem is to maximize πS = (p1 (x1S ) + p˜(x∗2 (x1S )) − c)x1S + (˜ p(x2S ) − c)(x2S − x1S ). Let (x∗1S , x∗2S ) denote the solution. The maximization program is different from that in the leasing scenario in the term p˜(x∗2 (x1S )), which represents the consumers’ rational in equilibrium, and thus we adopt the current formulation for simplicity. 17 In the absence of a growing market, the right hand side represents the lessor’s profits attained by leasing the same amount of product in both periods. Intuitively, Assumption 2 holds when the addition of group-2 consumers contributes significantly to the monopolist’s profits. Roughly speaking, it is true when either their willingness to pay 2 is high). We discuss such specific conditions in an is high enough (A2 is high), or the population is large enough ( A a2 online appendix.
10
expectation. The first order conditions are
M R1 (x1S ) = −
∂ p˜(x∗2 (x1S )) x1S , ∂x1S
M R2 (x2S ) = c + p˜0 (x2S )x1S .
(4)
The next assumption parallels Assumption 2. It ensures that the monopolist finds it more profitable selling to group-2 consumers, which implies that the solution to (4) attains the optimum. Assumption 3. The profits attained by the solution to (4) are higher than those attained by maximizing πS while excluding group-2 consumers. The mathematical expression for Assumption 3 is not provided here. Essentially, it requires either a high A2 (high willingness to pay) or a low a2 (a large population). Proposition 2. Suppose the growth of market is significant for the seller in the sense of Assumption 3. The solution to (4) maximizes the monopolist seller’s profits in the absence of penetrationpricing constraints. It is straightforward to compare the equilibrium profits: the strategy (x∗1S , x∗2S ) is accessible to the lessor, and thus her maximal profits must be weakly higher. The details for comparing the equilibrium quantities can be found in Online Appendix. The next proposition summarizes our findings. It is important to emphasize that these findings are consistent with Bulow’s (1982). Proposition 3. Suppose the growth of market is significant so that Assumptions 2 and 3 hold. In the absence of penetration-pricing constraints, the seller produces less than the lessor does in period 1, but the seller’s total quantity in period 2 is higher; that is, x∗1S < x∗1` and x∗2S > x∗2` . Moreover, leasing is more profitable than selling.
4.2
The Case with a Penetration-Pricing Constraint The monopolist’s pricing strategy will be much different when she takes into account the
penetration-pricing constraint. We start by reiterating some notations illustrated in Figure 1(ii): x ˆ1 ≡
A1 −A2 a1
is the critical value at which p1 (ˆ x1 ) = p˜(ˆ x1 ); while x ˆ2 ≡
A2 a ˜
leads to the same critical ˜
A1 price with p˜(ˆ x2 ) = p˜(ˆ x1 ). For the monopolist lessor, the range for feasible x02` s is [ 2a ,x ˆ1 ] ∪ (ˆ x2 , Aa˜ ]. 1
11
In other words, the lessor cannot penetrate the period-2 market unless the price is lower than pˆ ≡ p1 (ˆ x1 ).18 Therefore, the lessor’s problem is to maximize π` = (p1 (x1` ) + p˜(x2` ) − c) · x1` + ˜
A1 (˜ p(x2` ) − c)(x2` − x1` ), subject to x2` ∈ [ 2a ,x ˆ1 ] ∪ (ˆ x2 , Aa˜ ]. Let (xp1` , xp2` ) be the solution to this 1
problem. We are more interested in the case where the penetration-pricing constraint is binding, which requires the unconstrained solution x∗2` be lower than x ˆ2 : Assumption 4. x∗2` < x ˆ2 . As in Section 4.1, we further assume that the market enjoys significant growth so that the monopolist finds it more profitable leasing to some of the group-2 members. ˆ2 ) > maxx (2p1 (x) − c) · x. Assumption 5. π` (x∗1` , x ˜
Since Assumption 4 implies that π` is decreasing in x2` for x2` ∈ (ˆ x2 , Aa˜ ], we know that π` is ˆ2 . maximized at xp2` slightly above x Proposition 4. Suppose the penetration-pricing constraint is binding for the lessor (Assumptions 4) and that the market is growing in the sense of Assumption 5. The monopolist’s optimal ˆ2 . leasing strategy is xp1` = x∗1` and xp2` ≈ x For the monopolist seller, the restrictions imposed by the penetration-pricing constraint depend on how much she sells in period 1. If x1S is lower than the critical amount x ˆ1 , the seller faces a ˜
constraint similar to the lessor’s, so that the feasible x02S s are limited to (x1S , x ˆ1 ] ∪ (ˆ x2 , Aa˜ ]. In other words, when x1S < x ˆ1 , the seller has two options in the next period: she can either charge a high price (p2 ≥ pˆ) that only appeals to group-1 consumers, or a low price (p2 < pˆ) that reaches group-2 ˜
consumers. These options correspond to the intervals (x1S , x ˆ1 ] and (ˆ x2 , Aa˜ ], respectively. On the contrary, if x1S is greater than x ˆ1 , the seller has built a customer base large enough to penetrate the ˜
˜
A A market in period 2, so that she can select any x2S from [ 2˜ a, a ˜ ]. The penetration-pricing constraint
has no effect in this case. For the constraint to be binding in equilibrium, we assume Assumption 6. x∗1` < x ˆ1 ; 18
x∗2 (ˆ x1 ) < x ˆ2 .
