Optimal acquisition price management in a remanufacturing system

Abstract

Return items received from end customers for remanufacturing is uncertain in nature. In this paper, we considered that return demand function is uncertain and sensitive to the acquisition price and the availability of used products in the market. A mathematical model was developed to investigate the impact of the availability of return items in the market on acquisition price management. Here, three different methods of collection of return items from the market are considered: direct method, indirect method and coordinated method. A numerical study was conducted to illustrate the mathematical model, and an extensive sensitivity analysis was carried out for the three different methods of collection to examine the impact of market size and the available dependency factor on acquisition price and the channel profit. The results show that the total channel profit increases when the acquisition price increases for the direct method of collection. Furthermore, the results reveal that the total channel profit increases when the market size increases for the coordinated method of collection compared with the other two methods of collection.

Keywords::

• closed-loop supply chain
• acquisition price
• availability
• remanufacturing

1. Introduction

The gamut of activities of closed-loop supply chain management (CLSCM) revolves around the product take-back from end customers and the recovery of extra values by recapturing the entire product or some of its module. Users may return their product during or after the life cycle of the product due to a variety of reasons such as warranty, repair return, end-of-use return and end-of-life return. CLSCM is defined as the design, control and operation of a system to maximise value creation over the entire life of the product with dynamic recovery of value from different types and volumes of returns over time (Guide, Van Wassenhove, and Blackburn 2009). The cost of a remanufactured product is usually about 40–60% of the cost for manufacturing a brand new product with 20% of the manufacturing effort (Dowlatshahi 2000).

Considering the current competitive business environment and the concern of people about the adverse effects of used products on its environment, CLSCM has received wide attention from both academia and practitioners. Many authors have developed models to study the various aspects of CLSCM. Among others, Guide and Van Wassenhove (2009) and Pokhrel and Mutha (2009) have stated that there is a need to develop a new model to help manufacturers to optimise the closed-loop system in an integrated manner. CLSCM has enormous potential to deal with economic, social and environmental aspects. Once the product is returned to a manufacturer, it has several disposal options such as repair, remanufacturing, reuse, refurbish, recycle and landfill. The manufacturer can collect the used product either directly from the customer or from retailers or through a third party. Among all these methods of collection, the method that will maximise the profitability of CLSCM will be considered the best. A recoverable manufacturing system minimises the environmental impact of the industry by reusing materials, reducing energy use and reducing the need to landfill industrial by-products. The management of closed-loop supply chain activities can differ greatly from that of traditional manufacturing supply chain activities. At present, many original equipment manufacturers such as Hewlett-Packard (Jorjani, Leu, and Scott 2004; Shi et al. 2011a), Kodak (Savaskan and Van Wassenhove 2006), various automotive companies (Guide, Teunter, and Van Wassenhove 2003) and industrial equipment manufacturers, e.g. Caterpillar and Xerox (Atasu, Guide, and Van Wassenhove 2008), are being actively participating in remanufacturing business.

It has been observed that the determination of the optimal acquisition price, particularly based on the availability of used products in the market, is a major challenge for any remanufacturer. Few researchers have studied this problem by considering uncertain quantity (Ling et al. 2009; Shi et al. 2011; Qiang et al. 2013) and uncertain quality (Guide and Van Wanssenhove 2001; Tagaras and Zikopoulos 2008) of used products in the market. Cheng-Han (2012) has mentioned that the availability of used products in the market is an important factor since it is a major constraint for remanufacturing. However, to date, none of the studies has reported the effect of the availability of returned product on acquisition price in a stochastic return demand environment. In this paper, we considered that return demand function is uncertain and sensitive to the acquisition price and the availability of used products in the market; therefore, this study differs from the earlier work and contributes `to the knowledge of the existing literature. Here, the remanufacturer fulfils market demand by remanufacturing returned products similar to new products and through external purchasing of a brand new product in the similar line as has been considered in the earlier literature, and such type of scenario can be observed in the reuse of soft drink bottles and bullet jackets (see Kho et al. 2002; Choi, Hwang, and Kho 2007).

The rest of the paper is organised as follows. Section 2 provides a brief review of the literature related to CLSCM. Section 3 describes the background of the problem formulation. Section 4 presents a mathematical model for different methods of collection. Section 5 presents the numerical study and managerial insights drawn from the study. Finally, Section 6 presents the conclusion and future scope of the study.

