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supply chains with risk-averse

Int.J.Production Economics 114(2008)187–200

Price and service competition of supply chains with risk-averse

retailers under demand uncertainty

Tiaojun Xiao ?,Danqin Yang

School of Management Science and Engineering,Nanjing University,Nanjing 210093,China

Received 22May 2006;accepted 12January 2008

Available online 1February 2008

Abstract

We develop a price–service competition model of two supply chains to investigate the optimal decisions of players under demand uncertainty.Each supply chain consists of one risk-neutral supplier and one risk-averse retailer.We analyze the effects of the retailers’risk sensitivity on the players’optimal strategies.We ?nd that the higher the risk sensitivity of one retailer,the lower his optimal service level and retail price will be,while the effects of the rival’s risk sensitivity on his decisions depend on the substitutability of the two products.The optimal wholesale price of one supplier is ?rst increasing and then decreasing with the risk sensitivity of the two retailers if the substitutability is suf?ciently low,otherwise decreasing with the risk sensitivity.The expected equilibrium order quantity of one retailer is often increasing with his risk sensitivity.We also study the effects of the wholesale prices and the service investment ef?ciencies on the retail price–service level decisions of the retailers.We ?nd that the higher the service investment ef?ciency of one retailer,the lower the optimal retail price and service level of his rival will be.r 2008Elsevier B.V.All rights reserved.

Keywords:Service competition;Pricing;Risk sensitivity;Supply chain management;Game theory

1.Introduction

With the development of technology and the globalization of economy,the competition form of ?rms is evolving from the competition among ?rms to the competition among supply chains.For example,Microsoft (software supplier)and HTC (device manufacturer)constitute a supply chain that competes with the supply chain consisting of Symbian (software supplier)and Nokia (device manufacturer).How do the factors or the decisions

of the members in one chain affect the decisions of the members in the rival chain?This is an interesting topic focused by managers.In this paper,we will explore the competition between two supply chains.Pricing is important business behavior and competing ?rms often play a price war to attract customers.Besides price,service is also an impor-tant factor affecting the buying decisions of customers.For example,in auto industry,?nancial services such as auto loan,insurance,and main-tenance service play an important role in selecting a brand for customers.The customers of mobile,computer,and so on,often concern post-sale services.In the fast-food industry,some fast-food chains are seeking to attract customers by adding

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Corresponding author.Fax:+862583597501.

E-mail addresses:xiaotj@https://www.sodocs.net/doc/2a12373755.html, (T.Xiao),yangdanqin@https://www.sodocs.net/doc/2a12373755.html, (D.Yang).

gourmet sandwiches or premium breads to their menus,and it is also found that many of the new sandwich diners also want atmosphere,especially among the high-end sandwich eaters(Leung,2002). However,a?rm has to pay a cost for the service. Thus,the?rm must make a trade-off between the investment and the bene?t from providing service. What are the optimal retail prices and the service levels?How does the service investment ef?ciency of one retailer affect the rival’s decisions?

Owing to occasional factors or events,market demand becomes highly uncertain across many industries.The members of supply chain have to make the decisions such as price,production quantity,and investment based on the forecast for future demand.For example,in the case of Dell, Intel provides processors for Dell PCs.Intel takes several weeks to manufacture processors,but Dell cannot wait that long after a customer order.Intel must produce processors in advance.Thus,Dell and Intel have to make their ordering and production plans under demand uncertainty(Chopra and Meindl,2001).Retailers were often encouraged to place initial orders long before the products are introduced in the industries characterized by short product life cycles such as fashion apparel,toys,and computer hardware.The retail price,market de-mand,and production cost are often uncertain when a?rm determines the decisions in product selection and plant dimensioning(Alonso-Ayuso et al.,2005)

The risk attitude of one retailer towards demand uncertainty plays an important role in his decisions such as pricing,purchasing,and service investment. For example,companies were holding on to more cash because their risk aversion is still above average on the part of management due to the deep strike from the bankruptcies in the previous years (Zuckerman,2005).Economists alleged that the economy reversal is taking place exactly when business-investment spending has been much lower than anticipated,if there is an increase in risk aversion(Lp and Whitehouse,2007).A risk-averse retailer is trying to maximize his utility other than his expected pro?t.In general,the retailer’s utility is an increasing function of his expected pro?t but a decreasing function of the uncertainty and risk sensitivity(Holmstrom and Milgrom,1987;Gan et al.,2005).How do the uncertainty and the risk sensitivity of one retailer affect his and rival’s decisions?How do these two factors in?uence the optimal wholesale prices of his and rival’s suppliers?

To the best of our knowledge,few works consider uncertainty and risk sensitivity under the price and service competition structure of two independent chains.We consider two independent chains to address the above issues,where each supply chain consists of one risk-neutral supplier and one risk-averse retailer.We assume that two retailers compete in retail price and service investment whereas two suppliers compete in wholesale price under demand uncertainty.We?nd that the service investment ef?ciency of one retailer greatly affects the optimal price–service decisions of his rival.Risk sensitivity/demand uncertainty for one retailer greatly in?uences the optimal decisions of the members in the rival chain and the expected pro?t of the rival’s supplier;however,its effect on the expected order quantity of his rival is very small.We also show the importance of risk sensitivity for the players’bargaining power.

The reminder of the paper is organized as follows. The related literature is reviewed in Section2and then the basic model is presented in Section 3. Section4analyzes the effects of the risk aversions and the service investment ef?ciencies of the retailers and the wholesale prices of the suppliers on the retail price–service decisions of two compet-ing retailers.Section5concerns the suppliers’wholesale price decisions.Finally,in Section6we summarize the results and point out directions for future research.