For the parameters assumed in the example, one can verify that
12
A1 2a1
= 1, x ˆ1 = 32 , x ˆ2 = 92 , and
˜ A a ˜
= 6.
Given the time-consistency constraint, for any x1S , the seller maximizes (˜ p(x2S ) − c)(x2S − x1S ) subject to the penetration-pricing constraint corresponding to x1S . Let xp2 (x1S ) be the solution to the seller’s constrained problem in period 2. Along with Proposition 3 (x∗1S < x∗1` ), the first part of Assumption 6 implies that x∗1S < x ˆ1 . As we know that x∗2 (·) is an increasing function, the second part implies x∗2 (x1S ) < x ˆ2 ∀x1S < x ˆ1 . As such, one concludes that the unconstrained solution x∗2S (= x∗2 (x∗1S )) in Section 4.1 is lower than x ˆ2 . In short, Assumptions 6 implies a binding penetration-pricing constraint so that (x∗1S , x∗2S ) is not achievable. xp2 (x1S ) can now be derived as follows. For x1S < x ˆ1 , the interior solution x∗2 (x1S ) is not achievable, and thus xp2 (x1S ) = x02 (x1S ) or ≈ x ˆ2 .19 For x1S ≥ x ˆ1 , the seller is able to penetrate the market in period 2, and xp2 (x1S ) = x∗2 (x1S ). Given xp2 (x1S ), the seller’s optimization problem is p(x2S ) − c)(x2S − x1S ), max πSp = (p1 (x1S ) + p˜(xp2 (x1S )) − c)x1S + (˜
x1S ,x2S
s.t.
x2S ∈ (x1S , x ˆ1 ] ∪ (ˆ x2 ,
A˜ ] for x1S < x ˆ1 , a ˜
or
x2S ∈ [
A˜ A˜ , ] for x1S ≥ x ˆ1 . 2˜ a a ˜
(5)
ˆ1 and x1S ≥ x ˆ1 . In the former case, To solve (5), we maximize πSp separately over x1S < x ˆ2 . Provided Assumption 5, x ˆ2 yields a higher profit than x02 (x1S ). The xp2 (x1S ) = x02 (x1S ) or ≈ x ˆ2 ). In the latter case, xp2 (x1S ) = x∗2 (x1S ). Given the local maximizer in this scenario is thus (x∗1` , x ˆ1 and that ∂ 2 πS /∂x21S < 0, the objective function πSp is decreasing in x1S assumption that x∗1S < x for the relevant range and hence is maximized at x1S = x ˆ1 . Combining these scenarios, we need to compare πSp (x∗1` , x ˆ2 ) against πSp (ˆ x1 , x∗2 (ˆ x1 )) to determine the solution to (5). Proposition 5. Suppose the market is growing and the penetration-pricing constraint is binding for the seller (Assumptions 5 and 6). The monopolist’s optimal selling strategy, (xp1S , xp2S ), is either (x∗1` , x ˆ2 ) or (ˆ x1 , x∗2 (ˆ x1 )). When comparing (x∗1` , x ˆ2 ) and (ˆ x1 , x∗2 (ˆ x1 )), the seller trades off her profits in two periods. In the former strategy, the seller is bounded in period 2 by the penetration-pricing constraint, but she is able to maximize her period-1 profits. In the latter strategy, the seller sacrifices her period-1 19 0 x2 (x1S )
≡ arg maxx2S (p1 (x2S ) − c)(x2S − x1S ) maximizes the seller’s period-2 profits when she excludes group-2 consumers; see (A10) in Online Appendix.
