17－competition in remanufacturing

COMPETITION IN REMANUFACTURING*
PRANAB MAJUMDER AND HARRY GROENEVELT
Simon School of Business ,University of Rochester ,Rochester ,New York 14627,USA
We present a two-period model of remanufacturing in the face of competition.In our model,an
original equipment manufacturer (OEM )competes with a local remanufacturer (L )under many reverse
logistics configurations for the returned items.After establishing the Nash Equilibrium in the second
period sub-game,we use numerical experiments for comparative statics.OEM wants to increase L ’s
remanufacturing cost.Surprisingly,while L competes in the sales market,she has incentives to reduce
OEM ’s remanufacturing cost.A social planner who wants to increase remanufacturing can give
incentives to the OEM to increase the fraction available for remanufacturing,or reduce his remanu-
facturing costs.
(REMANUFACTURING;COMPETITION;REVERSE LOGISTICS)
1.Introduction
Remanufacturing is the process of disassembling used items,inspecting and repairing/reworking the components,and using these in new product manufacture.A product is considered remanufactured if its primary components come from a used product (after ensuring that they are functional).Consumer awareness,environmental activism,and legis-lative pressure are forcing firms to manufacture environmentally friendly products.Many firms now take back their products in order to recycle the material into new products.However,there may be greater economic benefit in remanufacturing these items instead of recycling them.
A Boston University study (Lund 1996)on the remanufacturing industry studied 11,000firms in eight industry sectors:automotive,compressors,electrical equipment,machinery,office furniture,tires,toner cartridges,and valves.It concluded that the remanufacturing industry is a $53billion industry in the U.S.alone,on par with the steel industry.The direct employment in 73,000remanufacturing firms in the U.S.was 480,000,equal to the consumer durables industry.
A number of large firms have highly visible recycling/remanufacturing programs.Kodak,the worldwide manufacturer of photographic media and equipment,has a very successful remanufacturing program for their family of single-use cameras (M2Presswire “KODAK:Kodak to receive 1999Gold Medal from World Environment Center”12/09/1998).Xerox,a well-known manufacturer of photocopy equipment,has an extensive program of recycling and remanufacturing for their lines of photocopiers and print toner cartridges (M2Presswire “XEROX:Xerox annual Environmental Report documents progress,sets goals”09/29/1998).*Received February 2000;revision received August 2000;accepted January 2001.
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126PRANAB MAJUMDER AND HARRY GROENEVELT
Manufacturers like Deere&Co.in agricultural equipment(Dow Jones News Service“Deere &co.In Venture to Remanufacture Diesel Engines”01/15/1998),Aftermarket Tech(Dow Jones News Service“Aftermarket Tech to Remanufacture Transmissions for Chrysler”05/21/1998),and otherfirms also actively remanufacture(PR Newswire“Frost&Sullivan—Engine Complexity Causing Shift from Rebuilding to Remanufacturing in Truck Engine Aftermarket”11/16/1998).
To remanufacture the used item,the manufacturer must retrieve them from the market where they are dispersed among consumers.This is accomplished by means of a reverse logistics chain(Lund1996)that is comparable to the new product distribution system in reverse.The agents in the reverse logistics chain are responsible for gathering the used items, classifying and segregating them,andfinally transporting them to the manufacturer.Often they may also take up some of the activities of remanufacturing like disassembly and cleaning.The manufacturer cannot maintain complete control over the entire chain;this may give rise to opportunistic behavior by some agents in the chain,particularly if the entire remanufacturing process can be duplicated.The reverse logistics chain is the nucleus of competition in many industries.
Smallerfirms often remanufacture a product even when the largerfirms have not started remanufacturing.Often a smallerfirm seizes the opportunity faster.In many cases,the smallerfirm uses different systems and technology.In some industries,like automobile and diesel engines,the manufacturer contracts with the smallerfirms to remanufacture their product(Lund1996).When the original equipment manufacturer(OEM)sets up a remanu-facturing program,it may compete with local small remanufacturers for the supply of used items.