2. Literature review

The literature on CLSCM is quite vast. Here, we provide a brief review of the literature related to our research work only. Many analytical and empirical studies have been carried out in different areas of CLSCM such as production planning and control, inventory management, methods of collection, product categorisation, acquisition management, pricing policy and network design (Fleischmann et al. 1997; Van der Laan, Salomon, and Dekker 1995; Savaskan, Shantanu, and Van Wassenhove 2004; Guide and Van Wassenhove 2009; Sasikumar and Kannan 2009; Shi et al. 2011; Toktay and Wei 2011; Ozkir and Basligil 2012).

When a manufacturer is involved in manufacturing as well as remanufacturing, the manufacturer faces the problem of inventory/production control since it is very difficult to manage the forward and backward flows of systems simultaneously. Van der Laan and Salmon (1997) presented a stochastic inventory planning control system for production, remanufacturing and disposal operations. The push-and-pull strategy was used to control the system in which all returned products were remanufactured. Koh et al. (2002) studied stationary demand that is satisfied by remanufacturing the product as well as by externally purchasing a brand new product. The model was validated by considering the examples of two remanufactured products: soft drink bottles and bullet jackets. Fleischmann et al. (2002) addressed the inventory planning problem of used products with independent return and considered the Poisson distribution. Fleischmann and Kuik (2003) investigated the inventory control system of manufacturing that is affected by returned products. Zhou et al. (2011) examined both manufacturing and remanufacturing systems using the inventory control strategy, which is an automatic order-based production control system. Choi, Hwang, and Kho (2007) studied deterministic demand that is satisfied by a newly purchased product and remanufactured product, and developed a mathematical model of joint economic order quantity and economic production quantity for inventory systems.

Furthermore, in the last two decades, a large number of environmental policies and legislations have been made on the product development process with a view to reducing the environmental impact of the product. CLSCM also considers additional design principles emanating from the requirement of sustainability. Kirkke, Harten, and Schuur (1999) addressed a network design principle for product recovery, rate of return, enhancing design for recycling and increasing the use of recycling. French and LaForge (2006) conducted an extensive study on reuse decisions made for components, products and materials through returns and sources. The CLSCM network design model can be categorised into a single objective (Kirkke, Harten, and Schuur 1999; Schultmann, Zumkeller, and Rentz 2006; Kannan, Sasikumar, and Devika 2010) and multiple objectives (Sheu, Chou, and Hu 2005; Pishvaee and Torabi 2010) during single and multiple periods. Alimoradi, Yussuf, and Zulkifli (2011) designed a closed-loop supply chain that consists of recovery options to find the best location for these facilities in a discrete space based on the opinion of decision makers. Ozkir and Basligil (2012) studied an optimal closed-loop supply chain network design for material recovery, component recovery and product recovery.

A few authors have emphasised on the categorisation of returned products, methods of collection and acquisition price management of used products. Guide and Van Wassenhove (2003) identified the common activities and types of return in CLSCM. Savaskan, Shantanu, and Van Wassenhove (2004) presented the categorisation of used products for remanufacturing, in which there is no distinction between the manufacturing and remanufacturing of the products. They have addressed the importance of remanufacturing of used products into new ones. It has been widely recognised in the literature and in practice that manufacturers can collect used products from the market in three ways: (i) manufacturers can collect directly from the customer, i.e. direct collection, (ii) manufacturers can provide suitable incentives to an existing retailer to induce collection and (iii) manufacturers can subcontract the collection activity with a third party. The second and third methods are called the indirect method of collection.