13
profits in order to penetrate the market and capture bigger gains in period 2. If the period-2 market is lucrative enough, as we assume in this paper, the latter strategy will be optimal. From Proposition 5, we observe that the leasing solution (x∗1` , x ˆ2 ) is feasible in the selling regime. It follows immediately that leasing is no longer more profitable than selling. The leasing profits are reduced because the lessor has to lower the period-2 price in order to reach group-2 consumers. In contrast, the seller’s loss occurs mostly in period 1. Thus, the constraint affects the lessor’s and the seller’s profits differently. Corollary 6. Following Proposition 5, one obtains π`p ≤ πSp in equilibrium. Therefore, selling is more profitable than leasing under the penetration-pricing constraint. The inequality in the corollary holds strictly when A2 is large enough or a2 is small enough. Recall that A2 /2 is the average willingness to pay by group-2 consumers, and A2 /a2 characterizes the size of population. Therefore, Corollary 6 implies that when the monopolist expects a significant growth of the market, she will prefer selling rather than leasing the product. Our analysis above employed a particular form of demand specification. In addition, we also made certain regularity assumptions to ensure that penetration-pricing constraints are binding. It can be shown that these presumptions do not place severe restrictions on our findings. A detailed mathematical analysis of these assumptions can be found in Online Appendix.
5
Concluding Remarks
In this paper, we study the pricing dynamics of a monopolist who produces durable goods with network effects. We demonstrate that introductory pricing constitutes the monopolist seller’s optimal strategy and that selling may be more profitable than leasing under certain market conditions. Our analysis relies on the specific criterion of equilibrium selection. That is, in face of multiple equilibria, we select the stable equilibrium that yields the smallest network size for any given price. This criterion entails that consumers are generally cautious about the prospect of network growth. Such presumption is plausible in reality if, for instance, the product/technology is novel and unfamiliar, or the firm is relatively unknown so that its success is in doubt. In contrast, for a 14
product like iPod by Apple Inc., a potential consumer will likely anticipate a very optimistic growth of its user network. Our criterion will hence be inapplicable in the latter scenario. This finding helps us to interpret the relative advantage of seller over lessor as predicated in our model, a result different from that of Bulow (1982). The advantage of the seller derives from her ability to establish a large installed base of committed users. In contrast, the lessor cannot demand renters committing to continuous renting since in each period the renters will always make independent leasing decisions. The seller’s strategic flexibility thus leads to higher profits. Note, however, that our conclusion is conflicting with past findings. Future empirical tests are needed to determine the correct theory.
References Bagnoli, S., S. Salant, J. Swierzbinski. 1989. Durable goods monopoly with discrete demand. Journal of Political Economy 97(6) 1459–1478. Bensaid, B., J. Lesne. 1996. Dynamic monopoly pricing with network externalities. International Journal of Industrial Organization 14(6) 837–855. Bond, E., L. Samuelson. 1987. The Coase conjecture need not hold for durable-goods monopolies with depreciation. Economics Letters 24(1) 93–97. Bruce, N., P. Desai, R. Staelin. 2006. Enabling the Willing: Consumer Rebates for Durable Goods. Marketing Science 25(4) 350-366. Brynjolfsson, E., C. F. Kemerer. 1996. Network externalities in microcomputer software: An econometric analysis of the spreadsheet market. Management Science 42(12) 1627–1647. Bucovetsky, S., J. Chilton. 1986. Concurrent renting and selling in a durable goods monopoly under threat of entry. Rand Journal of Economics 17(2) 261–278. Bulow, J. I. 1982. Durable-goods monopolists. Journal of Political Economy 90(2) 314–332.
