Differentiated pricing policy for new,remanufactured and refurbished products

Abstract

An original equipment manufacturer is studied, which makes new products and re-products (remanufactured and refurbished products) simultaneously. Facing the product diversity, the original equipment manufacturer needs to choose differentiated prices. The monopoly environment is analyzed in single-period and two-period (with invariable market and then variable market) planning horizons, and the optimal pricing strategy for the original equipment manufacturer is characterized. The results of the single-period planning horizon show that remanufactured products not only grab some market share from new and refurbished ones to form “bank run” but also maintain the total number constant and keep the profit increasing, while the refurbished products do it to remanufactured ones but not new ones and make the market growing and the profit increasing. Then, the results of the two-period planning horizon with the invariable market show that there is still “bank run” between new products and re-products. And in the variable market, as the market growth rate increases, the profit and the proportion of re-products increase simultaneously.

Keywords

Remanufacturing refurbishing differentiated products pricing policy market growth

Introduction

With depletion of energy and natural resource as well as increasing crisis in environment, recycling and reusing of end-of-life (EOL) products received widespread attention. Many countries have enforced environment legislation charging producers with responsibility for the whole life cycle of their products, in order to conserve resource and protect the environment. In addition, considering economic benefits, more and more enterprises are engaged in the recovery business, which realize value added from the EOL products for recycling.

As is known to all, remanufacturing is a common way of recycling. It is a process in which wasted products are disassembled, and their parts that still have residual life are treated as work blanks used in the production of remanufactured products. And these products often can match the quality and technical feature level of original products or even better.

Refurbishing is another important way of recycling, which is often confused with remanufacturing before. Actually, they are different. The basic technological process of refurbishing is the same as the process of repairing, but it is the industrial development mode of repairing, and the refurbished products will be sold again in consumer market. One benefit of refurbishing is to prolong the working life and average working performance of equipments; another is that refurbishing can relieve the high emissions and high pollution status of wasted equipments. Therefore, there are many potentials for saving energy and reducing emission during refurbishing.

This study is concerned with the remanufacturing and refurbishing operation. A monopoly environment is analyzed, in which the remanufactured, the refurbished and the original products are clearly distinguishable. First, demand functions of various market situations are derived. And the optimal strategies for a monopolist who offers three product types are characterized, unconstrained by the availability of used products. Then, the cases of the market with invariable scale and variable scale are both considered, constrained by used product availability. The analytical insights for all cases are provided.

The rest of this article is organized as follows: In section “Related research,” the related literature and the contributions of this article are discussed. In section “Problem description,” the variables and parameters are described, and the demand functions for three competing products are presented. In section “Single-period planning horizon,” the single-period planning horizon is studied. In section “Two-period planning horizon,” the two-period planning horizon with market scale invariable and market scale variable is studied, respectively. And the conclusions are put forward in section “Conclusion.”

Related research

There are numerous studies on pricing policy for remanufactured products in the current literature, which can be classified into two streams. One examines pricing decision-making under the circumstance of homogeneous demand. In which, the quality and price of remanufactured products are regarded as the same as new ones. And the other examines pricing decision-making under the circumstance of heterogeneous demand. In which, the remanufactured product is clearly differentiated from the new one.

Some of the literature in the first stream focused on the monopoly competition. Majumder and Groenevelt¹ considered a two-period model of remanufacturing, in face of competition between an original equipment manufacturer (OEM) and a local remanufacturer. They assumed that the OEM did not distinguish between new and remanufactured items, but distinguished the local remanufacturer’s items. The research results showed that it was more favorable to use the remanufacturing strategy.

Ferrer and Swaminathan² characterized the pricing and production strategies in monopoly and duopoly environments for two-period, multi-period and infinite-horizon settings. They assumed that the remanufactured product had no difference from the new one, and the used products could be reused many times. They showed that the optimal strategies obtained for the two-period problem are quite similar to the results of multi-period problems in most practical environments.

Bulmus et al.³ presented a two-period model with the competition between an OEM and an independent operator (IO) to characterize the optimal policy, with the assumption that there was no difference between remanufactured products and new ones. They proved that the price of the OEM’s products depended on its own cost structure. And then, when the cost of remanufacturing diminished, the IO had more chance to collect the available cores for remanufacturing. Furthermore, when consumers’ willingness to pay for the remanufactured products decreased, remanufacturing was no longer profitable.

The closed-loop supply chain management was the other research object in the first stream. Savaskan et al.,⁴ Savaskan and Van Wassenhove,⁵ Shi et al.⁶ and Huang et al.⁷ studied the recycling, pricing and production strategies for the remanufactured products from the perspective of closed-loop supply chain, which had seen important contributions in the study of remanufactured products’ pricing issue.