2.Literature review

Competition between two retailers is well studied in operation and marketing literature.For example, Trivedi(1998)found that the competitions at both retailer and manufacturer levels have signi?cant impacts on the members’pro?ts and prices.Choi (1996)focused on the intra-and inter-channel price competitions.Van Mieghem and Dada(1999) studied the production capacity decision in price competition and extended simultaneous price post-ponement duopoly to oligopoly.Dai et al.(2005) considered the pricing strategies of multiple?rms providing the same service and owing?nite capacity in a revenue management context.

Besides price,from the perspective of customers’behavior,service in?uences the customers’prefer-ences and their purchasing decisions,and hence market demand.A few papers regarded service as a new dimension for competition.Iyer(1998)studied multi-echelon coordination under price and non-

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200 188

price competition.The retailer in the traditional channel can compete against the e-tail channel by adding some value-added services(Yao and Liu, 2005).The retailer’s innovation stimulated by the supplier’s strategic commitment to a wholesale price is similar to service improvement in that both enhance market demand(Gilbert and Cvsa,2003). Dumrongsiri et al.(2008)studied the price–service competition between the two channels of the manufacturer(direct channel and retail channel), and found that an increase of the retailer’s service quality may increase the manufacturer’s pro?t and a larger range of customer service sensitivity may bene?t both parties.In our model,the two retailers play a price–service competition given the wholesale prices of suppliers.

In the economic literature on pricing,most assumed deterministic price-sensitive demand(e.g., Boyaci and Gallego,2002).A few papers added demand uncertainty to the pricing models(Deneck-ere et al.,1997;Mantrala and Raman,1999;Dana, 2001;Kunnumkal and Topaloglu,2008).But the attitude towards uncertainty is often ignored.In reality,risk attitude plays an essential role in the members’decisions(Webster and Weng,2000; Tsay,2002;Lee and Schwarz,2007).Webster and Weng(2000)addressed risk sensitivity on the manufacturer’s side and did not make explicitly a quantitative analysis of the manufacturer’s attitude towards uncertainty.Tsay(2002)investigated how risk sensitivity affects the manufacturer and retai-ler’s preferences for the distribution policies,i.e., return policy,under various scenarios of strategic power,and suggested that the penalty of ignoring risk sensitivity of a channel partner can be substantial.Yang et al.(2008)suggested a near-optimal algorithm to solve the EOQ equilibrium of a supply chain with one supplier and two risk-averse retailers competing in price,service,and lot size. Asplund(2002)showed that risk sensitivity impacts the competitive intensity.However,the above paid no attention to the interaction among multiple supply chains.Xiao and Yang(2008)developed an information revelation mechanism model of a one-manufacturer and one-retailer supply chain facing an outside integrated competitor to investigate the effect of the risk-sharing rule on the revelation mechanism under demand uncertainty,where the risk sensitivity of the retailer is private information and the retailer plays a price–service competition with the outside integrated manufacturer.We not only consider the price and service competition jointly,but also focus on the effects of the retailers’risk sensitivity on the optimal decisions of the players in the framework of two independent supply chains,in particular,on the optimal decisions of the players in the rival chain.

For service-level selection under an uncertain environment,Ishii(2000)established a Cournot duopoly competition model where risk-averse?rms choose optimal R&D levels under demand uncer-tainty.Ishii(2000)studied the R&D quantity competition between two?rms whereas we consider the price–service competition between two supply chains.He assumed that two?rms play an R&D competition in the?rst stage and a quantity competition in the second stage;however,two retailers simultaneously determine their retail prices and service levels in our model.

Our paper is closely related to Tsay and Agrawal (2000)and Bernstein and Federgruen(2004).Tsay and Agrawal(2000)studied a distribution system consisting of one manufacturer and two retailers competing in both retail price and service.But the decision structure is different because we consider the competition between two supply chains.We assume that all decisions were made before the demand uncertainty is resolved,while they did not incorporate uncertainty into the demand model. Moreover,our model explicitly considers the retailers’risk sensitivity and focuses primarily on the effects of risk sensitivity on the members’decisions.Bernstein and Federgruen(2004)devel-oped a general equilibrium model of oligopoly retailers competing in price and service under demand uncertainty.They did not consider the supplier’s decision and did not concern the retailers’risk sensitivity,which are mainly studied in our model.Our paper is also closely related to Jain (2007)for the investigation of the competition between two separate chains.Jain(2007)considered a supply chain consisting of two separate produc-tion-inventory systems to investigate the value of pooling capacity,where each consists of one inventory location and one production queue.

3.The basic model

Consider two competing supply chains facing uncertain demands,where each supply chain con-sists of one risk-neutral supplier and one risk-averse retailer.Two retailers compete in retail price as well as service investment.We assume that each retailer has a long-term relationship with his supplier,which

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200189

is assured by his individual rationality constraint. The products of two suppliers are partially differ-entiated and each supplier sells products to custo-mers through his retailer.

We have the following notations(i?1,2):

~a i the stochastic market base for retailer i,

with meanˉa i40,variance s2

i ;

c i the unit production cost of supplier i,

ˉa i X c i40;

d th

e substitutability coef?cient o

f the two

products,0o d o1;

p i the retail price of retailer i;

s i the service level of retailer i;

R i the reservation utility of retailer i,R i X0; w i the unit wholesale price of supplier i;

b the demand sensitivity of one retailer to his

own service level;

g the demand sensitivity of one retailer to the

rival’s service level,b4g40;

l i the constant absolute risk aversion (CARA)of retailer i,which is de?ned in

the Arrow–Pratt sense,l i X0;

Z i the service investment ef?ciency coef?cient of retailer i,Z i40.The larger the coef?cient

Z i,the lower the service investment ef?-

ciency(1/Z i)of retailer i will be.

