“Planned antiobsolescence” occurs when consumers engage in maintenance

Int. J. Ind. Organ. 28Organization (2010) 441–450 International Journal of Industrial 28 (2010) 441–450

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International Journal of Industrial Organization j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j i o

“Planned antiobsolescence” occurs when consumers engage in maintenance Hiroshi Kinokuni a,⁎, Takao Ohkawa a, Makoto Okamura b a b

Faculty of Economics, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan Economics Department, Hiroshima University, 1-2-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8525, Japan

a r t i c l e

i n f o

Article history: Received 1 June 2007 Received in revised form 17 September 2009 Accepted 28 October 2009 Available online 4 November 2009 JEL classification: L12 D21 M21

a b s t r a c t This paper examines whether the built-in durability chosen by a durable-good monopolist and the actual durability associated with consumer maintenance are excessive or insufficient. Our research indicates that only in situations where both the consumer maintenance market and the secondhand market exist can the monopolist select a socially excessive durability level: that is, planned antiobsolescence requires both markets. However, overinvestment in built-in durability does not necessarily lead to a socially excessive level of actual durability. We also show that the existence of maintenance may enhance built-in durability, even though maintenance activity can partially recover the depreciation in durability. © 2009 Elsevier B.V. All rights reserved.

Keywords: Durable goods monopoly Built-in durability Actual durability Maintenance Planned antiobsolescence

1. Introduction Durable goods industries often accompany secondhand markets. Automobile, housing, and computer industries have their own secondhand markets. Consumers in these markets can recover product durability by caring for, repairing, or maintaining their own products. An automobile is a typical product where maintenance matters. Automobile owners spend much time and effort on maintaining their vehicles. In the real world then, both the consumer maintenance market and the secondhand market affect the level of durability chosen by the supplier of the durable good. Most previous studies have focused on either of these markets.1 For instance, Waldman (1996) focused on the secondhand market. He established that the durable goods monopolist intentionally practices planned obsolescence to weaken the substitutability between new and used products. That is, lowering the quality of the used product

⁎ Corresponding author. Tel.: + 81 77 561 4991; fax: + 81 77 561 3947. E-mail addresses: [email protected] (H. Kinokuni), [email protected] (T. Ohkawa), [email protected] (M. Okamura). 1 Coase (1972) suggested that planned obsolescence gives the monopolist the ability to commit to the price of the new good in the future. Bulow (1986) and Bond and Samuelson (1984) modeled Coase's suggestion. They assumed that the used product is a perfect substitute for the new product. Therefore, the secondhand market is inactive in equilibrium. Waldman (2003) provides a comprehensive survey of the theory of durable goods. 0167-7187/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2009.10.008

raises the price of the new product and profits.2 Schmalensee (1974), Su (1975), and Rust (1986) concentrated on maintenance activity. They introduced maintenance into Swan's (1970) model, and investigated whether consumers discard the old product and purchase a new one, or extend the lifetime of the old product. They showed that low maintenance prices encourage excessive maintenance efforts, resulting in insufficient built-in durability.3 Because new and used goods in their model are perfect substitutes, the secondhand market is inactive, that is, used goods remain untraded in the secondhand market.4 Then we ask whether the monopolist chooses socially excessive or insufficient built-in durability when both the consumer maintenance market and the secondhand market exist. According to previous research, a plausible hypothesis naturally follows. Put simply, the 2 Waldman assumed that two types of consumers exist. Hendel and Lizzeri (1999) extended Waldman's model to an economy with continuous types of consumers and obtained planned obsolescence. 3 Rust (1986) showed that the monopolist sets an excessive built-in durability level when replacement rarely occurs. However, he did not examine the effect of built-in durability on the monopolist's profitability and consumer surplus through the secondhand market. 4 Another strand of research includes the aftermarket issue. Hendel and Lizzeri (1999), Kinokuni (1999), and Morita and Waldman (2004) analyzed the monopolists' incentives to monopolize the ‘aftermarket’, i.e., the repair (or maintenance) market. They assumed that built-in durability is exogenously determined and addressed the antitrust issue that the manufacturers of the principal product monopolize the aftermarket. Instead, we examine how the existence of maintenance activities affects the efficiency of the built-in durability.

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Fig. 1. Sequence of events.

monopolist always selects socially insufficient built-in durability, i.e., planned obsolescence necessarily occurs. The main purpose of this paper is to investigate the validity of this hypothesis. By introducing consumer maintenance activities into Waldman's (1996) durable goods monopoly model with two periods and two types of consumers, we develop a model that includes both the consumer maintenance market and the secondhand market. We will show that when both markets exist, the monopolist chooses a socially excessive built-in durability: that is, “planned antiobsolescence” can occur. Surprisingly, the above hypothesis is false. A brief explanation of the logic underlying this assertion is as follows. An increase in built-in durability has both positive and negative effects on the new product price via increasing the quality of the used product. The positive effect arises through an increase in the resale price in the secondhand market, while the negative effect arises through a rise in the value of retaining the used product. Since maintenance is a substitute for built-in durability, the existence of maintenance reduces both effects. Suppose that the difference in the maintenance recovery effects of two types of consumers, i.e., consumers with high valuation on a quality recovery and consumers with low valuation, is large. The maintenance recovery by the low-type consumers is small, while that by high-type consumers is large. This reduces the negative effect more than the positive effect. Therefore, the monopolist chooses a socially excessive level of built-in durability. This contrasts with Waldman's findings where the negative effect dominates the positive effect and planned obsolescence occurs. We will also examine whether the introduction of maintenance reduces the durability level, given the existence of a secondhand market. The introduction of the secondhand market may enhance the durability level, although maintenance may partly recover the durability depreciation. Furthermore, we examine the efficiency of the realized durability after consumer maintenance, so-called actual durability. We establish that even if planned antiobsolescence prevails, the actual durability level may become socially insufficient as a result of lower maintenance recovery. The outline of the paper is as follows. Section 2 develops a twoperiod model of a durable goods monopoly when both the consumer maintenance and the secondhand markets exist. Section 3 derives the equilibrium in this case and when either market exists. Section 4 shows how built-in durability affects new product prices. Section 5 investigates the efficiency of built-in durability. Section 6 examines whether the introduction of maintenance reduces the durability level, given the existence of a secondhand market. Section 7 addresses the efficiency of actual durability through consumer maintenance. Section 8 presents an example and examines the results. Section 9 concludes.