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Bulow, J. I. 1986. An economic theory of planned obsolescence. Quarterly Journal of Economics 101(4) 729–750. Cabral, L. M. B., D. J. Salant, G. A. Woroch. 1999. Monopoly pricing with network externalities. International Journal of Industrial Organization 17(2) 199–214. Chen, Y., J. Xie. 2007. Cross-Market Network Effect with Asymmetric Customer Loyalty: Implications for Competitive Advantage. Marketing Science 26(1) 52–66. Conlisk, J., E. Gerstner, J. Sobel. 1984. Cyclic pricing by a durable goods monopolist. Quarterly Journal of Economics 99(3) 489–505. Dean, J. 1976. Pricing policies for new products. Harvard Business Review 54(6) 141–153. Desai, P., D. Purohit. 1998. Leasing and selling: Optimal marketing strategies for a durable goods firm. Management Science 44(11) Part 2 of 2, S19-S34. Desai, P., D. Purohit. 1999. Competition in durable goods markets: The strategic consequences of leasing and selling. Marketing Science 18(1) 42-58. Dhebar, A., S. S. Oren. 1985. Optimal dynamic pricing for expanding networks. Marketing Science 4(4) 336–351. Economides, N. 2000. Durable goods monopoly with network externalities with application to the PC operating systems market. Quarterly Journal of Electronic Commerce 1(3) 193–201. Economides, N., C. Himmelberg. 1995. Critical mass and network evolution in telecommunications. In: Brock, G. W. (Ed), Toward a competitive telecommunication industry: Selected papers from the 1994 Telecommunications Policy Research Conference. Lawrence Erlbaum Associates, Mahwah, NJ. Farrell, J., M. L. Katz. 2005. Competition or predation? Consumer coordination, strategic pricing and price floors in network markets. Journal of Industrial Economics 53(2) 203-231.
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Farrell, J., G. Saloner. 1985. Standardization, compatibility, and innovation. RAND Journal of Economics 16(1) 70-83. Hauser, J., G. J. Tellis, A. Griffin. 2006. Research on Innovation: A Review and Agenda for Marketing Science. Marketing Science 25(6) 687-717. Kahn, C. 1986. The durable goods monopolist and consistency with increasing costs. Econometrica 54(2) 275–294. Katz, M. L., C. Shapiro. 1985. Network externalities, competition, and compatibility. American Economic Review 75(3) 424–440. Katz, M. L., C. Shapiro. 1986. Technology adoption in the presence of network externalities. Journal of Political Economy 94(4) 822–841. Lee, Y., G. C. O’Connor. 2003. New product launch strategy for network effects products. Journal of the Academy of Marketing Science 31(3) 241–255. Manes, S., P. Andrews. 1993. Gates. Doubleday, New York. Oren, S. S., S. A. Smith. 1981. Critical mass and tariff structure in electronic communications markets. Bell Journal of Economics 12(2) 467–487. Oren, S. S., S. A. Smith, R. B. Wilson. 1982. Nonlinear pricing in markets with interdependent demand. Marketing Science 1(3) 287–313. Rohlfs, J. H. 1974. A theory of interdependent demand for a communication service. Bell Journal of Economics and Management Science 5(1) 16–37. Rohlfs, J. H. 2001, Bandwagon Effects in High-Technology Industries, MIT Press, Cambridge. Shapiro, C., H. R. Varian. 1999. Information Rules. Harvard Business School Press, Cambridge. Sobel, J. 1991. Durable goods monopoly with entry of new consumers. Econometrica 59(5) 1455– 1485.
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Stokey, N. 1981. Rational expectations and durable goods pricing. Bell Journal of Economics 12(1) 112–128. Sun, B., J. Xie, H. H. Cao. 2004. Product strategy for innovators in markets with network effects. Marketing Science 23(2) 243–254. Varian, H. R., C. Shapiro. 2003. Linux adoption in the public sector: An economic analysis. See http://www.sims.berkeley.edu/∼hal/Papers/2004/linux-adoption-in-the-public-sector.pdf. Waldman, M. 1993. A new perspective on planned obsolescence. Quarterly Journal of Economics 108(1) 273–283. Xie, J., M. Sirbu. 1995. Price competition and compatibility in the presence of positive demand externalities. Management Science 41(5) 909-926.
18
Online Appendix Proof of Proposition 1 As we only consider the stable equilibria, the effective demand faced by the monopolist is the downward-sloping part of p1 (x) or p˜(x). Due to the double-hump shape of p˜(x), in the absence of penetration-pricing constraints, the effective inverse demand in period 2 consists of ˜ 2−a p = (Ax ˜x22 ) · x2 p = (A1 x2 − a1 x22 ) · x2 where x ˜1 is such that p1 (˜ x1 ) = max p˜(x) with x ˜1 ≥
A˜ A˜ , ], 2˜ a a ˜ A1 for x2 ∈ [ ,x ˜1 ), 2a1
for x2 ∈ [
A1 1 2a1 .