In most of the above-mentioned literature, it was assumed that remanufactured and new products were not distinguishable. However, remanufactured products were often offered as an alternative to new products at a low price or quality. Because information on remanufactured products was known by consumers constantly and related laws required that remanufactured products must be different from new ones in sale,⁸ the same pricing policy did not apply to the market gradually. So, some firms began to choose differentiated prices to sell remanufactured and new products, such as Groupe Michelin.⁹

Debo et al.¹⁰ formulated an infinite-horizon model to solve the pricing and production technology selection problem for remanufactured products under the condition of market segmentation. Ferguson and Toktay¹¹ formulated a two-period model to examine the recovery strategy and study the differential pricing strategy for new and remanufactured products under the conditions of heterogeneous consumer. Vorasayan and Ryan¹² constructed the different price combination for remanufactured and new products according to the heterogeneous demand of consumers.

Atasu et al.¹³ analyzed the effects of green segments, OEM competition and product life cycle on remanufacturing decisions. They thought that it was very effective to improve market competitiveness and protect market share by allowing the price discrimination for remanufactured products.

Mitra and Webster¹⁴ analyzed a two-period model with government subsidies, in the case of competition between a manufacturer and a remanufacturer. And then, they characterized pricing decision for the first and the second period with the condition that new and remanufactured products were clearly differentiated.

Ferrer and Swaminathan¹⁵ analyzed the monopoly environment in two-period, multi-period and infinite planning horizons to characterize the optimal pricing and production strategy, with new and remanufactured products clearly differentiated. And they selected the remanufacturing savings as the key parameter to discuss the conditions for implementing remanufacturing strategy.

Jung and Hwang¹⁶ studied the optimal pricing policy of an OEM and a remanufacturer with the cases of competition and cooperation. The results pointed out that the competition between the OEM and the remanufacturer could improve the collection yield of used products and the cooperation between them could improve their profits.

Dan and Ding¹⁷ studied the pricing strategy for new and remanufactured products in single-period and two-period planning horizons with supply of used products restricted, considering that consumers’ willingness to pay for remanufactured products is less than new ones. And then, they analyzed the “bank run” and “market growth” effects developed by remanufactured products.

Ding et al.¹⁸ further studied the optimal pricing for remanufactured products in the two-period planning horizon with market scale changed and discussed the change rules for the remanufacturing rate and profit. This article differs from the aforementioned studies in two aspects. First, we suppose that there is an OEM that makes new, remanufactured and refurbished products, and each product is differentiated from one another. The “bank run” between new products and re-products will be analyzed. Second, the variable market is considered in the two-period planning horizon with three competing products. If the product undergoes two life cycles, the size of the market probably changes due to the technical progress, the emergence of the substitutes and the market diffusion in the life cycle.¹⁹ And the impacts of market growth rate on the optimal policies will be discussed.

Problem description

Aiming at the heterogeneous environment with new, remanufactured and refurbished products, which differs from one another, the OEM can adopt differentiated production and pricing policy to explore potential profit space. So, a monopoly environment is modeled in this article to characterize the pricing policy of the OEM.

Variables and parameters

$Q$ Size of the potential market.

$p_{j}, q_{j}$ Price charged ( $p$ ) and quantity demanded ( $q$ ) for product $j$ . $j = n$ (new) or $r$ (remanufactured) or $f$ (refurbished), and $p_{n} > p_{r} > p_{f}$ .

$c_{j}$ Marginal cost to make a product $j$ , and $c_{n} > c_{r} > c_{f}$ .

$ρ$ Collection yield ( $ρ \leq 1$ ), defined as the fraction of new products made in period 1 that is available for remanufacturing and refurbishing in period 2. Hence, $q_{2 r} + q_{2 f} \leq ρ q_{1}$ .

$α, β$ Consumers’ tolerance for remanufactured products ( $α \leq 1$ ) and refurbished products ( $β \leq 1$ ), defined as a fraction of the utility provided by the new product.

$γ$ Capital return per period $(γ \leq 100 %)$ . It is used in the two-period planning horizon considering the time value of capital.

$μ$ Growth rate of the potential market. $μ > 1$ shows that the market is in growth phase, $μ < 1$ shows that it is in decline phase and $μ = 1$ shows that it is in stationary phase.

Three of these parameters, the cost parameter $c_{r}$ , $c_{f}$ and collection yield $ρ$ , characterize the firm’s ability to perform key remanufacturing and refurbishing activities. The cost parameter $c_{n}$ characterizes the firm’s ability to make new products. Consumers’ tolerance parameters $α$ and $β$ characterize the acceptance of consumer for remanufactured products and refurbished ones, respectively. The growth rate parameter $μ$ characterizes the changing situation for the size of the potential market.

Demand functions

In what follows, a useful lemma is presented to describe the self-selection quantities associated with the prices of new, remanufactured and refurbished products.

