Volume 3, Issue The Stackelberg equlbrum as a consstent conjectural equlbrum Ludovc A. Julen LEG, Unversté de Bourgogne Olver Musy EconomX, Unversté Pars Ouest-Nanterre La Défense Aurélen W. Sad Informaton & Operatons Management Department, ESCP Europe Abstract We consder a statc game wth conjectural varatons where some frms make conjectures whle others do not. Two propostons are proved. We frst show that there exsts a contnuum of conjectural varatons such that the conjectural equlbrum locally concdes wth the Stackelberg equlbrum (Proposton ). Second, we defne the condtons under whch a conjectural equlbrum s a locally consstent equlbrum (.e. such that conjectures are fulflled). The concept of (local) consstency s restrcted to frms makng conjectures. Two condtons on consstency are featured: consstency wthn a cohort and consstency among cohorts. The Stackelberg equlbrum fulflls only the latter condton (Proposton ). An example s provded. Ths paper nserts n the general CoFal research project devoted to cooperaton and coordnaton falures. We are especally grateful to an anonymous referee whom remarks and useful suggestons mproved the qualty of the paper. Remanng defcences are ours. Ctaton: Ludovc A. Julen and Olver Musy and Aurélen W. Sad, (0) ''The Stackelberg equlbrum as a consstent conjectural equlbrum'', Economcs Bulletn, Vol. 3 no. pp. 938-949. Submtted: Oct 9 00. Publshed: March, 0.
Introducton The Stackelberg (934) model s a sequental model embodyng two rms, one leader and one follower. They both have perfect nformaton about the market demand functon. The leader rm moves rst and makes ts decson takng nto account the reacton of the follower, whch s correctly perceved (Negsh and Okuguch (97)). The follower ratonally sets ts own output level accordng to any quantty set by the leader, wth the expectaton that the leader wll not counter-react. Smlarly, the leader may expect the follower to conform to hs best strategy. In a Stackelberg equlbrum, belefs are self-ful lled, and are therefore an essental feature of such a knd of economes. In ths note, we provde a conjectural nterpretaton of the Stackelberg equlbrum concept. We consder conjectural varatons n a statc determnstc model wth many rms. These conjectures capture the way a rm antcpates the reactons of ts rvals when t decdes to ncrease ts supply by one unt, n a smultaneous game (Bowley (94), Fredman and Mezzett (00), Fguères et al. (004)). In our model, we assume an asymmetry n the formaton of belefs: some rms (type ) make conjectures whle others (type ) do not. The asymmetry could be just ed for nstance by some d erences on costs that would put rms n asymmetrc postons, and thus cause them to formulate some asymmetrc conjectures. But, the foundatons of such an asymmetry s out of scope n ths paper. The dstrbuton of rms between each group s assumed to be exogenous. In our model, no sequental structure s assumed regardng the tmng of decsons. In the conjectural varatons lterature, sgn cant research has been devoted to local consstency of statc conjectures (Fguères et al. (004)). In our framework, a locally consstent equlbrum s such that each type rm s able to correctly perceve the slopes of the reacton functons of both type rms and ther compettors. Ths means that conjectures concde wth the true values of the slopes of reacton functons (Bresnahan (98), Perry (98), Ulph (983)). We consder symmetrc (wthn a cohort) constant conjectural varatons (Bowley (94), Fguères et al. (004), Perry (98)). It s known that under constant margnal costs, the compettve equlbrum turns out to be the only consstent equlbrum (Bresnahan (98), Perry (98)). The exstence of an asymmetry n the formaton of belefs complcates the modellng of conjectures snce t ntroduces ndrect e ects. Hence, the precedng result stll holds n our model. Wthn the same economy, we also study a sequental Stackelberg model à la Daughety (990), wthout assumng any spec c form for the demand functon and for the cost functons. We consder the relatonshp between the sequental Stackelberg equlbrum and the equlbrum of the smultaneous move game for Introducng an asymmetry n costs between rms could allow to endogenze ther type. Smlarly, the tmng of postons s often exogenous n Stackelberg competton, whle t can be grounded on costs asymmetres (Van Damme and Hurkens (999)). By constant conjectural varatons we mean that the formaton of conjectures do not vary wth a change n the strategy of a leader.
