Measuring the Impact of Increased Product Substitution on Pricing and Capacity Decisions under Linear Demand Models

Transcription

1 Measurng the Impact o Increased Product Substtuton on Prcng and Capacty Decsons under Lnear Demand Models Betul Lus Unversty o Massachusetts 160 Governors Drve, Amherst, MA Tel.: (413) Fax: (413) Emal: betullus@yahoo.com Ana Murel Unversty o Massachusetts 160 Governors Drve, Amherst, MA Tel.: (413) Fax: (413) Emal: murel@ecs.umass.edu 1

2 Measurng the Impact o Increased Product Substtuton on Prcng and Capacty Decsons under Lnear Demand Models We compare two alternatve measures o product substtutablty or lnear demand unctons that are commonly used n the lterature n a varety o prce/quantty decson and capacty nvestment problems, n monopolstc and compettve settngs. Whle the use o the ncorrect measure o product substtutablty leads to unrealstcally hgh prces and prots as products become more substtutable, the results obtaned usng the more approprate measure are n lne wth what s expected to happen n real markets. Usng the approprate measure o product substtutablty, we nd that the optmal nvestment n manuacturng lexblty or a rm tends to decrease as the products become closer substtutes; ths s because (1) prcng can be used more eectvely to balance supply and demand, and () the gans obtaned by shtng producton to the more protable product are reduced due to ncreased correlaton between the prce potentals o the substtutable products. The optmal nvestment n lexble capacty may ncrease, however, the correlaton between the market potentals o the two products sgncantly decreases, whle the correlaton between prce potentals remans stable as they become less derentated. The value o lexblty always ncreases wth demand varablty. We also show that, as long as the optmal nvestments n dedcated capacty or both products are postve, the optmal expected prces and producton quanttes do not depend on the cost o the lexble capacty. Manuacturng lexblty smply allows the rm to acheve those expected values at a lower nvestment cost. Keywords: Demand modelng, product substtutablty, capacty and lexblty plannng, prcng.

3 1. Introducton It s o great mportance to accurately dene the relaton between the demand and prces o the products n a market system. The resultng prce-demand equatons are a key nput to a wde varety o analytcal models rangng rom strategc capacty and lexblty determnaton to the operatonal prce settng common n revenue management. The valdty o the results obtaned rom these models wll o course be no better than that o the underlyng demand models consdered. To be o real value, these demand models must satsy two condtons: (1) represent customer behavor accurately, and () be smple enough to allow or soluton and analyss o the analytcal models they are ed nto. Consequently, the demand model to choose wll depend on the decson problem at hand. Whle consumer choce models, such as the multnomal logt model, are typcally used n the revenue management and product assortment lterature (Anderson et al. (199), McFadden (1986), Hanson and Martn (1996), Swann (1999), Aydn and Ryan (000), Jonard and Schenk (003)), more smplstc models may be approprate or strategc decsons to gan analytcal tractablty. Our objectve s to study smple lnear demand models to determne whether they provde a reasonable nput to capacty and prcng decson models, and to denty the mpact o product substtutablty on optmal capacty and lexblty nvestments. For that purpose, we consder two substtutable products and analyze the eect o product substtutablty on optmal prces, capactes and producton levels n varous decson models n monopolstc and compettve settngs. We wll use and compare alternatve measures o product substtutablty that are commonly used n the economcs lterature. Understandng the eect o product substtuton on capacty, lexblty and prcng decsons s mportant or many ndustres n plannng ther product assortment and/or the desgn o ther producton network. For nstance, consder a company wth a complex producton network, such 3

4 as General Motors, where multple products are produced n a number o plants. The perormance o the system s hghly dependent on the allocaton o products to plants and the use o lexblty (Jordan and Graves (1995) and Murel et al. (006)). Our results show that t sgncantly aects the optmal capacty and lexblty requrements, and s thus undamental n decdng whch products should share lexble resources. The lnear demand unctons are derved rom the maxmzaton problem o a representatve consumer wth a quadratc and strctly concave utlty uncton whch s dened as (Sngh and Vves (1984)) ( 1 ) = ( ) U Q, Q AQ A Q a Q bq Q a Q where j j Q s the amount o product, A > 0, a > 0 and a A ba > 0 (n order to obtan postve demand quanttes, as shown below) or = 1, and j. To ensure strct concavty o the utlty uncton we let a a 1 b 0 >. The parameter b captures the nteractons among the products and s dened as the measure o product substtutablty or complementarty; the products are substtutes, ndependent or complements when b > 0, b = 0 or b < 0, respectvely. The gven utlty uncton represents the act that the value o usng both substtutable (complementary) products s less (more) than the sum o the separate values o usng each product by tsel (Samuelson (1974)). Furthermore, the consumer utlty decreases (ncreases) as products become more substtutable (complementary),.e. as b (-b) ncreases, everythng else held constant. The maxmzaton o (, ) U Q1 Q PQ 1 1 P Q, where P s the prce o product, gves rse to the ollowng nverse and drect lnear demand unctons: P = A a Q bq,, j = 1, and j j 4