As recent events in the toner cartridge industry show,this is a matter of considerable concern for OEM s.Lexmark,the Kentucky,U.S.-based printer and toner cartridge manufac-turer,introduced the“Prebate”program in April1998(Narisetti,Raju;“Recycles Print-Toner Cartridges Trigger Legal War Among Rivals,”Wall Street Journal04/10/1998).This program allows customers to get a$30rebate off a$230toner cartridge(Optra-S)if they agree to return the used cartridge to Lexmark or destroy it(the cartridge is also available without Prebate).Simultaneously,Lexmark sent letters to hundreds of smaller cartridge remanufacturers stating that they would face legal action if they remanufactured Prebate cartridges.This triggered protests from remanufacturers(Hetter,Katia;“Seeing Red Over Cartridges—Lexmark ban on outside remanufacturers debated,”Newsday Nassau and Suf-folk06/17/1998).Legislation was considered in Long Island and Orange County that would mandate state agencies to buy only fromfirms that recycle/remanufacture their own products (Hetter,Katia;“Long Island Inc.—State Measure Would Set Remanufacturing Haven,”Newsday Nassau and Suffolk02/02/1999)(Norman,Jan;“Smallfirms worry about bigfirms’toner foray—Legislation:A bill would keep printer companies out of the cartridge business,”The Orange County Register Morning04/17/1999).The program certainly helped Lexmark get much higher cartridge returns(Lexmark claims50%returns,other sources estimate a conservative30%).In April1999,they announced expansion of the remanufacturing pro-gram(Business Wire“Lexmark announces expansion of remanufactured toner cartridges program in U.S.”04/23/1999).This made the Optra-S toner cartridge thefirst nationally available remanufactured cartridge.
In addition to this example of legal restriction of local remanufacturing,some OEM s may also use technology to achieve similar objectives.For instance,some toner cartridges have an encrypted counter that must be reset by the OEM in order to continue printing.Depending on the specifics of the situation,such methods can be considered to either restrict the local remanufacturer’s access to used items,or to increase their cost of remanufacturing(or both). This raises some questions about the competitive effect of the actions of each party.The OEM has invested in the design and manufacture of the item and may want to corner the cost benefits of remanufacturing.The local remanufacturers may be quicker and may make the
market more competitive.Finally,communities and legislatures may be interested in reduc-ing waste disposal problems and may want to increase net remanufacturing activity.It is not clear what impact each OEM strategy has on the total amount of remanufacturing.While competition drives down prices and more items are sold,the threat of remanufacturing by local firms may make OEM sell less earlier in the product life cycle in order to reduce the competition.How the market favors either the OEM or the local remanufacturer may also have a role to play.We will examine the effect of some strategies on the competitive prices and quantities in the market,as well as the players’profits.Our analysis should help us explain some actions that we see manufacturers take.Since we will examine these actions in a wide variety of scenarios,we will also prescribe some actions a social planner may take in order to increase net remanufacturing activity.
We formulate a two-period model to examine the effect of competition in remanufacturing.In the first period,only the OEM manufactures and sells new items.In the second period,a fraction of these items are returned for remanufacturing.However,OEM does not get all these returned items;some are used up by the local remanufacturer.These players now compete in the customer market to sell these items.This model captures the essential features of remanufacturing,a finite product lifetime,and competition in selling the product.The critical trade-offs for the OEM are between the lower cost of remanufacturing in the second period against the presence of increasing competition from the local remanufacturers.Our results show that the presence of competition causes the OEM to manufacture less in the first period and attempt to increase the local remanufacturer’s cost of remanufacturing.Both of these actions are aimed at reducing the second period competition.On the other hand,while the local remanufacturer competes with the OEM in selling the items,she helps the OEM reduce his remanufacturing cost.This is because any action that makes remanufacturing attractive to OEM induces him to manufacture more in the first period and hence makes it possible for the local remanufacturer to produce more in the second period.
This paper is organized as follows:Section 2discusses relevant research from different fields.Section 3describes the competitive model,and Section 4sets up the best response strategies of each player and solves different flavors of the second period sub-game.Section 5solves the first period of the game.Section 6presents two benchmark models for comparison with the competitive models.Section 7looks at the results of all the models and discusses insights from the paper.Section 8concludes the paper.
2.Existing Literature
Literature related to remanufacturing comes from many sources.The recycling literature includes consumer behavior about recycling,and competition in recycling.Remanufacturing is also related to secondary markets in durable goods.Many authors have documented the economics of remanufacturing and value recovery.Technical issues regarding remanufac-turing have been studied from the engineering perspective.Finally,the operations manage-ment perspective has examined remanufacturing with respect to production control,inven-tory,and multi-echelon/supply chain issues.
