Distribution Strategies for Durable Goods with Indirect Network Effects ∗
Sreekumar R. Bhaskaran †, Stephen M. Gilbert‡ December 5, 2003
Abstract In this paper we examine the influence that indirect network effects have on a durable goods manufacturer’s distribution strategy to lease or sell her product. Previous research has identified that a durable goods manufacturer can mitigate the potential for its own opportunistic behavior with respect to consumers by leasing instead of selling its product. However, we show that leasing creates an opportunity for a durable goods manufacturer to exploit firms that generate indirect network benefits by providing complementary products. As a result, the durable goods manufacturer faces a trade-off between leasing, which mitigates its potential to exploit consumers, and selling, which mitigates its potential to exploit producers of complementary products. In this paper, we examine this trade-off and show how a durable goods manufacturer can use a combination of leasing and selling to balance its strategic commitment across both its own market as well as the complementary market. We also analyze the impact of leasing with an option- to-buy contracts under these situations. (complementary markets, selling versus leasing, leasing with an option-to-buy)
∗
The authors are listed in alphabetical order
†
University of Texas at Austin, Management Department CBA 4.202, Austin, Tx-78712,
[email protected] ‡
University of Texas at Austin, Management Department CBA 4.202, Austin, Tx-78712,
[email protected]
1
Introduction
Many durable or semi-durable products are heavily influenced by indirect network effects that exist when the value of owning the durable product is enhanced by the availability of complementary goods or services. In many instances the lack of sufficient availability of complementary products can impede the success of a durable product. For example, high definition television (HDTV) has not been adopted as quickly as anticipated in spite of its technological advantages. A key reason for this has been the paucity of compelling programming, and it is projected that the launch of ESPN’s new high-definition channel may be the catalyst this industry has been waiting for (Business Week 2003). Another example involves adoption of fuel cell technology in the automotive industry. One reason why fuel cell cars have not been widely sold is the lack of availability of hydrogen fuel. Energy companies and automotive firms are trapped in a chicken-and-egg problem in which each industry could benefit from increases in the other’s output (Wall Street Journal 2003), but neither has an incentive to unilaterally increase its own production. In a recent work Nair et al. (2003) demonstrate and empirically measure the role that indirect effects from third-party compatible software played in the growth of Personal Digital Assistants (PDA) in the US. They suggest that when availability of complementary products is very important relative to price-quality effects, a firm should either consider investing in the complementary service providers or vertically integrate into the production of these complementary products. However, the competence that serves a firm well for introducing its own product could be quite different from the competence that is necessary to serve the complementary market. Thus, economies of scope may preclude attempts to centralize control of the durable and complementary products within a single firm. In such cases, we must understand how the complementary relationships among separate firms should be accounted for in strategic decision making. It has long been recognized that one of the strategic issues that a durable goods manufacturer (DGM) faces is that of mitigating the potential for its own opportunistic behavior with respect to its own consumers. This issue arises because a durable goods monopolist externalizes the impact that its current production has upon consumers who already own the product. As a result, it will produce at a rate that, over time, drives down prices and decreases the value of owning a used product. In anticipation of this opportunistic behavior, consumers are less willing to pay for ownership of the product. This
issue has been referred to as time-inconsistency, but because the threat of opportunistic behavior discourages consumers from investing in the durable product, it can also be considered as a form of a hold-up problem with respect to consumers. One well known way for a DGM to avoid the problem of time inconsistency (consumer hold-up) is to lease the product to the consumer instead of selling it. Leasing gives the DGM complete control over the market for used goods, eliminating its own incentive to produce at a rate that drives down prices. However, as much as leasing eliminates the manufacturer’s hold-up problem with respect to consumers, it can create another hold-up issue if there are significant indirect network effects. Specifically, by leasing its product, the DGM creates a situation in which it can inflate its prices to exploit investments that are made by firms that produce complementary goods and services. As a result, leasing can discourage the output of complementary products. By selling its product, the durable goods manufacturer relinquishes control of the used market, and eliminates much of her own ability to act opportunistically. Thus, the interaction between durability and demand complementarity creates a trade-off between a hold-up problem with respect to the consumer market versus another hold-up problem with respect to the complementary market. The contribution of this paper is to identify the conflict between these two interdependent hold-up issues, and to discuss how a durable goods manufacturer’s distribution strategy can be used to strike a balance between them. The remainder of the paper is organized as follows. In section 2, we review and analyze the relevant literature. In section 3, we develop a model to capture the interaction between the manufacturer of a durable product and a firm that produces a complementary product. We analyze how the strategic choice between leasing and selling can be used to the durable goods firm’s advantage. In Section 4, we extend our analysis to allow competition and entry in the complementary market. Finally, we summarize the managerial implications through a conceptual framework and point to directions of future research.
