Optimal Waste Disposal Fees When Product Durability is Endogenous: Accounting for Planned Obsolescence



Considering the Coase conjecture (J Law Econ 15:143–149, 1972), we explore the optimal disposal fee of solid waste through the choice of product durability by a producer. We find that introducing a disposal fee curbs planned obsolescence and increases product durability. This increased durability decreases the social cost of waste and increases the service flow from the durable product. Therefore, the optimal disposal fee is higher than the Pigouvian level in the closed-loop.


Solid waste management Design for environment Disposal fee Durability choice Planned obsolescence 

1 Introduction

The need to dispose of society’s solid waste results in environmental problems such as pollution from incineration and disposal facilities, depletion of landfill capacity, and damage to ecosystems and human health caused by exposure to waste and its byproducts. To address these challenges, the governments in many developed countries have introduced economic policies to reduce the generation of solid waste, such as disposal fees and subsidies for recycling. In parallel, durable goods such as cars, tires, furniture, batteries, appliances, and electronics that eventually require disposal at the end of their service lives have increasingly been the target of extended producer responsibility (EPR) regulations that hold manufacturers liable for end-of-life take-back, recycling, and final disposal of their products.1 A stated objective of the latter is to incentivize producers to “design for the environment” (DfE) by developing durable goods that provide more services and/or use fewer inputs over their lifecycle, while at the same time may be more easily taken back, recycled, and/or remanufactured at the end of their lives and ultimately generate smaller quantities of residuals over their lifecycles.

In this study, we investigate the impact of waste reduction policies on DfE, focusing on the level of product durability. Increased product durability is considered to be one of the outcomes of DfE, but has only specifically been analyzed by Runkel (2003) and Shinkuma (2007), as we discuss below.2 The policy setting we consider is pay-as-you-throw, where the households in a municipality pay a fee for the disposal of their solid waste based on its weight and/or volume. The quantity of waste depends on the durability of products, which is endogenously determined by a monopolistic producer. Hence, the more durable these goods, the smaller the amount of waste and the associated environmental cost of disposal. However, a monopolistic producer faces strong incentives to manufacture products that are less durable than would be socially optimal. This tension, and its implication for the design of waste reduction policy, are at the heart of the present study.

The Coase conjecture (1972) suggests that the producer aims to manufacture products that are less durable than would be socially optimal. This outcome is a consequence of its attempt to avoid a time-inconsistency problem whereby its ex post decisions diminish its overall profitability.3 Households expect that the monopolist has an incentive to sell additional units of the durable good at a lower price in a future period, which lowers their willingness to pay for the product in the present period. The latter erodes the producer’s monopoly pricing power, which crucially creates an incentive to manufacture a product that is less durable as a way of credibly committing to not reduce its future price. This phenomenon, termed “planned obsolescence” in the industrial organization literature, arises out of the monopolist’s desire to spur replacement demand, and was formalized by Bulow (1986).4 The question of how planned obsolescence might affect the environment and welfare remains unanswered.5 In particular, the concern is that the producer will seek to increase its long-run output by making a less durable product, which increases the quantity of waste and associated environmental damage.

We consider the situation in which a producer with monopoly power can choose its level of product durability and consider how a disposal fee affects environmental damage and overall welfare via durability choice. We construct a closed-loop model that shows that an increase in the disposal fee curbs planned obsolescence and decreases the overall level of production. A more durable product leads to a decline in the volume of waste, with reduced environmental costs. The novel result is that the optimal disposal fee exceeds the sum of the marginal disposal cost and the marginal cost of the environmental damage cost. The upshot is that with monopolistic production and endogenous product durability, the welfare-maximizing disposal fee should be set higher than the Pigouvian level.

Our result is a contribution to the theoretical literature on the design of waste policies. Several prior studies have examined the interplay between aspects of DfE and economic instruments for waste reduction. Focusing on recyclability, Calcott and Walls (2000) show that a disposal fee is inefficient if a market for recycling does not exist, while Ino (2011) derives the optimal disposal fee in the presence of a recycling market. Dubois and Eyckmans (2015) and Fleckinger and Glachant (2010) consider the effect of imperfect competition and indicate that market power affects the outcome of the waste policy. Dubois and Eyckmans (2015) consider imperfect competition on the recycling market and show that recycling taxes and other instruments can be influenced by strategic behavior. Fleckinger and Glachant (2010) assume imperfect competition in the product market and waste industry and show that EPR programs fail to implement the first-best optimum.

The closest studies to the present work are those by Shinkuma (2007) and Runkel (2003), who analyze durable goods. Shinkuma (2007) compares a disposal fee with an advance disposal fee and shows that the former is more desirable because the latter results in excess disposal and a shorter lifetime of durable products compared with the social optimum. Crucially, however, the durability of goods is not under the control of the firms that manufacture them; instead, products can be repaired at the end of their life and traded in secondhand markets. Runkel (2003) develops an infinite-horizon model with imperfectly competitive producers that choose the durability of their product. In contrast to our result, an EPR policy decreases the social cost of disposal by increasing product durability. The consequence is that the optimal tax on waste is lower than its marginal disposal cost, which is the opposite of what we find. However, the key difference is that Runkel does not consider the producer’s time-inconsistency problem and the incentive for planned obsolescence through reduced product durability. Here, we show that accounting for this crucial element generates the opposite result.

The rest of the paper is organized as follows. In Sect. 2, we present the model. Section 3 considers the effect of the disposal fee and optimal policy. Section 4 discusses how the other potential setups change our results in Sect. 3. Section 5 concludes.

2 Basic Setup

By developing Bulow’s (1986) two-period model, we propose a partial equilibrium model with a durable product and a numeraire good. The economy is composed of one monopolistic producer of the product, a continuum of identical households with a total mass normalized to one, and one municipality including a solid waste management (SWM) sector. The SWM sector’s operation is collecting and disposing solid waste. The representative household lives for two periods and has no offspring.