Guide and Jayaraman (2000) addressed a strategy for product acquisition management to monitor, coordinate and provide the interface between reverse logistics and production planning activities. Guide and Van Wassenhove (2001) developed a framework for analysing the profitability of reuse activities by considering the acquisition of used products. Guide, Teunter, and Van Wassenhove (2003) examined the optimal selling price and acquisition price that maximise the profit rate, and developed an analytical model for determining the optimal price of the used product that maximises the revenue. Bakal and Akcali (2006) studied a remanufacturing system for the end-of-life product in the automotive industry, and developed a mathematical model to determine the optimal value of the acquisition price and selling price in deterministic price sensitivity demand. Qu and Williams (2008) investigated the automotive shredder that generally balances the quality and quantity of incoming hulks by adjusting their acquisition price. Ling et al. (2009) developed a model to evaluate the acquisition price of used products. In this model, the acquisition price was determined based on the forecast sales of the remanufactured product, anticipated demand and the manufacturing cost. Anderson and Bao (2010) addressed price competition from supply chains to supply chains with a linear demand function, in which different manufacturers sell their product through exclusive retailers that compete for end customers. Galbreth and Blackburn 2010 examined the trade-off between acquisition cum scrapping costs and remanufacturing costs when the condition of used products varies widely and is uncertain. They found that when costs are linear and the optimal acquisition quantity has a closed-form solution and increases with the square root of condition variability. Shi, Zhang, and Sha (2011a) presented hybrid systems for manufacturing brand new products and remanufacturing returns similar to new ones with pricing and production decisions that satisfy the market demand. They developed a mathematical model to maximise the overall expected profit of the system by simultaneously considering the selling price, production quantities for a two-channel system and the acquisition price of the returned product considering uncertain situations. Zhou and Yu (2011) addressed management decision on acquisition price differentiation with inventory management. They analysed the behaviour of customers when they returned the products. Gu and Gao (2012) mentioned that the two competitive closed-loop supply chains focus on the management of the wholesale price, the retail prices and the collecting prices for the two competitive closed-loop supply chains. Shi, Zhang, and Sha (2011b) addressed a model to maximise the manufacturer's expected profit by jointly determining the production quantities of brand new products, the quantities of remanufactured products and the acquisition prices of used products, subject to a capacity constraint. Hahler and Fleischmann (2013) developed a mathematical model to determine the optimal acquisition price and collection quantities in a centralised and decentralised collection system. They compared the two collection strategies based on acquisition price differentiation in a reverse logistic system.

To maximise the overall profit of the remanufacturing system, the present study differs from the aforementioned studies in two ways. First, in this study, we introduce the issue of availability of used products in the market. Second, we develop a mathematical model in a stochastic return demand environment to evaluate three different methods of collection. This study is unique and makes contribution to the literature on CLSCM. To our knowledge, to date, none of the studies has investigated the effect of the availability of returned products on acquisition price in a stochastic return demand environment under different methods of collection.

3. Background of the problem formulation

Here, the model was developed in a line similar to that of Choi, Hwang, and Kho (2007) in which the remanufacturer fulfils the market demand by remanufacturing the returned products similar to new ones and through external purchasing of a brand new product. The remanufactured product can be sold at the same price in the same market, i.e. the remanufactured product can perfectly substitute brand new products. In the remanufacturing facility, used products are disassembled and inspected carefully, and components with good functions are reused while those with a lower quality are repaired, upgraded or replaced. Finally, the products are remanufactured as new ones; both defective and end-of-use products are used for remanufacturing. During the development of the model, it is assumed that the remanufacturer will sell the maximum quantity of used products collected from the market. After careful inspection for good conditions, all the components of the used products are remanufactured to meet the demand. Those products having no life are disposed off and demand of this portion of returned quantity is fulfilled through external purchasing.

For collection of returned products from the market, three methods were considered: direct method, indirect method and coordinated method. In the direct method of collection, the remanufacturer collects the used products and coordinates directly with the customer. The remanufacturer pays some value to the customer depending on the quality of the returned products. The main disadvantage of this method of collection is that the remanufacturer might not be able to collect more used products due to a lack of proper coordination. On the other hand, in the indirect method of collection, two stages are involved: in the first stage, the retailer collects the used products from the customer; in the second stage, the remanufacturer collects the returned products from the retailer. The remanufacturer is the Stackelberg leader and contracts some collection price (d) with the retailer. On the other hand, in the coordinated method of collection, the remanufacturer coordinates with the retailer to form a coordinated channel for collecting the used products from the market.

Here, we considered that the demand of the returned product is sensitive to the acquisition price as well as to the availability of the used product in the market. The relationship between the return demand and the acquisition price and the availability of the used product in the market can be represented as . The expected return quantity is defined as a function of acquisition price (f), market size (α ≥ 0) and available quantity (cQ), where c is the available dependency factor, with c ≥ 0, and price sensitive factor (β>0). The stochastic return quantity (ϵ) can be defined in the range [A, B] with the mean μ_r and the standard deviation σ_r. The value of A is usually zero and B is a value far larger than zero, and the probability distribution, e.g. uniform distribution, can provide an adequate approximation.