5 Note that the relaton a a j A baj a j b Q = P,, 1, and + P j j = j. a a b a a b a a b j j j > b or = 1, s also requred to ensure the ollowng two condtons: 1) demand or a product should be more senstve to changes n ts prce than to changes n the prce o the other product ) the total demand cannot ncrease wth an ncrease n product prces. When b 0, the normalzed quantty b a a measures the relatve degree o substtutablty 1 between the two products, rangng rom zero when b=0 (ndependent products) to one when A = A and b = a1 = a (perect substtutes), and s thus suggested as the measure o product 1 substtutablty. The above parameterzaton o lnear demand unctons, n whch b (or a uncton o b) s dened as the measure o product substtutablty, has been commonly used n the economcs lterature (Sngh and Vves (1984), Roller and Tombak (1990,1993), Tyag (1999), Bernhoen (001), Mukherjee (004)), but has rarely been used by the operatons management communty. Recent exceptons are the papers by Goyal and Netessne (005,007) and Bsh and Suwandechocha (006). and Lettng ε = ( a j A baj ) ( aa j b ), α = a j ( aa j b ), b ( aa j b ) j the drect demand unctons can be wrtten as Q = ε α P + β P,, j = 1, and j. j β = or, j = 1, In ths notaton the parameter ε > 0 s the demand ntercept whch represents the potental market sze or the product and α > 0 and β are the prce and cross-prce senstvty parameters, respectvely, wth α > β, = 1,. Smlarly, the products are substtutes, ndependent or complements when β > 0, β = 0 or β < 0, respectvely. The sum o the ntercepts, ε1 + ε, can be 5

6 nterpreted as the total market sze,.e. the maxmum total demand acheved when both prces are zero. Note that the market sze and prce and cross-prce senstvty parameters are all related and depend on the product substtutablty parameter b; as products become more substtutable (.e., as b ncreases), the customers become more senstve to changes n prces and the overall market sze decreases a A ba a A ba ε + ε = + = ( ) + ( ) a b A a b A a1a b a1a b a1a b s decreasng n b. These propertes relect the wdely accepted acts that (1) more derentated products reach a larger customer base and () consumers are less prce-senstve when purchasng a more unque tem (e.g. Tallur and Van Ryzn (005), pp ). However, n most o the Operatons Management and Marketng lterature the relatonshp among these parameters s gnored (except α > β ) and the parameter β, whch represents the cross-prce eect, s used as a measure o the product substtutablty (McGure and Staeln (1983), Cho (1991), Brge et al. (1998), Garca- Gallego and Georgantzs (001), Bsh and Suwandechocha (005)). When usng β nstead o b as the measure o product substtutablty n the lnear demand unctons, an ncrease n product substtutablty does not aect the product s own prce eect on ts demand or the total market sze; ths contradcts the accepted acts or derentated products mentoned above. A common thread n the results derved rom such models, whch use β as a measure o substtutablty, s that optmal prces and prots ncrease as product substtutablty grows. Ths counterntutve and unrealstc result, however, s oten not questoned. Only Cho (1991) argues that ths parameterzaton does not realstcally descrbe the relatonshp between demands and prces or substtutable products. He proposes a nonlnear demand model whch gves more realstc results, but s dcult to solve and analyze. McGure and Staeln (1983) also pont out that 6

7 cooperatvely set prces n a monopoly settng ncrease wthout bound as products become closer to perect substtutes and suggest a derent parameterzaton o lnear demand unctons. For completon whle mantanng the concseness o the current paper, we dscuss the parameterzaton o lnear demand unctons proposed by McGure and Staeln (1983) n the Appendx. More recently, Bller et al. (006) queston the ptalls o the β -measure, propose a couple o lnear demand alternatves and dscuss the problems arsng rom ther use. These results suggest that β s not the rght parameter to measure product substtutablty. Moreover, when the more approprate measure o product substtutablty, b, s studed n the above papers (by dong the approprate substtutons or the parameters) one can easly show that product prces decrease as products become more substtutable or all the models consdered. The prots may change n ether drecton, but the prots o all players do not ncrease at the same tme and the changes are reasonable. We should also pont out that usng the b-measure o product substtutablty n McGure and Staeln (1983) and Cho (1991) changes the trends o prces and prots, but does not change the other results derved, such as the choce o ndustry structure decentralzed vs. ntegrated- (McGure and Staeln (1983)) or the eects o power structure between the manuacturers and retalers (Cho (1991)). Brge et al. (1998), Bsh and Suwandechocha (005, 006) and Bller et al. (006) study the mpact o parameter β on the optmal capacty nvestment decsons and show that the (total) capacty nvestment level o each rm ncreases wth ths parameter. However, the study o parameter b results n decreased (total) capacty nvestment levels as products become closer substtutes (see secton 3 and Bsh and Suwandechocha (006) who also study both measures o product substtutablty). 7

8 Our study o lnear demand models or two substtutable products wthn derent system structures n compettve and monopoly settngs shows that the use o β as a measure o product substtutablty results n drastcally hgher prces and prots as product substtutablty ncreases n all the settngs studed, renorcng the prevously mentoned ndngs. We show, however, that the use o the rght measure o product substtutablty, where all o the coecents n the lnear demand unctons depend approprately on the level o product substtutablty, can provde results n lne wth what we expect to happen n practce. Lnear demand models wth the correct measure o product substtutablty provde analytcal tractablty whle leadng to realstc conclusons. The value o capacty lexblty or ndependent products has been extensvely studed n the operatons management lterature (e.g. Fne and Freund (1990), Jordan and Graves (1995), Van Meghem (1998), Van Meghem and Dada (1999), Bsh and Wang (004), Bller et al. (006)). More recently, attenton has shted to capacty nvestment decsons or substtutable products. Chod and Rud (005), Bsh and Suwandechocha (005, 006), and Goyal and Netessne (005, 007) study the choce between lexble and dedcated technology, not allowng nvestments n a mx o the two, or two substtutable products. Chod and Rud (005) dene the parameter β as the measure o substtutablty but do not analyze the senstvty o decsons wth respect to ths parameter. They show that the nvestment o a monopolst n lexble capacty ncreases n both demand varablty and correlaton. Bsh and Suwandechocha (006) study both measures o substtutablty, β and b, and show that the optmal lexble capacty level ncreases wth β whle decreases wth b. The study by Goyal and Netessne (005, 007) supports the results o Chod and Rud (005) and urther explores the mpact o product substtutablty on optmal capacty nvestments, usng the parameter b as the measure o product substtutablty, or compettve 8