Lemma

Suppose that three competing products, new ( $n$ ), remanufactured ( $r$ ) and refurbished ( $f$ ), are in the market. Let there be $Q$ potential consumers, characterized by the variable $ν \in (0, Q)$ uniformly distributed in this domain, according to their valuation of the new product. Let $α \in (0, 1)$ and $β \in (0, 1)$ denote the consumers’ tolerance for remanufactured and refurbished products, respectively. Also, $β$ is less than $α$ . The utility that a consumer of type $ν$ enjoys when buying the product is $U_{n} = v - p_{n}$ (new) or $U_{r} = α v - p_{r}$ (remanufactured) or $U_{f} = β v - p_{f}$ (refurbished). And the demand functions of three competing products can be described as follows:

If $α < (p_{r} / p_{n})$ and $β < (p_{f} / p_{n})$ , the demand functions of three products, respectively, are

{\begin{matrix} q_{n} = Q - p_{n} \\ q_{r} = 0 \\ q_{f} = 0 \end{matrix}

(1)

If $(p_{r} / p_{n}) \leq α \leq 1 - ((p_{n} - p_{r}) / Q)$ and $β < (p_{f} / p_{n})$ , the demand functions of three products, respectively, are

{\begin{matrix} q_{n} = Q - \frac{p_{n} - p_{r}}{1 - α} \\ q_{r} = \frac{α p_{n} - p_{r}}{α (1 - α)} \\ q_{f} = 0 \end{matrix}

(2)

If $α < (p_{r} / p_{n})$ and $(p_{f} / p_{n}) \leq β \leq 1 - ((p_{n} - p_{f}) / Q)$ , the demand functions of three products, respectively, are

{\begin{matrix} q_{n} = Q - \frac{p_{n} - p_{f}}{1 - β} \\ q_{r} = 0 \\ q_{f} = \frac{β p_{n} - p_{f}}{β (1 - β)} \end{matrix}

(3)

If $(p_{r} / p_{n}) \leq α \leq 1 - ((p_{n} - p_{r}) / Q)$ and $(α p_{f} / p_{r}) \leq β \leq α - (((1 - α) (p_{r} - p_{f})) / (p_{n} - p_{r}))$ , the demand functions of three products, respectively, are

{\begin{matrix} q_{n} = Q - \frac{p_{n} - p_{r}}{1 - α} \\ q_{r} = \frac{p_{n} - p_{r}}{1 - α} - \frac{p_{r} - p_{f}}{α - β} \\ q_{f} = \frac{β p_{r} - α p_{f}}{β (α - β)} \end{matrix}

(4)

Single-period planning horizon

Suppose that there are demands of new, remanufactured and refurbished products in the market at the same time. Based on equation (4), the optimal policy of a monopolist offering a portfolio of new, remanufactured and refurbished products for a single period can be defined. The profit function of the monopolist would be

\begin{matrix} π_{M} = \underset{p_{j}}{Max} \sum_{j = n, r, f} (p_{j} - c_{j}) q_{j} \\ Subject to q_{n} \geq 0, q_{r} \geq 0 and q_{f} \geq 0 \end{matrix}

(5)

The Hesse matrix of the objective function equation (5) is

H = [\begin{matrix} \frac{- 2}{1 - α} & \frac{2}{1 - α} & 0 \\ \frac{2}{1 - α} & \frac{- 2}{1 - α} - \frac{2}{α - β} & \frac{2}{α - β} \\ 0 & \frac{2}{α - β} & \frac{- 2 α}{β (α - β)} \end{matrix}]

And then, the determinants of the Hesse matrix are ${| H |}_{1} = - 2 / (1 - α) < 0$ , ${| H |}_{2} = 4 / ((1 - α) (α - β)) > 0$ and ${| H |}_{3} = - 8 / (β (1 - α) (α - β)) < 0$ . So, the objective function has one and only optimal solution.

The Kuhn–Tucker conditions of equation (5) can be described as

{\begin{matrix} \frac{\partial π_{M}}{\partial p_{n}} = Q - \frac{2 p_{n} - 2 p_{r} - c_{n} + c_{r}}{1 - α} = 0 \\ \frac{\partial π_{M}}{\partial p_{r}} = \frac{2 p_{n} - 2 p_{r} - c_{n} + c_{r}}{1 - α} - \frac{2 p_{r} - 2 p_{f} - c_{r} + c_{f}}{α - β} = 0 \\ \frac{\partial π_{M}}{\partial p_{f}} = \frac{2 β p_{r} - 2 α p_{f} - β c_{r} + α c_{f}}{β (α - β)} = 0 \end{matrix}

(6)

According to the Kuhn–Tucker conditions, the optimal prices are

{\begin{matrix} p_{n} = \frac{1}{2} (Q + c_{n}) \\ p_{r} = \frac{1}{2} (α Q + c_{r}) \\ p_{f} = \frac{1}{2} (β Q + c_{f}) \end{matrix}

(7)

Based on equations (4) and (7), the optimal quantities can be obtained, they are

{\begin{matrix} q_{n} = \frac{Q}{2} - \frac{c_{n} - c_{r}}{2 (1 - α)} \\ q_{r} = \frac{1}{2} (\frac{c_{n} - c_{r}}{1 - α} - \frac{c_{r} - c_{f}}{α - β}) \\ q_{f} = \frac{β c_{r} - α c_{f}}{2 β (α - β)} \end{matrix}

(8)

The total quantity is

q_{t} = q_{n} + q_{r} + q_{f} = \frac{β Q - c_{f}}{2 β}

(9)

Proof: See Appendix 1

According to case 4 of Lemma, the consumers’ tolerance $α$ satisfies $(c_{r} / c_{n}) < α < 1 - ((c_{n} - cr) / Q)$ and $β$ satisfies $(α c_{f} / c_{r}) < β < α - (((1 - α) (c_{r} - c_{f})) / c_{n} - c_{r})$ .