The monopolistic firm produces and sells the durable product in each period. The producer can choose not only the amount of the durable product but also its durability in period 1, and these are denoted by \(q_i\) (\(i=1,2\)) and \(D \in [0,1]\), respectively. Durability D represents the survival rate of the durable product in period 2. That is, a part of the used product, \((1-D)q_1\), is broken at the end of period 1. Therefore, the market stocks of the durable product in each period are \(Q_1 = q_1\) and \(Q_2 = Dq_1 + q_2\). The producer’s cost function in period i is given by
$$\begin{aligned} C_1 \equiv (1+ \delta D)cq_1 \;\;\;\; \text {and} \;\;\;\; C_2 \equiv cq_2, \end{aligned}$$
where c is the constant unit production cost and \(\delta \) is the discount factor. The cost function (1) satisfies the condition for Swan’s independence result (Swan 1970, 1971). Bulow (1986) shows that the condition for Swan’s independence result in the two-period model is given by
$$\begin{aligned} \frac{1}{q_1}\frac{\partial C_1}{\partial D} = \delta \frac{\partial C_2}{\partial q_2}. \end{aligned}$$
To ensure a certain level of the service flow of the product in period 2, the producer has two choices: to increase the durability in period 1 and to produce the new product in period 2. Condition (2) states that the two ways are equivalent from the viewpoint of saving cost. By using (1), the profit in period i is \(\pi _i = p_i q_i - C_i\), where \(p_i\) denotes the price in period i.
Then, the discounted present value of the overall profit in period 1 is
$$\begin{aligned} \Pi = \pi _1 + \frac{1}{R}\pi _2, \end{aligned}$$
where \(R \in [0,1]\) denotes the gross interest rate.
At the end of the lifecycle of the durable product, the used product becomes waste, which must be disposed by the economy. The disposal cost of waste services includes the costs of collection and final disposal such as incineration and landfilling. The SWM sector has a linear technology for SWM services. Thus, the exogenous disposal cost per unit is denoted by \(\tau \). Moreover, we assume that waste disposal generates environmental damage. We denote the marginal environmental damage cost by k.6 The amount of waste in each period is, respectively, \(G_1=(1-D)q_1\) and \(G_2=Dq_1+q_2\), where \(G_i\) is disposed at the end of period i. Thus, the discounted present cost of the waste disposal and environmental damage in the economy, in other words, the social cost of waste, is
$$\begin{aligned} \Gamma \equiv (\tau + k) (G_1 + \delta G_2). \end{aligned}$$
The SWM sector directly charges the household a disposal fee, denoted by f, for its disposal of solid waste based on the amount.7 The municipality will set the optimal disposal fee. If the optimal disposal fee leads to a deficit (a surplus) of the SWM sector, it is balanced by a transfer from (to) the household.
The representative household consumes the numeraire good \(x_i\) and service flow from the durable product, which equals the market stock of the durable product \(Q_i\). The utility function in period i is given by
$$\begin{aligned} u_i = x_i + Q_i \Bigl (a - \frac{Q_i}{2} \Bigl ), \end{aligned}$$
where a is a positive parameter. Then, the lifetime utility function is given by
$$\begin{aligned} U = u_1 + \delta u_2. \end{aligned}$$
\(\delta = R^{-1}\) should hold in equilibrium because of financial market perfection. We assume that the household receives income I at the beginning of period 1 only. The household’s budget constraint in each period is respectively
$$\begin{aligned} I= & {} x_1 + p_1q_1 + S + f G_1 \;\;\;\; \text {and} \end{aligned}$$
$$\begin{aligned} R S= & {} x_2 + p_2q_2 + f G_2, \end{aligned}$$
where S denotes the saving. From (7) and (8), the lifetime budget constraint is
$$\begin{aligned} I = x_1 + \frac{1}{R}x_2 + p_1q_1 +\frac{1}{R}p_2q_2 + f G_1 + \frac{1}{R} f G_2. \end{aligned}$$

3 Optimal Disposal Fee with Planned Obsolescence

The monopolist that sells durable goods faces the time-inconsistency problem known as the Coase conjecture (1972). Owing to the ex post incentive to supply additional units, the monopolist is unable to precommit to the future production path. The time-inconsistency problem reduces the monopolist’s overall profitability. The monopolist has an incentive to practice planned obsolescence to evade the Coase conjecture. Bulow (1982) proves the Coase conjecture and Bulow (1986) proves planned obsolescence by developing the two-period closed-loop model of the durable goods monopoly. We extend Bulow’s Bulow (1986) model by incorporating a disposal fee system.

In this section, we investigate the optimal waste disposal fee with planned obsolescence. The globally optimal disposal fee including the case without planned obsolescence is provided in Appendix B. The game is constructed by using a two-period process (see Fig. 1). The first period contains four stages. In stage 0, the municipality decides the disposal fee to maximize social welfare. In stage 1, the producer chooses the amount of product and its durability to maximize the discounted present value of the overall profit (3). In stage 2, the household purchases the numeraire good and durable product to maximize lifetime utility (6), subject to the lifetime budget constraint (9). In stage 3, the household discards a part of its used products \(G_1\) and the SWM sector collects and disposes the waste. The second period also contains the following three stages. In stage 4, the producer chooses the amount of product to maximize \(\pi _2\). In stage 5, the household purchases the numeraire good and product to maximize \(u_2\), subject to the budget constraint (8). In stage 6, the household discards all used products \(G_2\), and the SWM sector collects and disposes the waste. This game is solved by using backward induction. It is also a game of complete information. The solution satisfies the properties of the subgame perfect Nash equilibrium.
Fig. 1