There is an extensive body of recycling literature from socio-economics,environmental studies and resource management.While remanufacturing is different from recycling in terms of the specificity of use of the used items,some research merits mention.Clarke,Stavins,Breeno,and Bavaria (1994)as well as Daniel,Diakoulaki,and Pappis (1997)document the challenges of reengineering existing manufacturing and supporting processes to make them more environmentally friendly.Hornik,Cherian,Madansky,and Narayana (1995)summarize the determinants of recycling behavior from 67empirical studies.They conclude that the strongest predictors of recycling behavior are internal facilitators:consumer knowledge and commitment to recycling,followed by external incentives,monetary rewards,and social influence.
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Many authors have also studied the effect of competition on recycling.In these papers largefirms face competitive pressure from smaller recyclers who process scrap metal. Crandall(1996)empirically establishes the effect of mini-mill competition on the decline of large national integrated steel plants.In a recent paper,Grant(1999)proposes an economic model to explain data pertaining to the Alcoa antitrust case of1945and to explain conflicting interpretations of three earlier papers.He assumes that the small aluminum recyclers are price takers,and Alcoa does not recycle.In contrast,in this paper,remanufacturers face a demand curve,and OEM s remanufacture.
The durable goods literature is relevant because of the similar effect of endogenous competition due to used items that have been sold earlier.Anderson(1984)considers a monopolist selling goods to price-taking customers in the presence of a competitive second-ary market.Purohit(1992)considers the relationship between the markets for new and used automobiles from a marketing perspective.He concludes that depreciation of used cars is strongly influenced by the types of changes in the new models.Ferrer(1996d)has also looked at market segmentation in situations where a manufacturer sells both original and remanufactured goods.
Research on the specific cost structure of remanufacturing has been done by Ferrer(1996b) for tires and Ferrer(1997)for PCs.Ferrer(1996c)also formulated a generalized cost model drawing on remanufacturing costs in various industries in Europe.de Ron and Penev(1995) look at considerations in the recycling and remanufacturing of electronic components,while Ferrer(1996a)examines practices in remanufacturing for a variety of industries in Europe. From an engineering and design standpoint,the research has focused on the impact of engineering design on remanufacturability.Amezquita,Bras,and Hammond(1998)have sur-veyed remanufacturers about the issues in automotive parts remanufacturing.Amezquita,Bras, Hammond,and Salazar(1995)have attempted to characterize the remanufacturability of engi-neering systems.Bras and Hammond(1996)have studied the issue of designing a component for remanufacturability and formulating metrics for assessing remanufacturability.
From the operations management perspective,the literature includes production control, inventory policies,and capacity planning research.Guide and Srivastava(1997b)review models in repairable inventory from the60s onwards.They examine a number of papers that model repairable inventories in terms of classic theory in operations management including stocking policies,multi-echelon models and others.Guide(1994)looks at capacity planning for remanufacturing.Guide and Srivastava(1997a)look at order release strategies.Guide, Kraus,and Srivastava(1997)outline scheduling policies for remanufacturing.Van der Laan and Solomon(1997)consider production planning and inventory control for remanufactur-ing.Except for Toktay,Wein,and Zenios(1997),who consider inventory management of remanufacturable products in the entire supply and distribution chain,all the other operations management papers concentrate on the manufacturing facility.
In addition to these,in recent work Savaskan(1999)looks at the benefits of channel coordination between the OEM and the reverse logistics chain.This is relevant since we study the effect of competition between remanufacturers and OEM s.Anecdotal evidence from companies indicates that many local remanufacturers start out as reverse logistics chain operators,or operate with them to divert used items from the OEM.
This paper considers competition in remanufacturing.Research on competition in recy-cling differs from this;in recycling the manufacturer does not recycle,and the recyclers are price takers.In this paper,OEM may also remanufacture,and both players set prices.The determinants of consumer behavior for recycling are relevant,since similar determinants may influence the rate of returns for used items.Remanufacturing also differs from durable goods markets since the remanufactured items are functionally similar to original items and may be sold as such.The competition stems from other remanufacturers,not a secondary market created by goods sold earlier by the original manufacturer.The engineering and operations literatures do not consider competition at all.
Thus the main contribution of this paper is to model the effect of competition in remanufacturing and outline the impact of different parameters [such as (re)manufacturing costs and availability of returns to each player]on the competitive equilibrium.To our knowledge,this is the first paper to look at this issue.