In Section1,we have pointed that retail price and service level are two important factors affecting the market demand.Thus,similar to Tsay and Agrawal (2000),we assume that the demand function of retailer i is

~q i?~a iàp itdp jtb s iàg s j;i;j?1;2;j a i.

(1) The retailer with largerˉa i has a relative advantage of accessing customers due to a better brand, position,reputation,quality,and so on.The market demand of each retailer is an increasing function of his rival’s retail price and his own service level,but a decreasing function of his own retail price and the rival’s service level.We assume that the service

(investment)cost function of retailer i is1

2Z i s2

for

service level s i,i.e.,improving service level has a diminishing return on service expenditure(Gilbert and Cvsa,2003;Tsay and Agrawal,2000).Our model can be extended to the case where two retailers have different b and g because the effects of the difference between the service sensitivity of customers in two channels on the main result can be re?ected by the difference between the service investment ef?ciency coef?cients Z1and Z2. According to the above conclusion,the random pro?t of retailer i is

~p i?ep iàw iTe~a iàp itdp jtb s iàg s jTà1

Z i s2

, i;j?1;2;j a i.(2) Considering the risk sensitivity of the retailers,we assume that each retailer assesses his utility via the following Mean–Variance value function of his random pro?t(Agrawal and Seshadri,2000;Tsay, 2002;Gan et al.,2005;Lee and Schwarz,2007):

u ie~p iT?Ee~p iTàl i Vare~p iT,(3) where the second term is the risk cost of retailer i, and l i re?ects the attitude of retailer i towards uncertainty.Eq.(3)means that retailer i will make a trade-off between the mean and the variance of his random pro?t.The larger the CARA l i of retailer i, the more conservative his behavior will be.

The time order of this game is as follows: Stage1:Two suppliers determine their wholesale prices simultaneously;

Stage2:Two retailers jointly determine their retail prices and service levels,respectively.

This dynamic game can be analyzed by using backward induction technique.We will analyze the retailers’decisions given the suppliers’wholesale prices in the next section,and then investigate the suppliers’decisions in Section5.

4.The price and service decisions of the retailers

According to Eqs.(2)and(3),we know that given the wholesale prices,retailer i(i?1,2)jointly determines retail price p i and service level s i to maximize

u ie~p iT?ep iàw iTeˉa iàp itdp jtb s iàg s jT

à1

Z i s2

àl iep iàw iT2s2

.(4) Hessian matrix of u ie~p iTis

H i?

à2e1tl i s2

bàZ i

The utility function u ie~p iTis a concave function on (p i,s i)if and only if Hessian matrix H i is negatively

de?nited.De?ne B i?2e1tl i s2

Tàb2=Z i.

From Eq.(4),we derive the following. Proposition 1.If B1,B240,then the optimal retail price and the optimal service level of

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200 190

retailer i are

?M itw i,(5)

?b M i=Z i,(6) where M i??eˉa iàw itdw jTB jàeˉa jàw jtdw iTV j =eB i B jàV i V jTand V i?bg=Z iàd.

Proofs of all propositions are given in Appendix A.M i is the(gross)unit pro?t of retailer i,including his service investment.Retailer i will withdraw from the market such that retailer j monopolizes the market if M i p0.Thus,we assume M i40through-out this paper.b/Z i is the fraction of the service investment of retailer i in his gross pro?t.

From Eqs.(5)and(6),it follows that the higher the unit pro?t of retailer i,the higher his optimal service level will be.Noteˉa iàw itdw j?B i M itM j V j.From Proposition1and Eqs.(1) and(4),it follows that the optimal expected demand

of retailer i is Ee~q?

i T?A i M i and the optimal utility

of retailer i is u?

i e~p?

T?1

B i M2

,where A i?1t

2l i s2

i X1.In Proposition1,the condition B i40

assures that the solution satisfying the?rst-order

condition of u ie~p iTis optimal and retailer i obtains a

positive utility.Retailer i excessively invests in

service if B i p0,which incurs a negative utility to him.The condition B i40means that the service

investment should not be too inexpensive,which is

consistent with those made in Tsay and Agrawal

(2000)Gilbert and Cvsa(2003).Thus,we assume

B1,B240throughout this paper.

For convenience,we give the following lemma.

Lemma 1.If max f0;d?

àg o d o min f d?

;1g,then we

have B1B2àV1V240,where

When two retailers are symmetric and bebàgT=Z1p1t2l1s21(bebtgT=Z1p2e1tl1s21T),there exists at most one of the substitutability thresholds

d??lying in the open interval(0,1),denoted by d*.

Furthermore,we have B1B2àV1V240for all d4d* (d o d*).Lemma1means that(B1B2àV1V2)is positive only when the substitutability of the two products is medium;otherwise,(B1B2àV1V2)is negative.

From Lemma1and Proposition1,we derive the following propositions.Proposition2.Assume max f0;d?

g o d o min f d?

;1g. The optimal retail price and service level of retailer i satisfy

(i)If B j X1,then we have q p?

=q w j40and q s?

= q w j40;

(ii)If B j o1,then we have q p?i=q w j40and

q s?

=q w j40for0o d o bg=?Z je1àB jT ;q p?

=q w j?0and q s?i=q w j?0for d?bg=?Z je1àB jT ;q p?i= q w j o0and q s?i=q w j o0for bg=?Z je1àB jT o

d o1.