two periods at the most. We assume that the new product's quality is normalized to unity and that one-period usage depreciates quality by 1 − q (0 ≤ q b 1). Quality depreciation occurs at the end of a period. We refer to q as the level of built-in durability because the monopolist decides q by paying the cost when it supplies a new product. The unit cost of the new product with quality q in each period is given by c(q) N 0. We assume that c′(0) = 0, c′(q) N 0 and c″(q) N 0 for all q ∈ [0,1). Because we employ a two-period model, the monopolist has no incentive to choose a positive built-in durability in period 2. Therefore, the unit cost of the second-period new product is c(0) N 0. There are two types of consumers, type H and type L, both of whom live for two periods. Each individual consumes one unit of the product at most. The type H (L) consumer has a unit valuation for the quality vH (vL), with vH N vL N 0. The group size of type i consumer is given by ni (i = L, H). The monopolist and all consumers have a common discount factor, δ ∈ (0,1]. In the second period, consumers make maintenance effort m≥ 0 and can recover a fraction of the depreciated built-in durability, 1 −g(m), where 0 b g(m) ≤ 1, g′(m) b 0, g″(m) N 0, and g(0) = 1 hold. Actual (realized) durability θ, representing the reconditioned quality of the used product, is θðq; mÞ ≡ q + ð1−qÞð1−gðmÞÞ = 1−ð1−qÞgðmÞ:

ð1Þ

Note that θ(q, 0) =q. The function g(m) represents the recovery rate of depreciated built-in durability. We assume the maintenance market is competitive and the unit cost is r N 0.5 The price per unit of maintenance is r. Production and maintenance activities incur no fixed costs. The model consists of two periods: periods 1 and 2 (see Fig. 1). Period 1 includes two stages. In the first stage, the monopolist chooses a level of built-in durability. In the second stage, the monopolist decides the price of the new product. Period 2 also includes two stages. In the third stage, the monopolist sets the price of the new product. At the same time, a secondhand market opens and equilibrium is achieved. In the fourth stage, each consumer determines a maintenance level that maximizes her net surplus. We assume that in period 1, the monopolist is unable to commit to its second-period decision. We consider the consumer's choice of maintenance level in the fourth stage. In the second period, either the type H consumer or the type L consumer engages in maintenance. When the type i consumer (i = L, H) spends m on maintenance, and uses the maintained product, the resulting net surplus is given by NSi ≡ θðq; mÞvi −rm:

2. The basic model Following Waldman (1996), we consider a two-period model where in each period a monopolist supplies a new product that lasts

5 In many industries (e.g., automobiles, housing), maintenance markets are competitive. However, in some industries (e.g., computers), maintenance markets may be monopolized by the manufacturers of the principal product.

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443

Fig. 2. Built-in durability and maintenance.

The consumer chooses m to maximize the net surplus. The firstorder condition is −g ′ ðmÞð1−qÞvi − r = ð≤Þ0 if m Nð = Þ0:

ð2Þ

Condition (2) shows that the type i consumer sets a level of maintenance where the marginal utility from improving the used product equals the maintenance price. We denote the solution of Eq. (2) with mi(q). Fig. 2 depicts condition (2). The maintenance has an upper bound m̅i where g′(m̅i)vi + r = 0. If g′(0) is negative infinite, then limq→1 mi (q) = 0. If g′(0) is negative finite, there exists a unique threshold qi such that mi(q) N ( = )0 for q b ( ≥ )qi and 0 b qL b qH b 1. From Eq. (2), we have the following results.6 Lemma 1. (i) The maintenance level of both types decrease with built-in durability. (ii) The maintenance level of both types decrease with the maintenance price. (iii) The type H consumer has a stronger incentive to maintain the used product than the type L consumer. Lemma 1(i) means that built-in durability and maintenance are substitutes. Lemma 1(ii) states that the demand for maintenance is a decreasing function of the maintenance price. Lemma 1(iii) shows that the type H consumer will choose a higher maintenance level than would the type L one if the type H consumer decides to keep the used product in period 2. We define NSi(q) ≡ θ(q,mi(q))vi − rmi(q) and characterize this function in Lemma 2. Lemma 2. (i) Each consumer gains a positive net surplus from the used product when she chooses the optimal level of maintenance, i.e., NSi(q) N 0 for all q ∈ [0,1). (ii) The difference in the valuation of the new product and the used product for the type H consumer is larger than that of the type L consumer, i.e., vH − NSH(q) N vL − NSL(q) for all q ∈ [0,1). 3. The equilibrium We derive the equilibrium in three cases. (Case MS) Both the consumer maintenance and the secondhand markets exist.