Order of the game

Here, we illustrate the solution of the multi-stage game. The details of the solution are in “Appendix A”. First, we investigate the second-period decisions. In stage 6, the household discards \(G_2\) and the SWM sector collects and disposes it. In stage 5, the household maximizes \(u_2\) subject to (8) by choosing \(x_2\) and \(q_2\). Then, the second-period inverse demand function is
$$\begin{aligned} p_2 = a - (Dq_1 +q_2) -f. \end{aligned}$$
In stage 4, the producer chooses \(q_2\), which maximizes \(\pi _2=(p_2-c)q_2\), given the first-period decisions \(q_1\) and D. The first-order condition for the producer’s profit maximization in period 2 yields
$$\begin{aligned} \frac{\partial \pi _2}{\partial q_2} = p_2 - c + \frac{\partial p_2}{\partial q_2} q_2 = a - Dq_1 - 2q_2 -f -c =0, \end{aligned}$$
which gives the following reaction function:
$$\begin{aligned} q_2(q_1, D) = \frac{1}{2}(a - Dq_1 - f -c), \end{aligned}$$
which is a decreasing function of the residual durable product from the previous period.
Next, we consider the first-period decisions. In stage 3, the household discards \(G_1\) and the SWM sector collects and disposes it. In stage 2, the household maximizes (6) subject to (9) by choosing \(x_1\) and \(q_1\). The first-period inverse demand function is
$$\begin{aligned} p_1 = a - q_1 + \delta D(a-Dq_1 - q_2(q_1,D))-(1-D)f-\delta Df. \end{aligned}$$
The inverse demand functions (10) and (13) state that a high disposal fee f reduces demand for the durable product. In stage 1, the producer chooses not only \(q_1\) but also D, which maximizes (3), given (10), (12), and (13). The first-order conditions for the maximization of (3) subject to (10), (12), and (13) yield
$$\begin{aligned} \frac{\partial \Pi }{\partial q_1}= & {} p_1 - c(1+ \delta D)+ \frac{\partial p_1}{\partial q_1}q_1 + \delta \frac{\partial p_2}{\partial q_1} q_2(q_1,D) + \frac{\partial q_2(q_1,D)}{\partial q_1}\frac{\partial p_1}{\partial q_2} q_1 \nonumber \\&+\, \delta \frac{\partial q_2(q_1, D)}{\partial q_1}\frac{\partial \pi _2}{\partial q_2} \nonumber \\= & {} a - 2q_1 + \delta D(a-2Dq_1 - q_2(q_1,D)) - (1-D)f -\delta Df \nonumber \\&-\,(1+\delta D)c - \delta D q_2(q_1,D)+ \frac{1}{2}D^2 q_1 = 0, \end{aligned}$$
$$\begin{aligned} \frac{\partial \Pi }{\partial D}= & {} \left( \frac{\partial p_1}{\partial D} - \delta c\right) q_1 + \delta \frac{\partial p_2}{\partial D} q_2(q_1, D) + \frac{\partial q_2(q_1,D)}{\partial D}\frac{\partial p_1}{\partial q_2} q_1 \nonumber \\&+\, \delta \frac{\partial q_2(q_1,D)}{\partial D} \frac{\partial \pi _2}{\partial q_2} \nonumber \\= & {} \delta (a-2Dq_1 - q_2(q_1,D))q_1 + (1-\delta ) fq_1 - \delta cq_1 \nonumber \\&-\, \delta q_1q_2(q_1,D) + \frac{\delta }{2}D q_1^2 =0. \end{aligned}$$
Note that \(\frac{\partial \pi _2}{\partial q_2}=0\) from the envelop theorem.
We focus on the equilibrium where the monopolist practices planned obsolescence. From (12), (14), and (15), we have the following closed-loop equilibrium:
$$\begin{aligned} D^{CL}= & {} \frac{4f}{\delta (a-c-f)}, \;\;\;\; q_1^{CL} = Q_1^{CL} = \frac{a-c}{2} - \frac{f}{2}, \nonumber \\ q_2^{CL}= & {} \frac{a-c}{2} - \frac{2+\delta }{2\delta }f \;\; \text {and} \;\; Q_2^{CL} = \frac{a-c}{2} + \frac{2-\delta }{2\delta } f. \end{aligned}$$
The superscript CL denotes the closed-loop equilibrium. Note that \(D \in [0,1]\). Thus, if the disposal fee f is larger than \({\tilde{f}} \equiv \frac{\delta }{4+\delta }(a-c)\), durability reaches the upper bound, \(D^{CL}=1\), and then planned obsolescence disappears even in the closed-loop equilibrium. In this section, for simplification, we restrict our analysis to the case of planned obsolescence.8 In Appendix B, we discuss the closed-loop equilibrium where planned obsolescence does not occur. From (16), we obtain the following proposition.

Proposition 1

An increase in the disposal fee curbs planned obsolescence, that is, \(\partial D/ \partial f>0\). In particular, the absence of a disposal fee leads to excessive planned obsolescence, that is, \(D=0\).

In industrial organization, a monopolistic producer that supplies a durable product has an incentive to practice planned obsolescence. As Coase (1972) mentions, the monopolist faces the time-inconsistency problem. That is, the ex ante optimal decision is to keep the monopolistic price over an entire period, while the ex post optimal decision is to reduce the price to capture residual demand. However, because households expect a price reduction in the future, the producer cannot pursue the ex ante optimal strategy. To avoid the time-inconsistency problem, the producer has an incentive to reduce durability, leading to a credible commitment towards not lowering the second-period price. Therefore, the monopolistic producer chooses a level of insufficient durability.