3.1 Assumptions and notations

The assumptions made in the development of the model are as follows.

Here, a single cycle period was considered.

The remanufacturer satisfies the market demand based on return quantity received from the market and by external procurement.

where R_m is the purchasing cost of a brand new product, R_r is the remanufacturing cost of the returned product in direct collection and is the remanufacturing cost of the returned product in indirect collection.

Notations

Q	=	demand of the remanufactured product
p	=	selling price of the product per unit
f	=	acquisition price of the returned product per unit
w	=	wholesale price of the product per unit
d	=	collection price of the returned product per unit
s	=	shortage cost per unit
c	=	available dependency factor
z	=	Q − (α+βf+cQ) (see Petruzzi and Dada (1999), Shi et al. (2011a))
f(·)	=	probability density function of ϵ
F(·)	=	cumulative distribution function of ϵ
Rm	=	purchasing cost per unit of new product
Rr	=	remanufacturing cost of the returned product in direct collection per unit
Cr	=	remanufacturing cost of the returned product in indirect collection per unit

 

The remanufacturing cost in all the methods of collection includes the cost of dismantling, inspection, quality assurance, remanufacturing and other management costs. For this proposed model, a sum of such cost is assumed as the remanufacturing cost.

4. Development of the mathematical model

In this section, we developed a model for evaluating the direct, indirect and coordinated methods of collection. Let denote the profit function of member i in the model for different methods of collection, where i takes the value of m, r and ch, which denotes the remanufacturer, retailer and channel firm, respectively.

4.1 Model for the direct method of collection

Here, the used product is directly collected by the remanufacturer from the market. There are no intermediaries involved in the collection process. Figure 1 shows the flow of payments in a direct collection.

Figure 1 Flow of payments in the direct method of collection for a closed-loop supply chain.

The objective of the remanufacturer is to maximise the expected profit, which can be written as

Since is a concave function (for proof, see Appendix A), the simultaneous solution of the first-order condition gives

The optimal market demand can be calculated as

For proof, see Appendix B.

Proposition 1

Under the direct collection of the returned product from the market, the acquisition price increases with the increase of its sensitivity factor.

Proof:

Hence proved. ▪

Proposition 2

Under the direct collection of the returned product from the market, the acquisition price decreases with the increase in the available dependency factor.

Proof:

Hence proved. ▪

Proposition 3

Under the direct collection, the acquisition price decreases with the increase in the market size of the returned product.

Proof:

 ▪

Proposition 4

Under the direct collection, the value of z increases with the increase in the available dependency factor.

Proof:

Hence proved. ▪

4.2 Model for the indirect method of collection

In this case, initially, the retailer collects the used product from the end customer at a certain price and then sells it to the remanufacturer. Figure 2 shows the flow of payments in an indirect collection.

Figure 2 Flow of payments in the indirect method of collection for a closed-loop supply chain.

The profit function of the retailer can be written as

The simultaneous solution of the first-order condition gives

For proof, see Appendix C.

The profit function of the remanufacturer can be written as

By substituting the values from equations 6, 7 and 8 in the above equation, weget the ptimal profit i.e.,

Finally, the total profit of the channel in the indirect method of collection is the sum of the profit of the retailer and manufacturer given as

4.3 Model for the coordinated method of collection

In this section, we examine the problem in which the remanufacturer individually coordinates with the retailer to form a channel-coordinated system. Figure 3 shows the flow of payments in the coordinated collection.

Figure 3 Flow of payments in the coordinated method of collection for a closed-loop supply chain.

The objective of the coordinated model is to maximise the total channel profit.

The profit function of the remanufacturer can be written as

The profit function of the retailer can be written as

The profit function of the system after coordination can be written as

Again, the objective function π_ch (12) is solved by the first-order condition (see Appendix D) and the value of the wholesale price and the fraction of the received quantity are as follows:

The optimal value of f and z can be obtained by the iteration of computations. Then, Q can be written as

5. Numerical example and managerial insights

In this section, a numerical study was conducted to illustrate the model. The parameters are set as follows:

θ = 0.7, , , c = 0.4, s = 100, α = 500, β = 30, p = 230, w = 200, C_r = 32, d = (1.1) f and demand follows a uniform distribution [A, B] = [0, 100] with a mean of 50.