ths same basc economy n case of asymmetrc conjectural varatons. We then extend a lterature whch consders conjectural varatons as useful tools to easly capture n a general model varous olgopoly con guratons (perfect, Cournotan and collusve compettons). A synthess of ths approach has been provded by Dxt (986), n whch the Stackelberg equlbrum has not been studed. Two propostons are proved. We rst show that there exsts a contnuum of conjectural varatons such that the conjectural equlbrum locally concdes wth the Stackelberg equlbrum (Proposton ). Second, we de ne the condtons under whch a conjectural equlbrum s a locally consstent equlbrum (.e. such that conjectures are ful lled). The concept of (local) consstency s restrcted to rms makng conjectures. Two condtons on consstency are featured: consstency wthn a cohort and consstency among cohorts. The Stackelberg equlbrum ful lls only the latter condton (Proposton ). We provde an example llustratng these two propostons. Studyng a lnear economy, we notably show that the conjectural equlbrum may concde wth the multple leader-follower Stackelberg equlbrum model developed by Daughety (990). In addton, n case of asymmetry, the compettve equlbrum s locally consstent. The paper s organzed as follows. In secton the model s featured. Secton 3 provdes a de nton and a characterzaton of the conjectural equlbrum. Secton 4 states two propostons reagardng the Stackelberg equlbrum. In secton 5, we gve an example for a lnear economy to llustrate both propostons. In secton 6, we conclude. The model Consder an olgopoly ndustry wth n rms whch produce an homogeneous good. There are two types of rms, labeled and, so the ndustry ncludes n rms of type and n rms of type, wth n + n = n. Type rms make conjectures regardng the reacton of other type rms as well as type rms to a change n ther strategy, whle type rms are assumed not not to make any conjecture (see thereafter). Let p(x) be the prce functon for the ndustry (the nverse of the market demand functon), where X denotes ndustry output. We assume dp(x) < 0 and d p(x) 0. Let x and x j represent respectvely the amounts of good produced by rm, = ; :::; n, and j, j = ; :::; n. Assume X = X + X, where X = P n = x and X = P n j= xj, denote respectvely aggregate output of type rms and type rms. In what follows, we also denote X = P x (resp. X j = P j x j ) the output of all type (resp. ) rms but (resp. j). In addton, t s assumed that p(x) s contnuous. The cost functon of any rm or j s denoted by c (x ), = ; :::; n or c j (xj ), j = ; :::; n. We assume c (:) = c (:)for all and c j (:) = c (:) for all j. In addton, dc (x ) dx > 0 ( dcj (xj ) dx j > 0) and d c (x ) d(x ) 0 ( d c j (xj ) d(x j ) 0). The
pro ts functons of rms and j may be wrtten: = p(x)x c (x ), 8 = ; :::; n, j = p(x)xj c (x j ), 8j = ; :::; n. () Conjectural varatons formed by rm are denoted by (respectvely ) for = ; :::; n, and characterze the belefs of rm as for the reacton of type rms (respectvely type rms), to a one unt ncrease of ts output (Bresnahan (98), Perry (98)). Followng Perry (98), we only consder constant conjectures, so belefs of a rm are ndependent of the supply of the others. Other spec catons are concevable (Fguères et al. (004)). Formally, the conjectural varatons may be de ned as follows: = dx () = dx, = ; :::; n, where [ ; n ] and [ ; n ] represent ntracohort and ntercohort conjectures respectvely, that s conjectures formed by rm regardng the slope of the aggregate reacton functons of type rms and type rms. 3 In the followng, we wll only focus on the symmetrc case,.e. = and = for = ; :::; n. We must consder +. Among the market outcomes usually featured, three cases are of partcular nterest. Perfect competton corresponds to + =. In ths case, each rm of type expects ther drect rvals to absorb exactly ts supply expanson by a correspondng supply reducton, so as to leave the prce unchanged. When = 0 and = 0, rms expect no reacton to ts change n supply: ths s the Cournotan case. Fnally, when = n and = n, the market s collusve. Among the possble con guratons, t s worth consderng the cases for whch type rms form correct expectatons. In ths model, rms - whether of type or - play smultaneously. The players are the rms, the strateges are ther supply decsons and the payo s are ther pro ts. De nton An economy s a game correspondng to a vector of n components, ncludng n strateges and n payo s x ; x j ; ; j j=;:::;n. =;:::;n 3 Ths property holds n case on symmetry across rms (see Dxt (986)). 3