Theorem 1

When $α$ satisfies $(c_{r} / c_{n}) < α < 1 - ((c_{n} - c_{r}) / Q)$ and $β$ satisfies $(α c_{f} / c_{r}) < β < α - (((1 - α) (c_{r} - c_{f})) / (c_{n} - c_{r}))$ , the monopolist who adopts optimal policies as equations (7) and (8) can maximize his/her profits.

Theorem 1 shows that when $α$ and $β$ vary in certain domains, respectively, making three competing products (new, remanufactured and refurbished) at the same time is the optimal option of the monopolist. In these domains, as $α$ and $β$ increase, the value evaluation of re-products becomes higher, and they are more easily accepted by consumers. In this case, the monopolist who adopts the strategies of remanufacturing and refurbishing besides manufacturing can make profit maximization.

Corollary 1

$q_{r}$ is an increasing function of $α$ and remanufacturing cost savings $s_{r}$ , but $q_{n}$ and $q_{f}$ are opposite. $q_{f}$ and $q$ are increasing functions of $β$ and refurbishing cost savings $s_{f}$ , but $q_{r}$ is opposite. Also, $π_{M}$ is an increasing function of $α$ , $s_{r}$ , $β$ and $s_{f}$ , where $s_{r} = c_{n} - c_{r}$ and $s_{f} = c_{n} - c_{f}$ . (Appendix 1).

The following example illustrates these policies as the consumers’ tolerance $α$ and $β$ , remanufacturing savings $s_{r}$ and refurbishing savings $s_{f}$ vary in their domains. Printer cartridges of an enterprise are selected as a simulation example. The potential demand of the printer cartridges predicted by the enterprise is 1530 units in a single period, so consumers are uniformly distributed from 0 to 1530. And the cost to make a new product is 240, namely, $Q = 1530$ and $c_{n} = 240$ . Also, the cost to make a remanufactured product or a refurbished product depends on the quality of the used products. But according to some statistics of the enterprise, the average cost of a remanufactured product is 120 and the average cost of a refurbished product is 60.

Influence of $α$ on the optimal policy.

Suppose that $s_{r} = c_{n} - c_{r} = 120$ , $s_{f} = c_{n} - c_{f} = 180$ and $β = 0.62$ . According to case 4 of Lemma, $α$ satisfies $0.75 \leq α \leq 0.92$ . The results are shown in Figure 1.

Influence of $s_{r}$ on the optimal policy.

Suppose that $α = 0.86$ , $β = 0.62$ and $s_{f} = 180$ . According to case 4 of Lemma, $s_{r}$ satisfies $66.32 \leq s_{r} \leq 156.77$ . The results are shown in Figure 2.

Influence of $β$ on the optimal policy.

Suppose that $α = 0.86$ , $s_{r} = 120$ and $s_{f} = 180$ . According to case 4 of Lemma, $β$ satisfies $0.62 \leq β \leq 0.79$ . The results are shown in Figure 3.

Influence of $s_{f}$ on the optimal policy.

Suppose that $α = 0.86$ , $β = 0.62$ and $s_{r} = 120$ . According to case 4 of Lemma, $s_{f}$ satisfies $153.49 \leq s_{f} \leq 240$ . The results are shown in Figure 4.

Figure 1.

Influence of $α$ on the optimal policy for single-period planning horizon: (a) optimal sales and (b) optimal profits.

Figure 2.

Influence of $s_{r}$ on the optimal policy for single-period planning horizon: (a) optimal sales and (b) optimal profits.

Figure 3.

Influence of $β$ on the optimal policy for single-period planning horizon: (a) optimal sales and (b) optimal profits.

Figure 4.

Influence of $s_{f}$ on the optimal policy for single-period planning horizon: (a) optimal sales and (b) optimal profits.

There are some interpretations of these figures: As consumers’ tolerance $α$ (Figure 1) and remanufacturing savings $s_{r}$ (Figure 2) increase, the total quantity ( $q$ ) remains unchanged and the number of remanufactured units ( $q_{r}$ ) and the profit ( $π_{M}$ ) increase, but the number of new ( $q_{n}$ ) and refurbished ( $q_{f}$ ) units decrease. It shows that when three competing products are present in the market simultaneously, the increased parts of remanufactured products grab some market share from new and refurbished ones to form “bank run” while maintaining the total number constant. With the increase in $α$ and $s_{r}$ , remanufactured products become more profitable to keep the profits of the monopolist increasing.