The results showed that in the indirect collection, the channel profit is less than the profit observed in the direct and coordinated channels, and in the direct collection, the profit of the remanufacturer is more than that found in the other two channels (Table 1).

Table 1 Results of the different methods of collection.

Channel type	f^*	z^*	Q^*			Total profit,
Direct	99.94	93.71	5987	701,926	–	701,926
Indirect	13.12	39.13	1555	280,452	47,702	333,754
Coordinated	56.34	56.34	6217	510,286	250,141	766,027

5.1 Sensitivity analysis

Here, we carried out an extensive sensitivity analysis to examine the impact of various parameters on the solution.

5.1.1 Impact of market size

Here, the impact of market size (α) on the total channel profit () is studied. From Figure 4, it can be found that in all the three methods of collection, the total channel profit increases marginally as the market size increases. It was observed that as the market size increases, the coordinated method of collection makes more profit compared with the other two methods of collection. In the indirect method of collection, the total channel profit is less due to the presence of double marginalisation. In the direct collection system, since the remanufacturer collects the used product directly from the market, the total channel profit therefore lies in between the coordinated and indirect methods of collection.

Figure 4 Total channel profit versus market size (α) for the three methods of collection.

From Figure 5, it can be observed that the value of f marginally decreases as the market size increases in all the three methods of collection. In the direct method, the remanufacturer collects the used product directly from the customer with a lower acquisition price as the market size increases due to the availability of more returned products in the market. In the indirect method, the remanufacturer collects the returned product from the retailer at a higher acquisition price as the market size increases due to the lack of interest of the retailer to collect the used product from the market. In the coordinated method, the retailer collects the returned product directly from the customer at a higher acquisition price as the market size increases due to the coordination between the remanufacturer and the retailer.

Figure 5 Acquisition cost versus market size (α) for the three methods of collection.

The rate of change of the total profit in the direct channel is higher than that in the other two methods as the market size increases due to a higher acquisition cost. In the indirect method of collection, although the acquisition cost is low compared with the other two methods of collection, the remanufacturer will generate a less profit as the market size increases. The reason is that in the indirect method of collection, the remanufacturer will sell the product with a fixed wholesale price to the retailer and the retailer will sell that product to the market with a certain margin. Therefore, due to the presence of this double marginalisation, the total channel profit will be less in the indirect method than in the other two methods.

From the above analysis, it can be concluded that the value of f is negatively correlated with the total channel profit. Figure 6 shows the options that would help the remanufacturer to take decisions on the method of collection under different values of market size. If the remanufacturer wants to maximise his own profit under a threshold value (α < 2300), then the better way to collect the used product will be through the direct method. In this way, the remanufacturer will take decisions on the best method of collection under different values of market size for different profits.

Figure 6 Managerial decision matrix through the market size.

5.1.2 Impact of the coefficient of acquisition price (β)

Here, we examined the impact of the acquisition price sensitivity factor β on acquisition price (f) and expected channel profit () under different methods of collection. From Figure 7, it can be observed that the value of f increases as that of the acquisition price sensitivity factor increases until it reaches a threshold value of 52. When the value of β is greater than the threshold value, the value of f in all the three methods of collection increases marginally.

Figure 7 Acquisition cost versus coefficient of acquisition price (β) for the three different methods.

From Figure 8, it can be observed that the total channel profit increases as the value of β increases. In the direct method of collection, the value of the total channel profit increases following a parabolic curve as the value of β increases. The rate of change of the total channel profit in the direct channel is higher than that in the other two methods. As the acquisition price increases, the remanufacturer sells his product with a high wholesale price and, as a result, the channel revenue increases. However, it may not be feasible all the time due to the price-sensitive customer. The acquisition cost in the coordinated method of collection is higher than that in the other two methods of collection because both the remanufacturer and the retailer take decisions simultaneously to collect more number of used products from the market. In the indirect method of collection, when the value of β is less than 18, the value of f is not feasible as the acquisition price sensitivity factor increases. However, when the value of β is above 18, the value of f increases as the acquisition price sensitivity factor increases. It can be observed that under the direct method of collection, the channel profit increases at a constant rate when the value of β is less than 35, but above a value of 35, the value of the channel profit increases abruptly. The rate of change of the total channel profit in the indirect channel is less than that in both the direct and coordinated channels as the channel profit depends upon the acquisition price of the used products. In the direct method of collection, the remanufacturer will generate more revenue than that generated in the coordinated channel due to the direct selling of the remanufactured products to the customer with a fixed selling price. However, when the value of β is higher than 35, the remanufacturer generates less profit in the coordinated channel than that generated in the direct channel due to a higher acquisition cost. In the coordinated system, the total channel profit is shared by both the remanufacturer and the retailer.