Proposition2analyzes how the optimal retail price and service level of one retailer are affected by the wholesale price of the supplier in the rival chain. Proposition2means that if the service investment ef?ciency of the rival is suf?ciently low or his demand uncertainty is suf?ciently large such that B j X1,the increase of the wholesale price charged to the rival will lower the rival’s competitive advantage such that the retailer(i)would like to improve service to increase competitive advantage and raise retail price to make a higher unit pro?t.Otherwise, i.e.,B j o1,the result is contrary(similar)if the substitutability coef?cient is suf?ciently large (small).

Now we illustrate the effect of the wholesale price of one supplier on the optimal retail price and service level of his retailer.Since the positions of two chains are symmetrical,we only consider the effects of the wholesale price of supplier 1. For simplicity,we assume that two retailers are identical and the default values of parameters are as follows:

ˉa1?ˉa2?10;l1?l2?0:1;s1?s2?2,

Z1?Z2?1;g?0:6and w2?8.

From B1?B2?2.8àb2and V1?V2?0.6bàd,we have

B1B2àV1V2?e2:8àb2t0:6bàdT

?e2:8àb2à0:6btdT

which is positive for b?1,and equals zero only at d*?0.35for b?1.5.Furthermore,it is positive

d???

bgeZ1tZ2T?

?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

b2g2eZ1tZ2T2t4Z1Z2f b2?b2àg2à2Z2e1tl2s2

T à2Z1e1tl1s2

T?b2à2Z2e1tl2s2

T g q

2Z1Z2

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200191

for b ?1.5and d 40.35.So,the condition

max f 0;d ?àg o d o min f d ?

t;1g

(or B 1B 2àV 1V 240)holds for the values of parameters.According to Propositions 1and 2,we have Table 1.

From Table 1,we ?nd that the effect of the wholesale price of one supplier on the optimal retail price of his retailer depends on the relationship between the substitutability d and the service sensitivity parameter b for demand.In general,the higher the wholesale price of one supplier,the lower the optimal service level of his retailer will be.Under some conditions,the optimal service level of one retailer decreases whereas his optimal retail price increases as the wholesale price charged to him increases.It is most likely for this phenomenon to happen when the customers for one chain are relatively insensitive to service competition.Table 1also illustrates the effects of the wholesale price in one chain on the decisions of the rival retailer,which is consistent with Proposition 2.

Proposition 3.Assume max f 0;d ?àg o d o min f d ?

t;1g .We have

1.q s ?i =q l i o 0and q p ?

i =q l i o 0;

2.q s ?i =q l j o 0and q p ?

i =q l j o 0if bg =Z j o d o 1;

q s ?i =q l j ?0and q p ?

i =q l j ?0if d ?bg =Z j ;

q s ?i =q l j 40and q p ?

i =q l j 40if d o bg =Z j .

Proposition 3gives us the following insights:

The service level of a risk-averse ?rm is usually smaller than that of the risk-neutral ?rm.More-over,the higher the risk sensitivity (CARA)of one retailer,the lower the optimal service level of the rival will be if the substitutability of the two products is suf?ciently high,which reduces the service level of the whole industry.

Each retailer may reveal higher risk sensitivity than his real risk sensitivity to enhance his competitive advantage in service when the sub-stitutability is suf?ciently high.However,when the substitutability is suf?ciently low,the retailer would like to reveal lower risk sensitivity to attract customers from his rival’s market.

The increase of the risk sensitivity of one retailer results in the decrease of his service level such that the retailer should lower his retail price to compete with the rival retailer.

A better service should be compensated by a higher retail price because of the horizontal competition between the two retailers.

When the rival behaves more conservative,the retailer would like to undercut retail price to attract more customers and lower service level to save investment (i.e.,the retail price is more strategic than the service for the retailer)if the substitutability is suf?ciently high;otherwise,the retailer would like to improve his service to attract more customers and charge a higher retail price to make a higher unit pro?t (i.e.,the service is more strategic than the retail price for the retailer).

Similar to Proposition 3,demand uncertainty has the same effect on the equilibrium retail prices and service levels as that of the retailer’s risk sensitivity due to the speci?c relationship between them.

From Proposition 1,we can derive the following.

Proposition 4.If max f 0;d ?àg o d o min f d ?

t;1g and b B j tg V j 40,then the higher the service investment ef?ciency of one retailer (i),the higher his optimal retail price and service level will be .

Proposition 4implies that the more expensive the service investment of one retailer,the more strategic the retail price for the retailer if the substitutability of the two products is suf?ciently small (d p bg =Z j ).From f ?B 1B 2àV 1V 240and b 4g 40,it follows that the conditions of Proposition 4always hold when two retailers and their demand uncertainties are identical.

Figs.1and 2illustrate the effects of the service investment ef?ciency of the rival,where

Table 1

Optimal retail prices and service levels of the retailers No.d b w 1p ?1s ?1p ?2s ?210.4 1.069.78 3.7810.03 2.0320.4 1.0810.60 2.6010.60 2.6030.4 1.01011.43 1.4311.18 3.1840.4 1.56N N 50.4 1.5812.957.4312.957.4360.4 1.510N N 70.8 1.0612.28 6.2812.48 4.4880.8 1.0813.25 5.2513.25 5.2590.8 1.01014.23 4.2314.03 6.03100.8 1.5623.2325.8517.2313.85110.8 1.5820.9219.3820.9219.3812

0.8

1.5

18.62

12.92

24.62

24.92

Note :‘‘N’’represents that the retailer will withdraw from the

market.

T.Xiao,D.Yang /Int.J.Production Economics 114(2008)187–200

192

ˉa 1?ˉa 2?10,l 1?l 2?0.1,s 1?s 2?2,b ?1,g ?0.6,w 1?w 2?8,and Z 2?1.