3.Model Description
3.1.Introduction
Our model considers one OEM and one local remanufacturer.The local remanufacturer in our model is an abstraction for all non-OEM remanufacturing activity.In reality,there may be multiple players competing with each other and with OEM .Since in many situations the local remanufac-turers are constrained in their access to returned items,we expect competition between local remanufacturers themselves to be mainly focused in obtaining shells as opposed to price competition in selling remanufactured items.In addition,having only a single local remanufac-turer makes the model tractable while still allowing us to obtain interesting insights relating to the presence of local remanufacturing activity on the OEM ’s behavior.
We consider one product.The customer does not distinguish between remanufactured or original items sold by OEM ,but distinguishes between them and those sold by the local remanufacturer;i.e.,the brand is more important.This is true of many items,like single-use cameras and printer toner cartridges,where many OEM s themselves do not distinguish between new and remanufactured items.
We consider a two-period model.This is motivated by the finite lifetime for the products that we consider.While in reality each unit may go through multiple cycles of remanufac-turing (e.g.,up to at least 7for single-use cameras)and the lifetime of the product can be longer,the two-period model captures the essence of a finite lifetime and remanufacturing competition without sacrificing tractability.
In the first period,the OEM manufactures and sells new items.Only a fraction of these items is returned and available for the firms to remanufacture in the second period.This is due to various reasons:losses in the reverse logistics chain or wear and tear in course of usage.In the second period,the state of the world is determined by a single parameter:the quantity of returns from the first period.Hence,each player’s competitive response depends on this state variable.We utilize a demand function such that the quantity sold by each player depends on both his own price and the competitor’s price.The OEM can manufacture new items in the second period (in addition to remanufacturing old ones).We assume both players have complete information.
Once we have the competitive response in the second period,we can then look at the first period decision for the OEM .The OEM ’s decisions in the first period determine both the first period profits and the state variable for the second period.Hence his first period decision can capture the outcome of the entire game once the competitive equilibrium in the second period is mapped out as a function of the state variable.
The cost structure for each player is as follows.Neither player has any fixed costs of operation,and both have constant marginal costs of production.(Fixed costs can be easily considered after the total profits have been computed;they do not affect any price or quantity decision,but do affect whether each player would want to play the game.)The unit manufacturing cost for the OEM is greater than the unit remanufacturing cost.(While this is not always the case,the model loses its defining characteristics if the OEM ’s remanufacturing costs are higher than the manufacturing cost.)The OEM ’s costs do not change from the first period to the next.The local remanufacturer has only one cost,that of remanufacturing.
3.2.Shell Allocation Mechanisms for the Returns
When a used item comes back to the remanufacturer,it is often referred to as a shell;for example,a laser printer cartridge comes empty,or a single-use camera comes back sans the 129
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130PRANAB MAJUMDER AND HARRY GROENEVELT
film.These are often called shells,and we will use this term for used items that may be then remanufactured.
Typically,many players play a role in the reverse logistics chain,and the interactions between them can be very complex.In many cases there is competition for returned shells between OEM and the other parties in the reverse logistics chain,such as the local remanu-facturers.In addition,the local remanufacturer may have a position in the reverse logistics chain that enables her to capture a fraction of returned shells without purchasing the shells in a competitive manner.
We will not attempt to model the structure of the reverse logistics chain or competition for used items.Our general model framework allows for a number of different exogenously imposed“shell allocation mechanisms”(SAM)that,taken together,provide aflexible ap-proach to modeling the effect of the reverse logistics chain on shell acquisition by the players. This also helps us examine the characteristics of this model that are common to all these shell allocation mechanisms.
All the following shell allocation mechanisms are similar in that both players have access to a fraction of the shells returned from thefirst period(L gets␥,and OEM gets1Ϫ␥).They differ in whether any remaining shells from one player’s share are available to the other player.This leads to four shell allocation mechanisms.(1)OEM can use any shells left over by L,but L cannot do likewise.(2)L can use any shells left over by OEM,but OEM cannot do likewise.(3)Neither can use the shells left over by the other.If one player does not use his/her share completely,the other player still has no access to the remainder.(4)Both can use any shells left over by the other.
For each shell allocation mechanism,we formulate the two period model.Thus,there are four forms of the game that we will analyze.In all of them,OEM is the only player in thefirst period,and the quantity he manufactures and sells in thefirst period determines the quantity of used items available for remanufacturing in the second period(for both players).In the second period,both players obtain the shells as determined and then simultaneously deter-mine prices and quantities.
4.The Second Period Sub-Game
We will now look at the second period,when the state of the world is given by R,the quantity of used items returned from thefirst period.We analyze two parametric decision problems(one for each player)that will be used in Section5tofind the Nash Equilibrium for each of the shell allocation mechanisms.