(A1) (A2)
By solving the first order conditions:
M R1 (x1` ) = 2A1 x1` − 3a1 x21` = 0,
(A3)
˜ 2` − 3˜ M R2 (x2` ) = 2Ax ax22` = c,
(A4)
one obtains the solution in (3) as follows.
x∗1`
2 A1 = , 3 a1
x∗2`
A˜ 1 + = 3˜ a 3
s A˜ 2 3c − . a ˜ a ˜
(A5)
Note that the solution to (A4) only takes into account the segment of demand in (A1). An alternative strategy for the lessor is to lease to group-1 consumers exclusively in both periods so that the second-period demand is represented by (A2). To show that (A5) indeed maximizes π` , we have to ˜
˜
A A demonstrate (i) (A5) maximizes π` over x2` ∈ [ 2˜ a, a ˜ ], and (ii) (A5) generates higher profits than
the alternative strategy that excludes group-2 consumers. The first statement is easily verified geometrically: the objective function (˜ p(x2` ) − c) · x2` is a degree-three polynomial that has a root of zero and two positive roots. x∗2` in (A5) is the larger root to the quadratic equation (A4), and thus achieves the local maximum. It remains to show that the maximum is positive. It is equivalent to showing that p˜(x∗2` ) − c > 0. The inequality reduces 1
x ˜1 is not well-defined if max p1 (x) < max p˜(x), in which case (A2) vanishes and the effective demand consists of only (A1).
1
to c <
˜2 A 4˜ a,
which is assured by Assumption 1.
For the second part of the proof, we first derive the optimal strategy when the lessor excludes group-2 consumers. In what follows, we will show that when the demand is static, the lessor’s production in period 2 is zero so that x2` = x1` . For any leasing quantity x1` in period 1, the subsequent optimal strategy in period 2 depends on x1` . Thus, we will derive the optimum in two q A1 1 + ( Aa11 )2 − a3c1 . We consider the following three cases. For x1` ≤ x01 , stages. Define x01 ≡ 3a 3 1 the optimal strategy corresponding with x1` is x2` = x01 . For x1` ≥
2A1 3a1 ,
the optimal x2` is
2A1 3a1 .
1 Finally, for x1` ∈ (x01 , 2A 3a1 ), the optimal x2` is equal to x1` . One observes that π` is maximized
at x1` = x2` = x01 if x1` is confined to be weakly lower than x01 . Similarly, π` is maximized at x1` = x2` =
2A1 3a1
if x1` ≥
2A1 3a1 .
One concludes that x1` = x2` is necessary to maximize π` when the
demand does not change over time. In short, the lessor’s problem in this scenario is to maximize (2p1 (x) − c) · x. Note that the solution lies between x01 and
2A1 3a1 .
Comparing the profits and
π` (x∗1` , x∗2` ), one requires p(x∗2` ) − c) · x∗2` ≥ max(2p1 (x) − c) · x p1 (x∗1` ) · x∗1` + (˜ x
(A6)
for (A5) to be the solution. Define x0` ≡ arg maxx (2p1 (x) − c) · x. A sufficient condition for (A6) to hold is to assume that p1 (x0` ) ≤
˜2 A 4˜ a,
or equivalently, x0` ≥ x ˜1 . The left hand side is the price for the alternative strategy,
while the right hand side is the local maximum of the second hump of p˜(x). Given this inequality, (A6) would hold because there exists a strategy that is more profitable than (x0` , x0` ). Specifically, by choosing x02` such that p˜(x02` ) = p1 (x0` ), (x0` , x02` ) is more profitable than (x0` , x0` ). Note that x02` is well-defined if p1 (x0` ) ≤
˜2 A 4˜ a.
The condition (A6) involves the parameters such as A0g s, a0g s, and c. In the special case with zero marginal cost, it reduces to
˜3 A a ˜2
≥
A31 . a21
Proofs of Proposition 2 and Proposition 3 The proof is analogous to the previous proof. We first derive (x∗1S , x∗2S ) by maximizing πS over x2S ≥
˜ A 2˜ a.
Then we characterize the conditions under which this solution dominates those that 2
exclude group-2 consumers. Given the time-consistency constraint, for any x1S , the seller has to maximize her profits in period 2 as follows. ˜ 2S − a x∗2 (x1S ) ≡ arg max(Ax ˜x22S − c)(x2S − x1S ) x2S s ˜ ˜ 1 3c A 1 A A˜ 2 − x1S + x21S − . + x1S + = 3 a ˜ 3 a ˜ a ˜ a ˜
(A7)
Note that the above formulation for x∗2 (x1S ) is the larger root to the quadratic first order condition. For the above objective function, the coefficient of x32S is negative (−˜ a), and thus (A7) achieves the local maximum. The period-2 profits attained by x∗2 (x1S ) are positive if and only if the objective ˜ 2S − a function as a polynomial of x2S has three roots, or equivalently, (Ax ˜x22S − c) has two roots. Therefore, we require that A˜2 − 4˜ ac > 0 (or, Assumption 1: c < c ∀x1S , and x∗2 (x1S ) >
˜2 A 4˜ a ),
which implies p˜(x∗2 (x1S )) >
˜ A 2˜ a.