2
Literature Review
The study of durable goods has long been a central subject in the industrial organization literature. It is well established that the durability of a product can interfere with a 3
monopolistic manufacturer’s extraction of rents from consumers. The reason for this, is that after a manufacturer sells its durable product to some subset of the market, it has an incentive to continue production, selling its output at lower and lower prices. Coase (1972) conjectures that if rational consumers anticipate this behavior, then prices would fall down to competitive levels and the firm would lose complete control of the situation. This issue has been referred to in the literature as time-inconsistency to reflect the fact that for a durable product, the traditional monopolistic model is inconsistent with the passage of time. This line of thought is formalized by Bulow (1982) and Stokey (1981) who propose that, by leasing its product, a durable goods manufacturer can avoid the time inconsistency problem. Bulow (1982) also suggests that while leasing helps the firm to retain its monopoly power, selling also might have socially undesirable effects. Later research points out the conditions under which the Coase conjecture does not hold ground. Notable among these are the works by Conlisk, Gerstner, & Sobel (1984) who model constant inflow of new consumers, Bond & Samuelson (1984) who incorporate replacement sales, Kahn (1986) who studies increasing marginal production costs and Bagnoli, Salant, & Swierzbinski (1980) who use discrete demand. Research on durable goods has been pursued in 2 main directions. The first of these examines the relationship between durability and a firm’s incentive to innovate. Prominent among these are the works by Levinthal & Purohit (1989) and Waldman (1993) who argue that the incentive for a durable good manufacturer to make the existing product obsolete by introducing a newer version is high and this can intensify the time inconsistency effect. In a related work, Dhebar (1994) analyzes the case when a monopolist supplies a series of durable products of increasing quality to a heterogenous customer base and shows that intertemporal price discrimination issues in such circumstances could prevent a producer from credibly committing to future prices and qualities. This can result in a situation where there is no equilibrium strategy. Kornish (2001) points out that an equilibrium strategy exists if the monopolist does not offer upgrade pricing. Also, Subramaniam & Srinivasan (1998) demonstrate the role of an introductory product strategy to signal the trajectory of its cost curves to consumers. They show that firms can use a high introductory price to signal to consumers that cost reduction through learning experiences are low. The other main direction in durable goods research explores the interaction between a firm’s distribution strategy and the durability of its product. However, this line of
4
research tends to focus on competitive, rather than complementary, interactions. Bulow (1986) models the durability choice in an oligopoly and finds that the lease/sales ratio is dependent on the number of firms in the market. Bucovetsky & Chilton (1986) demonstrate that selling can dominate leasing when there is threat of entry from competitive firms or actual competition between firms. Competition is also examined by Desai & Purohit (1999), who show that in a duopoly, firms tend to prefer selling over leasing when their products are sufficiently substitutable. They also demonstrate that a combination of leasing and selling might be the optimal strategy. Subsequently, Desai & Purohit (1998) also show that firms can employ leasing and selling as a mechanism to differentiate between consumers. Our work also draws in from the literature on markets that involve either direct or indirect network effects. Katz & Shapiro (1994) categorizes such markets and identifies the issues which firms and consumers face while dealing with these markets. Farrell & Saloner (1986) investigate how the installed base for such products interacts with a firm’s incentive to innovate and looks at the welfare implications of certain strategies that firms might adopt. Similar issues are also considered by Katz & Shapiro (1985), Farrell & Saloner (1985) and Choi (1994). In another related work, Conner (1995) analyzes a market with direct network effects and suggests how an manufacturer can derive externality benefits from a lower quality version of its product that is produced by an imitator. Recent work by Parker & Van Alstyne (2001a), Parker & Van Alstyne (2001b) and Rochet & Tirole (2001) also examine how complementary interactions between products could be used to stimulate demand and handle competitive influences from other firms, and thus, provide a rationale for cross-subsidization of products. However, most of this literature focuses on network externality issues and ignores the effect of the durability of a product under such situations. Our research is also related to the stream of literature that deals with the hold-up problem when firms have to incur fixed costs, either for entering the market or for improving product quality. In anticipation of opportunistic behavior from other firms, they tend to under invest in quality or choose not to enter the market altogether. This issue has been dealt extensively in the industrial organization literature; for e.g. Williamson (1975) and Klein, Crawford, & Alchian (1978). An excellent overview of this stream of literature is provided by Katz (1989). In this context, when firms have to source from a competitive supplier industry, Subramaniam (1998) shows that an appropriate level of debt-financing 5
on the part of the manufacturer can mitigate the hold-up problem. In a recent work, Gilbert & Cvsa (2003) analyze the trade-off between mitigating hold-up by committing to future prices and the subsequent loss of flexibility. In contrast to much of the existing literature that emphasizes hold-up issues in vertical supply chain relationships, our research addresses the hold-up issues that involve consumers and firms that produce complementary products. Thus, while the effect of durability on the profitability of firms has been studied before, the ability of this class of goods to create markets in future periods has not received much attention. This aspect is important because durable goods have value beyond a single period and can thus stimulate demand for other products that are complementary in nature and can also benefit from the indirect network externalities that these goods offer. The contribution of our paper is to identify the trade-off that exists in the presence of both durability and indirect network effects and, show how a durable goods manufacturer can manage this trade-off with its distribution strategy.
3
The Model
Consider two complementary products, one of which is durable. For simplicity, we will assume that the other is a non-durable product or service. We assume that these two products are produced by different firms, to which we refer to as the Durable Goods Manufacturer (DGM) and the Complementary Product Provider (CPP). In order to represent durability, we adopt a variation of the two period linear demand model that Bulow (1982) proposed for durable goods. In this model, it is assumed that the durable good lasts for exactly two periods, and that there is no depreciation of its value between periods one and two. Assuming that the durable good lasts for two periods is not critical; it is only important to assume that it lasts for a finite amount of time, so that optimal decisions can be calculated in a recursive fashion. As in the Bulow (1982) model, we assume that consumers’ utility is defined by the value of the service provided by a product and that they are indifferent between leasing and owning the product. Based on this assumption, we represent the demand in terms of the value of the service that is provided by the product in each period. Let Ql be the quantity that the durable goods manufacturer leases in the first period, and let Qs be the
6
quantity that it sells. In the second period, there is no distinction between selling and leasing since, in either case, consumers obtain exactly one period of service from products they obtain at the beginning of period 2. Let Q2 be the quantity that the durable goods manufacturer introduces to the market in period 2. Thus, in a pure leasing operation, Q2 represents the only supply of the durable good that is available in period 2, whereas if any units are sold in the first period, then the total quantity available to consumers in period 2 is Qs + Q2 , i.e. the sum of the used items from period 1 and the new items released in period 2. We use α =
Ql Ql +Qs
to represent the fraction of the total goods that are leased in
