Market Share Dynamics and the Persistence of Leadership Debate

Transcription

1 Market Share Dynamcs and the Persstence of Leadershp Debate John Sutton * London School of Economcs ** Table of Contents. Introducton. The Man Idea A Model 8 4. The Data 6 5. The Scalng Relatonshp Passage Tmes I. 7. Passage Tmes II 8 8. Dggng Deeper Conclusons 37 References.. 4 Appendx. 4 Appendx. 47 EI/37 May 4 The Toyota Centre Suntory and Toyota Internatonal Centres for Economcs and Related Dscplnes London School of Economcs and Poltcal Scence Houghton Street London WCA AE Tel: () * The fnancal support of the Economc and Socal Research Councl s acknowledged. I would lke to thank Cara Whelan for her research assstance, and Yoshro Tama, Dasuke Tsuruta and Kunyosh Sato for helpng wth data collecton. ** STICERD, London School of Economcs, Houghton Street, London WCA AE, UK, Tel: , Fax: , Emal: j.sutton.@lse.ac.uk The authors. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 Abstract Ths paper ntroduces a novel analyss of the classc persstence of leadershp queston, and apples t to a newly constructed dataset for Japanese manufacturng. The analyss rests on an appeal to an emprcal scalng relatonshp between current market share and the varance of changes n market share. Ths relatonshp provdes a powerful model selecton crteron for canddate models of market share dynamcs. It also makes t feasble, even n small datasets, to test drectly for the propertes of the frst passage tmes correspondng to loss of leadershp. Keywords: market share, ndustry dynamcs, scalng, Japanese economy JEL: L, L6

3 . Introducton For how long does a typcal market leader n an ndustry mantan ts poston? Ths queston has attracted contnung attenton n the I.O. lterature over the past generaton. Two rval vews have emerged. The frst, assocated nter ala wth Alfred Chandler (99), asserts that leadershp tends to persst for a long tme. The rval vew, sometmes labelled Schumpeteran, emphasses the transence of leadershp postons; an explct verson of ths vew s spelt out n Frankln Fsher s (983), model of leapfroggng competton. The central problem wth ths debate s that no benchmark s proposed relatve to whch the duraton of leadershp mght be judged long or short. Thus, f t s observed that the typcal market leader stays n place for years ths can be nterpreted as long by wrters n the frst group, and as short by those n the second. Ths pont has not gone unnotced by contrbutors to the lterature; an unusually full and frank acknowledgement of the dffculty s set out by Mueller (986), who notes that hs concluson as to the degree of persstence rests on a subjectve judgement. Ths paper ntroduces a formal model of market share dynamcs, and uses t to provde a benchmark case, correspondng to a neutral stuaton n whch nether postve ( Chandleran ) effects or negatve ( Schumpeteran ) effects are present. Ths model provdes a natural benchmark aganst whch emprcally observed patterns of persstence can be gauged. Mueller s study relates to proft rates, whle the present paper relates to market shares; but the present pont apples equally to both measures.

4 What degree of persstence should we expect on the bass of theory? Game-theoretc models offer lttle gudance on ths queston. The ssue turns on the followng consderaton: suppose the market share gap between the leader and ts (nearest) rval narrows, then wll ths nduce an ncrease or a decrease n effort by the leader relatve to the rval? The factors that may nfluence outcomes here are numerous. One ( Chandleran ) vew emphasses the role played by the dynamc capabltes of frms. On ths vew, market leadershp s a correlate ( sgnal ) of superor capablty, whch s a slowly changng attrbute. Ths suggests a story n whch a short-run narrowng of the market share gap between leader and rval wll tend to be followed by a reverse movement as the gap reverts to the level correspondng to the frm s relatve capabltes. Another mportant factor relates to the detals of the underlyng technology, as represented by a stochastc mappng from R&D to product qualty. If, for example, ths mappng takes the (specal) form used by Ercsson and Pakes (995) for example, then the leadng frm may fnd t optmal to cease nvestng n R&D ( coastng ) even though ths leads to a greater probablty of beng leapfrogged by ts rval. Gven the rval perspectves on the ssues, how can we defne a useful benchmark case? One way forward s to begn wth the queston: f the gap between the leader and ts (nearest) rval narrows, does ths nduce a tendency for a further narrowng, or a tendency for a wdenng? The benchmark case proposed here s that n whch nether of these tendences s present; nstead, market share dynamcs follow a smple random walk (or frst-order Markovan process). Relatve to ths benchmark, we can consder two knds of bas, one of whch ( Chandleran ) leads to longer persstence of leadershp, whle the other ( Schumpeteran ) leads to shorter persstence. The dscussons of these two schools of thought n the lterature do not admt of any sharper defnton of ther respectve postons. 3

5 An dea that forms an mportant motvaton for ths exercse les n the classc observaton of Feller (95), to the effect that passage tmes n Markovan processes tend to be extremely long relatve to what we mght expect ntutvely. Feller dentfes ths as the most surprsng feature to emerge from the study of stochastc processes. In the lght of ths, t seems natural to nqure nto the degree of persstence that we would get n a smple Markovan model; for much of the dscusson the lterature pre-supposes that a long duraton of leadershp must mply that some economcally nterestng mechansm s at work that accounts for ths persstence. What Feller s nsght suggests, s that lookng for such explanatons may be napproprate. Even f leader and laggard are equally lucky or equally capable, then we wll stll see leadershp persst for what appears ntutvely to be a long tme; and for reasons whch are more a matter of arthmetc than economcs. The dea that some knd of Markovan model mght offer a useful frst approxmaton n modellng market dynamcs s not new; ndeed, wthn the dfferent but related growth of frms lterature t has a substantal hstory, begnnng from the semnal contrbuton of Lttle (96) and Lttle and Rayner (966). 3 Yet such models are often thought of as beng unsatsfactory, on the grounds that they do not treat changes n frms shares as an outcome of strategc nteractons (maxmzng behavour) n marketng, R&D, etc. but rather as the outcome of stochastc shocks. Here, I defend the usefulness of such models on the followng grounds: whle tradtonal dscussons between and among Chandlerans and Schumpeterans tactly assume that there s some sngle mechansm drvng (hgh or low) levels of persstence, the central message of the game-theoretc 3 The tests used n that lterature have been based on examnng correlatons between growth rates over successve perods. What s novel n the present paper, relatve to that knd of representaton, s the drect examnaton of the statstcs of frst passage tmes (see below). 4