Given x∗2 (x1S ), one derives (x∗1S , x∗2S ) by solving the following first order conditions as in (4): M R1 (x1S ) = 2A1 x1S − 3a1 x21S = −
˜ ∗ (x1S ) − a ∂(Ax ˜x∗2 (x1S )2 ) 2 x1S , ∂x1S
(A8)
˜ 2S − 3˜ M R2 (x2S ) = 2Ax ax22S = c + (A˜ − 2˜ ax2S )x1S . Note that the right hand side of (A8) is positive, given that x∗2 (x1S ) >
(A9) ˜ A 2˜ a
and
dx∗2 (x1S ) dx1S
> 0.
Comparing (A8) with (A3), one concludes that x∗1S < x∗1` since the marginal revenue M R1 is decreasing for the relevant range. Similarly, the right hand side of (A9) is less than c, and hence x∗2S > x∗2` . Similar to the leasing scenario, an alternative strategy for the seller is to sell to group-1 consumers exclusively. The following equation system is parallel to (A7)—(A9), and characterizes the
3
6 M R1
s
2 A1 A1 , 2a1 4a1
M R2
ss
M R2S (·; x∗1S ) ˜ A ˜2 A , a 2˜ a 4˜
s
s
pˆ
M R2S (·; x ˆ1 )
s ss
s
p1
p˜
r rr
MC
x∗2` 6 xp2S ∗ x2S
: y xp = xˆ X x∗1S 1 6 1S x∗1` = xp1`
xp2` = x ˆ2
˜ A a ˜
-
x
Figure 2: Optimal Dynamic Pricing alternative solution. x02 (x1S ) =
1 A1 + x1S 3 a1
2A1 x1S − 3a1 x21S = −
+
1 3
s
A1 a1
2
−
A1 3c x1S + x21S − , a1 a1
∂(A1 x02 (x1S ) − a1 x02 (x1S )2 ) x1S , ∂x1S
(A10)
2A1 x2S − 3a1 x22S = c + (A1 − 2a1 x2S )x1S . Let (x01S , x02S ) denote the solution to (A10). For (x∗1S , x∗2S ) to maximize the seller’s profits, it is sufficient to require2 p1 (x∗1S ) · x∗1S + (˜ p(x∗2S ) − c) · x∗2S ≥ p1 (x01S ) · x01S + (p1 (x02S ) − c) · x02S .
The condition would hold if, for example,
˜2 A 4˜ a
≥
(A11)
A21 4a1 .
The comparison of profits under different regimes is straightforward. Note that (x∗1S , x∗2S ) is feasible to the lessor, and thus the lessor’s profits must be higher than the seller’s. A graphical solution is provided in Figure 2. One observes that M R1 (x) (M R2 (x)) intersects ˜ A1 A 3 with p1 (x) (˜ p(x)) at x = 2a 2˜ a and 0, and that the marginal revenue function for the seller, 1 2 3
It is possible that x02S is greater than x ˜1 , and hence not feasible according to (A2). A1 M R1 (x) = p1 (x) implies p01 (x) · x = 0, which in turn implies either x = 2a or x = 0. 1
4
M R2S (x; x1 ) ≡ d(˜ p(x)(x − x1 ))/dx, intersects with p˜(x) at x =
˜ A 2˜ a
and x1 .4 To obtain the solution
to (4), one first derives x∗1S from the first equation, as the condition depends only on x1S . Note that the marginal revenue M R1 evaluated at x∗1S is positive, as shown in Figure 2. Given x∗1S , one can then plot the marginal revenue curve M R2S (x; x∗1S ) for the second period. The second equation can then be rewritten as M R2S (x2S ; x1S ) = c. Therefore, the intersection of the marginal revenue M R2S (·; x∗1S ) and the marginal cost M C = c determines the aggregate sales x = x∗2S . Proof of Proposition 4 Recall that π` = p1 (x1` ) · x1` + (˜ p(x2` ) − c) · x2` when x2` ≥ x1` . When the monopolist lessor ˜
A has to take into account the penetration-pricing constraint, x2` ∈ [ 2˜ ˆ2 ] is no longer feasible. We a, x
are more interested in the case when the constraint is binding, and thus we assume that x∗2` < x ˆ2 ˜
as in Assumption 4. It follows that π` is decreasing over x2` ∈ (ˆ x2 , Aa˜ ], and hence is maximized at x2` ≈ x ˆ2 for x2` in that interval. Meanwhile, the penetration-pricing constraint does not affect ˜