9 (007) and monopoly (005, 007) settngs. They show that as products become closer substtutes, the optmal nvestment n capacty decreases under both dedcated and lexble technologes. Moreover they conclude that as products become more substtutable rms are more avorably nclned towards nvestng n lexble technology. Ths s n contrast wth the ndngs o Bller et al. (006) n ther study o the β measure and other alternatve lnear demand unctons, whch shows that the need or manuacturng lexblty decreases as products become closer substtutes because demand can be shted through prcng to make better use o the avalable dedcated capacty. In ths study, we analyze the mpact o product substtuton on the optmal mx o dedcated and lexble capactes the rm should nvest n, usng the rght measure o product substtutablty,.e. the parameter b. Supportng the results o the exstng lterature, we show that the total capacty nvestment ncreases wth demand varablty and correlaton between market potentals. Whle the nvestment n lexble capacty ncreases wth demand varablty, t decreases wth the correlaton between demand or prce ntercepts. Furthermore, we show that the nvestment n lexble capacty may decrease or ncrease as the products become closer substtutes, dependng on the underlyng assumptons on the characterstcs o the market. Ths explans the conlctng results ound n the lterature. The paper s organzed as ollows. In secton, we study the prcng/quantty decsons o the rm(s) n monopoly and compettve settngs wth unlmted capactes and known lnear demand unctons. In secton 3, we assume random demand/prce ntercepts and study the capacty nvestment and producton/prcng decsons o a rm n two stages: In the rst stage, the capacty nvestment decsons are made under uncertan demand curves. In the second stage, the demand curve or each product s realzed and the rm makes producton and prcng decsons under the 9

10 capacty and lexblty constrants assocated wth the pror nvestment decsons. We rst study the problem or a monopoly rm, and then dscuss the olgopoly case where n symmetrc rms compete n quanttes (Cournot competton) n both markets. The nal secton dscusses conclusons and extensons. Throughout the paper, we ncrease the level o product substtutablty (.e., decrease product derentaton) by ncreasng β or b. To smply the exposton, we assume wthout loss o generalty that varable producton costs are zero (e.g. Sngh and Vves (1984), Chod and Rud (005)).. Unconstraned Prcng/Quantty Decsons In ths secton we study derent prce and quantty decson models n monopoly and compettve settngs or the two alternatve measures o product substtutablty/complementarty. In Sectons.1 and. we consder two substtutable products and analyze how optmal prce and producton decsons change as products become more substtutable,.e. as β or b becomes more postve. In Secton.3 we study the case o complementary products and dscuss how these decsons are aected as products become more complementary,.e. as β or b becomes more negatve. The prcng/quantty decson can be made beore or ater product demands are realzed. In both cases, the optmzaton problem at hand s essentally determnstc. Snce the prot uncton s lnear n the (random) demand/prce ntercept, maxmzng the expected prots n the ormer case s equvalent to solvng the determnstc problem wth the mean demand ntercepts. In the latter case, the determnstc problem s solved or each demand realzaton and the expected prots calculated usng the resultng prces and producton quanttes. 10

11 .1 Prce Decson Models In ths secton we study prcng decsons o the rms under unlmted capacty or three derent models: Monopoly, Bertrand and prce Stackelberg. The prot uncton Π o the rm whch oers product, s gven by: ( ) ( ε α β ) Π P, P = P P + P,, j = 1, and j. j j In the monopoly model, there s a sngle rm producng both products. The monopolst sets the prces o the products to maxmze the overall prots, Π 1 + Π. For the compettve models we assume that there are two rms each producng one o the products, wth rm oerng product, =1,. In the Bertrand model each rm chooses ts prot-maxmzng prce gven the prce o the rval rm and assumes that ts prce does not change the prce o the rval rm. In the prce Stackelberg model, one o the rms s the prce leader and the other one s the ollower. The leader denes ts prce and the ollower sets the prce ater observng the leader s prce. In our model we assume rm 1 s the leader and rm s the ollower. Table 1: Optmal prces and producton levels; unctons o β MODEL Prce Quantty α jε + βε j Monopoly P = Q = ε α α β ( j ) α jε + βε j Bertrand P = 4α α β Prce Stackelberg (Leader) Prce Stackelberg (Follower) P P = α ε + βε 1 1 4α 1α β ε βε + α ε = α 4α1α β j Q α α ε + α βε Q = j j 4α α j β α ε1 + βε Q1 = 4α ( + ) ε α βε α ε = α1α β 11