6 lterature n ths area s that we should not expect any sngle mechansm to play a domnant and systematc role n drvng market share dynamcs. Many patterns of nteracton may emerge between a leader and ts rvals, and these patterns wll reflect nter ala the belefs of agents as to rvals lkely responses to ther actons. The belefs of agents are among the several ndustry characterstcs that may nfluence outcomes, but whch are notorously dffcult to measure, proxy or control for n emprcal studes (Harrs (994)). Ths pont s developed n Secton 8 below, where we examne the pattern of market share dynamcs n selected ndustres. What emerges from these examples s: a) Very dfferent patterns may arse across ndustres wth apparently smlar characterstcs, b) Major shfts n the pattern of dynamcs may occur wthn an ndustry over successve tme perods. What ths suggests s that, whle t mght be possble to buld a satsfactory structural model of market share dynamcs for a sngle ndustry, or even a group of cognate ndustres, t s helpful n lookng across the general run of ndustres to begn by examnng the data aganst the background of a more modest, low level representaton of the knd proposed here. 5

7 . The Man Idea The man dea underlyng the method of analyss proposed here les n explotng two key features of the emprcal data, whch permt a very smple representaton of the stochastc process drvng the pattern of market shares. The analyss of market share dynamcs poses, n general, two serous challenges. Frst, snce market shares add to unty, shocks to dfferent frms shares are nterdependent. Second, the (dstrbuton of the) sze of shocks to each frm s share mght be expected to depend nter ala on that frm s current share. Ths mples that an approprate model mght be one n whch the dstrbuton of shocks to each frm s share would need to be condtoned on the full vector of market shares n the current perod. The role of the two emprcal features of the data on whch the present method of analyss rests s to permt a much smpler representaton of the underlyng stochastc process. The frst feature of the data on whch we rely s that, for all but four (hghly concentrated) ndustres among the 45 ndustres n the dataset, the shocks to the market shares of the ndustry s leadng frms dsplay an extremely low degree of correlaton, so that we may mpose, as a reasonable approxmaton, a model of ndependent shocks. The second feature of the data on whch we rely s that t exhbts a smple scalng relatonshp between a frm s market share and the varance (or standard devaton) of ts change n market share. 6

8 The nature of ths scalng relatonshp s as follows: the varance σ of the change m n a frm s market share m, measured n percentage ponts, ncreases n drect proporton to m; equvalently, the standard devaton of the fractonal change n m,.e. m/m, falls proportonally wth / m. The method of analyss used n what follows takes advantage of ths feature of the data. Essentally, t allows us to characterze the sze dstrbuton of annual shocks to market shares wthn each ndustry by reference to a pooled sample of all observatons (avodng the need to condton drectly on current market share, a procedure whch would not be practcable usng the small dataset nvolved here). Takng these two features together, the most basc persstence of leadershp queston,.e. that of analysng the tme elapsed untl the market leader s overtaken by any specfc rval, can be handled by reference to well-known propertes of a (smple) random walk. By appealng to the standard propertes of frst passage tmes for such processes, we can acheve a consderable smplfcaton n the analyss. 4 Before turnng to emprcal matters, t may be helpful to begn by settng out an llustratve theoretcal model. It s mportant to note, however, the emprcal analyss whch follows rests solely on a drect appeal to the two features of the data just 4 Whle the earler lterature has tested the null-hypothess of neutral or frst order Markovan property on whch we focus below, t has done so by lookng at (low-power) tests nvolvng comparsons of x t and x t +. A test of ths standard knd for the present dataset ndcates no sgnfcant correlaton(s) of ths knd, over any tmescale. By focussng drectly on the statstc of nterest (the frst passage tme), we can arrve here at a more powerful and drect test of the hypothess. A more fundamental problem wth the standard approach of examnng changes n each frm s sales as an (ndependent) stochastc process, as s done n the growth of frms lterature, s that ths approach s unsuted to examnng the persstence of leadershp queston: the counterhypothess aganst whch the null s tested, s that the sales of each frm form ndependent hgher order Markov processes. However, the economcally nterestng counterhypothess n the persstence of leadershp settng are ones n whch changes n the sales, or shares, of the frms depend nter ala on the current dfference n shares between the leader and ts (nearest) rval(s), and ths cannot be captured usng the standard methods. 7