From Figs.1and 2,we know that,in general,the higher the service investment ef?ciency of the rival retailer (i.e.,the smaller the Z j ),the lower both the optimal retail price and the optimal service level of the retailer (i )will be.That is,the retailer would like to offer a lower price to attract customers and provide a lower service level to save the investment due to his service investment disadvantage when the service investment ef?ciency of his rival becomes higher.For higher substitutability,there exists a larger effect of the rival’s service investment ef?ciency on the optimal price–service decisions of the retailer.

5.The suppliers’wholesale prices

Now,we study the optimal wholesale prices of the risk-neutral suppliers.Considering the two retailers’reactions in retail price and service,supplier i chooses wholesale price w i to maximize his expected pro?t,1i.e.,

max w i

E e~p ?S i T?A i ew i àc i TM i ,

(7)

subject to

u ?i e~p ?i T?1

B i M 2i X R i ,

(8)

where Inequality (8)is Individual Rationality constraint (IR)under which retailer i would like to accept the wholesale price (contract);otherwise,he will reject it and gain a reservation utility R i X 0through selling products of other suppliers.Inequal-ity (8)assures that the retailer in one supply chain would like to have a long-term relationship with his supplier.From B i ,M i ,R i X 0,it follows that Inequality (8)is equivalent to M _

i ew 1;w 2T?eˉa i àw i tdw j TB j

àeˉa j àw j tdw i TV j X F i ,

(9)

where F i ???????????????

2R i =B i p eB i B j àV i V j T.From B 1B 2àV 1V 240and B i 40,it follows that F i is an increasing function of the reservation utility R i .Owing to the fact that two suppliers are symmetrical,we only need to give the optimal wholesale price of supplier 1.Solving

the ?rst-order conditions of E e~p ?S i T,we derive the

following.

Proposition 5.If max f 0;d ?àg o d o min f d ?

t;1g and B i +dV i 40(i ?1,2),then the optimal wholesale

0.6

0.8

1.2

1.4

p 2

d =0.7

d =0.5

Fig.1.The optimal retail price p ?2versus the service investment

ef?ciency coef?cient Z 1.

0.6

0.8

1.2

1.4

s 2

d =0.5

d =0.7

Fig.2.The optimal service level s ?2versus the service investment

ef?ciency coef?cient Z 1.

Since the suppliers are risk-neutral,the results of this paper

hold for the case in which the unit production costs are uncertain

and independent of the stochastic market base ~a

i ,where c i is substituted by E (c i ).The uncertainty of the unit production cost of a supplier may come from her purchasing cost,holding cost,etc.When the unit production cost information is asymmetric,i.e.,each supplier only knows his own production cost,two suppliers play a static game with incomplete information.The solution of this game will be very complex in our framework,which is beyond the scope of this paper.Thus,for simplicity,we assume that the unit production costs are common knowledge to all players.

T.Xiao,D.Yang /Int.J.Production Economics 114(2008)187–200

193

price of supplier1is

Proposition5gives us the following insights:

The wholesale price of one supplier is a piecewise and decreasing function of the reservation utility of his retailer.That is,if the reservation utility (R i)is suf?ciently large,the supplier has to lower wholesale price to induce the retailer’s order when the retailer’s reservation utility increases; otherwise,the supplier keeps the original whole-sale price.

When the reservation utility of the retailer in the rival chain is suf?ciently large,the optimal wholesale price of the supplier is an increasing function of the reservation utility of the rival’s retailer if the demand sensitivity(g)to the rival’s service is very weak and the service investment ef?ciency of the rival retailer is suf?ciently high such that(V j+dB j)p0;otherwise,the optimal wholesale price of the supplier is a decreasing function of the reservation utility of the rival’s retailer. If the(IR)constraint of one retailer is binding, his supplier has to lower wholesale price to compensate the retailer for selling his product.

From Proposition5,we derive the following. Proposition 6.Assume max f0;d?

g o d o min f d?

;1g and dB i+V i40,i?1,2.We have

(i)q w00

=q l i o0for d4bg=Z i;q w00i=q l i?0for

d?bg=Z i;and q w00

=q l i40for d o bg=Z i;

(ii)q w00

=q l j o0for d4bg=Z j;q w00i=q l j?0for

d?bg=Z j;and q w00

=q l j40for d o bg=Z j.

When w00

is the optimal wholesale price of supplier i(i.e.,M

ew00

;w00

TX F i;i?1;2)and the conditions of Proposition6hold,Proposition6 gives us the following insights:

Retailer i will pay a lower wholesale price w00

i through behaving more conservative(increasing

w?1?

w00

if M

ew00

;w00

TX F i;i?1;2;

w01

if min f M

ew00

;w00

T;M

ew01

;w01

Tg X F1and M

ew00

;w00

To F2;

w10

if M

ew00

;w00

To F1and min f M

ew00

;w00

T;M

ew10

;w10

Tg X F2;

w11

otherwise;

>>>

>><

>>>

>>:

where(i,j?1,2,j?i)

w00 i ?

ˉa i?B je2B itdV iTàV i V j tˉa j?dB i B jàV jeB it2dV iT teB itdV iT?c jedB jtV jTt2c ieB jtdV jT

e4àdTB1B2t3dB1V2t3dB2V1te4dà1TV1V2

w10 1?

eB1tdV1T?c2eB2dtV2Tà2F1 tˉa1e2B1B2tdB2V1àV1V2Ttˉa2eB1B2dàB1V2à2dV1V2T

B1B2e2àdTtdB1V2tdB2V1te2dà1TV1V2

w10 2?