12 Table 1 presents the well-known expressons or the optmal prce and producton quantty o each rm or all three models as unctons o β (Sngh and Vves (1984), Brge et al. (1998)). The results show that when β s used to represent the degree o product substtutablty both product prces ncrease n all models as products become closer substtutes. Ths contradcts wth our ntuton and the observed act o hgher prces or hghly derentated products. In addton, the hgher prces or more substtutable products lead to hgher (Bertrand and prce Stackelberg) or unchanged (monopoly) producton levels, and hence to unrealstcally hgher prots or both rms. These results are drven by the act that or a xed set o prces the total system demand ncreases and customers senstvty to prce decreases as products become more substtutable accordng to the β measure: Q1 + Q = ε1 + ε ( α1 β ) P1 ( α β ) P. Table : Optmal prces and producton levels; unctons o b MODEL Prce Quantty A a A ba Monopoly P = Q = a a b Bertrand Prce Stackelberg (Leader) Prce Stackelberg (Follower) ( + ) a a A ba P = A 4a a b A P = j j a1a b j a ba 3A ba a a A P = a1 4a1a b j j ( j ) ( ) ( 4aa j b )( aa j b ) a j aa j b A aa jbaj Q = A1 a A1 ba Q = + 1 4a1 4( a1a b ) ( 4 3 ) ( ) Q = 4( a1a b )( a1a b ) a a a b A b a a b A In contrast, when b s used as a measure o product substtutablty, product prces do not ncrease wth product substtutablty (Table ). In the monopoly model, the optmal prce o each product does not change wth substtutablty. Ths s ntutve because n a monopoly settng one would expect the prce o a product (1) not to ncrease when t has a closer substtute, and () to 1

13 be dentcal n the two extreme cases,.e. when the product has no substtutes ( b = 0 ) and when the product has a perect substtute ( A1 = A and b = a1 = a ), snce both settngs are practcally equvalent to havng a unque product n the system. The optmal system-wde producton n the monopoly model decreases wth product substtutablty as a result o the decrease n total market sze. Moreover, as products become closer to perect substtutes ( A1 A and b a1 a ), the total producton level or the two products converges to the producton level o one o the products when they are ndependent (b=0). The same result ollows or prots, snce prces do not change wth product substtutablty. Then, havng two very close substtutes converges to the case o havng only one (commodty) product, whch s ntutve. Ths s n contrast wth our ndngs or the monopoly model wth the β - measure where, as products become closer to perect substtutes, both prces and prots ncrease to nnty. In compettve models, as products become less derentated, the buyers o each product become more senstve to ts prce,.e., the product s own prce senstvty ncreases; hence the competton strengthens and prces decrease. See Example 1 n the Appendx or llustraton. Under the assumpton o zero margnal costs, the optmal prces and prots decrease to zero as products become closer to perect substtutes. Ths s n lne wth the theory that when two rms producng homogeneous products wth the same margnal cost compete n prce, at equlbrum both prces are the same and equal to the margnal cost (Manseld and Yohe (004)). In the Bertrand model, the producton quanttes can move n ether drecton wth an ncrease n product substtutablty. Ths can be easly explaned n a symmetrc settng where A1 = A = A, a1 = a = a. For small values o b, an ncrease n b sgncantly reduces the demand ntercepts, 13

14 A ( a b) +, but has a relatvely small eect on each product s own prce senstvty, a ( a b ), resultng n lower producton levels. Conversely, or hgher values o b, an ncrease n b has a more pronounced eect on the products own prce eects than on the ntercepts. Ths leads to consderably reduced prces and ncreased producton levels as the two products become even closer substtutes. The optmal responses n the prce Stackelberg model exhbt smlar behavor, except that the producton level or the leader always decreases wth an ncrease n product substtutablty (see Example 1 n Appendx). In the prce Stackelberg model wth symmetrc demands, the ollower always has hgher benets: lower prces leadng to hgher producton levels and hgher prots. Thus, or dentcal rms under the descrbed lnear demand model, each rm preers ollowng rather than leadng and any sequental order s preerred over movng smultaneously (Gal-Or (1985), Dowrck (1986) and Damme and Hurkens (004)). In all the settngs analyzed, usng the parameter b as a measure o product substtutablty provdes a more accurate representaton o the eect o changes n the level o substtutablty on the closed market system. The results o the numercal example n the Appendx (see Tables A1 and A) hghlght the mportance o usng the rght parameter to measure the degree o product substtutablty. Table A1 shows the rapd ncrease n prces and prots as the products become closer substtutes when β s used as the measure o product substtutablty; n partcular, the monopoly case experences a 50-old ncrease n prces and prots when comparng β = 0 wth β = 49 n a symmetrc case where ε1 = ε = 500 and α1 = α = 50. Usng the b-measure, however, as the monopoly rm produces closer substtutes the prce remans stable but the system prots steadly decrease almost by hal as a result o decrease n the total market sze. For the compettve models, when β s used as the measure o product substtutablty, as the products become close to perect substtutes, each rm ncreases ts prots by at least a actor o 3.8 (e.g. 14

15 the ollower n the Stackelberg model ncreases ts prots by a actor o almost 6). Under the b- measure, n contrast, the competng rms see a steep reducton n ther prots by oerng close substtutes.. Quantty Competton Models We now turn our attenton to two derent quantty competton models, Cournot and quantty Stackelberg, and show that the study o quantty competton leads to very smlar results to that o prce competton. In the Cournot model the rms decde on the prot maxmzng quanttes smultaneously, whle n the quantty Stackelberg model rst the leader chooses ts producton and then ts compettor, the ollower, chooses ts producton level ater observng the leader s decson. The prot uncton o rm s gven by: ( ) ( ) Π Q, Q = Q A a Q bq,, j = 1, and j j j or equvalently, or the second parameterzaton o the lnear demand unctons, α jε + βε j α j β Π ( Q, Q j ) = Q Q,, 1, and Q j j = j. αα j β αα j β αα j β Table 3: Optmal prces and producton levels or quantty competton models; unctons o β MODEL Prce Quantty Cournot Quantty Stackelberg ( ) + ( 4α α j β )( αα j β ) j j j j P = α α α β ε α α βε ε1 αε1 + βε P = + 1 4α 4 (Leader) ( α α β ) 1 1 ( 4 3 ) + ( ) P = 4( α1α β )( α1α β ) Quantty Stackelberg (Follower) α α α β ε β α α β ε Q Q = ( ) α α β ε + α βε j j 4α α β Q = ε + α βε α1α β 3ε βε α α ε = α1 4α1α β j 15