9 mentoned, and does not depend upon the partcular model presented below. The reason for ntroducng the model s to provde an ntutve explanaton for three ponts whch mght otherwse seem puzzlng. These are:. the dea that market share shocks may be approxmately ndependent n ndustres where concentraton s low;. the scalng relatonshp It s natural to ask whether ths relatonshp has any theoretcal bass. An examnaton of the varous standard product dfferentaton models ndcates that the only type of model that appears to exhbt ths feature s a mult-product frm model that combnes a vertcal product attrbute of the standard knd wth a horzontal attrbute of the locatonal (Hotellng) type. In partcular, ths form of scalng relatonshp does not arse ether n sngle attrbute qualty models, whether of the vertcal product dfferentaton type (Sutton (99, 998)) or of the stochastc qualty jump type used by Ercson and Pakes (995) n ther model of market share dynamcs. Intutvely what drves the present scalng property s the dea that a large frm receves shocks of the same absolute sze, but that the expected number of such shocks occurrng n a gven tme nterval ncreases n drect proporton to the frm s sze. In order to provde a framework for the analyss that follows, we begn by ntroducng a (delberately smple) model of ths knd.. The model ntroduced here s a non-strategc one n whch changes n market shares are drven by exogenous shocks to product qualty. The motvaton for ntroducng a non-strategc model n ths context les n the argument that the approprate strategc model(s) would be hghly ndustry specfc, a pont on whch we elaborate n the fnal secton below. Ths rases the queston: what 8

10 of strategc nfluences that do not depend on hghly specfc ndustry characterstcs, but operate robustly across the general run of ndustres? It s well known that certan systematc strategc effects operate to place a lower bound on the level of concentraton that s sustanable as an ndustry equlbrum (for example, Sutton (99, 998). How does ths square wth the noton that market shares may fluctuate over tme, at least once some lower bound to ndustry concentraton s respected? The pont s addressed below (footnote 9). 3. A Model The model s a standard crcular road model, n whch frms offer products that are dfferentated by locaton. The products are located evenly around the crcumference of a crcle of unt dameter. Each (actve) frm owns a subset of these products. For smplcty, we confne attenton to the case where no frm owns two adjacent products; ths allows us to obtan a smple characterzaton of a Nash equlbrum n prces (t concdes wth the prce equlbrum for sngle product frms). We assocate wth each product a qualty ndex u. Consumers are located unformly along the crcle, the total sze of the populaton of consumers beng normalzed to unty. Each consumer buys exactly one unt of one of the goods on offer, the suppler beng chosen to maxmze the consumer s utlty, U(p, u) = u p td 9

11 where p s the prce s the prce of the chosen good and t s the (constant) unt cost of transport along the crcle. p u p u p + u + d / N d d / N d Fgure In what follows, the range of u wll be restrcted so as to ensure that the margnal consumers defnng the left and rght hand boundares of product s clentele wll le between product and ts mmedate neghbours. We can then wrte down the condtons defnng the dstance from frm to the margnal consumer on ts rght, whch we label, as follows: d p + u + td = p+ + u+ + t(/ N d ) whenced / N + [(p + p ) (u u ) / t () = + ] The dstance from frm to the margnal consumer on ts left, denoted by Fgure ), s calculated n the same way, vz. / N d (see

12 [(p p ) (u u )]/ t / N d = / N + () Addng () and () we obtan the quantty sold by frm, vz q / N + [(p + + p p ) (u+ + u u ) / t (3) = ] Settng cost to zero and wrtng the proft of frm as p q, we dfferentate wth respect to p to obtan the optmal reply (reacton functon) of frm, vz. p = t / N + ( p+ + p ) / 4 ( u+ + u u )/ 4 (4) Gven our assumptons that no frm owns two adjacent goods, and that the range of the qualty ndex s restrcted so as to ensure that the margnal consumer always les between the product and ts closest neghbour, t follows that the optmal reply (reacton functon) for each frm s to set the prce of each of ts products n accordance wth equaton (4).e. the frm s proft functon s addtvely separable nto a number of functons, correspondng to the proft earned from each product. In the specal case where all the u s are zero, the set of equatons defned by (4) collapse to those of the standard crcular road model: there s a symmetrc Nash equlbrum n prces, n whch all frms set the same prce p = t/n, as can be confrmed by nspecton of (4).

13 Our focus of nterest les n examnng the manner n whch exogenous shocks to (relatve) qualty levels of ndvdual products mpnge on the sales of the frm. It s shown n Appendx that a unt shock to the qualty of product, gven equlbrum prce responses by all frms, leads to a rse n the quantty (sales volume) of product of / 3 unts, and a fall n the sales volume of each other product. These mpacts on the sales of other products declne geometrcally as we move away from product ; for the k- th product to the rght or left of product the change n sales volume s ( 3) / 3. Gven our normalzaton of the total sze of the populaton of consumers to unty, the (change n) quantty sold by a frm equals ts (change n) volume market share. 5 k We do not restrct the pattern of shocks to qualtes n what follows. In each (short) perod, there s a small probablty p that a sngle shock to the qualty of some one (randomly chosen) product occurs, the sze of the shock beng drawn from some dstrbuton f ( u). Ths leads to a geometrcally declnng seres of shocks to the product n queston, and ts neghbours. We confne attenton throughout the case where the number of products s large; and gven the geometrcally declnng sze of mpact, we approxmate by neglectng all shocks beyond a certan radus vz. the lth product on the left to the lth product on the rght. So far we have gnored the possblty that a shock to the qualty of product k mght brng t outsde the range [,u]. We treat ths by settng u t+ = u f u t + u u, and u t+ = f u t + u. 5 We work for convenence n terms of volume market shares. The results for market shares by value are smlar, subject to an approxmaton.

