Mergers among leaders and mergers among followers John S. Heywood Unversty of Wsconsn - Mlwaukee Matthew McGnty Unversty of Wsconsn-Mlwaukee Abstract We are the frst to confrm that suffcent cost convexty n a Stackelberg model generates proftable mergers between two leaders and between two followers. Moreover, the degree of convexty requred for leaders to merge s generally far smaller than that requred for followers. Most mportantly, the structure of the stage game means that the convexty requred for ether two followers or two leaders to merge s less than that requred for two Cournot compettors. Dervatons and smulatons rely on Maple 8 and all programs are avalabe from the authors. Ctaton: Heywood, John S. and Matthew McGnty, (007) "Mergers among leaders and mergers among followers." Economcs Bulletn, Vol. 1, No. 1 pp. 1-7 Submtted: June 4, 007. Accepted: June 4, 007. UR: http://economcsbulletn.vanderblt.edu/007/volume1/eb-0710017a.pdf
1. Introducton Salant et al. (1983) examned a model of n dentcal Cournot compettors wth lnear costs of whch m merge. They demonstrate that only n the unlkely event that more than 80 percent of the market merges could the partcpants earn profts as a result of the merger. Ths demonstraton gave rse to a lterature on "the merger paradox" and to a prolonged effort to dentfy alternatve models n whch mergers can be proftable. 1 Huck et al. (001) retan lnear costs but adopt a Stackelberg model wth m leaders and n-m followers. They show that f a leader merges wth a follower, the frm earns more proft than ts two pre-merger component frms. Yet, for all other types of mergers, followers mergng wth each other or leaders mergng wth each other, two frm mergers wll never be proftable f there remans even a sngle excluded frm of the type mergng (leaders or followers). In ths paper, we modfy the Stackelberg model of Huck et al. (001) to allow for convex costs and focus on the possblty of proftable mergers between two followers and between two leaders. Perry and Porter (1985) show that wth suffcently convex costs two Cournot compettors out of n can proftably merge. Yet, Heywood and McGnty (007b) emphasze that the requred degree of convexty typcally remans unrealstcally large. For example, n order for two Cournot frms out of ten to proftably merge, a lnear margnal cost curve must have a slope more than tmes as steep as the demand curve. In the results that follow, we are the frst to show that n a Stackelberg model wth n frms, suffcent convexty generates proftable mergers between two leaders and between two followers. Moreover, the degree of convexty requred for two leaders to merge s generally far smaller than that requred for two followers to merge. Most mportantly, the structure of the stage game means that the degrees of convexty requred for ether two followers to merge or for two leaders to merge are each less than that requred for two of n Cournot compettors to merger. Thus, gven convex costs, proftable merger between smlar frms s more lkely n a market characterzed by Stackelberg leadershp. Ths mplcaton does not emerge from comparng Stackelberg to Cournot under the assumpton of lnear costs. In what follows, the next secton ntroduces the model presentng the equlbra. The thrd secton compares the nfluence of merger on proft solatng the central regulartes for mergers of two frms. A fnal secton concludes.. Model Set-up and Equlbrum Followng Huck et al. (001), we magne an ndustry of n frms facng a lnear demand curve: P = 1 Q, where Q = q 1 + q +... + q n. The frms play a quantty Stackelberg game wth m leaders and n m followers. Followng Perry and Porter (1985), all frms share the same convex cost schedule:c = f + ( 1/ ) cq generatng lnear margnal cost curves wth slope c. We consder a merger of two frms, ether two leaders or two followers, resultng n n -1 post merger frms. We do not consder the case of a follower mergng wth a leader because Huck et al. (001) have 1 These attempts to resolve the paradox nclude adoptng Bertrand competton (Deneckere and Davdson 1985), assumng two frms mergng creates a leader (Daugherty 1990), adoptng spatal competton (Rothschld et al. 000), and consderng frms that sequence output decsons across plants (Huck et al. 004; Creane and Davdson 004). 1