3 Conjectural equlbrum: de nton and characterzaton Before dealng wth the characterzaton of the conjectural equlbrum (CE), we provde a de nton of t. 3. De nton De nton A conjectural equlbrum for the economy s gven by a vector of strateges ~x ; ~x j j=;:::;n and a -tuple of conjectural varatons = ( ; ) =;:::;n such that for any = ; :::; n and any j = ; :::; n : () (~x (); ~x (); ~xj ()) (x (); ~x (); ~xj ()) for all x 6= ~x () j (~x (); ~x j j (); ~x ()) j (~x (); x j j (); ~x ()) for all xj 6= ~xj. A conjectural equlbrum s a non cooperatve equlbrum of game. In the equlbrum, each rm determnes a strategy ~x accordng to the conjectures and, such that no devaton s able to ncrease ts pro t when the strateges of the others reman unchanged. In addton, each rm j, j = ; :::; n determnes ts optmal strategy ~x j by takng as gven the strateges of all other rms. Frms of type do not make conjectures regardng unlateral devatons. Ths corresponds to a Nash equlbrum condtonal on expectatons formaton. We now provde a full characterzaton of a CE. 3. Characterzaton of the CE Gven (), programs of rms and j may be wrtten: x arg max p(x)x c (x ), = ; :::; n, x j arg max p(x)xj c (x j ), j = ; :::; n. The rst-order condtons are gven by: (3) p(x) + ( + + ) dp(x) x dc (x ) dx = 0, (4) p(x) + dp(x) xj dc (x j ) = 0, (5) dx j where @X @x @X + @X @x @X dx X and = dx = ( + ), wth @X = @X. It d ers from @x dx @X = @X @x because = dx from the symmetry of x and n the reacton functon of type rms (equaton 5). By rearrangng the equalty, t comes that = +. Fnally, as +, [ ; 0]. These optmal condtons above mplctly de ne the best response functon of both types of rms: x = (X ; X ; ) and x j = (X ; X j ). In the symmetrc equlbrum, x = x for = ; :::; n and x j = x for j = ; :::; n, 4
so one gets x = (n ; X ; ) and x = (n ; X ). The expected value of can be deduced from and by d erentatng X = n (n ; X ). By substtutng X n, we therefore obtan the equlbrum strategy of a type rm: ~x () = ~ (n ; n ; ). The equlbrum strategy of a type rm s then deduced: ~x = ~ (n ; n ; ). And the equlbrum pro ts can be derved: ~ = ~ (n ; n ; ) for any = ; :::; n and ~ j = ~ (n ; n ; ) for any j = ; :::; n. 4 Conjectural and Stackelberg equlbra We now prove under whch condtons the CE (locally) concdes wth the Stackelberg equlbrum (SE). We rst de ne a SE. Consder that type rms denote leaders, whle type are the followers. So, as n Daughety (990), the ndustry now ncludes n leaders and n followers, wth n + n = n. De nton 3 A Stackelberg equlbrum s gven by a (n + n )-tuple of strateges ^x ; ^x j j=;:::;n such that for any = ; :::; n and for any j = ; :::; n =;:::;n () (^x ; ^x ; ^xj (^x ; ^x )) (x ; ^x ; xj (x ; ^x )), for all xj (x ; ^x ) and x 6= ^x, () j (^xj ; ^x j ) j (xj ; ^x j ) for all xj 6= ^xj. In a Stackelberg equlbrum, all rms optmze ther pro t functon, and belefs are self-ful lled. Leaders and followers play a Cournot game wthn ther respectve cohort. The game s played under complete but mperfect nformaton among leaders and among followers. However, leaders have perfect nformaton about the reacton functon of the followers. Exstence and unqueness of the SE are skpped (see DeMguel and Xu (009), Sheral (984), Sheral et al. (983)). Proposton There exsts a contnuum of conjectural varatons such that the conjectural equlbrum locally concdes wth the Stackelberg equlbrum. Proof. Equatons (4)-(5) mplctly determne the equlbrum strateges when rm, = ; :::; n, conjectures the slopes of the aggregate reacton functons. In a SE, the reacton functon of rm j s obtaned from pro t maxmzaton, for gven strateges of the leader X, and de ned by x j = (X ; X j ). In the symmetrc case, the reacton functon becomes: x = (n ; X ). The program of leader s: x arg max p(x + X + X n (n ; X ))x c (x ), = ; :::; n. j= For any = ; :::; n and j = ; :::; n, the two rst-order condtons sequentally obtaned are gven by: p(x) + + dp(x) x p(x) + dp(x) xj 5 dc (x ) dx dc (x j ) dx j = 0, (6) = 0. (7)