the first period. The main difference between our model and existing ones is that we include the additional feature of allowing for a complementary product in the second period. Alternatively, we could allow for the complement to be available in both periods, like the durable one, but this complicates the analysis and adds no additional insights. The critical part of this assumption is that the availability of the complement in the second period must depend on the DGM’s decision about selling and leasing in the first period. Let us define y to be the quantity of the complementary product that is produced in period 2 only. We assume that there are a + M potential consumers for the durable good. In the absence of the complement, each consumer’s utility vd for one period of use from the durable good is uniformly distributed on the interval [−M, a]. Thus, without the complement, there are a consumers who derive positive utility from using the durable good for one period. We further assume that each unit of the complement that is available increases each consumer’s utility for the durable product by θ, where θ ∈ (0, 1) is a parameter representing the strength of the complementarity. Thus, in a period in which y units of the complement are available, each consumer’s utility for using the durable good in that period is uniformly distributed on the interval [−M + θy, a + θy]. Note that this approach is similar to that used by Conner (1995) in the context of direct network benefits. For the complementary product, which is non-durable, we assume that there are βa+N potential consumers, where β ≥ 0 is a parameter that reflects the relative size of the complementary market. In the second period, each of these consumer’s utility vc for the complement is uniformly distributed on the interval [−N + θ(Qs + Q2 ), βa + θ(Qs + Q2 )] where Qs is the quantity of the durable good that was sold in period 1 and Q2 is the additional quantity that is made available in period 2. Thus, Qs + Q2 represents the total
7
quantity of the durable good that is in use in period 2, and each unit of the durable good increases each consumer’s utility for the complement by θ. In order to ensure that additional output of one product increases the number of consumers who obtain positive utility from the other, we require that N ≥
a(1+θβ) 1−θ2
and M ≥
a(β+θ) . 1−θ2
Given these assumptions, the inverse demand function for the use of the durable good in period 1 can be represented as: p(q) = a − q, where q is the quantity of the durable good that is available in the market. Note that p(q) can also be interpreted as the utility that a marginal consumer derives from a period of service when there are exactly q units of the durable good in the market in period 1. In period 1, when there is no complementary product, the market lease price for a single period of service from the durable good is decreasing in the total quantity that is made available through both leasing and selling: p1d (Ql + Qs ) = a − (Ql + Qs )
(3.1)
In the second (i.e. final) period, there is no distinction between the value of one period of service from the product and the value of owning it. However, the availability of the complement can increase the value of owning the durable product. In this last period, market prices for the durable and complementary products depend on: the quantity (Qs ) that was sold in period 1 and is now available in the secondary market, the quantity (Q2 ) that is leased or sold in period 2, the quantity (y) of the complementary product, and the strength (θ) of the complementarity. Note that the quantity that was leased in period 1 has no bearing on the market prices in period 2. Thus, in period 2, the market price for the durable good can be represented as: p2d (Qs , Q2 , y) = a − (Q2 + Qs ) + θy
(3.2)
Based on our assumptions about the consumers’ utility for the complement, the market price for it can be represented as: pc (Qs , Q2 , y) = βa − y + θ(Qs + Q2 )
(3.3)
Although our assumptions about uniformly distributed utility for the complement and the use of the durable good lead to linear demand functions and simplify our analysis, the basic trade-off that we are examining requires only that the market price for the service provided by the durable good be decreasing in the quantity of this good that is available and increasing in the quantity of the complement, and that the market price for the 8
complement be decreasing in its own price and increasing in the quantity of the durable good. The assumption that there is no depreciation for the durable good simplifies the presentation of our results and helps to clarify the distinction between leasing and selling. Although our model can easily be generalized to allow for depreciation, this blurs the distinction between durable and non durable goods. Note that the assumption of no depreciation is consistent with that of Bulow (1982) and Bucovetsky & Chilton (1986). In all of our analysis, we assume that the marginal cost of production for the DGM is constant, and for ease of exposition we normalize this cost to zero. Although the assumption of linear production cost is simplistic, it is not unreasonable in a wide variety of settings. Moreover, the fundamental interaction between the selling vs. leasing strategy of the DGM and the complementary product requires only that the marginal production costs be non-decreasing. For now, let us assume that the CPP is a single, monopolistic firm that has a constant marginal cost. (We will later relax the assumption of a monopolistic CPP, and consider the entry decisions of multiple potential CPPs who have increasing marginal costs.) The profits of this monopolistic CPP can be expressed as: πc (Qs , Q2 , y) = ypc (Qs , Q2 , y) = y(βa − y + θ(Qs + Q2 ))
(3.4)
and the second period profits for the manufacturer of the durable product are: π2d (Qs , Q2 , y) = Q2 p2d (Qs , Q2 , y) = Q2 (a − (Q2 + Qs ) + θy)
(3.5)
In order to avoid the effects of one firm having more power than the other, we treat the quantity setting game in the second period as a Nash game in which both firms set quantities simultaneously. By applying first order conditions to (3.4) and (3.5), we have that the second period quantities will be the following function of the amount of durable goods that were sold in the first period: 2βa + aθ + Qs θ 4 − θ2 a(2 + θβ) − Qs (2 − θ2 ) Q2 (Qs ) = 4 − θ2 y(Qs ) =
(3.6) (3.7)
As can be seen above, the first period sales quantity, Qs , decreases the residual demand of the durable good in the second period and hence the additional quantity released in this period decreases with the first period sale quantity. However, the quantity of the complementary product is increasing in the first period sale of the durable good. 9
3.1
The Sales and Leasing Distribution Strategy
The first period decisions facing the firm are: (1) total quantity to be made available in the first period and (2) fraction of this quantity to be leased (α). Such mixed strategies are prevalent in many industries, including automobiles, software, electronic goods and home equipments. Let Q1 = Ql + Qs , i.e. the total output from the DGM in period 1. Thus, by substituting Qs = Q1 (1 − α) into (3.6) and (3.7), we obtain the following expressions for the second period quantities of the complement and of the DGM: 2βa + aθ + Q1 (1 − α)θ 4 − θ2 a(2 + θβ) − Q1 (1 − α)(2 − θ2 ) Q2 (Q1 , α) = 4 − θ2 y(Q1 , α) =
(3.8) (3.9)
Proposition 3.1 The second period output of the DGM and of the CPP are increasing and decreasing respectively in the fraction α of leasing employed in the first period. Moreover, as complementarity (θ) increases, a given increase in α (more leasing) causes smaller increases in the second period output of the DGM and larger reductions in the output of the CPP.
The intuition of this result is as follows: As the DGM leases a larger fraction of its products in the first period, there is greater residual demand in the second period, to which it responds with larger output. The presence of the complement increases this residual demand even further. However, because y(Q1 , α) is decreasing in α, leasing tends to discourage output of the complement. Thus, increasing the proportion of leasing has two opposing effects upon the second period demand for the durable product: It leaves more residual demand, but by discouraging the output of the complement, it does not allow as much demand stimulation from indirect network effects. This latter effect becomes stronger as the measure of complementarity becomes large, i.e. θ → 1. To fully understand why leasing discourages output of the complement, note that the DGM’s flexibility to control the second period market price of its product is directly related to the extent to which it employs leasing. By leasing its products, the DGM is signaling its intention to remain flexible in the second period. By doing so, it is able to appropriate more benefits of complementarity to itself. However, since the CPP anticipates this behavior from the DGM, it decreases its quantity in the second period, thus adversely 10
affecting the DGM. Alternatively, by shifting to a greater proportion of selling in the first period, the DGM effectively relinquishes its ability to expropriate the benefits of the output of the complementary product. It is also worth noting that, as the relative size (β) of complementary market increases, the second period quantities of both products increase. This is also a fairly intuitive observation. A larger value of β implies more second period demand and so optimal quantities increase in the second period.