As consumers’ tolerance $β$ (Figure 3) and refurbishing savings $s_{f}$ (Figure 4) increase, the total quantity ( $q$ ), the number of refurbished products ( $q_{f}$ ) and the profits ( $π_{M}$ ) increase, but the number of remanufactured ones ( $q_{r}$ ) decreases, while the number of new ones ( $q_{n}$ ) remains unchanged. It shows that the increased parts of refurbished products grab certain market share from remanufactured ones but have no impacts on new ones while keeping the total number and the profit of the monopolist increasing.

Two-period planning horizon

The pricing policy with adequate supply of used products has been studied in the previous section, but in reality, the supply is not always sufficient, and the number of re-products is affected by the sales of new products in the previous phase. In this section, we suppose that the monopolist has a two-period planning horizon. In which, the monopolist only produces new products in the first period, and new, remanufactured and refurbished products are produced simultaneously in the second period. And this section will be divided into two parts for studying: one is the market with invariable scale and the other is the market with variable scale.

The market with invariable scale

First, suppose that the market scale is invariable in the two-period planning horizon. And the time value of capital should be considered. If the first period is selected as the decision point, the discount coefficient of the second phase can be expressed as $1 / (1 + γ)$ ; therefore, the decision-making model for the two-period planning horizon is

\begin{matrix} π_{2 M} = \underset{p_{1}, p_{2 j}}{Max} ((p_{1} - c_{n}) q_{1} + \frac{1}{1 + γ} \sum_{j = n, r, f} (p_{2 j} - c_{j}) q_{2 j}) Subject to q_{1} = Q - p_{1} and q_{2 r} + q_{2 f} \leq ρ q_{1} \end{matrix}

(10)

Moreover, $q_{2 n}$ , $q_{2 r}$ and $q_{2 f}$ satisfy equation (4), and all variables satisfy the non-negativity constraint.

The Lagrange function of equation (10) is

L (p_{1}, p_{j}, λ) = π_{2 M} + λ (ρ q_{1} - q_{2 r} - q_{2 f})

(11)

where $λ (λ \geq 0)$ is the Lagrange multiplier.

According to the previous section, equation (11) also has one and only optimal solution. So, the Kuhn–Tucker conditions of equation (11) are

{\begin{matrix} \frac{\partial L (p_{1}, p_{j}, λ)}{\partial p_{1}} = 0 \\ \frac{\partial L (p_{1}, p_{j}, λ)}{\partial p_{j}} = 0, j = 2 n, 2 r, 2 f \\ λ (ρ q_{1} - (q_{2 r} + q_{2 f})) = 0 \\ ρ q_{1} - (q_{2 r} + q_{2 f}) \geq 0 \end{matrix}

(12)

The above Kuhn–Tucker conditions can be solved with two situations.

D₁₁ . When $ρ > (β (c_{n} - c_{r}) - (1 - α) c_{f}) / (β (1 - α) (Q - c_{n}))$ $(λ = 0)$ , the optimal prices are

{\begin{matrix} p_{1} = \frac{1}{2} (Q + c_{n}) \\ p_{2 n} = \frac{1}{2} (Q + c_{n}) \\ p_{2 r} = \frac{1}{2} (α Q + c_{r}) \\ p_{2 f} = \frac{1}{2} (β Q + c_{f}) \end{matrix}

(13)

And the optimal sales are

{\begin{matrix} q_{1} = \frac{1}{2} (Q - c_{n}) \\ q_{2 n} = \frac{Q}{2} - \frac{c_{n} - c_{r}}{2 (1 - α)} \\ q_{2 r} = \frac{c_{n} - c_{r}}{2 (1 - α)} - \frac{c_{r} - c_{f}}{2 (α - β)} \\ q_{2 f} = \frac{β c_{r} - α c_{f}}{2 β (α - β)} \end{matrix}

(14)

D₁₂ . When $ρ \leq (β (c_{n} - c_{r}) - (1 - α) c_{f}) / (β (1 - α) (Q - c_{n}))$ $(λ > 0)$ , the optimal prices are

{\begin{matrix} p_{1} = \frac{1}{2} (Q + c_{n}) - \frac{1}{2} ρ λ \\ p_{2 n} = \frac{1}{2} (Q + c_{n}) \\ p_{2 r} = \frac{1}{2} (α Q + c_{r}) + \frac{1}{2} (1 + γ) λ \\ p_{2 f} = \frac{1}{2} (β Q + c_{f}) + \frac{1}{2} (1 + γ) λ \end{matrix}

(15)

And the optimal sales are

{\begin{matrix} q_{1} = \frac{1}{2} (Q - c_{n}) + \frac{1}{2} ρ λ \\ q_{2 n} = \frac{Q}{2} - \frac{c_{n} - c_{r} - (1 + γ) λ}{2 (1 - α)} \\ q_{2 r} = \frac{c_{n} - c_{r} - (1 + γ) λ}{2 (1 - α)} - \frac{c_{r} - c_{f}}{2 (α - β)} \\ q_{2 f} = \frac{β c_{r} - α c_{f}}{2 β (α - β)} - \frac{(1 + γ) λ}{2 β} \end{matrix}