The notation used in Sections4and5is:p iϭprice charged by player i,z,q Oϭtotal quantity produced by OEM(including both new and remanufactured units),q ORϭquantity remanufactured by OEM,q Lϭquantity remanufactured by L,cϭunit manufacturing cost of a new item(for OEM only),r iϭunit remanufacturing cost for player i(this includes the unit acquisition cost of shells),Rϭquantity available for remanufacturing at the start of period 2,␥ϭfraction of shells allocated to L␥ʦ[0,1](the remaining fraction,1Ϫ␥,is allocated to OEM),␲iϭplayer i’s profit,D i(p O,p L)ϭDemand function for player i,⌰i(␳,p)ϭOptimization problem for player i’s best response when the quantity of returns he/she is allowed access to is␳,iʦ{O,L},and the other player charges a price p.
We make comparisons by structuring the demand function.For deriving analytic results, a linear demand function is used.We will assume the linear demand function
D O͑p O,p L͒ϭA OϪB O p OϩC O p L(1)
D L͑p O,p L͒ϭA LϪB L p LϩC L p O(2) Note that all parameters are positive.The total demand is D(p O,p L)ϭ(A OϩA L)Ϫ(B OϪC L)p OϪ(B LϪC O)p L.We will assume(B OϪC L)and(B LϪC O)to be
positive.In addition,we will assume that each player sets his price so that his/her own demand is never negative.[We need this assumption in order to have a well-defined demand function for the other player if one decides not to offer a product for sale.]
4.1.L ’s Problem (Second Period )
L ’s problem is to maximize her own profit using her own price and quantity as decision variables.The total quantity produced by her must be less than the minimum of the shells available to her and her demand.Hence the problem faced by L (when the quantity of returns available to her is ␳and the price set by OEM is p O )is
⌰L ͑␳,p O ͒:max q L ,p L q L ͑p L Ϫr L ͒(3)
s.t.q L Յ␳(4)
q L ՅD L ͑p O ,p L ͒(5)
q L Ն0,p L Ն0
(6)The feasible region is a convex region bounded by the four linear constraints 4–6.Constraint 4restricts the remanufactured quantity to a maximum of the available shells.Constraint 5restricts the remanufactured quantity to the demand;note that D L is decreasing in p L .
Proofs of the following results are in the Appendix.
L EMMA 1(L ’s best response ).Define p ៮L ϭ(A L ϩC L p O )/B L ,i .e .,D L (p O ,p ៮L )ϭ0.(i )If L makes a positive profit ,she always prefers to sell the quantity demanded ,i .e .,q L ϭD L (p O ,p L ).(ii )L ’s optimal profit is strictly positive if and only if p ៮L Ͼr L ,and ␳Ͼ0.P ROPOSITION 1(L ’s optimum strategy ).If p ៮L Ͼr L and ␳Ͼ0then L ’s unique optimal strategy is (i )if ␳Յ12D L (p O ,r L )then p *L ϭp ៮L Ϫ␳/B L and q *L ϭ␳.(ii )Otherwise p *L ϭ12(p ៮L ϩr L )and q *L ϭ12D L (p O ,r L ).If p ៮L Յr L or ␳ϭ0then p *L ϭp ៮L and q *L ϭ0is an optimal strategy for L .So ,in all cases ,p *L ϭmax [p ៮L Ϫ␳/B L ,12(p ៮L ϩmin {r L ,p ៮L })]and q *L ϭD L (p O ,p *L )is an optimal strategy for L .
L EMMA 2(L ’s best response continuity ).p *L and q *L as defined in Proposition 2are continuous functions of the parameters A L ,B L ,C L ,r L ,and ␳,and OEM ’s strategy p O .
4.2.OEM ’s Problem (Second Period )
OEM ’s problem is to maximize his profits using his price,total manufactured quantity,and remanufactured quantity as decision variables.Let the quantity of returns available to him be ␳.The price charged by L ,p L ,is also assumed given here.Obviously,OEM will not produce more than demanded.From definitions of the quantities,total manufactured quantity cannot be less than remanufactured quantity.Recall that z denotes the total OEM production,and q OR his remanufactured quantity.Then OEM ’s optimization problem is
⌰O ͑␳,p L ͒:max z ,q OR ,p O ͑p O Ϫc ͒z ϩ͑c Ϫr O ͒q OR
(7)
s.t.q OR Յ␳(8)
z ՅD O ͑p O ,p L ͒(9)
q OR Յz (10)
q OR Ն0,p O Ն0
(11)131
COMPETITION IN REMANUFACTURING
Note that we assume c Ͼr O ,since otherwise OEM will never remanufacture.