Figure 1: Decision making process of a consumer with valuation vd for 1 period of use
Having developed the second period response functions for both the DGM and the CPP, we now direct our attention to the first period. Note that we need to concern ourselves with the market prices for both leasing and sales. Since we have assumed that consumers are indifferent between leasing and owning the durable product, the lease price will be equal to the reservation price that a marginal consumer has for one period of use from the product when the total quantity available is Q1 = Ql + Qs . (Please refer to Fig. 1 for the consumer decision process). By substituting Q1 into (3.1), we have that the lease price is the following function of first period leasing and sales quantities: pl1d (Ql , Qs ) = a − Ql − Qs 11
(3.10)
Drawing again upon the assumption that consumers are indifferent between leasing and owning, the market price for units that the DGM sells in the first period is equal to the reservation price that a marginal consumer has for one period of use plus the anticipated discounted market price of the product in the second period. Recall that the market price of the durable good in period 2 is a function of the period 2 quantity responses of both the DGM and the CPP. Thus, if consumers are perfectly informed about the market dynamics and rationally anticipate the responses of the DGM and the CPP, then the market price of units sold by the DGM in period 1 can be represented as follows: ps1d (Ql , Qs ) = a − Ql − Qs + ρp2d (Qs , Q2 (Qs ), y(Qs )) a(2 + βθ) − Qs (2 − θ2 ) = a − Ql − Qs + ρ 4 − θ2 Using these derived implicit inverse demand functions and the anticipated second period responses of the DGM and the CPP, we can now express the total expected profits of the DGM as a function of the quantities that it sells and leases in period 1: πdM [Ql , Qs ] = ps1d (q, Qs )Qs + pl1d (Ql , Qs )Ql + ρπ2d (Qs , Q2 (Qs ), y(Qs )) = (a − Ql − Qs )(Ql + Qs ) ! 2(a + Qs ) + aβθ)(a(2 + βθ) − Qs (2 − θ2 )) +ρ (4 − θ2 )2
(3.11)
Proposition 3.2 a) Pure selling, i.e. no first period leasing, is optimal if and only if 4 ≤ θ2 (4 + βθ). b) If 4 > θ2 (4+βθ), then the profits of the DGM are maximized with the following combined quantity of leasing and selling in period 1: Q∗s + Q∗l = and the fraction of leasing is: α∗ =
a 2
(3.12)
4−4θ2 −βθ3 2(2−θ2 )
Corollary 3.1 The threshold level of complementarity, θth , above which pure selling (α = 0) becomes optimal decreases with β i.e.
∂θth β
< 0 (Refer Fig 2).
Note that if there are any indirect network effects, i.e. θ > 0, the DGM will always include some portion of selling in its optimal distribution strategy. So, pure leasing, i.e. no selling in the first period, is optimal if and only if θ = 0. On the other hand, it 12
Figure 2: Relationship between θth and β may be optimal for the DGM to exclude leasing from its distribution strategy and rely entirely on sales (Refer Fig 3). In order for a pure sales strategy to be optimal, we must have 4 < θ2 (4 + βθ), which will tend to occur when both complementarity and the size of the complementary market are large. It is of particular interest to observe that this condition cannot occur when β = 0 which would be the case if the market for the complementary product depends solely on the existence of the durable good. As the size of the complementary market increases, pure selling becomes optimal for even lower levels of complementarity. Proposition 3.2 and Corollary 3.1 extend the existing literature on durable goods by demonstrating that when there are indirect network benefits, a durable goods manufacturer can prefer selling to leasing. By leasing its product in the first period, the DGM has greater flexibility in the second period to take advantage of the indirect network benefits. But this opportunistic behavior from the DGM forces the CPP to decrease its quantity in the second period, thus decreasing the profits of the DGM in the process. In these circumstances, selling acts as a commitment against such opportunistic behavior. As the products become stronger complements, the marginal increase in profits due to such a commitment exceeds the time-inconsistency effect and a pure selling strategy becomes 13
Figure 3: Mixed Distribution Strategy optimal. It is also intuitive to note that the DGM’s preference for selling would increase if the CPP has the opportunity to invest in quality enhancing innovations or activities like advertising or promotion efforts that would improve the demand of its product.