6 All the proofs of the lemmas and propositions are available upon request from the author.

(Case S) The secondhand market exists, but the consumer maintenance market does not. (Case M) The maintenance market exists, but the secondhand market does not exist. Hereafter, let qeMS, qeS and qeM denote the equilibrium level of the built-in durability in Cases MS, S, and M. As in Waldman (1996), we focus on the equilibrium where the type H consumer purchases the new product from the monopolist in periods 1 and 2, and the type L consumer only purchases the used product from the type H consumer in period 2. This is the only case where the secondhand market opens in period 2. For the equilibrium to exist, the following conditions are assumed7: nL N nH ;

ð3aÞ

vL + δNSL ðqÞ b cðqÞ for all q ∈ ½0; 1Þ;

ð3bÞ

vH N cð0Þ = vL + ε; ε N 0 is small; and

ð3cÞ

vH + δð2vL −vH Þ≥cð1Þ:

ð3dÞ

Assumption (3a) guarantees that, when the type H consumer wants to sell her own used product, she can find buyers for the used product.8 Therefore, the used product is traded at a positive price if the secondhand market opens. Assumptions (3b) and (3c) are the sufficient conditions that the monopolist has no incentive to sell the new product to the type L consumer in both periods. Assumption (3c) also ensures that the monopolist has an incentive to sell the new product to the type H consumer in period 2. Assumption (3d) implies that selling the new product to the type H consumer in period 1 is profitable for the monopolist and the equilibrium profits are positive. We also assume that c ′ ð0Þ = 0bδð2vL −vH Þbc ′ ð1Þ:

ð3eÞ

This means that an interior solution for q exists. 7 Assumptions (3a) and (3b) are identical to Waldman (1996). Assumption (3c) is slightly stronger than Waldman (1996). Assumptions (3d) and (3e) are the same as those implicitly assumed in Waldman (1996). 8 We implicitly assume that there exist some type L consumers who buy nothing at the equilibrium.

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3.1. The equilibrium with both maintenance and secondhand markets (Case MS) This subsection considers an economy where both the consumer maintenance and the secondhand markets exist. We prove that an equilibrium exists where the monopolist sells the new product to the type H consumer in periods 1 and 2 and the type L consumer only purchases the used product in period 2 under assumptions (3a)– (3e).9 The proof is as follows: The equilibrium is derived using backward induction. The first step formulates the individual incentive constraint of each type of consumer in period 2. The second step considers the monopoly equilibrium of the new product market and the competitive equilibrium of the used product market in period 2. The third step derives the individual incentive constraint of each type of consumer in period 1. The fourth step considers the monopoly equilibrium of the new product market in period 1. The fifth step examines the monopolist's choice of built-in durability. Step 1. We derive the individual incentive constraint of each type of consumer in period 2 given that the type H consumers have purchased the new product in period 1. Since a consumer consumes one unit of product at most, if a type H consumer purchases a new product in period 2, she has an incentive to resell the used product to the type L consumer.10 The individual incentive constraint of a type H consumer who purchases the new product is11 vH + PU −P2 ≥ NSH ðqÞ:

ð4Þ

The LHS of Eq. (4) shows the type H consumer's net surplus when she purchases the new product at price P2 from the monopolist and sells the used product to a type L consumer at price PU. The RHS is the type H consumer's net surplus, which is positive from Lemma 2(i), when she continues to hold the used product (and engages in maintenance in the next stage). The individual incentive constraint of a type L consumer who purchases the used product is NSL ðqÞ−PU ≥ 0:

ð5Þ

The LHS of Eq. (5) states the type L consumer's net surplus when she purchases the used product at price PU. Note that NSL(q) N 0 from Lemma 2(i). The RHS is the type L consumer's net surplus when she purchases nothing. The individual incentive constraint ensuring a type L consumer does not purchase the new product is vL −P2 b 0:

ð6Þ

Step 2. We consider the second-period price formation of the new and used products in the third stage. Since the monopolist is unable to commit to its second-period choice in period 1, and given its concerns with second-period profitability, we need the constraint: P2 ≥ cð0Þ:

ð7Þ

Fig. 3. Decision on equilibrium prices.

consumer to purchase the new product. Even if the monopolist charges vL, it incurs a loss because (nH + nL)[vL − c(0)] b 0 from assumption (3c). The monopolist chooses the maximum of P2 that satisfies constraints (4), (5), (6), and (7). In Fig. 3, point E shows the combination of the equilibrium prices (P̂ 2, P̂ U). Thus, the equilibrium price P̂ U and quantity X̂ U in the secondhand market are P Û = P Û ðqÞ ≡ NSL ðqÞN 0; X ̂U = nH : The equilibrium price P̂ 2 and quantity X̂ 2 of the second-period new product are P 2̂ = P 2̂ ðqÞ≡vH + P Û ðqÞ−NSH ðqÞ = vH + NSL ðqÞ−NSH ðqÞN cð0Þ;

ð9Þ

X̂ 2 = nH :

Because the type H consumer's incentive to replace the used product with the new product strengthens as the price of the used product rises, it increases the price of the new product. For the type H consumer, retaining the used product is the opportunity cost of buying the new product. Fig. 4 shows the demand curve for the used product, given by constraint (5), and the supply curve of the used product, derived from constraint (4), and the competitive equilibrium price P̂ U. Fig. 5 depicts the demand curve for the second-period new product, given by constraint (4), and the monopoly equilibrium price P̂ 2.

In Fig. 3, the shaded area satisfies constraints (4), (5), (6), and (7) in the plane (P2, PU). Lemma 2(ii) guarantees that the point (c(0), c(0) − vH + NSH(q)), which is the intersect of P2 = c(0) and PU = P2 − vH + NSH(q), is below the point (c(0), NSL(q)) as long as c(0) is close to vL (assumption (3c)). The monopolist has no incentive to charge a second-period price below vL. This ensures condition (6) which does not induce the type L 9 We appreciate an anonymous referee's excellent comments that helped to greatly improve the argument on the existence of the equilibrium. 10 We assume that the used product's scrap value is zero. 11 The individual incentive constraint, and not that of the consumer group, should be examined.

ð8Þ

Fig. 4. Demand and supply curves for used product.

H. Kinokuni et al. / Int. J. Ind. Organ. 28 (2010) 441–450

At the first-period equilibrium, all the type H consumers are indifferent as to purchasing the new product in the first period or postponing the purchase of the new or used product in the following period. All the type L consumers are indifferent as to purchasing nothing in the first period and purchasing the used product in the following period, on the one hand, or purchasing the new product in the first period, on the other hand. As a result, no consumers have an incentive to deviate from this equilibrium. Fig. 6 shows the demand curve for the first-period new product, given constraint (10), and the monopoly equilibrium price P̂ 1. The first-period price does not necessarily exceed c(q). It is sufficient to show that the monopolist prefers to introduce the first-time product in period 1 rather than in period 2. The monopolist's overall profits are

Fig. 5. Demand and marginal cost curves for second-period new product.