where @ = n (n @x ; X ) s the actual slope of type rms aggregate reacton functon: In equlbrum, one gets @ = n (n @x ; ^X ) for = ; :::; n. Equaton (7) above s denttcal to equaton (5). Let + =, that s + + =, be the conjecture of type rms n a CE. The precedng system collapses to the system de ned by equatons (4)-(5). Thus, the CE locally concdes wth the SE. QED. Remark Notce that whle the two equlbra concde for = 0 and = =, they also concde for an n nte number of values for and such that + = or = + ( ). Proposton states that the equlbrum outcome of a smultaneous move game collapses to the equlbrum outcome a Stackelberg sequental game n whch a cohort of agents make expectatons regardng the mpact of ther decsons on the choces of another cohort of agents. We then complete the analyss of Dxt (986) by showng that conjectural varatons can also be useful to represent the Stackelberg market outcome n a statc game. When embodyng ntercohort conjectural varatons, the framework exhbts a large number of solutons correspondng to the Stackelberg market outcome. We now focus on locally consstent CE. 4 De nton 4 A locally ntercohort-consstent conjectural equlbrum for s a d CE wth = n (n dx ; X ~ ) and = +, where n s the aggregate reacton functon of type rms. A locally ntercohort-consstent CE restrcts the consstency to ntercohort conjectures,.e. to type rms conjectures regardng type rms reactons. But t presumes that must be correctly expected, wthout mplyng ful lled conjectures on. It de nes a partally consstent equlbrum. De nton 5 A locally ntracohort-consstent conjectural equlbrum for s a CE wth = d (n dx ; ~x ; X ~ ), where s the aggregate reacton functon of type rms but. A locally ntracohort-consstent CE restrcts the consstency to ntracohort conjectures;.e. to any type rm s conjectures regardng other type rms reactons. It also de nes a partally consstent equlbrum. De nton 6 A locally consstent conjectural equlbrum for and ntercohort consstent CE. s an ntracohort A locally consstent CE s an equlbrum strategy ~x for each such that no rm perceves an ncentve to change ts supply, whch s based on conjectural varatons, assumed to be a correct assessment of type and type rms. Each rm s then able to correctly perceve the equlbrum values of the slopes of the two aggregate reacton functons. 4 In an olgopoly framework, local consstency has been de ned by Bresnahan (98), Perry (98) and Fguères et al. (004). 6
Proposton Wthn the set of CE whch concde wth the SE, a necessary condton for a SE to be a locally consstent CE s = 0. In that case, = = (n ; ^X ) =. Then the SE converges toward the compettve equlbrum. n @ @x Proof. Accordng to def. 4 and 5, a locally consstent CE sats es = @ @x (n ; ~x ; X ~ @ ) and = n (n @x ; X ~ ). Equatons (5) and (7) beng dentcal, the aggregate reacton functon of type rms must be equvalent n both the CE and the SE. As a consequence, n the equlbrum, ther slopes must be @ equal;.e. = n (n @x ; ~ @ X ) = n (n @x ; X ~ ) =. To construct the locally consstent CE, we must determne the best responses of all type rms but and of all type rms. We follow a procedure gven by Perry (98) for the olgopoly case. The aggregate reacton functons of type rms but and type rms are mplctly obtaned from the rst-order condtons (4) and (5) of rms and j respectvely: p(x) + ( + + ) dp(x) p(x) + dp(x) X X n dc X n dx X dc n = 0, n dx = 0, where X = n (n ; X ), and X (n ; x ; X ). D erentatng mplctly the precedng equatons, one gets n the symmetrc equlbrum: dx dx = = h ( + ) dp(x) n (+)+ dp(x) n h dp(x) + d p(x) ~X n + dp(x) n + ( + + ) d p(x) + ( + )( + + ) d p(x) n + dx + d p(x) ~X n. d c n d(x j ) X ~ n X ~, d c n n d(x ) Accordng to De ntons 4 and 5, a consstent CE must satsfy the next three condtons: (C) dx = (C) dx = = (C3) = ( + ). Accordng to Proposton, the conjectural equlbra whch concde wth the SE, are de ned by + =. Ths equalty and condton (C) are jontly 7