3.2
Leasing with option-to-buy
In many durable goods markets, e.g. automobiles, musical instruments, etc., it is common for firms to offer consumers leasing contracts that include options to purchase the product at the conclusion of the lease period. GM’s Smart Buy, Chrysler’s Gold Key Plus, and Ford’s Red Carpet Option are examples of some such contracts. In this section, we consider a leasing contract in which the DGM specifies a lease price, which we denote by pl , for use of the product in the first period, as well as an exercise price, denoted by pe , at which the ownership of this good can be transferred to the consumer after one period of use. Although the DGM can commit in advance to pe , he cannot explicitly commit to the total quantity of the product that will be available in the second period. Let Q1 be the number of consumers who use the durable good in the first period. As in our previous analysis, the inverse demand function for the use of the product in the first period is pl (Q1 ) = a − Q1 . In the second period, after the complementary firm has entered the market, the first period users decide whether to exercise their option or to
14
Figure 4: Decision making process of a consumer with valuation v under leasing with option to buy either go without the product in the second period or purchase it at the market clearing price. As shown in Fig. 4, this decision depends on the relative values of their valuation, the exercise price, and the market clearing price. Denote by Qs the number of first period users who exercise their option to buy. (We use this notation to capture the notion that they have been converted to sales customers.) The market clearing price in period 2 can be expressed as: p2d (Q2d , y) = a − Q2d + yθ
(3.13)
where Q2d = Q2 + Qs denotes the total quantity available in the second period. As before, the CPP’s profit function in period 2 can be expressed as: πc (Q2d , y) = y(βa − y + θQ2d )
(3.14)
In period 2, the the total quantity available will depend on the relative values of Q1 , Q2 , y and pe , in the following way: Q2d
Q1 + Q2 = a − pe + θy Q2
pe ≤ a − Q1 − Q2 + θy a − Q1 − Q2 + θy ≤ pe ≤ a − Q2 + θy pe ≥ a − Q2 + θy 15
(3.15)
This can be explained as follows: If the exercise price is very low relative to the market clearing price associated with Q1 + Q2 units of the durable product and y units of the complement, then all Q1 first-period users exercise their option to buy and the DGM sells an additional Q2 units in period 2. At the other extreme, if the exercise price is very high relative to the quantity Q2 newly introduced units of the durable product and y units of the complement, then no first-period users exercise their option, and only the newly released units of the durable good are available in period 2. For intermediate exercise prices, each additional unit of the durable good that is introduced causes one more first-period user to purchase a newly released unit of the durable good instead of exercising his option to buy the unit that he leased in period 1. As a result, the marginal effect of increasing Q2 upon the availability (Q2d ) of the durable good in period 2 is zero. Given this, the DGM’s profit function in period 2 can be expressed as follows: π2d (Q2 ; Q1 , pe ) = Q1 pe + Q2 (a − Q1 − Q2 + θy)
=
pe ≤ a − Q1 − Q2 + θy a − Q1 − Q2 + θy ≤ pe ≤ a − Q2 + θy pe ≥ a − Q2 + θy
pe (a − pe + θy) Q2 (a − Q2 + θy)
(3.16)
Thus, the equilibrium in period 2 can be characterized as follows: Q∗2 pe ≤
a(2+βθ)−Q1 (2−θ2 ) 4−θ2
a(2+βθ)−Q1 (2−θ2 ) 4−θ2 a(2+βθ) 4−θ2
≤ pe ≤
y∗
2a−(2−θ2 )Q1 +aβθ 4−θ2
a(2+βθ) 4−θ2
2(a−pe )+aβθ 2−θ2 2a+aβθ 4−θ2
≤ pe
− Q1
2aβ+aθ+Q1 θ 4−θ2 aβ+aθ−pe θ 2−θ2 2aβ+aθ 4−θ2
∗ π2d
a(2+βθ)−Q1 (2−θ2 ) 2 2 4−θ pe (a(2+βθ)−2pe ) 2−θ2
a(2+βθ) 2 4−θ2
Given these second period equilibrium results, the DGM’s total profits for the two periods, denoted by πdO (Q1 , pe ), can be expressed as follows: πdO (Q1 , pe ) =
=
a(2+βθ)−Q1 (2−θ2 ) 2 Q1 (a − Q1 ) + ρ pe Q1 + 4−θ2
Q1 (a − Q1 ) + ρ
Q1 (a − Q1 ) +
pe (a(2+βθ)−2pe ) 2 2−θ 2 a(2+βθ) ρ 4−θ2
a(2+βθ)−Q1 (2−θ2 ) 4−θ2 2 a(2+βθ)−Q1 (2−θ ) (3.17) ≤ pe ≤ a(2+βθ) 4−θ2 4−θ2 a(2+βθ) ≤ pe 4−θ2
pe ≤
We can now characterize the DGM’s optimal strategy for implementing a lease with an option to buy contract. Proposition 3.3 The optimal strategy for the DGM to use in offering a lease contract with an option to buy can be characterized as follows: When 4 > (4 + βθ)θ2 : Q∗1 = Q1m and p∗e = pem , where: Q1m =
a 2 16
(3.18)
pem =
a(2 + (βθ)) 4
(3.19)
Alternatively, when 4 ≤ (4 + βθ)θ2 : Q∗1 = Q1s and p∗e = pes , where: a((4 − θ2 )2 + ρθ2 (2 + βθ)) 2θ4 + (8 − 4θ2 )(4 + ρ) a((4 − θ2 )(2 + 2βθ + θ2 ) + ρ(2 + βθ)(2 − θ2 )) = 2θ4 + (8 − 4θ2 )(4 + ρ)
Q1s = pes
(3.20) (3.21)
Corollary 3.2 Leasing with buy option replicates a mixed distribution strategy when 4 > (4 + βθ)θ2 and a pure selling strategy when 4 ≤ (4 + βθ)θ2 . It is easy to confirm this result by substituting the values (Q∗1 , p∗e ) for the optimal policy as defined in Proposition 3.3 into (3.15) to obtain the optimal total quantity, Q∗2d , that is available in period 2. For a given set of parameters, it can be confirmed that this quantity is identical to the sum of the optimal first and second period sales in the mixture of leasing and selling problem. In a similar manner, it can be confirmed that the optimal number of first period users and the quantity produced by the CPP are identical, regardless of whether we use a mixture of leasing and selling or a lease with an option to buy distribution policy.
Figure 5: Leasing with option: Analysis The impact of a leasing with an option to buy strategy is summarized in Fig. 5. Beyond a certain threshold of complementarity (4 ≤ (4 + βθ)θ2 ), the optimal first period 17
quantity of the DGM (Q∗1 ) and the second period quantity (y) of the CPP functions change, reflecting the fact that the constraint pe ≥ pe ≥
a(2+βθ)−Q1 (2−θ2 ) 4−θ2
is no longer binding. When
θ is above this threshold, the optimal quantity of the DGM in the first period is increasing in θ even though the complement is not available until the second period. Thus for higher values of θ, it becomes even more important for the DGM to signal to the CPP that the demand in the second period would be high. It does so by making the product available to a larger base of consumers in the first period itself. A common explanation for the existence of leasing with-option-to-buy contracts has been to mitigate moral hazard or prevent adverse selection (Hendel & Lizzeri 2002). Our analysis indicates that leasing with an option-to-buy can also serve as a commitment mechanism to mitigate hold-up even in the absence of moral hazard or adverse selection. Although this does not preclude the use of such contracts for other purposes, it is important to recognize these other strategic implications.