Figure 8 Total channel profit versus coefficient of acquisition price (β) for the three different methods.

From this analysis, it can be concluded that in order to maximise the total channel profit, it is better to compare the coordinated system with the direct method when the value of β is less than a certain threshold value of 35; otherwise, the case will be reversed. Figure 9 shows the ways that would help the remanufacturer to take decisions on the methods of collection under different coefficients of acquisition price. If the remanufacturer wants to maximise his own profit under a threshold value (β < 35), then the better way to collect the used products will be through the coordinated channel. Similarly, the remanufacturer will take decisions on the best method of collection under different values of β.

Figure 9 Managerial decision matrix through the coefficient of acquisition price (β).

5.1.3 Impact of available dependency factor (c)

Here, we studied the impact of available dependency factor (c) on acquisition price (f) and expected profit (π ^*). As the values of the available dependency factor increases, the value of f decreases marginally, as shown in Figure 10. The acquisition price of the used products decreases with the increase in the value of c as c is inversely proportional to f. Therefore, the price-sensitive customer takes interest to buy more number of new products and, as a result, the total channel profit will be increased exponentially, as shown in Figure 11. It can be found that as the available dependency factor increases, the profit gained by the remanufacturer is higher in the direct method of collection than in the other two methods of collection. However, the total channel profit is higher in the coordinated method than in the direct method of collection as the value of c increases. The values of f and π* follow similar patterns in both the direct and indirect systems as c is directly proportional to the profit.

Figure 10 Acquisition cost versus available dependency factor (c) for the three different methods.

Figure 11 Total channel profit versus available dependency factor (c) for the three different methods.

From this analysis, it can be found that if used products with more good quality are available in the market as c increases, then the remanufacturer will collect more used products by providing a low acquisition price in both the channels due to the competition in the market.

6. Conclusion and future work

This paper has investigated the closed-loop remanufacturing system by considering different methods of collection and the availability of used products in the market. A mathematical model was developed to determine the optimal acquisition price by considering the availability of returned products in the market under a stochastic random demand environment. Here, the optimal acquisition price of the used product was determined for direct, indirect and coordinated methods of collection to maximise the remanufacturer's profit.

The analysis highlighted an interesting trade-off among the direct, indirect and coordinated methods of collection. The remanufacturer can satisfy the market demand and maximise his expected profit when the market size is large. When the market size is small, the remanufacturer is able to collect only a few used products from the market and, as a result, his expected profit will be decreased. The extensive sensitivity analysis revealed the impact of the acquisition price sensitivity factor on both acquisition price and channel profit. It was found that the channel profit increases as the acquisition price decreases and the availability of used products in the market increases. Furthermore, the results revealed that the remanufacturer would chose to collect used products directly from the customer when the market size is large, the acquisition price is less and the demand of new products is matched with any used products he collects from the market.

The proposed model and solution approach provides an effective tool to gain some insights into the closed-loop system. Here, we have considered only the remanufacturing process and assumed that the remanufacturer only acquires used products that can be remanufactured. This can be extended by considering repair, refurbishing, cannibalisation and recycling. In addition, many industries collect the used product at different acquisition prices by considering different quality levels of the returned product, and the present study can be extended in that direction. That the remanufactured product can be sold in a competitive market with different prices provides another major opportunity for future extension of this work.

Appendix A

Since

We know that

So I can write

So we can write

The hessian matrix H ( f, z) is given s follows:

Since, from the operating conditions, D₁ and D₂ both are greater than zero and the members of the principal diagonal are negative. Hence, the objective function is a concave.

Appendix B

(B.1)

(B.2)The simultaneous solution of the first-order condition B.1 and B.2 gives

(B.3)Similarly we can derive z

(B.4)Hence proved.

Appendix C

Since

(C.1)

(C.2)The simultaneous solution of the first-order condition C.1 and C.2 gives

(C.3)Similarly we can derive z

Hence proved

Appendix D

(D.1)

(D.2)