B1?ˉa1B2dtc2eB2tdV2T tˉa2eB1B2àV1V2TtV1?c2eB2dtd2V2Tàˉa1dV2 àF1eB1dtV1T

B1B2e2àd2TtdB1V2tdB2V1te2d2à1TV1V2

w01 1?

B2?ˉa2B1dtc1eB1tdV1T tˉa1eB1B2àV1V2TtV2?c1eB1dtd2V1Tàˉa2dV1 àF2eB2dtV2T

B1B2e2àd2TtdB1V2tdB2V1te2d2à1TV1V2

w01 2?

eB2tdV2T?c1eB1dtV1Tà2F2 tˉa2e2B1B2tdB1V2àV1V2Ttˉa1eB1B2dàB2V1à2dV1V2T

B1B2e2àd2TtdB1V2tdB2V1te2d2à1TV1V2

w11 i ?

eˉa itˉa j dTàeB itdV iT

??????????????

2R i=B i

àeV jtdB jT

???????????????

2R j=B j

1àd2

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200

194

CARA l i )if the substitutability of the two products is suf?ciently high (d 4bg =Z i ).

In order to pay lower wholesale prices,the retailers will become more aggressive if the substitutability is suf?ciently small (d o bg =Z i ).That is,the supplier will bear a part of risk to encourage his retailer to order more products. Each retailer may have an incentive to conceal his real CARA and report a more bene?cial CARA to pay a lower wholesale price.

Each retailer can in?uence the pricing power of two suppliers by manipulating his CARA.For example,the CARA of one retailer has similar effects on the optimal wholesale prices of two suppliers when the substitutability is suf?ciently small,i.e.,the optimal wholesale prices will be simultaneously lowered if one retailer becomes more risk insensitive.

The effects of the retailer’s CARA on the wholesale prices depend on the substitutability of the two products.

Since the retailers have incentives to reveal false risk sensitivity information,the suppliers should design mechanisms to induce them to reveal actual information.

To capture the effects of risk sensitivity,we assume that the two retailers are symmetric in all parameters and the default values of these parameters in

Figs.3–10are R 1?R 2?4;ˉa 1?ˉa 2?10;b ?0:8;g ?0:5,Z 1?Z 2?0:8;

s 1?s 2?1:5

and

c 1?c 2?1:0.

0.250.5

0.75

1 1.25 1.5

d = 0.60

d = 0.30

w ?1

Fig.3.The optimal wholesale price of supplier 1versus the risk sensitivity of retailer 1with l 2?0.5.0.25

0.5

0.75

1.25

1.5

5.5

6.5

7.5

d = 0.60

d = 0.30

w ?1

Fig.4.The optimal wholesale price of supplier 1versus the risk sensitivity of retailer 2with l 1?0.5.

0.250.5

0.751 1.25 1.5

d = 0.60

d = 0.30

E ( ?

S 1)

～Fig.5.The expected pro?t of supplier 1versus the risk sensitivity of retailer 1with l 2?0.5.

T.Xiao,D.Yang /Int.J.Production Economics 114(2008)187–200

195

Given these values,the conditions of Propositions 1and 5hold.In the following,Figs.3–8illustrate how the optimal wholesale price and expected pro?t of one supplier depend on the risk sensitivity of his retailer/the rival’s retailer,and Figs.9and 10illustrate how the expected order quantity of one retailer jointly depends on the risk sensitivity of the two retailers.From Figs.3and 4,we know that the optimal wholesale price of one supplier is ?rst increasing and then decreasing with the risk sensitivity of the two retailers if the substitutability of the two products is

0.25

0.5

0.751

1.25

1.5

d = 0.60

d = 0.30

E ( ?S 1

)～Fig.6.The expected pro?t of supplier 1versus the risk sensitivity of retailer 2with l 1?0.5.

1.5

E ( ?

S 1)

～Fig.7.The expected pro?t of supplier 1versus the two retailers’

risk sensitivity with d ?

0.60.

1.5

E ( ?

S 1)

～Fig.8.The expected pro?t of supplier 1versus the two retailers’

risk sensitivity with d ?0.30.

1.57

7.5

E (q ?

～Fig.9.The expected order quantity of retailer 1versus the two

retailers’risk sensitivity with d ?0.60.

E (q ?1)

～Fig.10.The expected order quantity of retailer 1versus the two

retailers’risk sensitivity with d ?0.30.

T.Xiao,D.Yang /Int.J.Production Economics 114(2008)187–200

196

suf?ciently low,otherwise,decreasing with the risk sensitivity.That is,if the substitutability coef?cient and the risk sensitivity of the two retailers are small enough,both suppliers will raise their wholesale prices when the risk sensitivity of one retailer increases;otherwise,both suppliers will lower their wholesale prices when the risk sensitivity of one retailer increases.In general,the decreasing slope is higher than the increasing slope.

From Figs.3–6,we know that the effect of the risk sensitivity of the rival’s retailer on the equilibrium expected pro?t of the supplier is similar to its effect on his optimal wholesale price;however, the equilibrium expected pro?t of one supplier is ?rst increasing and then decreasing with the risk sensitivity of his retailer,also see Figs.7and8. Figs.9and10mean that the effect of the rival’s risk sensitivity on the expected order quantity of the retailer is very small.However,the expected order quantity of one retailer is often increasing with his risk sensitivity due to the effects of the retailer’s risk sensitivity on the wholesale prices.