14 When a product qualty falls to, we treat ths as an ext event. 6 We assume that such an event s followed by the entry of a new product by some frm, at ntal qualty. The probablty that the new product s entered by frm j s set equal to the proporton of products currently owned by frm j. 7 We wll not be drectly concerned n what follows wth the long run steady state propertes of the model; 8 here, t suffces to remark that a frm s expected market share, condtonal on ts havng n out of N products, equals / N. j n j We consder the mpact on the pattern of market shares of a qualty shock that affects a sngle randomly chosen product. In what follows, we focus on the largest and second largest frm n the ndustry; ther respectve numbers of products are denoted as / N = p and n / N p, and we denote by p3 p p n = products owned by all other frms. = the combned share of 6 Ths representaton of ext events s chosen purely for convenence; a more sophstcated model would nvolve a consderaton of the sunk cost ncurred n enterng a product, and would nvolve the determnaton of an optmal threshold u* at whch a product would be deleted. 7 Agan, ths feature of the model s chosen n order to brng the model nto lne wth the emprcally observed scalng property. 8 In a model of ths type, there wll, once entry and ext are modelled as optmzng decsons, be a lower bound to the level of concentraton (specfcally, to the market share of the largest frm; see Sutton (99, 998)). Ths comes about as follows: suppose we allow frms to choose the qualty of ther products optmally, subject to some fxed cost schedule. Then, f the number of frms becomes suffcently large, so that the maxmum market share falls below some crtcal level, t be optmal for one frm to devate, ether by rasng the qualty of (at least one) of ts products, so as to capture a greater market share. The dea behnd the present model s that the number of frms that are actve n the market has been arrved at by some earler (unmodelled) process of entry, and t s not so large as to volate the lower bound to concentraton. The focus of nterest here les n askng, how do market shares fluctuate wthn the regon permtted by these bounds? 3

15 It s ntutvely clear that, n examnng the behavour of the market share gap m m (or the gap p p whch concdes wth the expected value of m m ), that there are two polar cases of nterest, vz. where p 3 s large, so that frms and are small and where p 3 s close to zero, so that p p. In the latter case, there s close (negatve) correlaton between changes n the market shares of frm and frm. In the former case, ths correlaton s close to zero and we can approxmate shocks to m m by treatng and m as ndependent. An emprcal examnaton of the present dataset m ndcates that the correlaton between m and m s very close to zero (see Secton 5 below). Wth ths n mnd, we focus on the case where p 3 s large, where we may analyse the mpact of a sngle unt shock to the qualty of some randomly chosen good by representng the probablty that frm (or frm ) receves the assocated quantty shock of order k as (or p respectvely), and gnore all multple events. In ths case, the p expected change n m can be approxmated as: s kp = p k k s k where s k s the change n quantty (volume market share) for a product dervng from a unt shock to the qualty of a product at the k-th locaton to the rght, or left assocated wth a shock of order k, and p s the share of products owned by frm I, and where the sum s taken over k = l,...,,,,..., l. Now consder any (dscrete) dstrbuton of qualty shocks: let f j denote the probablty that a shock of sze j occurs. Then, recallng that the derved quantty changes are 4

16 drectly proportonal to the sze of the qualty shocks, the varance of changes to m can be represented as var ( m ) = = p j j k p f ( s k j j j j k f ( s k ) ) Notng that the double sum n ths last expresson s a constant, the varance of proportonal to, whch we can proxy emprcally by m. p m s It follows that the standard devaton of changes to market shares satsfes σ ( m ) constant. m It follows that, f we replace the market share may wrte m by m, then for small changes we m m m whence σ( m ) constant 5

17 so that we have a measure of volatlty that s constant over. We can now construct an ndustry-specfc measure of the degree of volatlty by poolng all observatons of m for all frms over some perod; whence we defne the volatlty measure m vol = σ ( ) m In the case under consderaton, where m and m are treated as ndependent, we may now proceed as follows. Denote by g( m ) the (symmetrc) p.d.f. of (small) changes to m. Defne the gap between frm and frm as g m m Gven the ndependence of changes to m and m, we may model the evoluton of g as a smple random walk whose ncrements are drawn from the dstrbuton g o g. If the dstrbuton of shocks to m s, for example, normal wth standard devaton σ, then changes to m m are normal wth standard devaton σ + σ = σ can therefore normalze by defnng the gap. We g = m. σ( m m ) whose evoluton can be modelled as a smple random walk, whose movements are drawn from the standard normal dstrbuton N(,). In the next secton, we follow ths procedure wth one modfcaton; the dstrbuton of shocks s better represented as a t- dstrbuton, and we modfy the procedure slghtly to reflect ths. 6

18 4. The Data The dataset conssts of annual observatons of market shares for leadng frms n 45 narrowly defned ndustres n Japanese manufacturng over the 5 year perod (Appendx ). These data were compled usng the annual volumes publshed by Yano Company. Ths source covers a large number of ndustres, but occasonal changes n coverage and presentaton occur, and t was possble to construct farly long and consstent seres only for these 45 ndustres. Specfcally, t was possble to comple a hstory of 3 years or longer for each of the 45 ndustres. The startng data for ths hstory s 974 for the large majorty of ndustres, but t s between 975 and 977 n a small number of cases. The man tests descrbed n the next secton are carred out by reference to the -year hstory of these 45 ndustres. A seres of ntervews wth selected companes was used to check ssues of nterpretaton and relablty of data. Data of ths knd would be very dffcult to comple for a broad cross-secton of ndustres n other countres; the avalablty of the Yano data was a prmary reason for focussng on Japan. The second, equally mportant, reason for ths focus les n the rarty of mergers and acqustons. For U.S. or U.K. data, for example, t would be dffcult to study the dstrbuton of frst passage tmes over an extended tme perod wthout havng to confront the confoundng nfluence of M&A events. In the present data-set, only one merger nvolvng leadng frms occurs over the 5 year perod n these 45 ndustres. 7