the lessor’s strategy in period 1. Hence, if x2` is limited to (ˆ x2 , Aa˜ ], a plausible solution would be ˆ2 . x1` = x∗1` and x2` slightly above x The lessor’s alternative strategy is to exclude group-2 consumers in period 2 and select x2` from A1 [ 2a ,x ˆ1 ]. As shown in the proof of Proposition 1, one obtains another candidate for the solution by 1
ˆ2 ) to maximize π` , one requires maximizing (2p1 (x) − c) · x. For (x∗1` , x p(ˆ x2 ) − c) · x ˆ2 ≥ max(2p1 (x) − c) · x, p1 (x∗1` ) · x∗1` + (˜ x
(A12)
which represents the assumption of a fast growing market. In sum, Assumptions 4 and (A12) ensure that xp1` = x∗1` and xp2` ≈ x ˆ2 under the penetration-pricing constraint. Proof of Proposition 5 Recall that (x∗1S , x∗2S ) maximizes πS in the absence of penetration-pricing constraints, and x∗2 (x1S ) = arg maxx2S (˜ p(x2S ) − c)(x2S − x1S ). We want to characterize the conditions under which (ˆ x1 , x∗2 (ˆ x1 )) maximizes xpS . Note that x∗2 (x1S ) is increasing in x1S given Assumption 1. Thus, Assumption 6 (x∗2 (ˆ x1 ) < x ˆ2 ) implies x∗2 (x1S ) < x ˆ2 , ∀x1S < x ˆ1 . Consequently, x∗2 (x1S ) is not 4
M R2S (x; x1 ) = p˜(x) implies p˜0 (x)(x − x1 ) = 0, which in turn implies either x =
5
˜ A 2˜ a
or x = x1 .
feasible for any x1S < x ˆ1 . In particular, x∗2S = x∗2 (x∗1S ) is not feasible as x∗1S < x ˆ1 , which is implied by Assumption 6 (x∗1` < x ˆ1 ). In sum, Assumption 6 implies a binding penetration-pricing constraint. To solve the maximization problem in (5), we first derive xp2 (x1S ) that maximizes the seller’s period-2 profits subject to the penetration-pricing constraint. For x1S ≥ x ˆ1 , the constraint is not binding, and hence xp2 (x1S ) = x∗2 (x1S ). Meanwhile, for x1S < x ˆ1 , the constraint limits the ˜
feasible x02S s to either [x1S , x ˆ1 ] or (ˆ x2 , Aa˜ ]. For the latter interval, the seller’s period-2 profits are maximized at x2S slightly above x ˆ2 since x∗2 (x1S ) < x ˆ2 . For the former interval, it is maximized at min(x02 (x1S ), x ˆ1 ), where x02 (x1S ) ≡ arg max(p1 (x2S ) − c)(x2S − x1S ) maximizes period-2 profits when the seller excludes group-2 consumers (see the proof of Proposition 2). Apparently, x2S = x ˆ1 is less profitable than x2S = x ˆ2 as the prices are the same. One concludes that xp2 (x1S ) is either ˆ2 for x1S < x ˆ1 . x02 (x1S ) or slightly above x ˆ1 , Given xp2 (x1S ), one obtains three candidates for (xp1S , xp2S ) that maximizes πSp . For x1S ≥ x xp2 (x1S ) = x∗2 (x1S ). One can show that ∂ 2 πSp /∂x21S is negative. Therefore, πSp is decreasing over x1S ∈ [ˆ x1 , Aa11 ] as x x1 , x∗2 (ˆ x1 )) if x1S is limited to [ˆ x1 , Aa11 ]. ˆ1 > x∗1S , and hence πSp is maximized at (ˆ For x1S < x ˆ1 , xp2 (x1S ) is either equal to x02 (x1S ) or approximately x ˆ2 . In the former case, πSp is maximized at (x01S , x02S ) that solves (A10). In the latter case, πSp is maximized at xp1S = x∗1` because ˆ2 ) generates higher xp2 (x1S ) is a constant and independent of x1S . Equation (A12) implies that (x∗1` , x x1 , x∗2 (ˆ x1 )) profits than (x0` , x0` ), which in turn dominates (x01S , x02S ). In sum, (xp1S , xp2S ) is either (ˆ or approximately (x∗1` , x ˆ2 ). A graphical solution is illustrated in Figure 2. The monopolist lessor must set the period-2 price slightly below pˆ due to the penetration-pricing constraint, and hence xp2` ≈ x ˆ2 . For the monopolist seller, xp1S is equal to x ˆ1 in order to penetrate the market.5 Consequently, xp2S is determined by ˜
˜2
A A intersecting M C = c with the parabola that passes through (ˆ x1 , pˆ) and ( 2˜ a , 4˜ a ).