16 Table 3 lsts the optmal prces and producton quanttes as a uncton o β or the two models. When β s used as the measure o product substtutablty, we observe that the prces o both products ncrease as products become closer substtutes resultng n hgher prots or both compettors. At the same tme, although the producton quanttes o each product can move n ether drecton, the system-wde producton ncreases wth product substtutablty. In contrast, when parameter b s used as the measure o substtutablty or quantty competton models, both the prce and producton quantty or each product decrease wth product substtutablty (the only excepton s the producton level or the leader n the quantty Stackelberg model whch may change n ether drecton; see Table 4) resultng n lower prots or each manuacturng rm. Moreover as products converge to perect substtutes the soluton converges to the stuaton where rms produce a homogeneous product. Table 4: Optmal prces and producton levels or quantty competton models; unctons o b MODEL Prce Quantty a ( a j A baj ) a A ba Cournot P = Q = 4a a b 4a a b Quantty Stackelberg (Leader) Quantty Stackelberg (Follower) a A ba P1 = 4a j 1 ( ) A a a A ba P = a1a b Q j j a A ba Q = 1 1 4a1a b A a A ba = a 4a1a b j Observe that or dentcal rms, each rm preers leadng rather than ollowng and smultaneous movement s preerred to ollowng, but not to leadng. Ths s n contrast wth the case where the rms strateges are dened n terms o prces, or whch we prevously showed 16

17 that rms preer ollowng n a sequental game and any sequental order s preerred over movng smultaneously. As n prce competton models, whle the results or the β measure are nconsstent wth common sense, the results when usng b to measure product substtutablty are n lne wth our ntuton (see Tables A1 and A or a numercal llustraton)..3 Lnear Demand Models or Complementary Products Economdes and Vard (006) (also see Stanord Knowledgebase (004)) suggest that the sales o a product ncrease as the number o complementary products created or t ncreases, and the rm has an ncentve to make the products more compatble. Davs and Murphy (000) also menton that the demand or a product ncreases n the presence o a complementary product. Venkatesh and Kamakura (003) study prcng o two complementary products under a monopoly and show that the optmal prces o products are monotoncally ncreasng n the degree o product complementarty. The monopolst gans by chargng hgher prces or complementary products whle stmulatng more consumers to buy both products. They measure the degree o complementarty/substtutablty as the relatve ncrease n reservaton prce ganed through purchasng both products, as compared to the sum o the reservaton prces o the two products. For lnear demand unctons, b s used to measure the degree o product complementarty, then the results or the varous prce/quantty decson models studed n the prevous sectons support the above assertons; that s, as products become more complementary the market potental or each product ncreases leadng to an ncrease n producton (sales) o each product and resultng prots. Furthermore, the optmal product prces tend to ncrease wth product complementarty. 17

18 When β s used to measure the degree o product complementarty, on the other hand, the market potental (demand ntercept) o a product does not change when t has a complement; ths contradcts the percepton that addtonal customer segments are reached when oerng complementary products. Furthermore, the expressons gven n Tables 1 and 3 show that the prot maxmzng prce and sales o a product both decrease n the presence o a complementary product, ether produced by the rm or a compettor, whch s not n lne wth what we expect to happen n practce (see Table A3 or numercal llustraton). These contradctory results show that the parameter β does not properly capture the complementarty eect between the products. Also note that α ε + βε > 0 s requred or =1, and j when β < 0, so that the nverse j j demand unctons have postve ntercepts. We must pont out, however, that under the b-measure the ncrease n producton and prots may become unrealstcally large at hgh levels o complementarty. For the unconstraned prcng models shown n Table, or example, observe that when a1 = a the producton o each product and the rms prots ncrease unboundedly as products become more complementary,.e. as b a1 ( = a). For quantty competton models (Table 4 and Table A4 n Appendx), ths undesrable eect s much less pronounced. Observe also that when products become closer to perect complements one would expect the sales or the products to become more smlar and converge to the same value (Wang (006)), whch s not satsed or the prce and quantty Stackelberg models, and satsed or the others only when the demand unctons or the products are dentcal. Note that although one would expect the own and cross prce eects to be smlar or almost perect complements ( b a1 a ), the prce ntercepts can take any values dependng on the characterstcs o the products. 18

19 In concluson, our analyss or complementary products shows that b s a much better measure o complementarty than β. It needs to be used wth cauton, though, or strong complements,.e. when b a1 a, snce t may lead to unrealstcally hgh prces and sales, partcularly or prce decson models, and to sgncantly derent sales o products wth dstant prce ntercepts or quantty competton models. 3. Capacty Investment Decsons under Uncertanty In ths secton, we address the capacty nvestment problem o a monopolst that aces uncertan demand or two products, and analyze the mpact o product substtutablty on the rm s decsons. For that purpose, we use the b-measure, snce t was shown to be a more approprate measure o substtutablty or lnear demand unctons. We consder a two-stage decson problem that has been extensvely studed n the operatons management lterature under varous assumptons (Fne and Freund (1990), Chod and Rud (005), Goyal and Netessne (005, 007), Bller et al. (006), Bsh and Suwandechocha (005,006). In the rst stage, the rm needs to determne the levels o lexble and dedcated capactes to nstall under hgh demand uncertanty. The uncertanty n demand at ths stage s modeled by assumng random prce/demand ntercepts, as done n much o the prevous work (e.g., Fne and Freund (1990), Van Meghem and Dada (1999)). In the second stage, demand s realzed and prot maxmzng prces and producton quanttes are determned. Whle the rm s decsons are made n two stages, we ormulate and solve the two stages smultaneously usng a scenaro-based stochastc programmng approach as n Fne and Freund (1990) and Bller et al. (006). Our goal s to understand how product substtuton mpacts the optmal mx o dedcated and lexble capacty the rm should nvest n. 19