(16)

The total sale of the second period is

q_{2 t} = q_{2 n} + q_{2 r} + q_{2 f} = \frac{β Q - c_{f} - (1 + γ) λ}{2 β}

(17)

where

λ = \frac{β (c_{n} - c_{r}) - (1 - α) c_{f} - ρ β (1 - α) (Q - c_{n})}{(1 + γ) (1 - α + β) + (1 - α) β ρ^{2}}

Theorem 2

There is one critical value $\bar{ρ}$ . If $ρ > \bar{ρ}$ , the monopolist adopts an optimal policy as D₁₁, and if $ρ \leq \bar{ρ}$ , the monopolist adopts an optimal policy as D₁₂, where

\bar{ρ} = \frac{β (c_{n} - c_{r}) - (1 - α) c_{f}}{β (1 - α) (Q - c_{n})}

Theorem 2 shows that the decision-making of the two-period planning horizon with the invariable market depends on the value of the collection yield $ρ$ . If the collection yield is higher $(ρ > \bar{ρ})$ , the optimal policy adopted by monopolist is similar to the single period. If the collection yield is lower $(ρ \leq \bar{ρ})$ , the monopolist adopts a dynamic optimal policy for the two-period planning horizon.

The optimal policy of the two-period planning horizon is affected by not only marginal cost $c_{j}$ $(j = n, r, f)$ and consumers’ tolerance $α$ and $β$ but also the collection yield $ρ$ and the discount rate $γ$ . So, the following corollaries can be obtained.

Corollary 2

When $ρ \leq \bar{ρ}$ , $p_{1}$ is an increasing function of $γ$ as well as $p_{2 r}$ and $p_{2 f}$ , $p_{1}$ is a convex function of $ρ$ and $p_{2 r}$ is a decreasing function of $ρ$ as well as $p_{2 f}$ . (Appendix 1).

Corollary 3

When $ρ \leq \bar{ρ}$ , $q_{1}$ is a concave function of $ρ$ , $q_{2}$ is an increasing function of $ρ$ as well as $q_{2 r}$ , $q_{2 f}$ and $π_{2 M}$ , while $q_{2 n}$ is a decreasing function of $ρ$ . (Appendix 1).

The previous example is used to illustrate the above conclusions. Hence, $Q = 1530$ and $c_{n} = 240$ . Moreover, suppose that $c_{r} = 120$ , $c_{f} = 60$ , $α = 0.86$ , $β = 0.62$ and $γ = 5 %$ . Accordingly, $ρ < \bar{ρ} = (β (c_{n} - c_{r}) - (1 - α) c_{f}) / (β (1 - α) (Q - c_{n})) = 0.59$ ; it can be seen that the collection yield $ρ$ satisfies $0 < ρ < 0.59$ . The results are shown in Figure 5.

Figure 5.

Influence of $ρ$ on the optimal policy for two-period planning horizon: (a) optimal prices, (b) optimal sales and (c) optimal profits.

Figure 6.

Influence of $μ$ on the optimal policy for two-period planning horizon: (a) optimal prices, (b) optimal sales and (c) optimal profits.

If the collection yield satisfies $ρ \leq \bar{ρ}$ , as it increases, in the first phase, the optimal price of the new product decreases first and then increases, while the number of new ones increases first and then decreases. And in the second phase, the prices of the remanufactured and refurbished products as well as the number of the new products decrease, but the number of the remanufactured and refurbished ones increase so as to the total quantity and the profit of the monopolist. It shows that when the collection yield is lower, the monopolist reduces the price of new ones in the first phase in order to increase sales, ensuring sufficient supply for remanufacturing and refurbishing in the second phase. With the increase in the collection yield, the marginal profit of a re-product decreases, building a little bit contribution for the monopolist’s profits. Therefore, the monopolist begins to increase the price of new products in the first phase for the sake of maximizing the overall profits. Moreover, the increased quantities of re-products grab certain market share from new products, which leads to the decrease in new ones and the increase in the total number and the profits of the monopolist.

The market with variable scale

Suppose that the size of potential market in the second period is variable, which can be described as $μ Q$ . Then, based on equation (4), the demand functions of the second period are

{\begin{matrix} q'_{2 n} = μ (Q - \frac{p_{2 n} - p_{2 r}}{1 - α}) \\ q'_{2 r} = μ (\frac{p_{2 n} - p_{2 r}}{1 - α} - \frac{p_{2 r} - p_{2 f}}{α - β}) \\ q'_{2 f} = \frac{μ (β p_{2 r} - α p_{2 f})}{β (α - β)} \end{matrix}

(18)

Therefore, the decision-making model is

\begin{matrix} π'_{2 M} = \underset{p'_{1}, p'_{2 j}}{Max} ((p'_{1} - c_{n}) q'_{1} + \frac{1}{1 + γ} \sum_{j = n, r, f} (p'_{2 j} - c_{j}) q'_{2 j}) Subject to q'_{1} = Q - p'_{1}, q'_{2 r} + q'_{2 f} \leq ρ q'_{1} \end{matrix}

(19)

Moreover, $q'_{2 n}$ , $q'_{2 r}$ and $q'_{2 f}$ satisfy equation (18), and all variables satisfy the non-negativity constraint. And the calculation process of equation (19) is similar to equation (10).