Note that in the price competition model with linear demand function,we cannot carry out the sensitivity analysis of the substitutability coef?cient of the two products because the increase of the substitutability will result in aggregate demand ampli?cation(Ingene and Parry,2004;Xiao et al., 2007).Thus,in this paper,we only analyze the effects of the other factors on the results given the substitutability and do not investigate how the change of the substitutability affects the results. 6.Conclusions

We consider the competition between two supply chains in a demand uncertainty environment.Each supply chain consists of one risk-neutral supplier and one risk-averse retailer,where two retailers compete in retail price and service.For the members in the same chain,the supplier is a leader and the retailer is a follower.We mainly discuss the effects of the retailers’sensitivity(CARA)/demand uncer-tainties,the service investment ef?ciencies and the charged wholesale prices on the players’decisions, in particular,on those of the members in the rival chain.

We?nd that the service investment ef?ciency of one retailer greatly affects the optimal price–service decisions of his rival,i.e.,the higher the service investment ef?ciency of one retailer,the lower the optimal retail price and service level of his rival will be.This paper also suggests that,in general,the optimal retail price and service level of one retailer is decreasing with his own CARA,while the effects of his CARA on the price–service decisions of his rival depend on the substitutability of the two products.The suppliers should not ignore their retailers’risk attitudes/demand uncertainties and the effects of these factors on the optimal wholesale prices and the expected pro?ts of the suppliers depend on the substitutability to a large degree.The expected order quantity of one retailer is increasing with his risk sensitivity.

There are several directions for future research. Firstly,we assume that two suppliers are risk-neutral.The case with risk-averse suppliers is challenging and interesting.Secondly,it can be extended to the case where retailers lie in more powerful positions,i.e.,two retailers act as Stack-elberg leaders.Thirdly,this paper assumes that the risk sensitivity of the two retailers is common knowledge;however,CARA of one retailer may be private information.It is interesting but challen-ging to investigate how suppliers design incentive mechanisms that induce retailers to reveal their private information.Fourthly,this paper only considers the single period problem with multiple stages in a single selling season.A multi-period problem in which the repeated game is played will be very interesting,but this problem can be studied by the aid of computer due to its complexity. Finally,we assume that the retailer in one supply chain owns a long-term relationship with his supplier.The problem that both retailers can replenish their products from both suppliers is also very interesting,where the substitutability of the products will increase when two retailers sell the products of the same supplier.Ingene and Parry (2004)and Xiao et al.(2007)pointed out that modeling the change of competitive substitutability with a linear demand function in the price competi-tion model has the weakness such as aggregate demand ampli?cation while the quantity competi-tion model can avoid the weakness.Thus,it is better for one to adopt the quantity competition model to study this problem.

Acknowledgments

We would like to thank the anonymous referee and the editor for their many helpful suggestions and insightful comments,which have signi?cantly improved the content and presentation of this

T.Xiao,D.Yang/Int.J.Production Economics114(2008)187–200197

paper.This research was supported in part by:(i)the National Natural Science Foundation of China under Grants 70301014,70671055,and 70731002;(ii)Program for ‘‘New Century Excellent Talents in University’’of the Ministry of Education,China;and (iii)the Fund for ‘‘Study on the Evolution of Complex Economic System’’at ‘‘Innovation Center of Economic Transition and Development of Nanjing University’’of the Ministry of Education,China.Appendix A

Proof of Proposition 1.From Hessian matrix H i of u i e~p

i T,we know that u i e~p i Tis jointly concave on (p i ,s i )if B i 40.Thus,from B 1,B 240,we know that the solution satisfying the ?rst-order conditions of u i e~p

i Tis optimal.The ?rst-order conditions are for i ,j ?1,2,j ?i

q u i e~p i T=q p i ?eà2à2l i s 2i Tp i tdp j tb s i àg s j tˉa

i te1t

2l i s 2i Tw i

?0,

(A.1)q u i e~p

i T=q s i ?b ep i àw i TàZ i s i ?0.(A.2)

From Eq.(A.2),we have s i ?b ep i àw i T=Z i .Insert-ing it into Eq.(A.1),we obtain

B 1p 1tV 2p 2?ˉa 1teB 1à1Tw 1teV 2td Tw 2,

(A.3)

V 1p 1tB 2p 2?ˉa 2teV 1td Tw 1teB 2à1Tw 2,

(A.4)

where V i ?bg =Z i àd ,i ?1,2.

From Eqs.(A.3)and (A.4),it follows that the optimal retail price of retailer i is p ?i ?M i tw i ;

i ?1;2,

where M i ??eˉa i àw i tdw j TB j àeˉa j àw j tdw i TV j =eB i B j àV i V j T.Furthermore,the optimal service

level of retailer i is s ?i ?b ep ?

i àw i T=Z i ?b M i =Z i ,i ?1,2.&

Proof of Lemma 1.Let f ed T?B 1B 2àV 1V 2?e2t

2l 1s 21àb 2=Z 1Te2t2l 2s 2

2àb 2=Z 2Tàebg =Z 1àd Tebg =Z 2àd T,which is a concave function of the substitutability coef?cient d .Solving the square equation f (d )?0for d ,we have d ????bg eZ 1tZ 2T?????