Our analysis has employed a particular form of demand specification. In addition, we also made certain regularity assumptions to ensure that penetration-pricing constraints are binding. In what follows we show that these presumptions do not place severe restrictions on our findings. 5
In Figure 2, we assume that πSp (ˆ x1 , x∗2 (ˆ x1 )) > πSp (x∗1` , x ˆ2 ).
6
A2 A1
6 ( 13 , 13 )
t A6(i) A4
A6(ii) A5
H
j H
A7
0
a2 a1
Figure 3: Graphical Representation of Assumptions For the demand specification, the most crucial property is the inverted-U shape illustrated in Figure 1(i). Suppose the first period demand correspondence is given by p1 (x) = q1 (x)x, where (x) q1 (x) is non-linear. Define 1 (x) ≡ − qq01(x)x . Then p1 (x) exhibits an inverted-U shape if and only if 1
there exists a threshold x b such that 1 (x) > 1 for x < x b and 1 (x) < 1 for x > x b.6 Note that 1 (x) can be interpreted as the price elasticity of the generic demand without network effects. Therefore, the aforementioned condition means that the product is elastic in terms of generic demand when the price is relatively high (q > q1 (b x)), and vice versa. In other words, the generic demand must be more sensitive to price changes when the network size is small, and less sensitive when the network gets large. Our analysis can be extended to a more general demand specification with multiplicative network effects, so long as q1 (x) and q2 (x) satisfy this condition. With the same argument, we can also extend our analysis to include additive network effects.7 The conclusion that selling is more profitable than leasing requires certain regularity conditions (Assumptions 1 to 6 and πSp (ˆ x1 , x∗2 (ˆ x1 )) > πSp (x∗1` , x ˆ2 )). These conditions amount to requiring that 6 0 p1 (x) 7
= −q10 (x)x · (1 (x) − 1), and thus the slope of p1 (x) is positive if and only if 1 (x) > 1. Suppose p(x) = q(x) + f (x), where f (x) represents the network effect. For p(x) to exhibit an inverted-U shape, it must be true that p0 (x) > 0 initially and p0 (x) < 0 for larger x. This condition will hold if, for example, f (x) = x, q(x) is concave, and there exists x0 such that q(x0 ) = −1. Another example would be q(x) = A − ax, f (x) is concave, and there exists x00 such that f (x00 ) = a. Some numerical simulations are available from the authors on request.
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(i) the monopolist expects to benefit from the market growth; (ii) the early adopters’ willingness to pay must be significantly higher than the majority consumers’. For part (i), the aggregate market appears lucrative if either the majority consumers’ willingness to pay is sufficiently high (high A2 ), or the population size is large (high
A2 a2 ).
Meanwhile, part (ii) implies that
A2 A1
must be bounded
from above. Otherwise the majority consumers have similar preferences as early adopters, and the aggregate demand will have a single hump instead of double humps. A concrete characterization of these conditions is illustrated in Figure 3, which assumes zero production costs (c = 0). Curve Ai for i = 1, · · · , 6 in the diagram represents Assumption i, while A7 characterizes the condition under which selling is more profitable than leasing: πSp (ˆ x1 , x∗2 (ˆ x1 )) > πSp (x∗1` , x ˆ2 ). Recall that Assumption 1 provides a upper bound for c, which is satisfied when c = 0. We omit A2 and A3 since Assumptions 2 and 3 are implied by the remaining assumptions for any c. For all the A2 assumptions to hold, ( aa21 , A ) must fall in the wedge-shaped area bounded by the axis of 1
well as A5, A6(i), and A7 curves.
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A2 A1 ,
as