20 Flexble capacty has been shown to be a very valuable tool or the rm to balance supply and demand, and thus ncrease ts prots. The gans stem rom the ablty to sht producton ether to the product wth hgher demand (Fne and Freund (1990)) or to that wth hgher premum (Van Meghem (1998)). As a result, t s hghly valuable when the correlaton between the demands or the prces o the products s low. As products become closer substtutes (as b ncreases), however, customers become more senstve to changes n product prces. Furthermore, a unt ncrease n the prce o one product results n a larger porton o demand dverted to the other product, snce b/a ncreases. Hence, a smaller ncrease n a product s prce results n a larger porton o demand shted to ts substtute. In other words, t becomes nexpensve to sht demand rom one product to the other through prcng. Thus, the benets o usng lexble (costly) capacty to sht producton to the product wth hgher market potental are sgncantly reduced. Consequently, or hgh levels o substtuton, the value o lexblty les n the ablty to sht producton to the hgher premum product, the product wth hgher prce ntercept. Observe, however, that as the products become less derentated they should command smlar prces n the marketplace and, havng smlar premums, the benets o capacty lexblty are very lmted. Our computatonal work ully supports these arguments and shows that the need or lexblty dmnshes as the products become less derentated, unless we assume that the correlaton between prce ntercepts remans low. 3.1 Model and Assumptons Our two-stage plannng model can be descrbed as ollows. The prce and demand ntercepts, ( A, A ) and (, ) 1 ε ε respectvely, are assumed to be dscrete random varables where 1 ( a A ba ) ( a a b ) j j j ε =,, j = 1,, j, and b s dened as the measure o product substtutablty. At the tme o the capacty nvestment decsons we assume that the monopolst ex- 0

21 pects S possble prce/demand realzatons or scenaros, denoted by A s and ε s, =1,, s=1,,s, each wth an occurrence probablty q s wth S qs = 1. We also assume that the possble prce s= 1 ntercepts satsy the condtons a A ba > 0 or, j = 1,, j and s = 1,, S, to ensure j s js postve demand ntercepts. In ths rst stage, the rm decdes the level o dedcated capacty or product, K, =1,, and the level o lexble capacty, K, gven unt nvestment costs c, =1,,, where c1 c < c and c < c1 + c. Ater observng the realzed scenaro s at the second, stage, the rm decdes on prot maxmzng quanttes, prces, P s, or the products. Q s, whch also determne the optmal Usng ths notaton, the two-stage decson problem can be ormulated as a sngle problem as ollows: subject to Q1 s, Q s, s= 1,, S K1, K, K S ( ) ( ) Max q Q A a Q bq + Q A a Q bq c K s 1s 1s 1 1s s s s s 1s s= 1 = 1,, 0 Q1 s K1 K + s (1) 0 Q s K K + s () Q + Q K + K + K s (3) 1s s 1 A a Q bq s (4) 1s 1 1s s 0 A a Q bq s (5) s s 1s 0 K, K, K 0 (6) 1 1

22 Ths s a concave quadratc problem that can be easly solved wth o-the-shel solvers such as CPLEX. In the ollowng secton we derve propertes o the optmal prces and producton quanttes analytcally. In Secton 3.3 we carry out extensve numercal experments to determne the eect o ncreased product substtutablty on the optmal nvestments n lexble and dedcated capactes, snce t s dcult to nd closed orm solutons to the decson varables. 3. Propertes o the Optmal Soluton The eect o product substtutablty on expected product prces and producton quanttes can be derved rom the Kuhn-Tucker optmalty condtons and s gven n the Theorem below. Theorem 1: Let * * * K1, K and K denote the optmal dedcated and lexble capacty nvestments and P, Q, or =1,, the optmal prces and producton levels under each demand scenaro s, s* s* or s=1,, S, n the soluton to the stochastc program. Let * E P and * E Q denote the assocated optmal expected prce and producton quantty and E[ A ] the mean prce ntercept or product, =1,. 1. I the rm makes an nvestment to produce both products,.e. ether K, K > 0 or * * 1 K > 0, then * * E P = [ ] E A + c or =1, [ ] a E A be A a c bc * E Q + = j j j j ( aa j b ) or,j=1, and j where c = c * K > 0 and c = c * K = 0.

23 . I the rm nvests n a sngle product, say product, then the expected prce and producton or ths product are [ ] E A + c E P = * and * E Q = [ ] E A a c, respectvely. See Appendx or the proo. The nterestng case s the one n whch at optmalty the rm nvests n producton capacty, dedcated or lexble, or both products; ths wll typcally be the case n practce. I we urther assume that both products are always manuactured by the rm (as n assumpton (A1) o Goyal and Netessne (007)), then clearly the optmal nvestment n dedcated capactes wll be postve and t ollows rom the above theorem that the expected prces and producton quanttes do not depend on the cost o lexble capacty. Flexblty smply allows the rm to oer the same expected prces and quanttes wth a lower nvestment cost (see Lus (008) or numercal llustratons). Furthermore, the expected prce or each product only depends on the mean prce ntercept and the unt cost o dedcated capacty, and ncreases wth them. Thus, the optmal expected prce or product does not depend on the parameters a and b o the nverse demand unctons, and hence on the level o product substtuton. Note that the unconstraned optmal prces ( P = A ) or the monopoly model do not depend on these parameters. The average optmal prces or the capacty nvestment problem exhbt smlar behavor, wth an addtonal term proportonal to the unt cost o capacty. The producton quantty, however, s aected by all parameters except the cost o lexble capacty. The optmal expected producton quantty ncreases wth the mean prce ntercept o the product and decreases wth that o the other product. Smlarly, t decreases as the unt cost o dedcated capacty or the product ncreases whle t ncreases wth the unt capacty cost o the other product. Fnally, we should pont out that the expected 3