ΠðqÞ = nH ½P 1̂ ðqÞ−cðqÞ + δnH ½P 2̂ ðqÞ−cð0Þ: At the second-period equilibrium, all the type H consumers are indifferent as to selling the used product and purchasing a new product, on the one hand, or maintaining the used product and keeping it, on the other hand. All the type L consumers are indifferent as to purchasing the used product or purchasing nothing. As a result, no consumers have an incentive to deviate from this. Step 3. We examine the individual incentive constraint of each type of consumer in period 1 given the second-period equilibrium. The individual incentive constraint of a type H consumer who purchases the new product in period 1 is vH −P1 + δ½vH + P Û −P 2̂ ≥ maxfδ½vH −P 2̂ ; δ½NSH ðqÞ−P Û g:

ð10Þ

The LHS of Eq. (10) shows the type H consumer's net surplus when she purchases the first-period new product. The RHS is the type H consumer's net surplus when she postpones consumption and purchases the new product at price P̂ 2 from the monopolist or the used product at price P̂ U from the type L consumer in period 2. The individual incentive constraint of a type L consumer who does not purchase the new product in period 1 is δ½NSL ðqÞ−P Û N vL −P1 + maxfδNSL ðqÞ; δP Û g:

ð11Þ

The RHS of Eq. (11) shows the type L consumer's net surplus when she purchases the first-period new product. The LHS is the type L consumer's net surplus when she purchases the used product at price P̂ U from the other type L consumer in period 2. Step 4. We consider the first-period price decision of the new product in the second stage. Combining Eqs. (10) and (11) yields vL + δP Û ðqÞbP1 ðqÞ≤vH + δP Û ðqÞ:

ð12Þ

The monopolist has no incentive to charge a second-period price below vL + δNSL(q). This ensures condition (11) which does not induce the type L consumer to purchase the new product. Even if the monopolist charges vL + δNSL(q), it incurs a loss because (nH + nL)[vL + δNSL(q) − c(q)] b 0 for all q ∈ [0,1) from assumption (3b). The monopolist charges the maximum price that satisfies constraint (12). The equilibrium price P̂ 1 and quantity X̂ 1 of the first-period new product are P 1̂ = P 1̂ ðqÞ≡vH + δP Û ðqÞ = vH + δNSL ðqÞN 0;

445

We rewrite Eq. (14) as Π(q) = η1(q) + η2, where η1(q) ≡ nH[vH + δ{2NSL(q) − NSH(q)} − c(q)] and η2 ≡ δnH[vH − c(0)]. The monopolist earns a profit η2 if the monopolist does not sell the first-period new product and sells the new product only in period 2. Assumption (3d) means that η1(1) ≥ 0 and assumption (3e) implies that η′1(1) b 0. These assumptions guarantee that there exists q ∈ [0,1) such that η1(q) N 0. The monopolist then has an incentive to sell the new product to the type H consumer in period 1. Step 5. We investigate the monopolist's decision on built-in durability in stage 1. The monopolist determines q to maximize its overall profits. The profit-maximizing condition is ′ ′ ′ P1̂ ðqÞ + δP2̂ ðqÞ−c ðqÞ = 0;

or equivalently, δgðmL ðqÞÞvL + δ½gðmL ðqÞÞvL −gðmH ðqÞÞvH −c ′ ðqÞ = 0:

ð15Þ

Condition (15) shows that the monopolist sets built-in durability where marginal revenue equals marginal cost. The marginal revenue consists of the marginal changes in the first-period and second-period new product prices. Equilibrium built-in durability in Case MS, qeMS, is the solution of Eq. (15). Because Π′(0) N 0 and Π′(1) b 0 from assumption (3e), we have qeMS ∈ (0,1). From assumption (3d), we have Π(qeMS) ≥ η2 N 0, meaning that the monopolist can enjoy positive profit at the equilibrium. By inserting qeMS into Eqs. (2), (8), (9), (13), and (14), we obtain the equilibrium level of the remaining variables. We summarize the above arguments in the following proposition. Proposition 1. Suppose that both the consumer maintenance market and the secondhand market exist. If assumptions (3a)–(3e) hold, an equilibrium exists where the monopolist chooses qeMS ∈ (0,1) and sells the new product to the type H consumer in periods 1 and 2, and the type L consumer purchases the used product in period 2.

ð13Þ

X̂ 1 = nH : A consumer who purchases the first-period new product takes account of the benefit of reselling it in the future. The type H consumer's incentive to purchase the new product in period 1 strengthens as the used product's price rises, raising the first-period new product's price.

ð14Þ

Fig. 6. Demand and marginal cost curves for first-period new product.

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3.2. The equilibrium with a secondhand market and without a maintenance market (Case S) This subsection considers an economy where a secondhand market exists but a maintenance market does not. We examine the equilibrium where the monopolist sells the new product to the type H consumer in both periods and the type L consumer only purchases the used product in period 2 under assumptions (3a)–(3e).12 When there exists no consumer maintenance, the equilibrium does not depend on condition (2). Instead, we use mL = mH = 0, g(0) = 1, and NSi = qvi, i = H, L. Since Cases MS and S are connected at mL = mH = 0, the continuity is guaranteed. Thus, the profit-maximization condition (15) reduces to δvL + δðvL −vH Þ−c ′ ðqÞ = 0:

ð16Þ

The equilibrium built-in durability in Case S, qeS ∈ (0,1), is the solution of Eq. (16). Condition (16) corresponds to that in Waldman (1996, p. 508). Corollary 1. Suppose that a secondhand market exists but the maintenance market does not. If assumptions (3a)–(3e) hold, an equilibrium exists where the monopolist chooses qeS ∈ (0,1) and sells the new product to the type H consumer in periods 1 and 2, and the type L consumer purchases the used product in period 2. 3.3. The equilibrium with a maintenance market and without a secondhand market (Case M) This subsection considers an economy where the consumer maintenance market exists but the secondhand market does not.13 We examine the equilibrium where the monopolist sells the new product to the type H consumer only in period 1 and the consumer continues to hold the used product in period 2. For the equilibrium to exist, instead of assumptions (3a)–(3e), we assume as follows: ð1 + δÞvH ≥ cð1Þ;

ð3dÞ′

c ′ ð0Þ = 0 b δvH b c ′ ð1Þ; and

ð3eÞ′

mH Þ + r ― mH b cð0Þ: vH gð―

ð3fÞ

Assumption (3d)′ ensures that selling the new product to the type H consumer in period 1 is profitable for the monopolist and the monopolist can earn positive profit at the equilibrium. Assumption (3e)′ guarantees that there exists an interior solution for q. Assumption (3f) implies that the monopolist has no incentive to provide the new product in period 2.14 Thus, the monopolist sells the new product only in period 1. The first-period new product price is P 1̂ ðqÞ = vH + δNSH ðqÞ:

ð17Þ

The monopolist's profits are ΠS ðqÞ = nH ½vH + δNSH ðqÞ−cðqÞ:

ð18Þ

The first-order condition of profit maximization is δgðmH ðqÞÞvH −c ′ ðqÞ = 0:

ð19Þ

The equilibrium level of built-in durability in Case M, qeM ∈ (0,1), is the solution of Eq. (19). Corollary 2. Suppose that the consumer maintenance market exists but the secondhand market does not. If assumptions (3d)′, (3e)′, and (3f) hold, an equilibrium exists where the monopolist chooses qeM ∈ (0,1) and sells the new product to the type H consumer in period 1. 4. How built-in durability affects prices This section analyzes how the built-in durability affects product prices. We consider Case MS. Using Eq. (2), a marginal change in the used product's price is ′

P Û ðqÞ = NSL ðqÞ = gðmL ðqÞÞvL : ′

ð20Þ

As built-in durability increases, the type L consumer's willingness to pay for the used product increases via the improvement in quality, and pushes up the used product price in the secondhand market.15 Using Eqs. (13) and (20), the marginal change in the first-period product price is ′ ′ P 1̂ ðqÞ = δP Û ðqÞ = δgðmL ðqÞÞvL ;

ð21Þ

which corresponds to the first term of Eq. (15). Using Eq. (21), P̂ 1 increases with q because the resale price (i.e., the used product's price in the secondhand market) also increases. Using Eqs. (2), (9), and (20), the marginal change in the second-period product's price is ′ ′ ′ P 2̂ ðqÞ = P Û ðqÞ−NSH ðqÞ = gðmL ðqÞÞvL −gðmH ðqÞÞvH ;

ð22Þ

which yields the second term of Eq. (15). From Eq. (22), an increase in built-in durability has a positive effect on the second-period new product price—the first term of Eq. (22)—because it increases the resale price and encourages type H consumers to buy the new product. An increase in built-in durability also has a negative effect on the price—the second term of Eq. (22)—because it improves the type H consumer's surplus associated with retaining the used product and reduces her willingness to pay for the new product. When the difference between g(mL(q)) and g(mH(q)) is sufficiently large (small), the positive effect dominates (is dominated by) the negative effect, resulting in P̂ 2′ (q) N (b)0. We show the effects of built-in durability on the product price in all cases in Lemma 3 and Table 1. Lemma 3. (i) When both the consumer maintenance and the secondhand markets exist, then P̂ 1′ (q) N 0 and P̂ 2′ (q) N (b)0 if g(m) is strictly log-concave (strictly log-convex) for all m ≥ 0. (ii) When a secondhand market exists, but not a consumer maintenance market, then P̂ 1′ (q) N 0 and P̂ 2′ (q) b 0. (iii) When a consumer maintenance market exists, but not a secondhand market, P̂ 1′ (q) N 0. 5. Efficiency of built-in durability

12

Without consumer maintenance, we rewrite assumption (3b) as vL(1 + δq) b c(q) for all q 2 [0,1). 13 This corresponds to Schmalensee (1974), Su (1975), and Rust (1986). They defined durability as the product lifetime and developed continuous-time and infinitehorizon models. Although our model is quite different, it encompasses their fundamental character. 14 Without the secondhand market, a consumer who replaces the used product with the new one in period 2 will dispose of the used product. In order to sell the new product in period 2, the monopolist has to lower the price to vH − NSH(q). This price must be quite low. Assumption (3f), i.e., maxq[vH − NSH(q)] b c(0), is a more reasonable assumption when no secondhand market exists.

We investigate the efficiency of equilibrium built-in durability in three cases. We consider the second-best solution in the sense that the social planner can set built-in durability, but not intervene in ⁎ denote the consumer maintenance decisions. Let q⁎MS, qS⁎, and qM efficient level of built-in durability in Cases MS, S, and M, respectively. 15 The effect of a price change on consumer maintenance choices through the effect of built-in durability is negligible because consumers always optimally adjust their maintenance efforts (Envelope Theorem).

H. Kinokuni et al. / Int. J. Ind. Organ. 28 (2010) 441–450 Table 1 Price changes (marginal revenue) with respect to q in periods 1 and 2.

Case MS Case S Case M

Table 2 Secondhand market and reservation utility effects.

