Physics

Procedía

2012 International Conference on Applied Physics and Industrial Engineering

Price Decisions of New Product Based on Subsidy-price-depending and Payment-sharing

Gu Qiaolun1, Gao Tiegang2

'School of Information Technology Engineering Tianjin University of Technology and Education Tianjin, P.R. China, 300222 2College of Software Nankai University Tianjin, P.R. China, 300071

Abstract

In order to stimulate private spending and curb pollution, the Chinese government has decided to offer financial subsidy to carry out home-appliance replacement from June 1, 2009 to May 31, 2010. Because the financial subsidy can be offered until May 31, 2010, the manufacturer must face the problem: if the government stops to offer the financial subsidy, whether to continue the home-appliance replacement or not? If continue to carry out the home-appliance replacement, the manufacturer should give the payment to the replacement consumer. In this case, there is a potential risk for the manufacturer: the unit cost of the new product will increase. In this paper, we studied the price decisions for the manufacturer and the retailer under two cases: case I, the government offers the financial subsidy which is depended on the retail price; case II, the government stops to offer the financial subsidy, the manufacturer will share the payment with the retailer. At the same time, we analyze the impacts of the subsidizing rate and the sharing ratio on the optimal results by a numerical example.

© 2011 Published by Elsevier B.V Selection and/or peer-review under responsibility of ICAPIE Organization Committee.

Keywords:price decision, new product, subsidy-price-depending, payment-sharing

1. Introduction

Nowadays, the home-appliance replacement has become a hot topic. The Chinese government has decided to offer financial subsidy to carry out home-appliance replacement from June 1, 2009 to May 31, 2010. It is the subsidy of the home-appliance replacement that increases the replacement amount and the new product demand, and the manufacturer can get more profit. But the situation will be changed after May 31, 2010. So, the manufacturer must face a problem: if the government stops to offer the financial subsidy, whether to continue the home-appliance replacement or not? If continue to carry out the home-

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Physics Procedía 24 (2012) 1073 - 1080

1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of ICAPIE Organization Committee. doi:10.1016/j.phpro.2012.02.160

appliance replacement, the manufacturer must be responsible for the subsidy by giving the payment to the replacement consumer. In this case, there is a potential risk for the manufacturer: the unit cost of the new product will increase. How to control the risk is an important problem, too.

In order to solve these problems, we will give the price decisions for the manufacturer and the retailer in two cases: case I, the government offers the financial subsidy which is depended on the retail price; case II, the government stops to offer the financial subsidy, the manufacturer will share the payment with the retailer. At the same time, we analyze the impacts of the subsidizing rate and the sharing ratio on the optimal results by a numerical example.

This paper is organized as follows. In section II, we review the literature about the price decisions for product replacement. In section III, we will show and analyze the price decisions for the new product under two cases. In section IV, we will give a numerical example and analyze the optimal results. We summarize our results in section V.

2. Literature Review

The literature related to our study is about the trade-in rebates and cost allocating. For the trade-in rebates, Saibal et al studied the optimal pricing/trade-in strategies for durable, remanufacturable products. The model they gave can help the managers to determine the optimal price for new customers and the optimal trade-in rebate for replacement customers, for quite a general class of age-profile distributions [1]. Fudenberg and Tirole, Levinthal and Purohit studied the issues related to monopoly pricing and/or production policies of successive generations of a product. Their study is based on a two-period framework, and in the second period, the manager should decide to offer trade-in rebates for upgrades to repeat purchasers or to buyback some of the old models [2][3]. Fei analyzed and compared the trade-in strategy which manufacturers carry out and the trade-in subsidies the government carries out, concluded that the policy of the government is more conductive to the expansion of consumer demand for electrical household appliances [4]. In the work of [5-7], the authors addressed how individual customers make their replacement decisions based on the "mental book value" of their current product and their utility for the new product.

For the cost allocating, Wei gives a scheme by allocating part of the cost of the investment to other divisions [8]. Toktay and Wei assume that manufacturing and remanufacturing operations are undertaken by two separate divisions, the new and remanufactured products are sold to different market segments that do not overlap. They propose a mechanism that achieves optimality in a decentralized system with two self-interested decision-makers who are responsible for the two processes respectively. This mechanism is a cost allocation mechanism that allocates a portion of the initial production cost of the product to each of the two stages of the product life-cycle [9]. In reference [9], they focus on the new products and the remanufactured product. In the work of [10], Gu and Gao analyzed the investment risk for upgrade-products, gave the control strategy for the investment risk by allocating the investment cost at two stages: manufacturing stage and remanufacturing stage. Based on the strategy, the price decisions of the upgrade-product's sale price and the used-product's collecting price, the demand of the upgrade-products and the supply of the used-products, and the profits of the two stages are presented. The reasonable investment cost allocation ratio is shown to ensure the growth of the total profit.