D p =e2Z 1Z 2T,where

D ?b 2g 2

eZ 1tZ 2T2

t4Z 1Z 2f b 2

?b 2

àg 2

à2Z 2e1t

l 2s 22T

à2Z 1e1tl 1s 21T?b 2à2Z 2e1tl 2s 2

2T g .Since f (d )is a concave function of d ,it follows that

B 1B 2àV 1V 240if d ?ào d o d ?

t.Furthermore,Lem-ma 1follows from 0o d o 1.&

Proof of Proposition 2.Differentiating p ?i with respect to w j ,we have q p ?

i =q w j àá?edB j tV j T=àeB i B j àV i V j TT.Note dB j tV j ?d eB j à1Ttbg =Z j .If B j X 1,we have dB j tV j 40.If B j o 1,then we have dB j tV j 40for 0o d o bg =?Z j e1àB j T ;dB j +V j ?0for d ?bg =?Z j e1àB j T ;and dB j +V j o 0for bg =?Z j e1àB j T o d o 1.From Lemma 1,we have B i B j àV i V j 40.Thus,we have q p ?i =q w j 40if B i 40,B j X 1and max f 0;d ?àg o d o min f d ?

t;1g .Similarly,we can show the other parts of Proposition 2.&Proof of Proposition 3.Part (i):From Lemma 1and that B i is an increasing function of l i ,it follows that s ?i

is a decreasing function of l i if max f 0;d ?àg o d o min f d ?t;1g ,i.e.,q s ?

i =q l i o 0.Part (ii):Differentiating s ?i with respect to l j ,we have

q s ?i j ?2bs 2j ?eˉa j àw j tdw i TB i àeˉa i àw i tdw j TV i V j Z i eB i B j àV i V j T?

2bs 2j M j V j

Z i eB i B j àV i V j T

.(A.5)

From Lemma 1,we know B 1B 2àV 1V 240.Further-more,from Eq.(A.5)and M j 40,it follows that

q s ?i =q l j

has the same sign as V j ,i.e.,sign eq s ?

i =q l j T?sign eV j T.

Thus,Proposition 3follows.&Proof of Proposition 4.Note that the numerator of M i is independent of Z i .From Proof of Lemma 1and b B j tg V j 40,we have f ?B 1B 2àV 1V 240,eq f =q Z i T?bZ à2

i eb B j tg V j T40,and q ef Z i T=q Z i àá?eq f =q Z i TZ i tf 40.Proposition 4follows from Lem-ma 1,Proposition 1,and M i 40.&

Proof of Proposition 5.Differentiating E e~p ?S i Ttwice with respect to w i ,we have

q 2E e~p ?S i

T2i

?à2A i eB j tdV j

1212

.From Lemma 1,we know B 1B 2àV 1V 240.Thus,

E e~p ?S i Tis a concave function of w i if B j +dV j 40,i.e.,

the second-order conditions of E e~p ?S i Tare satis?ed.

Solving the ?rst-order conditions of E e~p n S i Twithout

considering (IR)constraint (9),we obtain the

wholesale prices w 001and w 002.If M _

1ew 001;w 00

2TX F 1and M _

2ew 001;w 002TX F 2,i.e.,both (IR)constraints are

satis?ed,the optimal wholesale price of supplier i is

T.Xiao,D.Yang /Int.J.Production Economics 114(2008)187–200

198

w 00i .If M _

ew 001;w 002TX F 1and M _

ew 001;w 00

2To F 2,sup-plier 2has to lower wholesale price until M _

2ew 1;w 2T?F 2.

Thus,

solving the equations

q E e~p ?S 1T=q w 1?0and M _2

ew 1;w 2T?F 2for (w 1,w 2),

we obtain ew 011;w 012T.If M _

1ew 011;w 012TX F 1is also

satis?ed,then the optimal wholesale price of supplier i is w 01i .Otherwise,solving M _

1ew 1;w 2T?F 1and M _

ew 1;w 2T?F 2for (w 1,w 2),we can obtain

the optimal wholesale prices w 111and w 11

2.Similarly,we can consider the other cases.&

Proof of Proposition 6.We only need to consider w 001due to the symmetry.From 0o d o 1,B i 40and dB i tV i 40,we have B i tdV i 40.Part (i):Differentiating w 001with respect to B 1,we get

q w 001

1?V 1edB 2tV 2TD 1eV 2Tf B 1?e4àd TB 2t3dV 2 tV 1?3dB 2te4d à1TV 2 g ,where

D 1eV 2T?3ˉa 1dV 2t3ˉa 2dB 2tˉa 1e2td 2TB 2tˉa 2e1t2d 2TV 2

àc 2e1àd 2TedB 2tV 2Tà2c 1e1àd 2TeB 2tdV 2T.

Differentiating D 1(V 2)with respect to V 2,we have

q D 1eV 2T=q V 2?eˉa 2àc 2Ttd e3ˉa 1à2c 1Tt2ˉa 2d 2t2c 1d 3tc 2d 2,

which is positive because ˉa i X c i .Furthermore,it follows from ˉa i X c i ,0o d o 1,B 240,and dB 2tV 240that

D 1eV 2T4D 1eàdB 2T?2B 2e1àd 2Teˉa 1àc 1tˉa 2d tc 1d 2T40.

Note that B 1is an increasing function of l 1and

eq w 001=q l 1T?eq w 00

1=q B 1Teq B 1=q l 1T.Thus,we have sign eq w 001=q l 1T?sign eV 1T.Part (ii):Differentiating w 001with respect to B 2,we get

q w 001

q B 2?2V 2eB 1tdV 1TD 2eV 1Tf B 1?e4àd 2TB 2t3dV 2 tV 1?3dB 2te4d 2à1TV 2 g 2

,where

D 2eV 1T?B 1f d ?3ˉa 1àc 1e1àd 2T à2c 2e1àd 2Tg

tV 1?ˉa 1e1t2d 2Tàec 1t2c 2d Te1àd 2T tˉa 2?B 1e2td 2Tt3dV 1 .

Similar to Part (i),we have D 2(V 1)is an increasing

function of V 1.From dB 1+V 140,B 140,ˉa 2X c 2,and 0o d o 1,it follows that D 2eV 1T4D 2eàdB 1T?2B 1e1àd 2T?eˉa 2àc 2Ttˉa 1d tc 2d 2 40.Thus,we have

sign e@w 001=@l 2T?sign e@w 00

1=@B 2T?sign eV 2T.&

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