P̂ 1′ (q)

P̂ 2′ (q)

δg(mL(q))vL δvL δg(mH(q))vH

g(mL(q))vL − g(mH(q))vH vL − vH –

5.1. Efficiency with both maintenance and secondhand markets (Case MS) Assume that both the consumer maintenance and the secondhand markets exist. Social welfare W(q) consists of the type H consumer's surplus from purchasing the new product in period j, j = 1,2, CSjH(q), the revenue from reselling the used product to the type L consumer in period 2, R2H(q), the type L consumer's surplus from purchasing the used product in period 2, CS2L(q), and the overall profits of the monopolist, Π(q), expressed as follows16: WðqÞ = CS1H ðqÞ + CS2H ðqÞ + R2H ðqÞ + CS2L ðqÞ + ΠðqÞ;

ð23Þ

where CS1H ðqÞ = nH ½vH −P 1̂ ðqÞ = −δnH P Û ðqÞ;

ð23aÞ

CS2H ðqÞ = δnH ½vH −P 2̂ ðqÞ;

ð23bÞ

R2H ðqÞ = δnH P Û ðqÞ; and

ð23cÞ

CS2L ðqÞ = δnH ½NSL ðqÞ−P Û ðqÞ:

ð23dÞ

From Eqs. (23a) and (23c), it can be deduced that the type H consumer's surplus from purchasing the first-period new product, including the resale value, is exploited by the monopolist, i.e. CS1H(q) + R2H(q) ≡ 0. Eq. (23d) represents that the pricing of the used product leaves the type L consumer no surplus from purchasing the used product, i.e., CS2L(q) ≡ 0. Thus, evaluating the first-order derivative of social welfare W′(q) at q =qeMS shows it to be identical with CS′2H(qeMS) and yields ′

e

e

e

W ðqMS Þ = −δnH gðmL ðqMS ÞÞvL + δnH gðmH ðqMS ÞÞvH :

ð24Þ

Eq. (24) shows that when the monopolist chooses q, it fails to fully internalize the effects on consumer surplus. What are the effects that the monopolist cannot internalize? Note that Eq. (23b) shows that the effect of the built-in durability on CS2H works opposite to the effect on P̂ 2. As shown in Lemma 3(i), the built-in durability has a positive effect on P̂ 2 through raising the resale value of the used product in the secondhand market. This effect is negative for consumer surplus. We call this the secondhand market effect, shown by the first term of Eq. (24). The built-in durability has a negative effect on P̂ 2 because it enhances the value of the option that type H consumer retains the used product. This effect is positive for consumer surplus. We refer to this effect as the reservation utility effect, shown by the second term of Eq. (24). In Table 2, we show the secondhand market effect and the reservation utility effect in Case MS. If the secondhand market effect dominates (is dominated by) the reservation utility effect, the total effect is negative (positive), and the monopolist chooses the socially excessive (insufficient) level of built-in durability. We establish our main result as follows. Proposition 2. (i) If g(m) is strictly log-concave (strictly log-convex) for all m ≥ 0, then the monopolist chooses the socially excessive (insufficient) level of built-in durability. (ii) If g(m) has both strictly log-concave and 16

We assume that W(q) is a single-peaked function.

447

Case MS Case S Case M

Secondhand market effect

Reservation utility effect

− δnHg(mL(qeMS))vL − δnHvL − δnHg(mH(qeM))vH

δnHg(mH(qeMS))vH δnHvH δnHg(mH(qeM))vH

strictly log-convex regions in [mL(qeMS), mH(qeMS)], whether the monopolist chooses the socially excessive level of built-in durability is ambiguous. The efficiency of the built-in durability depends on whether the secondhand market effect or the reservation utility effect dominates. Each effect relies on each type of consumers' valuation for quality weighted by the recovery rate. Let us intuitively explain why planned antiobsolescence occurs when g(m) is strictly log-concave, i.e., g(m)g″(m)−[g′(m)]2 b 0. Log-concavity means that the elasticity of g(m) with respect to m is elastic.17 If g(m) is strictly log-concave, the difference in the maintenance recovery between both types is large, that is, g(mL(q)) is considerably larger than g(mH(q)). Due to the small recovery for the type L consumer, the type L consumer's marginal valuation for built-in durability is large because maintenance is a substitute for built-in durability. This leads to the large secondhand market effect. Owing to the large recovery for the type H consumer, the type H consumer's marginal valuation for builtin durability is small. This leads to the small reservation utility effect. Accordingly, if g(m) is strictly log-concave, the secondhand market effect dominates the reservation utility effect and planned antiobsolescence occurs. For example, g(m)=(1−m)/(1+m)(0≤mb 1) is a strictly log-concave function. Alternatively, if g(m) is strictly log-convex, the difference in the maintenance recovery between both types is small; that is, g(mL(q)) is not much different from g(mH(q)). In this case, the reservation utility effect tends to dominate the secondhand market effect and planned obsolescence occurs. For example, g(m) = 1/(1+m) (m ≥ 0) is a strictly log-convex function. If g(m) =e− bm(b N 0, m ≥ 0), which is neither strictly log-concave nor log-convex, then the secondhand market effect exactly cancels out the reservation utility effect, so the monopolist necessarily selects the socially efficient level of built-in durability. 5.2. Efficiency with a secondhand market and without a maintenance market (Case S) Assume that a secondhand market exists, but not a consumer maintenance market. In this case, g(0) = 1 holds. Evaluating W′(q) at q = qeS yields ′

e

WS ðqS Þ = −δnH gð0ÞvL + δnH gð0ÞvH = −δnH vL + δnH vH N 0:

ð25Þ

Without consumer maintenance, − δvL gives the secondhand market effect and δvH gives the reservation utility effect. Thus, we obtain Waldman's (1996, p. 497) result: The reservation utility effect necessarily dominates the secondhand market effect, and planned obsolescence necessarily occurs, i.e., qeS b qS⁎ (see Table 2). 5.3. Efficiency with a maintenance market and without a secondhand market (Case M) Assume that a consumer maintenance market exists, but there is no secondhand market. The first-order derivative of social welfare is 17

−

Strictly speaking, strictly log-concavity is equivalent to: mg ′ ðmÞ gðmÞ

!

 dm v N 1; dv m

using dm/dv = − g′(m)/vg″(m). Unless both g and m with respect to v are inelastic, g(m) is strictly log-concave.