Comparing with the above literatures, our study in this paper has the following features: For the price decisions, we consider the subsidy which is depended on the retail price and the payment sharing between the manufacturer and the retailer.

3. Price Decisions

In this section, we will give the price decisions under two cases: In case I, the manufacture produce new products and the retailer sell the new products in the consumer market, the government will offer the financial subsidy to stimulate the home-appliance replacement. In case II, the manufacture makes new products and the retailer sells the new products in the consumer market, the government stops to offer the financial subsidy. In order to maintain the consumer market, the manufacturer will continue the home-appliance replacement by offering a payment to the replacement consumer.

3.1 Assumptions and Notations.

In order to facilitate the analysis, the following assumptions are postulated. Assumption 1, the manufacturer has the same unit manufacturing cost of a new product in each case and the retailer has the same unit operating cost when the retailer sells a new product. Assumption 2, the manufacturer has sufficient channel power over the retailer to act as a Stackelberg leader in each case. Assumption 3, the manufacturer should offer the payment that equals to the subsidy when the government stops the subsidy.

The notations used in this paper are shown as bellow.

Variables:

p1m, p2m : The unit wholesale price of the new product in case I and case II respectively.

p1r, p2r: The unit retail price of the new product in case I and case II respectively.

Costs:

cm : The unit manufacturing cost of a new product used in each case.

cr : The unit operating cost when the retailer sells a new product in each case.

Parameters:

Dj(p1r): The demand for new products used in case I, D1(p1r) = <- ¡plr + as . It is a decrease function of the retail price and an increase function of the subsidy s which depends on the retail price, s = ap1r ,0 < a < 1, a is the subsidized rate. c means the sensitivity coefficient of the consumer for the

subsidy (or payment used in case II) . < and 3 being positive parameters and < > ¡cm . < means the potential market of the original products and 3 means the sensitivity coefficient of the end-customer for the sale price.

D2(p2r) : The demand for new products used in case II, D2(p2r) = <- ¡p2r +asl. It is a decrease function of the retail price and an increase function of the payment s1 which depends on the retail price, here s1 = ap2r.

Y: The manufacturer's share ratio of the payment used in case II.

n 1m, n2m : The total profit of the Manufacturer used in each case respectively.

n1r, n2r: The total profit of the Retailer used in each case respectively.

3.2 Price Decision for Case I.

In this case, the manufacturer's problem is to maximize his profit by deciding the wholesale price and the retailer's problem is to maximize her profit by deciding the retail price, their problems are shown as equation (1) and (2).

Max n 1m = (p1m - cm )« - faxr +cap1r ). (1)

Max n ir = (pir - cr - pim - Mr + aapir )• (2)

Here, P-aa should not be zero.

In terms of the assumption 2, it is easy to get the optimal results of the wholesale price and the retail price of case I:

p*m = (0 + (P-aa)(cm - cr))/(2(P-aa)). (3)

p*r = (30 + (P-aa )(cm + cr )) /(4(p - aa )). (4)

Moreover, we can get D*, n*m and n*r by substituting pm and p1r inn 1m , n 1r andD1 with p*m

and p*r,

D* = (0- (P-aa)(cm + cr))/4 (5)

n 1m = (0 - (P - aa)(cm + cr))2 /(8(P - aa)) (6)

n*r = (0 - (P - aa)(cm + cr))2 /(16(P - aa)) (7) From equations (3)-(7), we can get the proposition 1 directly.

Proposition 1 In case I, i) If a < (P / a), then p*m > 0, p*r > 0, D* > 0, n*m > 0 and n*r > 0. ii) If a > (P / a), then p*m < 0, p*r < 0, n*m < 0 and n*r < 0, butD* > 0.

3.3 Price Decision for Case II.

In this case, when making the price decision, the manufacturer will minus the payment which he should share in his profit function, and the retailer will minus the payment which her should share in her profit function, namely,

Max n 2m = (p2m - cm - Yap2r )(0 - P2r + aap2r ). (8)

Max n 2r = (p 2r - cr - p 2m - (1 - ï)ap 2r )(0 -p2r (9)

/3p2r +aap2r ).