19 The level of aggregaton n ths dataset corresponds roughly to the 5-dgt SIC classfcaton for the U.S. The ndustres nclude, for example, margarne, photographc flm, beer and cash regsters. The number of frms ncluded vares across ndustres, the typcal case beng half a dozen or so. Excluded frms generally have very small shares. Ther excluson does not affect the computaton of frst-passage tmes, snce f one of these frms grows to become a leadng suppler, t s ncorporated n the data-set. There are no nstances n whch such a newly entered frm overtakes the market leader durng the perod covered by the data. 5. The Scalng Relatonshp We begn wth a descrptve account of some basc features of the data. a. We begn by takng the top two frms n some reference year (year 5), and we examne, whch we label hereafter frm and frm respectvely. We examne the annual change n market share for frm, versus the change for frm, n each year. The resultng scatter for the pooled sample of all ndustres s shown n panel (a) of fgure. The correlaton coeffcent s., ndcatng that the data s well represented by the frst lmtng case descrbed n the precedng secton. To explore ths further, the exercse was repeated by excludng successve groups of ndustres, usng as a crteron the combned market share of the top two frms n the reference year. Only when the crtcal value of ths combned market share was set to exclude all but 4 ndustres dd a clear negatve correlaton appear. Wth ths n mnd, we proceed to explore the full dataset usng the model based on the frst lmtng case, as developed n the precedng secton. (Excludng these 4 8

20 ndustres from the analyss whch follows has no materal effect on our conclusons). 9 b. To nvestgate the relatonshp between current market share, and the dstrbuton of change s n market share, a pooled sample of all annual observatons was formed, and parttoned nto groups (bands) by market share,.e. all pars m t, m ) for whch m m t m + fall n group, and so on. For each band, the ( t standard devaton of m t was estmated. Fnally a regresson of ln σ( m t / m t ) aganst l n mt llustrated n Fgure 3, yelds a slope of.53, whch s not sgnfcantly dfferent to /, and whch suggests that the data s well represented by the frst lmtng case (as opposed to the second lmtng case) of the model set out earler. c. To nvestgate the dstrbuton of the sze of shoc ks to market share (whch s not restrcted wthn the above model), we may take advantage of the scalng relatonshp to examne the dstrbuton of m / m, whch should be t t m t ndependent of. Ths ndcates that the dstrbuton s represented by a t- dstrbuton wth a coeffcent of about.3,.e. of the form a f (x) =. 3 ( + x ). Ths s llustrated n Fgure 4. Ths descrpton does not however extend to the tals of the dstrbuton; there are no observatons outsde a range of about 3 standard devatons from the orgn, a pont to whch we return below. 9 It mght seem surprsng prma face that the lack of correlaton holds even n moderately concentrated ndustres. Ths may reflect the fact that, n some ndustres, the two leadng frms do not compete head- to-head, so that ther gans (or losses) of market share mpnge more on lower ranked frms, than on each other. By repeatng ths exercse for subsets of the data correspondng to dfferent bands of m t, t s confrmed that the form of ths dstrbuton does not vary notceably wth m t, as expected. 9

21 MS + MS > 5% Change MS Change MS Change MSI (a) Change MSI (b) MS + MS > 8% MS + MS > 9% Change MS - Change MS Change MSI (c) Change MSI (d) Fgure. The annual change n market share for the top rankng frm (horzontal axs) versus the change for the second rankng frm (vertcal axs). (The two frms are those ranked and n year 5 of the dataset). Panel (a) shows the data for all 45 ndustres, whle panels (b), (c) and (d) show data for those ndustres n whch the combned market share of the top frms n year 5 exceeded 5%, 8% and 9% respectvely.

22 d. Gven the scalng property, t s natural to beg n the nvestgaton of passage tmes by focussng attenton on the top two frms, vz: we take some reference date t = and label frms n descendng order of market share at that date. We now examne the frst date at whch the market share of frm exceeds that of frm (labelled t n what follows). The advantage of begnnng wth an t analyss of s two-fold. Frst, we may take advantage of the theoretcal results developed above to reduce the problem to the study of a smple random walk, thus allowng us to draw some standard results for passage tmes. Second, ths allows us to place an upper bound on the passage tme for loss of leadershp. Fgure 3: The Scalng Relatonshp ln MS σ MS ln MS t- It mght seem natural to begn by checkng whether changes n the market share gap between the two frms exhbt any (postve or negatve) seral correlaton over successve years. A seres of checks, usng dfferent tme perods (lags), ndcated no sgnfcant correlaton of ths knd. Ths, however, does not exclude more subtle forms of departure from the null hypothess explored here, as was noted n footnote 4 above.

23 zero_changes postve_changes negatve_changes - f ( MS ) ln (.3 + MS ) Fgure 4. The form of the dstrbuton of m / m 6. Passage Tmes I Ths secton looks at two methods of testng n relaton to the passage tme t. The theoretcal model of Secton 3 leads to an approxmate representaton n whch m m can be modelled as a smple random walk n whch the jump n value from t to t+ s represented by a draw from some dstrbuton g o g. To control for the dfferent levels of market share volatlty across ndustres, we represent the gap as ( )/ σ m m, where σ s an ndustry specfc volatlty parameter. Our way of predctng the dstrbuton of the passage tme t, therefore, would be to begn by estmatng the volatlty parameter σ ( m ) for each ndustry, and usng ths