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W′M(q) = nH[δg(mH(q))vH − c′(q)], which is equivalent to Eq. (19). This means that the monopolist chooses an optimal level of built-in durability. The negative secondhand market effect then cancels out the positive reservation utility effect (see Table 2). Even if consumer maintenance exists, planned antiobsolescence never occurs when a secondhand market does not exist.

exceeds that earned without the secondhand market (represented by the sum of the first term in Eq. (19)) or not. The following proposition summarizes our result.

6. Does the existence of maintenance or the secondhand market reduce built-in durability?

7. Efficiency of actual durability

In this section, we examine how the introduction of consumer maintenance changes the monopolist's choice of built-in durability as well as investigate how the existence of the secondhand market affects it.18 First, we consider how the introduction of consumer maintenance activity changes the choice of built-in durability. A seemingly plausible hypothesis is as follows: because consumer maintenance is a substitute for built-in durability, consumer maintenance will reduce built-in durability. Comparing Eq. (15) with Eq. (16), we can see that the introduction of consumer maintenance alters marginal revenue but does not change marginal cost. Table 1 shows the first-period and secondperiod marginal revenue representing the marginal changes in P̂ 1 and P̂ 2, respectively. The existence of maintenance reduces the marginal valuation since maintenance substitutes in part for built-in durability. Thus, the introduction of consumer maintenance reduces P̂ 1′(q). The above hypothesis pays attention only to the effect of maintenance on P̂ 1′(q). The effect of q on the second-period marginal revenue P̂ 2′ (q) has both a positive effect through the secondhand market and a negative effect through the reservation utility (see Eq. (22)). The positive (negative) effect involves the type L (the type H) consumer's marginal valuation for the built-in durability. The introduction of maintenance reduces both effects due to the substitution of maintenance for built-in durability. As the resulting gap in the recovery rate between the two types of consumers expands—that is, g(mL(qeMS)) is considerably larger than g(mH(qeMS))—the introduction of consumer maintenance greatly diminishes the negative effect when compared with the positive effect. As a result, the introduction of maintenance increases P̂ 2′(q). If the effect of maintenance on P̂ 2′(q) is far larger than the effect on P̂ 1′(q), the introduction of consumer maintenance increases built-in durability. This tends to occur when the difference between vL and vH is large. The following proposition explains this.

Proposition 4. If g(m) is strictly log-concave (strictly log-convex), the existence of the secondhand market encourages the monopolist to produce a more (less) durable product.

This section examines whether the actual (realized) durability is socially excessive or insufficient when both the consumer maintenance market and the secondhand market exist. We focus on the second-best solution in the sense that the social planner can determine the built-in durability, but not intervene in the consumers' maintenance decisions. The actual durability is given by θ(q) = 1 − (1 − q)g(mL(q)). The realized durability does not depend on just the built-in durability, but also on the maintenance level. We examine the efficiency of consumers' maintenance choices. Proposition 5. Suppose that consumers undertake positive maintenance. The equilibrium level of maintenance is socially insufficient (excessive) when the monopolist sets a socially excessive (insufficient) level of built-in durability. Proposition 5 implies that the distortion direction of the maintenance choice is opposite to that of the built-in durability choice. In Proposition 6, we examine the efficiency of the actual durability. Proposition 6. Even if the built-in durability is socially excessive, the actual durability can be socially insufficient. While actual durability increases along with built-in durability, an increase in built-in durability indirectly depresses actual durability because it reduces consumers' maintenance efforts (see Eq. (1)). When g(m) is strictly log-concave, the elasticity of the recovery rate becomes more elastic. Therefore, the indirect negative effect outweighs the direct positive effect, resulting in dθ/dq b 0. When g(m) is strictly log-convex, then the positive effect dominates the negative effect, resulting in dθ/dq N 0. From Proposition 2, if g(m) is strictly log-concave (strictly logconvex) for all m ≥ 0, then the monopolist sets a socially excessive (insufficient) built-in durability. Therefore, when g(m) is strictly log-

Proposition 3. When the difference between vH and vL is large (small), the existence of consumer maintenance encourages the monopolist to produce a more (less) durable product. Next, we examine how the existence of the secondhand market affects the choice of built-in durability. In the M-case equilibrium, the new product's introduction occurs only in the first period. Type H consumers purchase the new product in period 1, and keep it in period 2, whereas type L consumers do nothing in both periods. Comparing Eq. (15) with Eq. (19), the existence of the secondhand market alters marginal revenue but does not change marginal cost. The M-case first-period marginal revenue is higher than that of the MS-case one, i.e., δnHg(mH(q))vH N δnHg(mL(q))vL for any q. While the M-case second-period marginal revenue is zero, the MS-case one is δnH{g(mL(q))vL − g(mH(q))vH}, whose sign depends on the built-in durability of the MS-case equilibrium. Consequently, the secondhand market's influence on the monopolist's choice of built-in durability depends on whether marginal revenue with the secondhand market (represented by the sum of the first and second terms in Eq. (15)) 18 We are grateful to an anonymous referee for the discussion on improving this section. The comparison of social welfare is complicated. However, under reasonable assumptions, we can show that social welfare at the MS-case equilibrium is largest. This discussion is available upon request from the authors.

Fig. 7. Efficiencies of built-in durability.

H. Kinokuni et al. / Int. J. Ind. Organ. 28 (2010) 441–450

449

Table 3 Numerical examples.