4
Entry in the Complementary Market
In this section, we relax the assumption that the complementary market consists of a single firm and allow for competition and entry. To keep the analysis simple, we continue to assume that this market opens only in the second period. However, we generalize the cost function for the complementary products providers (CPPs) to allow for a fixed cost, F that is incurred at the time of entry as well as a production cost function, c(y), where y is the production quantity. We assume that c(y) is convex to reflect diseconomies of scale or scope. Firms make their entry decision sequentially and they continue to enter the market as long as they make non-negative profits. As at the beginning of Section 3, we focus on a mixture of leasing and selling rather than a lease with an option to buy. However, as illustrated in the previous section, our results can be easily adapted to the lease with an option to buy context. The sequence of decisions by the DGM and the CPPs can be viewed as the following stages. In the first period, the DGM decides her distribution strategy and makes her production decision accordingly. In response to the first-period distribution by the DGM, the CPPs enter the market in a sequential fashion after incurring a fixed cost, F. Finally, based on the number of firms that have entered, the DGM and all of the CPPs determine
18
their second period quantities simultaneously and the profits are realized. The equilibrium is found by backward induction to ensure subgame perfection and sequential rationality; i.e. at each stage, the decision-maker’s choice is not only optimal in that stage, but also reflects rational anticipation of the optimal actions of the other players in the subsequent stages. Let yi be the quantity of the complementary product that is produced by the ith CPP. The total quantity of the complementary product available in the second period is Y =
N X
yi
i=1
where N is the number of firms that have entered the complementary market. In the second period, the market prices for the durable and complementary goods depend on several things: the quantity (Qs ) sold in period 1 and is now available in the secondary market, the quantity (Q2 ) that the DGM releases in period 2, the quantity (Y ) of the complement. In period 2, the market price for the durable good is: p2d (Qs , Q2 , Y ) = a − Q2 − Qs + θY
(4.22)
while we represent the market price for the complement as: pc (Qs , Q2 , Y ) = βa − Y + θ(Qs + Q2 )
(4.23)
Thus the second period profits of the CPPs can be represented as follows: πc (Qs , Q2 , yi ) = yi (βa − yi − Y−i + θ(Qs + Q2 )) − F − c(yi ) ∀i = 1 ... N
(4.24)
where Y−i = Y − yi , and the second period profits for the manufacturer of the durable product are: π2d (Qs , Q2 , Y ) = Q2 (a − (Q2 + Qs ) + θY )
(4.25)
Firms set quantities simultaneously and competitively. Since firms can enter the complementary market sequentially, we assume that firms continue to enter as long as they can make a non-negative profits. Thus, in equilibrium, every firm in the complementary market makes only enough profit to recover his fixed cost of entry. πc (Qs , Q2 , yi ) = ye (βa − N ye + θ(Qs + Q2 )) − F − c(ye ) = 0 19
(4.26)
where ye satisfies the first order conditions for (4.24). Note that this approach, which is consistent with Subramaniam (1998), implicitly treats N as continuous, relaxing the requirement that there be an integer number of firms that enter the market. By applying first order conditions to (4.24) and (4.25) and using (4.26), we can find the second period quantities of all the firms as function of the DGM’s distribution strategy and the number of CPPs who enter in equilibrium. Since all the CPPs are symmetric and have the same costs for entry and marginal costs, the quantity produced by each of these firms, denoted by ye , would be the same and would satisfy the following equation. ye (ye + c0 (ye )) − F − c(ye ) = 0
(4.27)
The best response function of the DGM can be represented as a function of this equilibrium quantity and the number of firms who have entered the complementary market. Q2 (Qs , N ) =
a + θN ye − Qs 2
(4.28)
The number of firms who enter are given by the equation: aβ − (N + 1)ye +
θ(a + N θye + Qs ) − c0 (ye ) = 0 2
(4.29)
Solving explicitly for N , the number of firms that enter can be expressed as a function of the first period sale quantity (Qs ) and the equilibrium production quantity of the complementary firms (ye ).
a(2β + θ) + Qs θ − 2(ye + c0 (ye )) (4.30) ye (2 − θ2 ) Note that N is increasing in Qs . As the DGM sells a larger fraction of the products, N=
the number of firms who enter the complementary market increases. This demonstrates the role of selling when there is competition in the complementary market. Here, selling provides a credible commitment to greater availability of the durable product (and greater demand for the complement). Since the complementary market has no barriers to entry except a fixed cost, this increased demand translates into a greater number of firms entering the complementary market. Now we focus our attention on the first period. In the first period, the DGM makes her decision on the quantity and distribution strategy anticipating what effect these would have in the second period. The total profits from the two periods are: π1d (Ql , Qs ) = ps1d .Qs + pl1d Ql + ρπ2d (Qs , Ql (Qs ), Y (Qs )) (a + θN ye )2 − Q2s = (a − Ql − Qs )(Ql + Qs ) + 4 20
(4.31)
where N is given by (4.29) and ye by (4.28). We can obtain the optimal level of lease and sale quantity from first order conditions. Proposition 4.4 a) If a + θ3 (ye + c0 (ye )) > a(θ2 + βθ3 ), then the optimal sale and lease quantities are: a(1 − θ2 − βθ3 ) + θ3 (ye + c0 (ye )) 2(1 − θ2 ) ! 0 2 a(1 + βθ) − θ(ye + c (ye )) = θ 2(1 − θ2 )
Q∗∗ = l Q∗∗ s The fraction of leasing is: α∗∗ =
a(1 − θ2 − βθ3 ) + θ3 (ye + c0 (ye )) a(1 − θ2 )
(4.32)
b) If a + θ3 (ye + c0 (ye )) ≤ a(θ2 + βθ3 ), then pure selling is optimal (α∗∗ = 0). Pure leasing (α = 1) is never optimal for any θ > 0. Corollary 4.3 The DGM’s preference for selling increases as the size of the complementary market or the degree of complementarity between the two products increases. ∂(α∗∗ ) ∂(α∗∗ ) ≤ 0, and ∂θ ∂β ≤ 0. These results confirm that when we allow for entry and competition in the complementary market our results are