24 estmate to normalze the sze of the ntal gap ( m m ). To do ths, we use the data fo r some ntal perod [,t ] to estmate σ ( m ) ; take the gap at the end of ths perod as the ntal gap, and predct the probablty for each ndustry of a crossng t durng the remanng tme perod [ t,t]. Fnally, these probabltes are summed across ndustres to obtan a predcted number of crossngs n these 45 ndustres over the perod [ t,t]. The dsadvantages of ths method are:. It uses up several years of data n estmatng ( ) σ ; m. Even f σ s constant over the entre perod, the precson of the estmate obtaned from some short ntal perod may be low.. It depends upon the emprcally estmated form of f ( ) m. Whle a sharp characterzaton of ths dstrb uton s possble, a dffculty les n specfyng the range of observatons,.e. the tals of the dstrbuton. Results may be senstve to the specfcaton used here, and ths nvolves some arbtrarness. Ths method was mplemented usng the frst 5 years of data to estmate the ndustry specfc volatlty parameter σ. However n vew of problem (), the results are not reported here (they are broadly consstent wth the results reported below). Instead, two alternatve methods are used, as follows: The frst method of testng takes advantage of the propertes of the smple random walk, n order to avod the need to estmate the volatlty parameter. Ths property s as follows: let t o be defned as the frst date at whch m crosses m, so that the gap at ths date 3

25 equals (approxmately) zero. Now take the nterval from t o onwards. Dvde ths nto two equal sub-ntervals, the second sub-nterval beng [ t ( T t )/, T] [ (T t )/,T ] + accordng as t s odd or even respectvely. t o o + or o If the gap follows a random walk, then the probablty that a crossng m m t occurs durng the second sub-nterval equals. Ths result holds ndependently of the dstrbuton of shocks, and so of the volatlty parameter. The ntuton s as follows: f we make the process more volatle, then the probablty that the gap drfts upwards (or downwards) by a large amount durng the frst subnterval rses; but ts probablty of returnng thereafter from a dstant value ncreases to the same degree. We can nterpret the null hypothess beng nvestgated here n terms of the crcular road model of secton 3: we nfer from the equalty of market shares at rate t o that the two frms have an equal number of products at that date (.e. each has the same expected number of products condtonal on ts observed share, vz, t owns a fracton m of the products). The dynamcs of market share then follows a random walk, wth no (postve or negatve) drft. Under ths null hypothess, t follows that, when we observe a crossng at tme t, the probablty that we wll observe a crossng n the second sub-nterval defned above s ½. What f the hypothess fals? Say, for example, that frm had some underlyng capablty superor to that of frm, and that ths wll make t (more) lkely that frm wll pull ahead, and stay ahead, of frm n the future. The presence of such a bas n favour of frm would cause the stochastc process descrbng the market share gap to 4

26 exhbt postve drft. Ths s consstent wth the presence of (a reduced number of) crossngs of the frms market shares, but the observaton of a crossng does not mply an equalty between the frms future fortunes; the expected number of crossngs n the second sub-nterval s now less than ½. The man dsadvantage of ths test s that t requres us to dscard all ndustres n whch no crossng occurs pror to the last two years of the data; ths leaves us wth only 8 ndustres out of 45. The number of second perod crossngs n the set of 8 ndustres equals 5, as compared wth an expected level of 9 under the null hypothess. 3 The null hypothess s rejected at the 5% level (one-tal test, see footnote ). Ths suggests that, once a frm moves ahead of ts rval(s), the degree of persstence of leadershp may be greater than predcted under the null. Turnng to the second method, we agan appeal to the scalng property to justfy the poolng of all observatons of m [,T ] m for each ndustry for the full perod, and the modellng of the evoluton of m m as a smple random walk. Now, however, we use the set of pooled observatons of m for all frms and for all perods, wthn each ndustry, to predct the dstrbuton of the frst passage tme for that To see what s nvolved here, t s useful to ask what analogous argument would be for the gap n scores n a basketball game. (I am grateful to Barry Nalebuff for suggestng ths analogy.) At tme, the teams scores are equal, but the abltes of the teams wll, n general, dffer. Only f abltes (scorng possbltes) are equal, does the present model apply. If abltes dffer, the gap n scores follows a random walk wth (postve or negatve) drft, and whle scores may at some tme(s) concde, the probabltes of a second perod crossng s less than ½.. 3 A devaton of 4 or more from the expected level of 9 wll occur wth probablty 4.8% under the null hypothess. Thus we ju st fal to reject the null at the 5% level (-tal text). 5

27 ndustry. We then sum the probablty of observng a crossng n each ndustry by tme T n order to arrve at the expected number of crossngs over our full set of ndustres. 4 The advantage of ths procedure s that t avods the need to ft some dstrbuton to the observatons of market share changes, and t allows us to focus drectly on the ssue of nterest,.e. whether the evoluton of the market share gap exhbts some subtle form of behavour that dstngushes t from our Markovan benchmark. The results of the procedure are shown n Table and Fgure 5. The predcted values, and the assocated 95% confdence nterval, are obtaned by smulatng the random walk (Monte Carlo estmates). The results are shown for a seres of alternatve startng dates (year, 5, and 5, respectvely). 5 The observed values show a tendency to le below predcted values. Th ey le wthn a 95% confdence nterval n two of the four cases, whle fallng outsde n two. 4 Ths procedure s equvalent, under our ndependence assumpton, to modellng m m as a smple random walk. 5 The results predcted values for these alternatve startng dates were obtaned by drawng market share shocks from the pooled sample for all perods. The procedure was repeated sung the sample of shocks for the correspondng tme perod (perod 6 to fnal perod, etc.). The results were closely smlar to those shown n Fgure 5. 6

28 Years after ntal year (year 5) Actual number of crossngs Predcted number of crossngs Mean 95% Confdence Interval Table Actual and Predcted Crossng Tmes for the Two Leadng Frms, over 45 Industres. 7