D₂₁ . When $μ < (ρ β (1 - α) (Q - c_{n})) / (β (c_{n} - c_{r}) - (1 - α) c_{f})$ , the optimal prices are

{\begin{matrix} p'_{1} = \frac{1}{2} (Q + c_{n}) \\ p'_{2 n} = \frac{1}{2} (Q + c_{n}) \\ p'_{2 r} = \frac{1}{2} (α Q + c_{r}) \\ p'_{2 f} = \frac{1}{2} (β Q + c_{f}) \end{matrix}

(20)

And the optimal sales are

{\begin{matrix} q'_{1} = \frac{1}{2} (Q - c_{n}) \\ q'_{2 n} = μ (\frac{1}{2} Q - \frac{c_{n} - c_{r}}{2 (1 - α)}) \\ q'_{2 r} = μ (\frac{c_{n} - c_{r}}{2 (1 - α)} - \frac{c_{r} - c_{f}}{2 (α - β)}) \\ q'_{2 f} = \frac{μ (β c_{r} - α c_{f})}{2 β (α - β)} \end{matrix}

(21)

D₂₂ . When $μ \geq (ρ β (1 - α) (Q - c_{n})) / (β (c_{n} - c_{r}) - (1 - α) c_{f})$ , the optimal prices are

{\begin{matrix} p'_{1} = \frac{1}{2} (Q + c_{n}) - \frac{1}{2} ρ λ' \\ p'_{2 n} = \frac{1}{2} (Q + c_{n}) \\ p'_{2 r} = \frac{1}{2} (α Q + c_{r}) + \frac{1}{2} (1 + γ) λ' \\ p' 2 f = \frac{1}{2} (β Q + c_{f}) + \frac{1}{2} (1 + γ) λ' \end{matrix}

(22)

And the optimal sales are

{\begin{matrix} q'_{1} = \frac{1}{2} (Q - c_{n}) + \frac{1}{2} ρ λ' \\ q'_{2 n} = μ (\frac{Q}{2} - \frac{c_{n} - c_{r} - (1 + γ) λ'}{2 (1 - α)}) \\ q'_{2 r} = μ (\frac{c_{n} - c_{r} - (1 + γ) λ'}{2 (1 - α)} - \frac{c_{r} - c_{f}}{2 (α - β)}) \\ q'_{2 f} = μ (\frac{β c_{r} - α c_{f}}{2 β (α - β)} - \frac{(1 + γ) λ'}{2 β}) \end{matrix}

(23)

The total sale of the second period is

q'_{2 t} = q'_{2 n} + q'_{2 r} + q'_{2 f} = \frac{μ (β Q - c_{f} - (1 + γ) λ')}{2 β}

(24)

where

λ' = \frac{(β (c_{n} - c_{r}) - (1 - α) c_{f}) μ - ρ β (1 - α) (Q - c_{n})}{μ (1 + γ) (1 - α + β) + (1 - α) ρ^{2} β}

Theorem 3

There is one critical value $\bar{μ}$ ; if $μ < \bar{μ}$ , the monopolist adopts an optimal policy as D₂₁, and if $μ \geq \bar{μ}$ , the monopolist adopts an optimal policy as D₂₂, where

\bar{μ} = \frac{ρ β (1 - α) (Q - c_{n})}{β (c_{n} - c_{r}) - (1 - α) c_{f}}

Theorem 3 shows that the decision-making of the two-period planning horizon with the variable market depends on the value of the growth rate $μ$ . If the growth rate is smaller $(μ < \bar{μ})$ , the optimal prices adopted by the monopolist are similar to the single period. If the rate of increase is bigger $(μ \geq \bar{μ})$ , the monopolist adopts a lower price for new products $(((Q + c_{n} - ρ λ') / 2) < ((Q + c_{n}) / 2))$ in the first phase and adopts a higher price for remanufactured products $(((α Q + c_{r} + (1 + γ) λ') / 2) > ((α Q + c_{r}) / 2))$ and refurbished ones $(((β Q + c_{f} + (1 + γ) λ') / 2) > (β Q + c_{f} / 2))$ in the second phase.

Corollary 4

When $μ \geq \bar{μ}$ , $p'_{1}$ is a decreasing function of $μ$ , and $p'_{2 r}$ is an increasing function of $μ$ as well as $p'_{2 f}$ , $q'_{1}$ , $q'_{2 (r + f)}$ and $π'_{2 M}$ , where $q'_{2 (r + f)} = q'_{2 r} + q'_{2 f}$ . (Appendix 1).