Example 1 Example 2 Example 3

vH

vL

α

β

r

δ

qeMS

q⁎ MS

θ(qeMS)

θ(q⁎MS )

qeS

qeM

13.5 18.0 17.0

9 12 9

10 13 10

9.6 11.4 13.0

7.5 11.5 8.8

0.3 0.4 0.6

0.175 0.252 0.284

0.135 0.258 0.258

0.583 0.544 0.546

0.584 0.545 0.539

0.141 0.211 0.046

0.068 0.271 0.110

concave or strictly log-convex for all m ≥ 0, the actual durability is necessarily socially insufficient. However, if g(m) has both regions where g(m) is strictly log-concave and strictly log-convex, the actual durability can be socially excessive. In Section 8, we will demonstrate that this case exists using a numerical example. 8. An example We provide an example with the particular recovery function g(m) and derive the monopolist's choice of built-in durability and the equilibrium level of actual durability when both the consumer maintenance and the secondhand markets exist.19 We consider the following recovery function: pffiffiffiffiffi gðmÞ = 1− m; 0 ≤ m b 1:

ð26Þ

This function is strictly log-concave (log-convex) for m N (b)1/4. We also specify the cost function: cðqÞ = α +

β 2 q ðα; β N 0Þ: 2

ð27Þ

Under Eq. (26) and Eq. (27), if v H N (b)β(2r − v L)/(β − δv L), then planned antiobsolescence (planned obsolescence) occurs, i.e., ⁎ . This result corresponds to Proposition 2. Fig. 7 compares qeMS N (b)qMS 20 e qMS with q⁎ In region I, the monopolist selects MS on the plane (vL, vH). the socially excessive built-in durability, while in region II, it chooses the socially insufficient built-in durability. Table 3 provides the numerical examples. Example 1 represents the situation where built-in durability is socially excessive but actual durability is socially insufficient, and Example 2 represents the situation where both built-in durability and actual durability are socially insufficient. Example 3 represents the situation where both built-in durability and actual durability are socially excessive. When comparing built-in durability of the MS-case equilibrium with that of the S-case equilibrium, all examples show that qeMS N qeS. When comparing built-in durability of the MS-case equilibrium with that of the M-case equilibrium, Examples 1 and 3 show that qeMS N qeM and Example 2 shows that qeMS b qeM. 9. Conclusion The existing literature on planned obsolescence only separately considers the secondhand market and the consumer maintenance market. We develop a model that can analyze the consumer maintenance market and the secondhand market at the same time by introducing consumer maintenance decisions into Waldman's (1996) model. In Waldman's model, actual durability just equals built-in durability because consumer maintenance does not exist. In our model actual durability differs from built-in durability because maintenance recovers part of the built-in decay. We have examined the efficiency of both the level of built-in durability and realized durability through consumer maintenance. A 19

The detailed derivations in this example are available upon request from the author. 20 ⁎ b 1 and 0 b m L (qeMS), In Fig. 7, we focus on parameters such that 0 b qeMS, qMS ⁎ ) b 1 hold. m L(qMS

change in built-in durability generates both a negative effect (the secondhand market effect) and positive effect (the reservation utility effect) on consumer surplus. The monopolist fails to internalize these effects. When consumers engage in maintenance activities, the efficiency of the built-in durability depends on how maintenance activities influence these effects. In turn, these effects rest on the recovery function of the maintenance effort. As the elasticity increases, the gap in the recovery rate of the two types of consumers becomes larger. Accordingly, the secondhand market effect dominates the reservation utility effect, and the monopolist chooses socially excessive built-in durability. Thus, in a world with consumer maintenance, the monopolist may practice planned antiobsolescence. This phenomenon requires both the consumer maintenance and the secondhand markets. When consumer maintenance substitutes for built-in durability, does the introduction of consumer maintenance discourage the builtin durability the monopolist choose? We have found that when a secondhand market exists, the monopolist may produce more durable goods with consumer maintenance than without it. The reason is as follows. Because an increase in built-in durability raises the value of the used product, it increases the price of the used product in the secondhand market and the value of retaining the used product. The former effect gives consumers incentive to purchase the new product while the latter discourages them. The former has a positive effect on the monopolist's profitability but the latter has a negative effect. The existence of maintenance reduces both effects. It encourages the builtin durability the monopolist chooses when it largely diminishes the negative effect compared to the positive effect. Even if the monopolist chooses a socially excessive built-in durability, the actual durability level can be socially insufficient when the excessiveness of the built-in durability drastically discourages consumers' maintenance efforts. How the distortion of built-in durability affects actual durability depends on the recovery function of the maintenance effort. Our results suggest policies concerning consumer maintenance. The appropriate maintenance regulation depends on whether the monopolist initially practices planned obsolescence. Suppose that the monopolist initially practices planned obsolescence. The reservation utility effect dominates the secondhand market effect; so tightening the minimum standard of maintenance widens the gap between the effects and worsens social welfare. Suppose that the monopolist engages in planned antiobsolescence. The secondhand market effect dominates the reservation utility effect; so raising the minimum standard narrows the gap between the effects, resulting in an improvement in social welfare. The result obtained in the two-period model is robust with extension to a multiperiod model. In the two-period model, an increase in q has positive effects on the first- and second-period new product prices through increasing the resale price, and a negative effect on the second-period new product price through increasing the reservation utility. This mechanism works in a multiperiod model.21 Acknowledgments We would like to thank the editor, David Martimort, and anonymous referees for their insightful comments and suggestions. Helpful

21

A more formal argument is available upon request from the author.

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comments and suggestions from Kazuharu Kiyono, Motonari Kurasawa, Toshiriho Matsumura, and Atsuro Utaka, along with participants at a 2005 COE seminar at Kyoto University and the 2006 Annual Meeting of the Japanese Economic Association at Fukushima University are appreciated. This research was supported by MEXT.OPENRESEARCH (2004-2008) and Japan Society for Promotion of Science (No. 19203015). References Bond, E., Samuelson, L., 1984. Durable good monopolies with rational expectations and replacement sales. Rand Journal of Economics 15, 336–345. Bulow, J., 1986. An economic theory of planned obsolescence. Quarterly Journal of Economics 101, 729–749. Coase, R., 1972. Durability and monopoly. Journal of Law and Economics 15, 143–149.

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