Proofs of the following results are in the Appendix.
L EMMA 3(OEM ’s best response ).Define p ៮O ϭ(A O ϩC O p L )/B O ,i .e .,D O (p ៮O ,p L )ϭ0.(i )If OEM makes a positive profit ,OEM always prefers to sell the quantity demanded ,i .e .,z ϭD O (p O ,p L ).(ii )OEM makes positive profit if and only if {␳Ͼ0and p ៮O Ͼr O }or {p ៮O Ͼc }.P ROPOSITION 2(OEM ’s global optimum strategy ).If {␳Ͼ0and p ៮O Ͼr O }or {p ៮O Ͼc },then the unique optimal strategy for OEM is given by the following cases (i )if ␳Յ12D O (c ,p L )then p *O ϭ12(p ៮O ϩc ),z *ϭ12D O (c ,p L )and q *OR ϭ␳.(ii )If 12D O (c ,p L )Ͻ␳Յ12D O (r O ,p L )then p *O ϭp ៮O Ϫ(␳/B O )and z *ϭq *OR ϭ␳.(iii )If 12D O (r O ,p L )Ͻ␳then p *O ϭ12(p ៮O ϩr O )and z *ϭq *OR ϭ12D O (r O ,p L ).If {␳ϭ0and p ៮O Յc }or {p ៮O Յr O }then p *O ϭp ៮O ,and z *ϭq *OR ϭ0is an optimal strategy for OEM .So ,in all cases ,p *O ϭmin {p ៮O ,max [12(p ៮O ϩr O ),min {12(p ៮O ϩc ),p ៮O Ϫ␳/B O }]},q *OR ϭmin {␳,D O (p *O ,p L )},and z *ϭD O (p *O ,p L )is an optimal policy for OEM .
L EMMA 4(OEM ’s best response continuity ).p *O ,z *,and q *OR as defined in Proposition 2are continuous in the parameters A O ,B O ,C O ,c ,r O ,and ␳,and L ’s strategy p L .
4.3.The Second Period Nash Equilibrium
We have looked at the optimization problem for each player in the second period.Given the state of the world,the parameters,and the other players’strategy,we have solved for each player’s optimum strategy.These two must now be combined in order to get a Nash Equilibrium.A Nash Equilibrium is a set of strategies (one for each player)such that given the strategies of the other player,each player has no incentive to change his/her strategy.This means solving the optimum strategy equations for both players simultaneously.For shell allocation mechanism 1,L ’s optimization is ⌰L (␥R ,p O ),since she gets access to a fraction ␥of the shells.OEM ’s optimization is ⌰O (R Ϫq L ,p L ),since he gets access to all remaining shells.Note that for OEM ,R Ϫq L is still a parameter in his optimization problem.By lemma 4the optimal strategies for OEM are now continuous functions of A O ,B O ,C O ,r O ,and R ,and L ’s strategy (p L ,q L ).
The players’optimizations for all shell allocation mechanisms (SAM s)are summarized in (12).
SAM
L solves OEM solves 1
⌰L (␥R ,p O )⌰O (R Ϫq L ,p L )2
⌰L (R Ϫq OR ,p O )⌰O [(1Ϫ␥)R ,p L ]3
⌰L (␥R ,p O )
⌰O [(1Ϫ␥)R ,p L ]
4⌰L (max [␥R ,R Ϫq OR ],p O )⌰O {max [(1Ϫ␥)R ,R Ϫq L ],p L }(12)
For SAM 3the fractions each player has access to add up to 1.While the fractions may,in general,add up to less than 1,we can incorporate that in our model by increasing the loss of items from one period to another,and normalizing the sum of the fractions to 1.The following results apply to all SAM s.The proofs for SAM 1are in the Appendix;the proofs for the other SAM s are similar.
T HEOREM 1(Pure Strategy Nash Equilibrium Existence and Uniqueness ).A unique pure strategy Nash Equilibrium exists in the second period game .
L EMMA 5(Pure Strategy Nash Equilibrium continuity ).The Nash equilibrium prices and quantities in the second period game are continuous functions of the parameters c ,r L ,r O ,and R .
132PRANAB MAJUMDER AND HARRY GROENEVELT。