Let the monopolist in the previous examples operate with the potential market variable. Hence, $Q = 1530$ , $c_{n} = 240$ , $c_{r} = 120$ , $c_{f} = 60$ , $α = 0.86$ , $β = 0.62$ , $γ = 5 %$ and $ρ = 0.53$ , so $μ \geq \bar{μ} = (ρ β (1 - α) (Q - c_{n})) / (β (c_{n} - c_{r}) - (1 - α) c_{f}) = 0.89$ .

When the market growth rate $μ$ (when $μ \geq \bar{μ}$ ) increases, in the first phase, the optimal price of the new product decreases, and its sales volume increases. In the second phase, the optimal prices of the remanufactured and refurbished product increase, and their total sales and the manufacturer’s profit increase. It shows that when the market growth rate increases gradually, the market has expanded rapidly, and the aggregate demand of re-products grows rapidly in the second phase. However, this will cause that the supply of used products in the first stage cannot meet the needs of remanufacturing and refurbishing. Therefore, the manufacturer prefers to adopt a lower price policy of the new product in the first phase to increase sales, ensuring abundant supplies of used products for the second phase. However, the marginal profit of the new product decreases with lower price in the first phase, which may lead to the loss of total profits. To make up for the loss, the manufacturer adopts a higher price policy for the remanufactured and refurbished products in the second stage, increasing the marginal profit of them to ensure the total profit maximization.

Corollary 5

When $μ < \bar{μ}$ , the reutilization ratio ( $η$ ) of used products satisfies $η < ρ$ . When $μ \geq \bar{μ}$ , it satisfies $η = ρ$ , where $η = (q'_{2 r} + q'_{2 f}) / q'_{1}$ . It can be expressed as

η = {\begin{matrix} \frac{μ (β (c_{n} - c_{r}) - (1 - α) c_{f})}{β (1 - α) (Q - c_{n})} < ρ, & μ < \bar{μ} \\ ρ, & μ \geq \bar{μ} \end{matrix}

(25)

Corollary 5 illustrates that the market growth rate has an impact on the reutilization ratio of the used products. When the market growth rate is lower $(μ < \bar{μ})$ , the size of the market in the second stage is smaller, and the total demand for re-products becomes less, which lead to a lower reutilization ratio of waste products, namely, $η < ρ$ . But when the market growth rate is higher $(μ \geq \bar{μ})$ , the market scale and the total demand for re-products become larger, which lead to a higher reutilization ratio of waste products, namely, $η = ρ$ .

Conclusion and future research

Conclusion

This article analyzes a monopoly in which the new, remanufactured and refurbished products are clearly differentiated. The optimal pricing strategies under finite (one and two periods) planning horizons of an OEM are characterized to identify the benefits of remanufacturing and refurbishing for the enterprise.

In the single-period planning horizon, the results show that:

Remanufactured products grab some market share from new and refurbished ones to form “bank run” and maintain the market constant and keep the profits increasing.

Refurbished products form “bank run” on remanufactured ones but not new ones and make the whole market growing and the total profits increasing.

In other words, the monopolist can get more profits, increase the reuse ratio of used products and reduce waste pollution of the environment from remanufacturing and refurbishing. In the two-period planning horizon, the results of the invariable market show that:

There is also “bank run” between the new products and the re-products, deriving from the collection yield $ρ$ (when $ρ \leq \bar{ρ}$ ).

The total profits increase with the increase in the collection yield. And the firm can intensify the recycling efforts by constructing recycling channels and developing recycling marketing strategy to improve its profits.

When the size of the market is variable, the results show that different market growth rates lead to different impacts on decision-making for optimal price. The firm can take appropriate pricing strategy through the forecast of market growth situation to ensure profit maximization:

If the market growth rate is smaller, the prices of re-products are not affected, but the firm’s optimal profits and the reutilization ratio of used products increase with the increase in the market growth rate.

If the market growth rate is higher, the used products can be fully reused, but the firm’s profits are constrained by the quantity of used products. So, in order to ensure profit maximization, it needs to adopt a lower price policy of new products in the first phase to increase sales, ensuring abundant supplies of used products for remanufacturing and refurbishing in the second phase.

Future research

This article assumes that there is an oligopoly market, without considering the influence of competitors for pricing strategies of re-products. But in reality, the complete monopoly market is rare, and the firm often faces the competition derived from the third-party remanufacturers, such as a local remanufacturer,¹ an independent operator^2,3 and so on. Therefore, it is significant to study the pricing strategies for re-products considering the competition.

In addition, this article also assumes that there are only two phases for the recycling of used products that are only reused one time. But in practice, it is a multi-stage process to recycle used products, and they can also be reused many times. So, another future research direction is to consider the multi-stage for pricing strategy of re-products. Finally, the optimal pricing strategy of the OEM with three competing products can be studied in closed-loop supply chain systems.

Footnotes

Appendix 1 Acknowledgements

The authors thank the editors and anonymous referees who commented on this article.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This study was financially supported by the Chongqing Science and Technology Research Program (no. cstc2012ggC00001 and cstc2012gg-yyjs00010) and the Fundamental Research Funds for the Central Universities (no. CDJZR12118801) in China.

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