As shown in (16), a rise in the disposal fee increases the durability chosen by the producer. We present the mechanism as follows. An increased disposal fee decreases the household’s willingness to pay for the durable product due to the increased waste cost. The term \(-f\) in condition (11) and the terms \(-(1-D)f -\delta Df\) in condition (14) show them. Therefore, a rise in the disposal fee reduces each period’s output. These effects give the producer an incentive to enhance durability since increased durability replaces the output reduction. An increase in the disposal fee also makes the household delay the disposal of the waste and discount the disposal cost. The term \((1-\delta )fq_1\) in condition (15) shows that this effect makes the producer increase durability. Thus, the disposal fee is an instrument to commit to the future price and thus it curbs planned obsolescence.9

Now, we analyze the effect of the disposal fee on the social cost of waste. From (4) and (16), we have
$$\begin{aligned}&G_1^{CL} = \frac{a-c}{2} - \frac{4+\delta }{2\delta }f, \;\;\;\; G_2^{CL} = \frac{a-c}{2} + \frac{2-\delta }{2\delta }f \nonumber \\&\quad \text {and} \;\;\;\; \Gamma ^{CL} = \frac{\delta (1+\delta )}{2\delta } (a-c) - \frac{4-\delta + \delta ^2}{2\delta }f. \end{aligned}$$
The effects of the disposal fee on the amounts of waste and social cost of waste are summarized by the following proposition.

Proposition 2

The disposal fee decreases the amount of waste in period 1 and increases that in period 2. Therefore, this policy reduces the social cost of waste.

The disposal fee increases the amount of waste in period 2, \(G_2\), while it decreases \(G_1\). This result stems from the effect of durability. From Proposition 1, the disposal fee improves durability. In other words, this policy reduces the probability that the product becomes waste in period 1, and then the unbroken product in period 1 becomes waste in period 2. Therefore, the effect raises the amount of waste in period 2. Moreover, from (16), the disposal fee decreases \(q_1\) and \(q_2\). In conclusion, the disposal fee reduces the social cost of waste.

Propositions 1 and 2 imply that the “downstream” waste policy affects the “upstream” product design, that is, the disposal fee encourages DfE. Most studies have paid attention to recyclability as DfE in the context of waste policy. Fullerton and Wu (1998) and Calcott and Walls (Calcott and Walls 2000, 2005) show that a “downstream” disposal fee promotes “upstream” DfE because a higher disposal fee faced by the consumer increases his or her willingness to pay for a product with greater recyclability. In the case of durability, by contrast, a higher disposal fee decreases the willingness to pay for a durable product. The reduction in production in period 1 curbs planned obsolescence, thereby reducing the social cost of waste.

Here, we define social welfare and consider the optimal disposal fee for waste services. In our model, from (5), (6), and (9), the consumer surplus in the durable product market CS is defined as the lifetime utility that the representative household enjoys minus its expenditure for two periods:
$$\begin{aligned} CS= & {} U - \left( x_1 + \frac{1}{R}x_2 + p_1q_1 +\frac{1}{R}p_2q_2 \right) \nonumber \\= & {} x_1 + \left( a - \frac{Q_1}{2} \right) Q_1 + \delta \left\{ x_2 + \left( a-\frac{Q_2}{2} \right) Q_2 \right\} - I \nonumber \\&+ \left( G_1 f + \frac{1}{R} G_2 f \right) . \end{aligned}$$
Recall that a deficit or surplus of the SWM sector is balanced by a transfer to the household’s income I.
Social welfare consists of the consumer surplus, producer surplus, and social cost of waste. From (3), (4), (5), (6), (18), and \(R^{-1}=\delta \), we have
$$\begin{aligned} W\equiv & {} CS + \Pi - \Gamma \nonumber \\= & {} \left( a - \frac{Q_1}{2} - c \right) Q_1 + \delta \left( a-\frac{Q_2}{2} - c \right) Q_2 - \Gamma . \end{aligned}$$
We derive the second-best disposal fee, that is, the optimal disposal fee given the closed-loop equilibrium of (16). From (16) and (19), the first-order condition of welfare maximization yields
$$\begin{aligned} \frac{d W^{CL}}{df}= & {} (a-q_1^{CL}-c)\frac{dq_1^{CL}}{df} \nonumber \\&+\, \delta \{ a-(D^{CL}q_1^{CL} + q_2^{CL} ) -c \} \frac{d(D^{CL}q_1^{CL} + q_2^{CL})}{df} \nonumber \\&-\, (\tau +k)\frac{d}{df} \{ (1-D^{CL})q_1^{CL}+\delta (D^{CL}q_1^{CL} + q_2^{CL} ) \} \nonumber \\= & {} -\,\frac{1}{4}(a-c+f) + \frac{1}{4\delta } (2-\delta ) \{ \delta (a-c)-(2-\delta )f \} \nonumber \\&+\, \frac{1}{2 \delta } (4-\delta + \delta ^2) (\tau + k) = 0. \end{aligned}$$
The optimal disposal fee with planned obsolescence, \(f^{CL}\), is
$$\begin{aligned} f^{CL} = \frac{\delta (1-\delta ) (a-c) +2 (4-\delta + \delta ^2)(\tau +k)}{4-3 \delta + \delta ^2}, \end{aligned}$$
which is an interior solution for \(f^{CL} < {\tilde{f}} \equiv \frac{\delta }{4+\delta }(a-c)\). This condition is rewritten as
$$\begin{aligned} \tau + k < \frac{\delta ^3}{(4+\delta ) (4-\delta +\delta ^2)}(a-c) \equiv {\tilde{K}}. \end{aligned}$$
Under condition (22),10 we find that the optimal disposal fee is higher than the Pigouvian level since
$$\begin{aligned} f^{CL} - (\tau + k) = \frac{\delta (1-\delta ) (a-c) + (4+\delta +\delta ^2)(\tau +k)}{4-3\delta +\delta ^2} >0. \end{aligned}$$
From (23), the optimal disposal fee is larger than the marginal disposal cost \(\tau \) plus the marginal environmental damage cost k when the producer practices planned obsolescence. Therefore, the municipality should implement a disposal fee higher than the Pigouvian level. From (23), we have the following proposition.