29 Two-frm Frst Crossng Tmes Cumulated No. of Crossngs t Actual Upper Mean Low er Two-frm Frst Crossng Tmes (as of year 6) Cumulated No. of Crossngs t Actual Upper Mean Low er Two-frm Frst Crossng Tmes (as of year ) 8 Cumulated No. of Crossngs Actual Upper Mean Low er t Two-frm Frst Crossng Tmes (as of year 6) 8 6 Cumulated No. of Crossngs Actual Upper Mean Low er t Fgure 5 The top panel shows the number of crossng of the leadng frm by ts closest rval n the ntal perod, by elapsed tme t that occur wthn the 45 ndustres. The second, thrd and fourth panels repeat ths exercse, begnnng from the sxth, eleventh and sxteenth year of the 3-year run. (The closest rval s defned as the second largest frm n the correspondng ntal year). The expected number of crossngs, and the confdence nterval, for our benchmark process are also shown. 8

30 Total Frst Crossng Tmes Cumulated No. of Crossngs Low er Mean Upper Actual t 5 5 Total Frst Crossng Tmes (Largest Frms Excluded) Cumulated No. of Crossngs t Low er Mean Upper Actual Fgure 6 The top panel shows the number of crossngs of the leadng frm by any rval, by elapsed tme t, that occur wthn the 45 ndustres. The lower panel shows the results for the 4-ndustry dataset n whch 4 hghly concentrated ndustres are omtted. 7. Passage Tmes II The analyss can be extended from the study of crossng tmes of the form t to the study of general crossng tmes (.e. the ntal leader s overtaken by any frm). Here, the reducton of the problem to one n whch the dstrbuton of crossng tmes s determned by a sngle number ( normalzed gap ) as no longer possble. We need nstead to specfy the full vector of ntal market shares, and the volatlty parameter for the ndustry. Monte Carlo estmates were constructed n ths way, whch specfy the probablty of any crossng durng the nterval [, ], and the probabltes were summed, as before, over 9

31 ndustres to obtan an expected number of crossngs. The results are shown n the top panel of Fgure 6. Most crossngs are, n practce, made by the second largest frm n the ndustry, and the results shown n Fgure 6 are closely smlar to those shown n the correspondng (top) panel of Fgure 5. It was noted above that the assumpton of ndependence underlyng the null hypothess s nvald for the four most hghly concentrated ndustres n the dataset. Wth ths n mnd, the exercse was repeated usng the remanng 4 ndustres only. The results are shown n the lower panel of Fgure 6, and are closely smlar to those shown n the upper panel. (No crossngs occur n these 4 ndustres, and they all feature a large ntal gap n shares, and low volatlty, so that the expected number of crossngs under the null hypothess s close to zero). The overall concluson s that there appears to be a tendency for fewer crossngs (longer persstence of leadershp) than predcted by the benchmark model. Ths observaton rases an obvous queston: could ths tendency to be drven by some systematc (strategc) mechansm that operates across the general run of ndustres? Is there some Chandleran mechansm at work, for example, whch could be nterpreted by sayng that current annual market shares are not a suffcent statstc for the (superor) level of capablty employed by (leadng) frms? Or, on the other hand, does ths tendency merely represent the overall average behavour of a seres of ndustres, each drven by ts own dosyncratc features? To nvestgate these questons, we turn to some case studes of ndustres n the sample. 3

32 8. Dggng Deeper A central argument of the present paper s that any smple stochastc model can easly be bettered as a representaton of any one ndustry, by ncorporatng ndustry-specfc features whch wll nclude a strategc representaton of frms compettve responses to market share changes. Once we am at constructng a rcher model of ths knd, however, we meet the problem that strategc effects wll turn on varous features, some of whch are ntrnscally unobservable as far as the outsde economst s concerned. A more sophstcated model would retan exogenous shocks to underlyng technology and tastes parameters, but would extend frm s reactons beyond the prce-quantty adjustments allowed for above, to deal wth changes n marketng and/or R&D outlays amed at rasng (perceved) qualty, and wth the entry and ext of products. It s n respect of these latter adjustments that subtle dfferences appear across dfferent ndustres, whch seem to be drven by varous factors, some of whch are very dffcult to measure, proxy or control for n emprcal studes. Most mportantly n the present context, they nclude the belefs of agents as to ther rvals prvate nformaton, and strategc responses. In consderng a model that allows for strategc responses on the marketng or R&D sde, the key queston of nterest s: how do these strategc responses mpnge on the degree of volatlty of market shares, and on the evoluton of shares over tme? One obvous factor that mght mpnge on market share volatlty s the degree to whch exstng products are dsplaced by new products. We begn wth two ndustres n whch 3

33 the rate of product dsplacement was very hgh, but n whch market share patterns were very dfferent. In the cash regster ndustry, the technology changed contnuously and dramatcally over the 5-year perod, as free-standng electromagnetc regsters were frst replaced by electronc types, and as these electronc types were n turn dsplaced by store-wde or company-wde computer Year Cash Regsters 3

34 Year Margarne Year Beer Fgure 4(a). Market Shares 33

35 Year Photocopers Year Colour Flm Fgure 4(b). Market Shares 34

36 Year Pocket Calculators Fgure 4(c). Market Shares lnked networks. The market share pattern was extremely volatle, as successve frms ganed a relatve techncal advantage. (Fgure 4(a)) A contrastng pattern arses n the margarne ndustry. Here, the ndustry was characterzed (perhaps surprsngly) by a very rapd rate of ntroducton of new varetes (one manufacturer s 99 brochure contaned scores of varetes, whch dffered n form of packagng, choce of flavourng, hardness and texture, etc.). In spte of the hgh degree of new product ntroductons, market shares remaned remarkably stable, as each successful nnovaton by any frm was mmedately countered wth a response by rvals, who quckly mtated successful products. 35