The above corollary follows from the
qualitatively similar to those that we obtained for a single CPP. That is, in the presence of any complementarity, some amount of selling is always optimal for the DGM, and there exists a threshold above which pure selling becomes optimal. This is because selling acts as a commitment mechanism that induces more firms to enter the complementary market. Now we can analyze the influence of cost structure of the complementary market on the distribution strategy of the DGM. Proposition 4.5 As the fixed cost for entry into the complementary market (F) decreases, the DGM sells a larger proportion of her first period output. As the fixed cost for entry decreases, the potential for the DGM to benefit from indirect network benefits increases. This increases the DGM’s need to signal that it will not behave opportunistically with respect to the CPPs in the second period. By selling a greater portion of the durable good, the DGM sends a stronger signal and thus encourages a larger number of CPPs to enter the market. 21
5
Concluding Remarks
In this paper we examine the effect that a complementary product has on a durable goods manufacturer’s distribution strategy to lease or sell her products. We show that in the case that the complementary product is produced by another firm(s) and the extent of complementarity is sufficiently strong, the durable goods manufacturer’s preference for leasing will shift to selling. In the absence of complete contracts, selling can act as a commitment mechanism that can mitigate opportunistic behavior and encourage a larger supply of the complementary product. From a managerial standpoint, we can represent the optimal strategy of the DGM as function of the relative market sizes and the level of complementarity. This can be depicted as follows in a 2 X 2 matrix. (Please refer to Fig. 6). The demand enhancement from the complementary market is induced through these two factors:, the level of complementarity between the two products and the relative size of the two markets. When a DGM anticipates a large market for a strongly complementary product, it is better off pursuing a pure selling strategy. Conversely, when complementarity is weak due to either a small market for the complementary product or due to low level of interaction between the two products, the firm would do well to use a pure leasing strategy. In all the other cases, mixed distribution strategy involving partial leasing and selling would be optimal for the DGM. We also analyze a situation in which there is competition in the complementary market and fixed costs for entry. Here, selling encourages a greater number of complementary firms to enter the market and effects greater demand stimulation of the durable good. Thus in all the cases we have considered, selling acts a Pareto-improving mechanism making all the players involved, better off. The firms are better off because they are able to generate higher profits and the consumers are better off because greater quantity of higher quality products are available. The recent developments on HDTV programming underlines the impact of a distribution strategy of a DGM on complementary products or services. While leasing was initially the dominant model for distribution of HDTV set-tops used by cable service providers, the last few years have seen an increasing tendency for these firms to sell their HDTV set-tops (Dow Jones Newswires 2003). Subsequently, there has been a significant
22
Figure 6: Distribution Strategy Matrix increase in the number of broadcasters who are planning to air HDTV programming. Since 2001, Fox, HBO and ESPN have announced that they would be entering this medium of communication in a big way (Business Week 2003, The New York Times 2003). Our analytical model may explain this interplay between the distribution strategy of the HDTV set-tops and the fixed costs that firms have to incur to enter the HDTV programming market. Technological advances in programming have made production and broadcasting costs for the digital media lower and thus reduced the fixed costs of entry for HDTV programming. These developments may have played a role in prompting the cable companies to shift their distribution strategy towards selling. Future work could look at other mechanisms that could induce greater level of investments from complementary firms. Committing to prices and quantities to mitigate hold-up is prevalent when firms are related vertically. However, these mechanisms need explicit contracts and hence enforceability could be an issue when firms are not directly related. It would also be interesting to consider cases when firms do not have full information about the other firms’ costs and opportunities of investment.
23
References Bagnoli, S., S. Salant, & J. Swierzbinski (1980). Durable goods monopoly with discrete demand. Journal of Political Economy 97, 1459–1478. Bond, E. & L. Samuelson (1984). Durable goods monopolies with rational expectations and replacement sales. Rand Journal of Economics 17 (3), 336–345. Bucovetsky, S. & J. Chilton (1986). Concurrent renting and selling in a durable goods monopoly under threat of entry. Rand Journal of Economics 17, 261–278. Bulow, J. (1982). Durable goods monopolists. Journal of Political Economy 90 (2), 314– 332. Bulow, J. (1986). An economic theory of planned obsolescence. Quarterly Journal of Economics 51, 729–750. Business Week (2003, Apr). HDTV: Coming into focus? Choi, J. P. (1994). Network externality, compatibility choice and planned obsolescence. Journal of Industrial Economics 42 (2), 167–182. Coase, R. (1972). Durability and monopoly. Journal of Law and Economics 15, 143–149. Conlisk, J., E. Gerstner, & J. Sobel (1984). Cyclic pricing by a durable goods monopolist. Quarterly Journal of Economics 99, 489–505. Conner, K. R. (1995). Obtaining strategic advantage from being imitated: When can encouraging clones pay? Management Science 41 (2), 209–225. Desai, P. & D. Purohit (1998). Leasing and selling: Optimal marketing strategies for a durable goods firm. Management Science 44 (11), 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. (1994). Durable-goods monopolists, rational consumers, and improving products. Marketing Science 13 (1), 100–120. Dow Jones Newswires (2003, June). Coming soon, cable companies to hawk services at retail stores. Farrell, J. & G. Saloner (1985). Standardization, compatibility and innovation. Rand Journal of Economics 16, 70–83. Farrell, J. & G. Saloner (1986). Installed base and compatibility: Innovation, product preannouncements, and predation. The American Economic Review 76 (5), 940–955. Gilbert, S. M. & V. Cvsa (2003). Strategic commitment to price to stimulate downstream innovation in a supply chain. European Journal of Operational Research, forthcoming. Hendel, I. & A. Lizzeri (2002). The role of leasing under adverse selection. Journal of Political Economy 110 (1), 113–143. 24