Proposition 3

When the durable goods producer practices planned obsolescence, the optimal disposal fee is higher than the marginal social cost of waste, \(\tau + k\).

In our model, the municipality should decide the optimal fee by considering three effects of the disposal fee on the economy. First, if the municipality took care of the social cost of waste alone, the disposal fee would equal the marginal social cost, \(\tau +k\). Second, the municipality has an incentive to reduce the disposal fee to raise the consumer surplus because the monopolistic producer underproduces. Previous studies focus on the trade-off of the disposal fee between environmental improvement and the underproduction in imperfect competition.11 Third, the municipality has an incentive to raise the disposal fee to reduce the producer’s inducement of planned obsolescence. From Proposition 1, the disposal fee increases durability, resulting in increasing the service flow in period 2. As mentioned in Proposition 3, the third effect dominates the second effect.

The result of Proposition 3 is in contrast to Runkel (2003). By using a durability choice model, Proposition 5 of Runkel claims that if the disposal fee reduces the service flow from durable products, the optimal disposal fee is lower than the marginal disposal costs under imperfect competition. Runkel sets up a model where firms can precommit to the amount of service flow in the future (i.e., they use open-loop strategies). Therefore, planned obsolescence does not occur. On the contrary, the monopolistic producer in our model adopts closed-loop strategies; in other words, the producer is unable to precommit to decisions in the future. The disposal fee has an additional effect that curbs planned obsolescence. This effect leads to a result different from Runkel’s.

4 Discussion

In the previous section, we conclude that the optimal disposal fee is higher than the marginal social cost of waste when the producer practices planned obsolescence. The features of our model are the closed-loop equilibrium where the time-inconsistency problem occurs, the monopoly that leads to the deadweight loss from the underproduction of durable products, and the two-period structure that allows backward induction. In this section, we examine how these assumptions of the model affect our results.

4.1 The Open-Loop Model

We consider the open-loop model of the durable goods monopoly (i.e., the producer can precommit to the second-period production).12 Runkel (2003) shows that EPR policies such as a disposal tax increases durability and that the optimal tax level is lower than the marginal disposal cost under imperfect competition by developing the open-loop model. We check whether removing the time-inconsistency problem from the previous section’s model brings into Runkel’s result.

Because the producer can commit to the future production schedule, it simultaneously chooses \(q_1\), D, and \(q_2\) in period 1 to maximize the overall profit. By maximizing (3) subject to (10) and (13), we find that the solution crucially depends on the value of the disposal fee. If \(f>0\), we obtain
$$\begin{aligned} D^{OL}=1, \;\;\; q_1^{OL} = \frac{1}{2}(a-c) -\frac{\delta }{2 (1+\delta )} f, \;\;\; q_2^{OL} =0 \nonumber \\ \text {and} \;\;\;\; \Pi ^{OL} = \frac{1}{4(1+\delta )} \{ (1+\delta ) (a-c) - \delta f \}^2, \end{aligned}$$
where the superscript OL denotes the open-loop equilibrium. In the open-loop model, the monopolist does not face the time-inconsistency problem and has no incentive to practice planned obsolescence. Instead, it has an incentive to increase durability to enhance the household’s willingness to pay for the product by reducing the disposal cost and delaying the disposal of the waste.
If \(f=0\), on the contrary, we have \(q_1^{OL}\), which is in the following (25), but the values of \(D^{OL}\) and \(q_2^{OL}\) are not determined. The equilibrium is described as follows:
$$\begin{aligned} q_1^{OL} = \frac{1}{2}(a-c), \;\;\; q_2^{OL} = \frac{1}{2} (a-c)(1-D) \;\;\; \text {and} \;\;\; \Pi ^{OL} = \frac{1}{4} (1+\delta )(a-c)^2. \end{aligned}$$
The derivations of (24) and (25) are described in Appendix C.

From assumption (2), leading to Swan’s independence result, the producer chooses durability by minimizing the production cost in open-loop models. In addition, a linear cost function (1) results in indeterminacy between D and \(q_2\). Once the government imposes the disposal fee, however, the producer stops supplying the product in the second period and sets the maximal durability \(D^{OL}=1\) because the extra production in the second period harms the producer’s profitability by increasing the household’s waste cost. The characteristics of the open-loop equilibrium (24) are in sharp contrast to those of the closed-loop equilibrium (16), in which \(D^{OL}<1\) for \(0 \le f < {\tilde{f}}\).

By taking the limit of the disposal fee as \(f \rightarrow 0\), we find that the equilibrium (24) corresponds to (25). Therefore, by using (24), we can derive the optimal disposal fee in the open-loop equilibrium. Social welfare in the open-loop equilibrium is given by
$$\begin{aligned} W^{OL} = (1+\delta ) (a - \frac{q_1^{OL}}{2} -c) q_1^{OL} - \delta (\tau + k) q_1^{OL}. \end{aligned}$$
The first-order condition of welfare maximization yields
$$\begin{aligned} \frac{d W^{OL}}{d f} = \{ (1+\delta ) (a-q_1^{OL} -c) - \delta (\tau + k) \} \frac{d q_1^{OL}}{d f} = 0. \end{aligned}$$
From (27), we have
$$\begin{aligned} (1+\delta ) (a-q_1^{OL}) - (1+\delta )c-\delta (\tau +k) =0. \end{aligned}$$
(28) implies that the market price equals the sum of the marginal production cost and marginal waste cost. From (24) and (28), the optimal disposal fee is
$$\begin{aligned} f^{OL} = 2 (\tau + k) - \left( 1+\frac{1}{\delta }\right) (a-c). \end{aligned}$$
Comparing \(f^{OL}\) with the marginal waste cost shows
$$\begin{aligned} f^{OL} - (\tau +k) = (\tau + k) - \left( 1+\frac{1}{\delta }\right) (a-c) < 0. \end{aligned}$$
The inequality sign in (30) results from our implicit assumption that the household’s maximum willingness to pay the product exceeds the marginal cost of production and marginal social cost at least; \(a>c+ \tau +k\).