37 One nterpretaton of the dfferent outcomes n these two ndustres les n the dfferent strategc responses of frms to rvals successes. Ths dfference mght, however, smply reflect dfferences n the ease wth whch nnovatons can be mtated by rvals. To explore ths latter dea, t s nterestng to look at the evoluton of market shares n the beer ndustry. Here, there are two perods of nterest. The late 97s was marked by what came to be known n the ndustry as the packagng wars. Frms ved wth each other n ntroducng new forms of packagng (bottles and cans of new szes; plastc contaners n odd and unusual shapes, and so on). Throughout ths perod, market shares remaned qute stable. The second perod s the 98s, when a beer was marketed by the Asah company, then the ndustry s fourth largest frm, under the name Asah Dry. Despte ts ntal success, rvals were slow to respond, and Asah Dry propelled the Asah company to second place n the ndustry. (The market leader Krn eventually mtated ths strategy by marketng ts Krn Dry product, whose sales remaned below those of Asah Dry over the next decade). The queston rased by ths s: f we constructed a fully specfed, strategc model of the Japanese beer market, what varables accessble to the researcher could have predcted the non-mpact of the packagng wars, as aganst the substantal mpact of the Dry beers marketng campagn? It would seem that the speed and effectveness of rvals responses dffered n the two cases because of dfferent belefs on the part of rvals as to the probable effectveness of the nnovator s strategy. What ths suggests s that, just as the lterature on dynamc olgopoly suggests, the sze of the market share response, and so the level of market share volatlty n the ndustry, wll depend nter ala on the belefs of agents a factor that we must perforce treat as an unobservable n most settngs. 36

38 What I want to suggest, then, s that whle dfferences n frms strategc responses to rvals actons may be one of the factors that account for the dfferent patterns of evoluton of shares n dfferent ndustres, these dfferences n strategc responses wll depend delcately on factors that are dffcult to account for on the bass of stable and observable ndustry characterstcs. To llustrate ths pont, t s of nterest to consder two ndustres whch dsplay qute dfferent patterns of market share dynamcs at dfferent perods. The photocoper ndustry, for example, offers an example n whch a techncally nnovatve follower (the thrd frm n 974) gradually overtakes the leader; but the ndustry then moves to a stable settng for 5 years, wth the nnovator sharng frst place wth the orgnal market leader. (Fgure 4(b)). In the colour flm market, there are agan two phases; over the decade, the market leader s share rses at the expense of ts nearest rval, but then shares stablze and show lttle volatlty over the next 5 years. (Fgure 4(c)). In contrast to these two cases, consder the market for pocket calculators, an ndustry that was marked, lke the photocopers and colour flm ndustres, by a fast rate of techncal nnovaton and new product ntroductons. (Fgure (4(c)). Here, the top two frms escalated ther nnovatory actvtes and both ther shares ncreased steadly, and n step, as the shares of all the smaller frms declned. It s easy to specfy a sutable game theoretc model whch has these features (see for example, Sutton (99), Chapter 5); but t s not clear what observable ndustry characterstcs for 974 could have predcted that ths ndustry s market share pattern would have dffered n ths way from those seen n photocopers or colour flm beyond attrbutng t to dfferent stochastc realzatons of outcomes to the dfferent frms early nnovatory efforts, and argung that the ntal accdental successes nduced dfferent strategc choces thereafter. 37

39 These examples, taken together, suggest a serous caveat regardng the tradtonal persstence of leadershp debate. That debate has been conducted on the premse that there mght be some general mechansm(s), ether of a Schumpeteran or Chandleran knd, that operate(s) across the general run of ndustres. What these examples suggest s that there are many mechansms, some operatng n one drecton, others n another drecton, so that when we test for some bas n ether drecton we are (at best) assessng some knd of average outcome that wll be hghly senstve to our selecton of ndustres. It s n ths (rather cautous) sprt that any conclusons as to a possble Chandleran bas n the present set of Japanese ndustres should be drawn. 9. Conclusons Ths paper makes three ponts. The frst relates to the lmtatons of game theoretc (strategc) models. The second relates to the use of scalng relatonshps for the varance of frm growth rates, and market shares. The thrd relates to the persstence of leadershp debate. a. Game Theoretc Models I have argued elsewhere (Sutton (99, 998)) that game theoretc models can lead us to a small number of robust predctons, whch allow us to place lmted restrctons (bounds) on market structure. Beyond these few robust results, however, outcomes wll depend delcately on factors that are dffcult to measure, proxy or control for n cross- 38

40 ndustry studes. In ths settng, t can be of nterest to examne low-level representatons of the data, of the knd attempted here. b. Scalng Relatonshps The recent lterature regardng scalng relatonshps on frm growth rates, has focussed on the descrpton of relatonshps, and on dfferences n vews as to canddate explanatons for such relatonshps (Stanley et al. (996), Sutton ()). Lttle attenton has been pad to the queston of whether the characterzaton of such relatonshps s emprcally useful. In ths paper, I have argued that the characterzaton of a smple scalng relatonshp between a frm s market share and the varance of changes n market share, permts a useful smplfcaton n the descrpton of market share dynamcs, allowng the (lmted) crossng-tme problem addressed here to be reduced to the study of a smple random walk. Ths scalng relatonshp also provdes a useful crtera for model selecton n the area of market share dynamcs, as t s a feature of only one of the several standard models n the current economcs lterature. 6 c. The persstence of leadershp problem The clam of ths paper n regard to the persstence of leadershp debate s a modest one. We explore the propertes of a benchmark model n whch there s no systematc bas, 6 It s worth notng, however, that t s also consstent wth certan models, popular n the marketng lterature, n whch ndvdual consumers are attached to frms over successve perods, but where each consumer has a (small) probablty of shftng allegance to a new frm n any perod. These models are purely statstcal n nature, and do not rest on optmsng by agents. For a model of ths type based on maxmzng behavour by frms and consumers, see Sutton (98). 39