24 prces and producton quanttes do not depend on the varablty n demand or the correlaton between the demand/prce ntercepts. Chod and Rud (005, Proposton 8) also show that the expected output prces are unaected by resource lexblty, but n a very derent settng where the rm nvests n a sngle technology, the costs o lexble and dedcated capacty are assumed dentcal and the soluton s restrcted to an approxmate clearance strategy (capacty must be ully utlzed and the nonnegatvty o the producton quanttes s gnored). 3.3 Computatonal Study Our computatonal study shows that the mpact o product substtutablty on the optmal lexble capacty requrements depends heavly on the underlyng assumptons on the dstrbuton o the random product demand and prce ntercepts, and how they change as b ncreases. For ths purpose, we study two extreme cases: n Secton 3.3. (3.3.3) we assume that the correlaton o the demand (prce) ntercepts remans constant as substtutablty ncreases. Beore that, n the next secton, we characterze the relatonshp between the correlatons o demand and prce ntercepts and dscuss the merts o each o the two extreme cases to capture realty Relatonshp between Demand and Prce Intercept Dstrbutons The ollowng theorem characterzes the relatonshp between the correlatons o demand and prce ntercepts. Theorem : Assume that the prce and demand ntercepts o the lnear demand unctons, aa1 ba a1 A ba 1 ( A1, A ) and ( ε1, ε ), respectvely, wth ( ε1, ε ) =, a1a b a1a b The ollowng relatons hold between the demand and prce ntercepts:, are random varables. 4

25 1) I Corr( ε1, ε ) = 1 or Corr( A1, A ) = 1, we have Corr( ε1, ε ) = Corr( A1, A ) or all levels o product substtutablty. ) As products become more substtutable, the correlaton between the demand ntercepts s kept constant, the coecent o varaton or the two ndvdual demand ntercepts dentcal and also constant, and Corr( ε1, ε ) 1, then the correlaton o the prce ntercepts ncreases to 1. 3) As products become more substtutable, the correlaton between the prce ntercepts and the varances o the ndvdual prce ntercepts are kept constant, and Corr( A1, A ) 1, then the correlaton o the demand ntercepts decreases to -1. See Appendx or the proo. Note that or the case o constant correlaton between the demand ntercepts n ), we requre equal coecents o varaton o the two product demand ntercepts. Extensve computatonal experments show that the result holds n general, except when the coecent o varaton o one product s drastcally larger than the other (n the order o ten tmes). In the rst part o the computatonal study we assume that the correlaton between the demand ntercepts s kept constant as the products become closer substtutes; ths constant correlaton can take any value rom -1 to 1. Ths s a reasonable assumpton snce any level o correlaton seems to be possble n practce between the demand ntercepts or all levels o product substtutablty. For example, consder the extreme case where the products are perect substtutes and equally prced. In ths case, the total demand o the two products, whch we can reer to as the demand or the general commodty, depends only on that one prce. In a partcular settng, the correlaton among the demand ntercepts o the two products could be very negatve the demand curve or the general commodty s arly well known but there s hgh uncertanty as to 5

26 whch o the two substtutable products consumers wll choose. In a derent settng, the two products mght be dentcal n the eyes o the consumer and n that case the market potentals would be perectly postvely correlated. Furthermore, under the assumpton o constant correlaton o demand ntercepts as products become more substtutable, the correlaton between the prce ntercepts ncreases to 1 (Theorem ). Ths makes sense at an ntutve level snce one would expect the maxmum possble prces or two substtutable products to become more smlar as the degree o substtuton ncreases and equal when the products are perect substtutes. The above argument also suggests that t may not be approprate to x the correlaton o the prce ntercepts to study the mpact o ncreasng b, at least or hgh levels o b. Theorem shows that, n that case, as b ncreases the demand ntercepts become more and more negatvely correlated, whch does not seem practcal n general. For completeness we also study the case o xed correlaton o the prce ntercepts, see Secton 3.3.3, n order to understand how the results n Goyal and Netessne (007) extend to the case where nvestments n a mx o dedcated and lexble technology are possble. In general, both demand and prce ntercept correlatons may be aected by the product substtutablty parameter. Such cases are beyond the scope o our study, snce they requre the speccaton o the ntercept dstrbutons as unctons o b, or whch we have no practcal data/evdence Constant Correlaton o Demand Intercepts For the computatonal study, we consder 100 possble uture demand scenaros, whch are combnatons o 10 scenaros obtaned rom dscretzng a normal dstrbuton wth a gven mean and standard devaton, as n Bller et al. (006). We let c1 = c = 4, c = 4., a1 = a = 0.0, E[ A ] = E[ A ] = 50, and let b 0 represent the level o substtuton. 1 6