In the open-loop equilibrium, the optimal disposal fee is lower than the Pigouvian level. Because there is no effect on curbing planned obsolescence, the government only addresses the trade-off between the social cost from waste and the monopolistic distortion. The result is in contrast to Proposition 3 in our closed-loop model, while this is essentially similar to Proposition 5 of the Runkel’s (2003) open-loop model.

4.2 The Extension to an Oligopoly Model

In Sect. 3, we investigated the optimal disposal fee by using the durable goods monopoly model developed by Bulow (1986). In our monopoly model, a rise in the disposal fee has opposite effects. While it makes the monopolist reduce the output amounts and harms social welfare, it makes the monopolist increases its durability and improves social welfare. Proposition 3 indicates that the latter effect dominates the former effect. These effects exist when the producer has market power. If the producer loses market power, the optimal disposal fee must converge to the Pigouvian level.

In this section, we extend our monopoly model to the symmetric Cournot oligopoly model and prove our conjecture by using the limit theorem. Bulow (1986) provides a durable goods symmetric oligopoly model in Example 2. Our model is based on this. We assume that there exist n producers who have the same technology. Producer j’s cost function in period i is given by
$$\begin{aligned} C_{1j} \equiv (1+\delta D_j)cq_{1j} \;\;\; \text {and} \;\;\; C_{2j} \equiv cq_{2j}, \;\;\;\;\; j=1,...,n., \end{aligned}$$
where \(q_{ij}\) is producer j’s output in period i and \(D_j\) is producer j’s product durability. We solve the game by backward induction to obtain the closed-loop equilibrium. The second-period inverse function is
$$\begin{aligned} p_2 = a - \sum _{j=1}^{n} (D_j q_{1j} + q_{2j} ) -f. \end{aligned}$$
Producer j’s second-period problem is to choose \(q_{2j}\), which maximizes \(\pi _{2j} = (p_2 -c)q_{2j}\), given other producers’ second-period outputs and all producers’ first-period choices of \(q_{1j}\) and \(D_j\). The reaction function for producer j is, using symmetry, \(q_{2j} \equiv q_2\) for all j,
$$\begin{aligned} q_2 = \frac{a-c-f-\sum _{j=1}^{n}(D_j q_{1j})}{n+1}. \end{aligned}$$
To solve producers’ first-period profit maximization problems, we use the symmetry of competitors as with Bulow’s procedure. Supporting that \(n-1\) competitors choose \({\bar{q}}_1\) and \({\bar{D}}\) in period 1, the first-period inverse demand function producer j faces is given by
$$\begin{aligned} p_1= & {} a - q_{1j} - (n-1) {\bar{q}}_1+ \delta D_j \{ a- D_j q_{1j} - (n-1){\bar{D}}{\bar{q}}_1 - nq_2 \} q_{2} \nonumber \\&-\, (1-D_j)f-\delta D_j f. \end{aligned}$$
Eventually, producer j’s profit maximization problem in period 1 is given by
$$\begin{aligned} \max _{q_{1j}, D_j} \Pi _j= & {} \{ p_1 - (1+\delta D_j)c \} q_{1j} + \delta (p_2 - c) q_2 \\&\text {s.t.} \;\;\; q_2 = \frac{a-c-f-D_j q_{1j}-(n-1){\bar{D}}{\bar{q}}_1}{n+1}. \end{aligned}$$
From the assumption of symmetric producers, \(q_{1j} \equiv q_1\) and \(D_j \equiv D\) for all j, we obtain the closed-loop equilibrium as follows:
$$\begin{aligned} D= & {} \frac{(n+1)[ \delta (a-c)(n-1) + \{ 1+\delta +n(2-\delta +n) \} f]}{\delta (n^2 +1)(a-c-f)}, \nonumber \\ q_{1}= & {} \frac{a-c-f}{n+1} \;\; \text {and} \;\; q_{2} = \frac{\delta (a-c) - \{ \delta + n(n+1) \} f}{\delta (n^2 +1)}. \end{aligned}$$
Since \(\frac{\partial D}{\partial n} >0\) and \(D \in [0,1]\), the threshold of the number of producers is given by
$$\begin{aligned} {\bar{n}}(f) = \left[ \frac{2\delta (a-c-f)}{f} \right] ^{\frac{1}{3}} -1. \end{aligned}$$
If \(n < {\bar{n}}(f)\), then \(D < 1\); that is, planned obsolescence occurs.

Now, suppose that the disposal fee is absent, that is, \(f=0\). Because \({\bar{n}} (f) \rightarrow \infty \) for \(f \rightarrow 0\), planned obsolescence necessarily occurs in our oligopoly equilibrium if the number of producers is finite. This result corresponds to the example of the symmetric oligopoly in Bulow (1986).