shown t to be proftable n the lnear case, a result that carres over to the case wth convex costs. We take the orgnal number of frms n to be exogenous whch allows us to gnore the fxed cost and set f=0 n the cost schedules. Indeed, as Perry and Porter (1985) make clear, adoptng a postve fxed cost does not change n any way the ncentves for merger because the frm would retan the fxed costs from each of ts consttuent parts. The crtcal comparson determnng the proftablty of merger n our model wll be the sum of profts earned by two of the n pre merger frms and the proft earned by the post merger frm. The pre-merger equlbrum prce, quanttes and profts are: q q F (1 + = =1 to m m + 1+ c + cn + c (1 + c + cn + c m = =m+1 to n ( m + 1+ c + cn + c )(1 + n m + 3 (1 + 3c + cn + 3c + nc + c mc mc ) P = (1) ( m + 1+ c + cn + c )(1 + n m + 3 (1 + ( + 5c + cn + (4 + n) c + c mc(1 + ) = =1 to m ( m + 1+ c + cn + c ) (1 + n m + 3 F (1 + c + cn + c m( + 5c + cn + (4 + n) c + c mc( + ) = =m+1 to n ( m + 1+ c + cn + c ) (1 + n m + In the post-merger equlbru the n- frms excluded from the merger have cost functons C = 1/ ) cq but the frm retans plants each wth that same cost functon. ( The resultng composte cost functon of the two-plant frm s C = ( 1/ 4) cq. Ths functon reflects the underlyng advantage of beng able to drect output across multple plants. Note, however, that f the output of the frm remaned dentcal to that of ts consttuent pre-merger frms, q = q, the total cost to produce that output would be unchanged. The merger by tself does not mmedately provde cost savngs. The pont of the merger remans to reduce output to explot market power. We frst consder the case of two leaders mergng. There wll now be three types of frms: the leader, the m - excluded leaders and the n - m followers. The equlbrum resultng from n- frms wth C and one frm wth C can be easly characterzed but the expressons are very long (avalable from the authors upon request). As a pont of reference, f c=1 the pre-merger proft of a leader from (1) can be compared wth the proft of the leader: ( n + 6 m) ( c = 1) = ( m + 4 + n) ( n + m) 4( n + 4 m)(40 + 14( n m) mn + n + m ) ( c = 1) = () (16 m + m + 10n + n )( n + m) Whle t can be establshed that the proft s larger, the crtcal comparson for any c recognzes that the proft of two pre-merger leaders stands as the opportunty cost: g ( n, = (3) Only when ths expresson s postve s there proft from merger. Whle the full expresson for () for any c s also avalable from the authors, we note that g ( n, c = 0) collapses to exactly
the proft expresson for a merger among leaders as presented by Huck et al. (001) for the case of lnear costs: g ( n, c = 0) = (1 + m m ) / m ( n m + 1)( m + 1). To take an example wth convex costs that can be calculated from () and (3), when n=6, m=3, c=1, the pre-merger leader proft,, s.013 and the proft of the leader,, s.049. Thus g equals.003 and the merger s proftable. An analogous equlbrum and expresson can be derved for the proft assocated wth two followers mergng. The three types of frms now nclude the follower, the n-m- excluded followers and the m leaders. The crtcal dfference s: F F F g ( n, = (4) Whle the full expresson for (3) s avalable from the authors, g F (n, c = 0) collapses to the expresson for a merger among followers as presented by Huck et al. (001) for the case of lnear g F costs: (n, c = 0) = (n n + mn m m + 1/( m + 1) ( n m) (n + 1 m). 3. Identfyng Crtcal evels of Cost Convexty By pluggng n specfc values of n and () can be solved for the crtcal value of c such that the proft gan from the merger of two leaders s zero. For all of the values of n and m there exsts only a sngle postve root. Values above that root return postve proft and values below that root return negatve proft. As an llustraton, g ( n = 6, m = 3, crosses the X-axs from negatve to postve only once at the value c = 0.55. Ths s shown n Fgure 1. Also shown n Fgure 1 are the graphs retanng three leaders but decreasng n to 4 (ncreasng the crtcal and ncreasng n to 8 (decreasng the crtcal. In both both cases m=3.whle such smulaton lacks elegance, t s requred by the complexty of the expresson n () whch nvolves powers of c up to tenth order. Fgure 1: The proft from two leaders mergng, g ( n,. Proft n=8 n=6 n=4 As an alternatve, g ( n, = 0 can be solved for c n general as a functon of n and m. Ths yelds fve roots and whle the relevant root s not analytcally tractable, t can be plotted n three dmensons and shown to be postve. 3
Smlarly, pluggng n specfc values of n and m nto (3) allows solvng for the crtcal value of c such that the proft gan from the merger of two followers s zero. There exsts only a sngle postve root for all values of n and m except when n - m =. In the case of only two followers there are two roots requrng a bt of care n dentfyng crtcal regons. The results of our smulatons are presented n Table I. The Maple program that generated the table and other results s avalable upon request. Table entres are the values of c suffcently hgh that a two frm merger wll be proftable. The top entry s for a merger between leaders and bottom entry s for a merger between followers. We collect our observatons nto a seres of regulartes that flow from the smulatons llustrated n the table. Table I: Mnmum Condtons for a Proftable Smultaneous Merger Crtcal Cost Parameter, c, Stackelberg m= m=4 M=6 m=8 m=10 n-m= 0 3.00 8.63 14.53 0.47 0.15 3.60* 0.10 9.30* 0.09 15.