27 In the rst case, we assume that the correlaton o the demand ntercepts ε 1 and ε s xed, and generate demand scenaros rom a multvarate normal dstrbuton ( ε1, ε ) wth mean a E[ A ] be[ A ] E[ ε ] =, coecent o varaton 10%.e. Var[ ε ] = 0.10 E[ ε ] or =1,, j, j j aa j b and the gven correlaton, Corr( ε1, ε ) = ρ. As noted n Bller et al. (006), 10% varablty o the demand ntercepts n a lnear demand model translates nto a much hgher demand coecent o varaton once you set the prce to a realstc value. Table 5 shows the optmal prces, producton quanttes, capacty nvestment levels and the resultng prots or the gven example wth ρ = 0. We see that as products become closer substtutes, the producton levels and total capacty nvestment decrease, due to a reducton n total market sze, and so do prots. Furthermore, nvestment n lexble capacty becomes less attractve as products become more substtutable because o the ncreased correlaton between the maxmum prces a customer s wllng to pay or the products. As a result, not as much can be ganed rom shtng producton to the hgher premum product. Table 5: The eect o b on optmal decsons and prots; demand ntercepts are ndependently dstrbuted wth coecent o varaton 10% b Expected Prce Expected Producton Dedcated Capacty Flexble Capacty Total Capacty Prots Cor( A1, A ) , , , , , , Numercal experments to study the senstvty o the nvestment decsons to demand varablty and correlaton (Table 6), and to the cost o a unt o dedcated capacty, c, and the relatve 7

28 cost o lexble capacty, c c, (Lus (008)) show the robustness o our conclusons and lead to the ollowng results: (1) Consstent wth the prevous lterature, the total capacty nvestment and the prots o the rm ncrease wth demand varablty and lexble capacty becomes more benecal (Chod and Rud (005), Goyal and Netessne (005, 007)). () Smlar to the results o Chod and Rud (005), whle the total capacty nvestment ncreases wth the correlaton between the demand ntercepts, lexble capacty becomes less valuable. (3) The rm s prots tend to decrease as the correlaton o the demand ntercepts ncreases. Ths s because lexble capacty cannot be used eectvely to reduce nvestment costs. (4) For hgh levels o substtutablty, however, prots ncrease wth the correlaton between the demand ntercepts. Ths s because capacty lexblty has lttle value due to the hgh correlaton o the prce ntercepts; hence, the rm manly gans rom the ncreased probablty o hgher realzatons o both product demands assocated wth the ncreased correlaton between them. (5) The study o the senstvty to the cost rato shows that ncreasng the cost o lexblty nduces the rm to nvest n hgher total capacty but less lexble capacty n order to acheve the same average prces and producton levels wth lower prots (a clear consequence o Theorem 1). 8

29 Table 6: The eect o b on optmal decsons and prots or derent levels o demand varablty and correlaton between demand ntercepts (DC: Dedcated Capacty, FC: Flexble Capacty, TC: Total Capacty, ρ = Corr( ε1, ε ), γ = Corr( A1, A ) ) Coecent o Varaton or Demand Intercepts 5% 10% 15% 0% ρ b DC FC TC Prots DC FC TC Prots DC FC TC Prots DC FC TC Prots γ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

30 Table 7: The eect o b on optmal decsons and prots or derent levels o demand varablty and correlaton between prce ntercepts (DC: Dedcated Capacty, FC: Flexble Capacty, TC: Total Capacty, ρ = Corr( ε1, ε ), γ = Corr( A1, A ) ) Coecent o Varaton or Prce Intercepts 5% 10% 15% 0% γ b DC FC TC Prots DC FC TC Prots DC FC TC Prots DC FC TC Prots ρ , , , , , , , , , , , , , ,743 NA NA NA NA NA NA NA NA ,669 NA NA NA NA NA NA NA NA NA NA NA NA , , , , , , , , , , , , , ,541 NA NA NA NA NA NA NA NA ,576 NA NA NA NA NA NA NA NA NA NA NA NA , , , , , , , , , , , , , ,33 NA NA NA NA NA NA NA NA ,479 NA NA NA NA NA NA NA NA NA NA NA NA

31 3.3.3 Constant Correlaton o Prce Intercepts We now assume that the correlaton o the prce ntercepts A 1 and A s xed, and generate demand scenaros rom a multvarate normal dstrbuton ( A1, A ) wth mean E[ A ] = 50 and coecent o varaton 10%,.e. Var[ A ] = 0.10 E[ A ], or = 1,, and the gven correlaton, Corr( A1, A ) = γ. The values o the other parameters are taken as gven n the prevous secton. In ths case, we only consder b values or whch all scenaros are easble (.e. a A ba > 0 j s js or all s=1,,100, =1,, j ) n order to analyze the sole eect o b, and report NA on the others. The results gven n Table 7 show that the need or lexble capacty ncreases as products become more substtutable, n agreement wth the results n Goyal and Netessne (005,007). We observe that as the parameter b ncreases the market potentals become more negatvely correlated and the lexble capacty becomes more economcally attractve, because t enables the rm to reap handsome prots by satsyng the most popular and protable product. Interestngly, the optmal total capactes reported n Table 6 and Table 7 are very smlar or the same level o correlaton between demand ntercepts (Table 6) or prce ntercepts (Table 7). Ths s because or symmetrc cases, such as the ones tested, the varance o total demand s dentcal n both cases and so s the amount o total capacty requred to acheve the same expected producton levels and expected prces. When the correlaton o prce ntercepts s xed, ths s acheved by usng capacty lexblty eectvely snce demand ntercepts tend to be negatvely correlated. When the correlaton o demand ntercepts s xed, on the other hand, ths s acheved almost entrely wth dedcated capacty by shtng demand rom one product to the other through prcng. Not surprsngly, hgher prots are reaped n the case o xng prce ntercept correlaton as a result o eectvely usng a larger amount o lexble capacty to acheve 31