In contrast to Bulow, our model has a disposal fee. If \(n \ge {\bar{n}}(f)\), planned obsolescence does not occur because durability reaches the upper bound. In this case, we have
$$\begin{aligned}&D = 1 \equiv {\widehat{D}}, \;\;\; q_{1} = \frac{ \{ (n+1)^2+\delta (n-1) \} (a-c)-\delta (n-1)f }{1+\delta +n \{ n^2+(3+\delta )n+3 \} } \equiv {\hat{q}}_1 \nonumber \\&\quad \text {and} \;\;\; q_{2} = \frac{ (1+\delta +n)(a-c) - \{ 1+\delta +n(n+2) \} f }{1+\delta +n \{ n^2+(3+\delta )n+3 \} } \equiv {\hat{q}}_2. \end{aligned}$$
We investigate the optimal disposal fee in the perfectly competitive market for durable goods by using the Cournot limit theorem. When the number of producers is sufficiently large, that is, \(n \ge {\bar{n}}(f)\), the equilibrium is given by (37). Social welfare is given by
$$\begin{aligned} {\widehat{W}} = \left( a-c-\frac{n {\hat{q}}_1}{2} \right) n {\hat{q}}_1 + \delta \left\{ a-c-\frac{n({\hat{q}}_1 +{\hat{q}}_2)}{2} -(\tau + k) \right\} n({\hat{q}}_1 +{\hat{q}}_2). \end{aligned}$$
Since \(\lim _{n \rightarrow \infty } (n {\bar{q}}_1) =a-c\) and \(\lim _{n \rightarrow \infty } (n({\hat{q}}_1 +{\hat{q}}_2)) =a-c-f\), we obtain
$$\begin{aligned} \lim _{n \rightarrow \infty } {\widehat{W}} = \frac{(a-c)^2}{2} + \delta \left( \frac{a-c}{2} + \frac{f}{2} - (\tau +k) \right) (a-c-f). \end{aligned}$$
The first-order condition for welfare maximization is
$$\begin{aligned} \frac{d}{df} \left( \lim _{n \rightarrow \infty } {\widehat{W}} \right) = \delta ( \tau + k - f) =0. \end{aligned}$$
Thus, the optimal disposal fee in the perfectly competitive market is \(\tau + k\). A rise in the number of producers curbs planned obsolescence and eliminates the deadweight loss. Thus, the optimal disposal fee converges to the Pigouvian level in the perfectly competitive market.13

4.3 Extension to an Infinite-Horizon Model

As mentioned above, Runkel (2003) analyzes the optimal disposal fee, termed the tax rate on waste, in an infinite-horizon open-loop model with durable products. In this subsection, we discuss the possibility of extending our analysis to an infinite-horizon model.

Because open-loop models permit a monopolistic producer to commit to the future production schedule, the producer can choose its production and durability in all periods at the present time. Thus, the open-loop setups do not essentially differ between finite- and infinite-horizon models. In the closed-loop model, however, the producer chooses current production and durability subject to its own future activities. Thus, an extension to an infinite-horizon model is complicated because backward induction is not allowed.

Bond and Samuelson (1984) discuss that the time-inconsistency problem would also lead to under-durability in the infinite-horizon environment. To our best knowledge, however, only Fethke and Jagannathan (2002) extend Bulow (1986) to an infinite-horizon model. Assuming that the monopolistic producer can commit to the durability of the products in the future but cannot commit to the production path, they explicitly show that the degree of durability is less than the efficient level in their infinite-horizon model. To obtain this result, instead of using backward induction, they guess a stationary and linear function of the future resale price rules and then verify that the rules are optimal.

To investigate whether the optimal disposal fee is higher than the Pigouvian level in a closed-loop infinite period model, one way would be to introduce waste, an externality, and a disposal fee into Fethke and Jagannathan’s (2002) model. However, this extension exceeds the scope of our study and is left to future research.

5 Concluding Remarks

While a durable goods monopolist has an incentive to practice planned obsolescence, less durable products generate more waste and harm the environment. This study examines how introducing the disposal fee policy, which is a type of downstream waste policy, affects durability choice by a durable goods producer. In particular, it investigates a second-best disposal fee policy when considering an endogenous durability choice by a monopolistic producer. We find that the increased disposal fee curbs planned obsolescence by developing the two-period and closed-loop model. Introducing the disposal fee reduces waste via DfE. Accounting for planned obsolescence, the optimal disposal fee is higher than the Pigouvian level even if a monopoly distortion exists in the market. This result sharply contrasts with those of previous works.

The findings of this study provide a new implication for environmental policy under imperfect competition. However, we pay little attention to other waste policies and the respective processes of SWM services, that is, the collection of discarded products, sorting of recyclable components and materials, and incineration and/or landfilling of non-recycling parts. These avenues are left to future research.


  1. 1.

    Waste policies have been set for a wide variety of durable products. For example, European battery producers are required to collect 45% of used batteries. In Japan, the Automobile Recycling Law, established in 2002, demands that consumers pay the disposal fees at the time of purchase of cars. In Canada, when buying new tires, consumers are responsible for paying a fee per tire under a waste disposal scheme called the “tire stewardship program,” which is transferred to the provincial tire recycling agency and is used to help cover the cost of collecting, transporting, and recycling scrap tires.

  2. 2.

    Most previous studies have paid attention to the recyclability as DfE. For example, see Calcott and Walls (2000, 2005) and Fullerton and Wu (1998).

  3. 3.

    In this study, we assume that excess waste generation is due to low product durability, which then raises demand and increases the amount of waste. However, the energy consumption of highly durable products may be less efficient than that of new durable products. Since this study focuses on product durability as a feature of DfE, to simplify our analysis, we ignore the energy efficiency of durable products.

  4. 4.

    See also Bond and Samuelson (1984), Bulow (1982), Hendel and Lizzeri (1999), Kinokuni (1999), Kinokuni et al. (2010) and Waldman (1996). Waldman (2003) provides a comprehensive survey of the theory of durable products.

  5. 5.