* 0.09 1.* 0.08 7.1* n-m=4 0 10.10 1.88 15.84 6.93 1.67 1.69 7.56 18.59 33.47 n-m=6 0 16.0 1.1.01 7.87 10.90 33.76 16.7 39.67 n-m=8 0.4 0.84 8.10 4.0 33.98 9.18 39.88 14.89 45.80 n-m=10 0 8.6 0.63 34.15.95 40.05 7.56 45.96 13.09 51.89 Crtcal Cost Parameter, c, Cournot** n= n=4 n=6 n=8 n=10 0 4.50 10.4 16.40.38 *Wherever there are two followers all values below the frst entry and above the second entry are proftable. **The values for Cournot come from Heywood and McGnty (007b). Regularty 1: As the number of followers (leaders) ncreases, the crtcal degree of convexty for two leaders to proftably merge decreases (ncreases). Thus, leaders are more lkely to be able to proftably merge when there are relatvely few of them but relatvely many followers. In these crcumstances, the leaders have few same stage compettors (ncreasng output n response to ther reducton) and they enjoy leadershp over many frms. Ths regularty provdes nsght nto merger dynamcs as well. As the number of leaders falls because of merger, the crtcal degree of convexty falls wth t. Thus, f the convexty allows the orgnal merger, addtonal mergers between leaders wll be proftable. Regularty : As ether the number of leaders ncreases, or as the number of followers ncreases, the degree of convexty for two followers to merge ncreases. Followers are placed at a dsadvantage for mergng both my more same stage compettors and more leaders that ncorporate ther desred quantty reducton. Agan, smlar mplcatons result 4
for merger dynamcs. If the convexty s suffcent to allow the orgnal merger of two followers, ths regularty suggests that further mergers between followers wll be proftable as well. Regularty 3: The convexty requred for two leaders to proftably merge exceeds that necessary for two followers to proftably merge (wth the excepton of when there are only two followers). Ths reflects the tmng advantage of the leaders. Merged followers wll have ther desred output reducton taken nto account n the frst stage quantty settng of the leaders resultng n a smaller declne n market output than that assocated wth a merger among leaders. Regularty 4: Holdng the total number of frms constant, the degree of cost convexty for ether two leaders or two followers to merge s lower than for two Cournot compettors to merge. The results for Cournot n Table I show, for nstance, a crtcal value of 16.40 for of 8 frms to merge. Ths exceeds the crtcal level for followers or leaders when there are 8 frms n a market wth Stackelberg leadershp (for nstance, compare to 4 leaders and 4 followers). Thus, separatng the stages of quantty choce blunts the reacton to the quantty reducton undertaken by the frms resultng n a lower needed level of cost convexty. An mportant porton of the merger paradox s the free-rder aspect. It s often more advantageous for frms to reman excluded rvals rather than. Whle t s possble to model mergers that harm rvals, these have been the excepton rather than the rule (Farrell and Shapro 1990; Heywood and McGnty 007a). The ntroducton of the stage game even wth convex costs does not change ths as excluded rvals contnue to beneft from merger. Regularty 5: Mergers of ether two leaders or of two followers wll result n ncreased profts for the excluded frms at both stages. The proft of an excluded leader post merger s subtracted from the pre-merger leader proft. Ths s set equal to zero and solved for a crtcal c n terms of m and n. Whle there are two real roots, both are negatve and all values of the proft dfference are postve for c above the crtcal level. Ths can be repeated for followers. As the degree of convexty s constraned to be nonnegatve, excluded frms beneft from the merger of two leaders. The leader reduces ts output compared to ts pre-merger consttuent frms. Whle t can earn proft from dong so wth suffcent convexty, excluded frms gan for any degree of convexty. Examnng the merger of two followers requred returnng to our smulatons. Nonetheless, all our trals show both excluded followers and excluded leaders beneftng from the merger of two followers. 4. Conclusons Suffcent convexty generates proftable mergers between two leaders and between two followers. The degree of convexty requred for two leaders to merge s generally far smaller than that requred for two followers to merge. Most mportantly, the structure of the stage game means that the degrees of convexty requred for ether two followers to merge or for two leaders to merge are each less than that requred for two of n Cournot compettors to merger. Thus, gven convex costs, proftable merger between smlar frms s more lkely n a market characterzed by Stackelberg leadershp. 5
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