sats ed provded = 0 or = + <. As [ ; n ], a consstent SE must satsfy: = 0 = =. From Def. 4 and 5, consstent conjectural equlbra are xed ponts of: dx dx = =. Snce d P (X) 0, the condton = dx = 0 requres =. Ths corresponds to a spec c case of perfect competton n whch type rms do not react to any move of another type rm, whle type rms determne ther strategy wthout modfyng the equlbrum prce. QED. Proposton puts nto lght the consstency of the CE and ts correspondence wth the SE: consstency of the SE requres that the rms to correctly perceve the true slope of the aggregate reacton functon emanatng from the rms whch do not form conjectures. For 6=, none of the ntercohortconsstent CE that concde wth the SE can be ntracohort-consstent. 5 An example It s assumed that p(x) s contnuous, lnear and decreasng wth X and that t may be wrtten: p(x) = max f0; a bxg, a; b > 0. (8) The cost functons of any rm and j are assumed to be lnear,.e. c x, 8 = ; :::; n and c x j, 8j = ; :::; n. We assume c = c = c, 8, 8j. The pro ts of any rms and any rm j may be wrtten = a c b(x + X + X ) x for = ; :::; n and j = (a c b(x + x j + X j ))xj for j = ; :::; n. Ths economy has a unque symmetrc compettve equlbrum gven by x = x j = a c bn, = ; :::; n and j = ; :::; n, wth = 0 for = ; :::; n and j = 0 for j = ; :::; n. Assume symmetry among all rms of each type,.e. x = x for any = ; :::; n and x j = x for any j = ; :::; n. The best response functon of all type rms but may be deduced from (4) and the best response of any type rm s gven by (5): 5 5 One cannot use the rst-order condton to rm s pro t maxmzaton problem to de ne how all rms but respond to a one-unt ncrease n s output, because dong so would gnore the ndrect e ects of a one-unt ncrease n s output.on ths pont, see notably Kamen and Schwartz (983) and Perry (98)). 8
X (a bx) b( + + ) n c = 0, (9) x = a c b(n + ) X n +. (0) From (0), one deduces the slope of the reacton functon of type rms: 6 dx = + dx, wth = n n +. () In addton, d erentatng (9) leads to: (n ) + dx dx =. () n + + The ntracohort-consstency condton dx = yelds: + + n + + = 0. (3) n From the second consstency condton = ( + ), one deduces after rearrangement: + ( + )n + ( + )(n ) = 0. (4) From equaton (4), the locally consstent conjectural equlbrum s unque and de ned by: 7 n = n +, (5) q = ( + )n [( + )n ] 4(n )( + ) +, = ( + ). It has been shown n proposton that wthn the set of CE equlbra that concde wth the SE, a consstent equlbrum must satsfy = 0. In the above equatons, there are only two cases for whch = 0: ether n = (the leader s alone and behaves as a monopolst) or = (perfectly compettve behavors are expected for type agents, whch corresponds to n! 6 When n =, one has =, whch s the true slope of the reacton functon n case of a lnear Stackelberg duopoly. 7 (+)n As + one gets ( + ). Thus for n, one has p [(+)n ] 4(n )(+) < (+)n s = (+)n + p [(+)n ] 4(n )(+). ( + ). One deduces that the only possble value 9
). The SE s only partally consstent, accordng to de ntons 4 and 5. n The SE s a locally ntercohort-consstent CE f = n + for any feasble values of and, whle t s a locally ntracohort-consstent CE when p [(+)n ] 4(n )(+) = (+)n +, for any feasble values of and. n It s worth notng that proposton s also sats ed for = 0 and = Fnally, from (9)-(0), the equlbrum strateges are: n +. a c ~x ( ; ) = b[n + (n + )( + + )], (6) ~x ( ; ) = When = 0 and = (a c)( + + ) b[n + (n + )( + + )]. (7) n n +, these equlbrum strateges become: ~x = ~x = a c b(n + ), (8) (a c) b(n + )(n + ). (9) These equlbrum strateges concde wth the strateges obtaned n the multple leader-follower Stackelberg equlbrum developed by Daughety (990). In addton, when = and = = 0, ()-() yelds: dx dx =, (0) = 0. So, the asymmetrc compettve equlbrum s locally consstent. Note that t does not mply that s correctly expected (snce = n +n ). It corresponds to compettve equlbrum strateges, where type rms share the market. Ths result con rms that perfect compettve equlbrum s consstent when agents form compettve conjectural varatons under constant margnal costs (Bresnahan (98), Perry (98)). 6 Concluson We determne the condtons under whch the Stackelberg equlbrum concdes wth the equlbrum of a smultaneous move game n whch rms form asymmetrc conjectures. We also precse the de ntons of consstent conjectural varatons n an asymmetrc framework. 0
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