Kahn, C. (1986). The durable goods monopolist and consistency with increasing costs. Econometrica 54 (2), 275–294. Katz, M. L. (1989). Vertical Contractual Relations. North-Holland, Amsterdam: Handbook of Industrial Organization. Katz, M. L. & C. Shapiro (1985). Network externality, competition and compatibility. The American Economic Review 75 (3), 424–440. Katz, M. L. & C. Shapiro (1994). Systems competition and network effects. Journal of Economic Perspectives 8 (2), 93–115. Klein, B., R. Crawford, & A. Alchian (1978). Vertical intergration, appropriable rents, and the competitive contracting process. Journal of Law and Economics 26, 297–326. Kornish, L. J. (2001). Pricing of a durable goods monopolist under rapid sequential innovation. Management Science 47 (11), 1552–1561. Levinthal, D. A. & D. Purohit (1989). Durable goods and product obsolescence. Marketing Science 8 (1), 35–56. Nair, H., P. Chintagunta, & J. P. Dub (2003). Empirical analysis of indirect network effects in the market for personal digital assistants. University of Chicago, Working Paper . Parker, G. G. & M. Van Alstyne (2001a). Information complements, substitutes and strategic product design. mimeo: University of Michigan. Parker, G. G. & M. Van Alstyne (2001b). Unbundling in the presecnce of network externalities. mimeo: University of Michigan. Rochet, J. C. & J. Tirole (2001). Platform competition in two-sided markets. mimeo: University of Toulouse. Stokey, N. (1981). Rational expectations and durable goods pricing. Bell Journal of Economics 12, 112–128. Subramaniam, B. & K. Srinivasan (1998). Modifying customer expectations of price decreases for a durable product. Management Science 44 (6), 776–786. Subramaniam, V. (1998). Efficient sourcing and debt financing in imperfect product markets. Management Science 44 (9), 1167–1178. The New York Times (2003, June). Fox television to add HDTV to primetime by Fall 2004 . Waldman, M. (1993). A new perspective on planned obsolescence. The Quarterly Journal of Economics 108 (1), 273–283. Wall Street Journal (2003, April). Fuel-cell car concept raises a tricky question: Who will pay for it? Williamson, O. (1975). Markets and Heirarchies: Analysis and Anti-trust Implications. New York: Free Press. 25
Appendix Proof of Proposition 3.1 Proof. By differentiating (3.8) and (3.9) with respect to α, it is easy to confirm that Q2 (Q1 , α) is increasing in α and y(Qa , α) is decreasing in α. For the second part of the proposition, note that ∂ 2 y(α, Q1 ) Q1 (4 + θ2 ) = − Q1 (a − Q1 ) + ρ (4 − θ2 )
where the right hand side is equal to πdO (Q1 , pe ) when pe ≥
a(2+βθ) . 4−θ2
!2
♦
Proof of Proposition 4.4 It can be verified that the Hessian matrix for (4.31) is negative definite, so first-order conditions are sufficient for optimality. 27
a) After substituting 4.29 into 4.31 and employing first order conditions, it is easy to confirm that partial derivatives of (4.31) with respect to Ql and Qs are equal to zero at the 3 0 2 3 ∗∗ point (Q∗∗ l , Qs ). Both of these are non-negative so long as a+θ (ye +c (ye )) > a(θ +βθ ),
and the optimal leasing fraction, α∗∗ follows immediately. b) From (4.32) it can be seen that the requirement that Ql ≥ 0 becomes binding when a + θ3 (ye + c0 (ye )) ≤ a(θ2 + βθ3 ). So when this is satisfied, pure selling is optimal. To see that pure leasing is never optimal for θ > 0, observe that Q∗∗ s > 0 unless a(1 + βθ) ≤ θ(ye + c0 (ye )). We can use 4.27 to substitute into this inequality to obtain: 1 a +β ≤ θ
F + c(ye ) ye
!
For θ < 1, we have that: 1 a (θ + β) < a +β ≤ θ
F + c(ye ) ye
!
The left hand side of this inequality is an upper bound on the maximum price for which there is positive demand for the complement, and the right hand side represents the average per-unit cost of production. Since we cannot have an equilibrium in which the average production cost exceeds the price, it follows that we cannot have Q∗∗ s ≤ 0. Proof of Proposition 4.5 Proof. We can explicitly solve for the number of firms in equilibrium, Ne using (4.29). The number of firms are: Ne =
a(2β + θ − βθ2 ) − (2 − θ2 )(ye + c0 (ye )) 2ye (1 − θ2 )
(5.34)
To find the relationship between (1 − α) and F, we first rearrange 5.34 as follows: Ne y e =
a(2β + θ − βθ2 ) − (2 − θ2 )(ye + c0 (ye )) 2(1 − θ2 )
Differentiating this expression, we have: Ne + y e
∂N (2 − θ2 )(1 + c00 (ye )) =− < 0 ∀ θ (0, 1) ∂ye (1 − θ2 )
where the inequality follows from the convexity of c(y) which implies c00 (ye ) > 0. From eqn. 4.32, we can see that ∂α 2θ2 ∂N ∂ye =− ye + Ne 2 ∂F a(2 − θ ) ∂ye ∂F !
28
(5.35)
∂y We can find ∂Fe by differentiating eqn 4.27 1 ∂ye = ∂F 2ye + c00 (ye ) Since the production costs are convex and increasing, we know that c00 (ye ) is greater than ∂y ∂N y zero. So ∂Fe > 0. This property in conjunction with the fact that ∂ye e < 0 shows that e ∂α ∂F > 0 and so the manufacturer’s preference for selling increases as fixed costs for entry (F) decreases. ♦
29
Mixed Strategy(4 ≥ θ2 (4 + βθ))
Pure Selling (4 < θ2 (4 + βθ))
Sale Qty. (Qs )
aθ2 (2+βθ) 4(2−θ2 )
a(16+θ4 −2θ2 (4−ρ)+βθ3 ρ 2(θ4 +4(4+ρ)−2θ2 (4+ρ))
Lease Qty. (Ql )
a(4−4θ2 −βθ3 ) 4(2−θ2 )
0
a(2+βθ) 4
a(4−θ2 )(2+2βθ+θ2 )+a(2+βθ)(2−θ2 )ρ 2(θ4 +4(4+ρ)−2θ2 (4+ρ))
a(2θ+β(4−θ2 )) 4(2−θ2 )
a(β(4−θ2 )(4+ρ)+θ(2(6+ρ)−3θ2 ) 2(θ4 +4(4+ρ)−2θ2 (4+ρ))
Lease Fraction (α)
4−4θ2 −βθ3 2(2−θ2 )
0
DGM Profits(Πd )
a2 (4βθρ+4(1+ρ)−θ2 (2−β 2 ρ) 8(2−θ2 )
a2 (θ4 + 2βθ3 ρ + 4(2 + ρ)2 + 4βθρ(4 + ρ)
2nd period Qty. (Q2 )
30
CPP Qty.(y)
+θ2 (4(1+β 2 )ρ+β 2 ρ2 −8)) 4(θ4 +4(4+ρ)−2θ2 (4+ρ)) CPP Profits(Πc )
a2 (2θ+β(4−θ2 )2 16(2−θ2 )2
a(θ(3θ2 −2(6+ρ))−β(4−θ2 )(4+ρ)) ρ 2(θ4 +4(4+ρ)−2θ2 (4+ρ)
Table 1: Quantities and Profits under different distribution strategies
!2