    As Guiltinan (2009) mentions, the environmental forms of planned obsolescence include limiting functional life design (or “death dating”), designing for limited repair, and choosing design aesthetics that lead to reduced satisfaction.

  6. 6.

    For simplicity, we assume that the environmental damage cost is a linear function of the amount of waste.

  7. 7.

    A central issue of our study is the time-inconsistency problem for the producer. If we assume that the municipality also adopts a different disposal fee in each period in addition to the producer, the time-inconsistency problems of both players complicate our model. To avoid these complications and obtain clear results, we assume a time-independent disposal fee.

  8. 8.

    The parameter condition for this restriction is given by (22).

  9. 9.

    Runkel (2003) demonstrates that a waste tax has a waste-reducing effect and a waste-delaying effect in his open-loop model. Our closed-loop model shows that the disposal fee has corresponding effects and alleviates the time-inconsistency problem the monopolistic producer faces.

  10. 10.

    Appendix B extends our analysis to the case of \(\tau + k \ge {\tilde{K}}\).

  11. 11.

    For a review of theoretical studies of environmental policy under imperfect competition, see Requate (2006).

  12. 12.

    If the monopolist supplies durable goods to consumers by a rental contract, it can commit the future stock of the durable goods in the market. Therefore, the equilibrium of the rental case of the durable goods monopoly corresponds to the open-loop equilibrium of the sales case. See Bulow (1982; 1986).

  13. 13.

    Shinkuma (2007) shows that an exogenous disposal fee, which equals the marginal disposal cost, attains the social optimum in the perfectly competitive durable goods market by using an infinite-horizon model where product durability is affected by consumers’ repairing rather than producers’ decisions.



We are grateful to the editor, the two anonymous referees, Masanobu Ishikawa, Tomomichi Mizuno, Akihisa Shibata, Takayoshi Shinkuma, Kenji Takeuchi, Tomoki Fujii, and Makoto Yano. This research was supported by MEXT/JSPS KAKENHI Grant Numbers 26380338 and 26380344.


  1. Bond E, Samuelson L (1984) Durable good monopolies with rational expectations and replacement sales. RAND J Econ 15:336–345CrossRefGoogle Scholar
  2. Bulow J (1982) Durable-goods monopolists. J Polit Econ 90:314–332CrossRefGoogle Scholar
  3. Bulow J (1986) An economic theory of planned obsolescence. Q J Econ 101:729–749CrossRefGoogle Scholar
  4. Calcott P, Walls M (2000) Can downstream waste disposal policies encourage upstream “design for environment”? Am Econ Rev Pap Proc 90:233–237CrossRefGoogle Scholar
  5. Calcott P, Walls M (2005) Waste, recycling and ‘Design for environment’: roles for markets and policy instruments. Resour Energy Econ 27:287–305CrossRefGoogle Scholar
  6. Coase R (1972) Durability and monopoly. J Law Econ 15:143–149CrossRefGoogle Scholar
  7. Dubois M, Eyckmans J (2015) Efficient waste management policies and strategic behavior with open borders. Environm Resour Econ 62:907–923CrossRefGoogle Scholar
  8. Fethke G, Jagannathan R (2002) Monopoly with endogenous durability. J Econ Dyn Control 26:1009–1027CrossRefGoogle Scholar
  9. Fleckinger P, Glachant M (2010) The organization of extended producer responsibility in waste policy with product differentiation. J Environ Econ Manag 59:57–66CrossRefGoogle Scholar
  10. Fullerton D, Wu W (1998) Policies for green design. J Environ Econ Manag 36:131–148CrossRefGoogle Scholar
  11. Guiltinan J (2009) Creative destruction and destructive creations: environmental ethics and planned obsolescence. J Bus Ethics 89:19–28CrossRefGoogle Scholar
  12. Hendel I, Lizzeri A (1999) Interfering with secondary markets. RAND J Econ 30:1–21CrossRefGoogle Scholar
  13. Ino H (2011) Optimal environmental policy for waste disposal and recycling when firms are not compliant. J Environ Econ Manag 62:290–308CrossRefGoogle Scholar
  14. Kinokuni H (1999) Repair market structure, product durability, and monopoly. Aust Econ Pap 38:343–353CrossRefGoogle Scholar
  15. Kinokuni H, Ohkawa T, Okamura M (2010) Planned antiobsolescence occurs when consumers engage in maintenance. Int J Ind Organ 28:441–450CrossRefGoogle Scholar
  16. Requate T (2006) Environmental policy under imperfect competition. In: Tietenberg T, Folmer H (eds) The international Yearbook of Environmental and Resource Economics 2006/2007, Cheltenham, UK and Northampton, MA, USA: Edward Elgar, pp 120–207Google Scholar
  17. Runkel M (2003) Product durability and extended producer responsibility in solid waste management. Environ Resour Econ 24:161–182CrossRefGoogle Scholar
  18. Shinkuma T (2007) Reconsideration of an advance disposal fee policy for end-of-life durable goods. J Environ Econ Manag 53:110–121CrossRefGoogle Scholar
  19. Swan P (1970) Durability of consumption goods. Am Econ Rev 60:884–894Google Scholar
  20. Swan P (1971) The durability of goods and regulation of monopoly. Bell J Econ Manag Sci 2:347–357CrossRefGoogle Scholar
  21. Waldman M (1996) Durable goods pricing when quality matters. J Bus 69:489–510CrossRefGoogle Scholar
  22. Waldman M (2003) Durable goods theory for real world markets. J Econ Perspect 17:131–154CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of EconomicsRitsumeikan UniversityKusatsuJapan
  2. 2.Faculty of InformaticsKansai UniversityTakatsukiJapan
  3. 3.Kobe City University of Foreign StudiesKobeJapan

Personalised recommendations