Use the same method as case I, the optimal results of the wholesale price and the retail price in case II are obtained:

p*m = (2A20 + (P - aa )(cm - 2(L-ya)cr ))/ (2(2A - a)(P - aa)).

p * r = ((6 A-a) A0 + (P-aa)( cm + 2 Acr ))/ (4A(2A - ya)(P - aa)).

Here A = 1 - (1 - y)a .

From the above equations, we can get the following propositions easily.

Proposition 2 In case II, when a is fixed, p *m and p *r decrease as Y decreases, D*, n *2m and n *2r increase as Y decreases.

Proposition 3 In case II, when Y is fixed, i) If a < (P / a) , then p*m > 0 , p*r > 0 , D* > 0 , n *2m > 0 and n *2r > 0 . ii) If a > (P / a), then p *m < 0, p *r < 0 , n *2m < 0 and n 2r < 0, butD2* > 0.

4. Numerical Analysis

In this section, we present the numerical analysis of the optimal results via a numerical example. Here, let <j> = 10000,P = 2, cm = 100,cr = 20,a = 3.

4.1 Impacts of a on the Optimal Results of Case I.

If the subsidy depends on the retail price, the subsidized rate a will affect the price decisions and the profits. The following figures show us the changing of the wholesale price, the retail price, the new product demand, the profit of the manufacturer, and the profit of the retailer with different a .

FIGURE 2. Changing of the new product demand with different a in case I

FIGURE 3. Profits changing of the manufacturer and retailer with different a in case I

From the above figures, we can find the results: when a < (fi / m), namely, a < 2/3, the wholesale price, the retailer price, the profit of the manufacturer and the profit of the retailer are positive; when a > 2/3, the wholesale price, the retailer price, the profit of the manufacturer and the profit of the retailer are negative; but the new product demand increases when a increases.

4.2 Impacts of Y on the Optimal Results of Case II.

Here, let a = 0.4 . In this situation, the share ratio of the payment 7 will affect the price decisions and the profits. The following figures show us the changing of the wholesale price, the retail price, the new product demand, the profit of the manufacturer, and the profit of the retailer with different 7 .

11000 10000 I 9000 ! 8000 I 7000 i 6000 ! 5000 ! 4000 î 3000 2000 1000

----wholesale price

—O— retail price —*— demand

FIGURE 4. Changing of wholesale price, retail price and demand with different Y in case II

• manufactuer's profit - retailer's profit

FIGURE 5. Profits changing of the manufacturer and the retailer with different Y in case II

The above figures tell us the results: if a = 0.4 , when the manufacturer's share ratio of the payment decreases, the wholesale price and the retailer will decrease, the new product demand, the profits of the manufacturer and the retailer will increase. It means the manufacturer and the retailer can get more profit with the payment sharing method.

4.3 Comparing the Optimal results of Two Cases.

In order to compare the optimal results of two cases, we assume Y is fixed. Here, letY = 0.5. The changing of the optimal results of two cases are shown as figure 6 and figure 7.

FIGURE 6. Changing of wholesale price, retail price and demand with different a in two cases

FIGURE 7. Profits changing of the manufacturer and the retailer with different a in two cases

Figure 6 and figure 7 show us the results: i) if Y is fixed, when a < (ft / a>), namely, a < 2 / 3 , the wholesale price, the retailer price, the profit of the manufacturer and the profit of the retailer in two cases are positive; when a > 2 / 3 , the wholesale price, the retailer price, the profit of the manufacturer and the profit of the retailer in two cases are negative; the new product demand in case I will increase when a increases while the new product demand in case II will decrease. ii) if Y is fixed, when a < 2 / 3 , the wholesale price in case I is not smaller than the wholesale price in case II; the retail price in case I is not larger than the retail price in case II; the manufacturer's profit in case I is not smaller than the manufacturer's profit in case II; the retailer's profit in case I is not smaller than the retailer's profit in case II.

5. Conclusions.

In this paper, we studied the price decisions for the manufacturer and the retailer under two cases: case I, the government offers the financial subsidy which is depended on the retail price; case II, the government stops to offer the financial subsidy, the manufacturer will share the payment with the retailer. At the same time, we analyze the impacts of the subsidizing rate and the sharing ratio on the optimal results by a numerical example.

Acknowledgment

This work is supported by the National Nature Science Foundation under Grant No. 70871089. References

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