Sales Distribution of Consumer Electronics

Sales Distribution of Consumer Electronics Ry ohei Hisano a ∗ , T ak a yuk i Mizu no b a Gr aduate Scho ol of Ec onomics, Hitotsub ash i University, 2-1 Naka, Kunitachi City, T okyo 186-8603, Jap an b Institute of Ec on omic R ese ar ch, Hitotsub ashi University, 2-1 Naka, Kunitachi City, T okyo 186-8603, Jap an (Dated: August, 30 2010) Abstract Using the uniform most p ow erful un biased test, w e observ ed the sales distr ibution of consumer electronics in Japan on a daily b asis and rep ort th at it follo ws b oth a lognormal distr ib ution and a p ow er-la w d istribution and dep ends on the state of the market. W e sho w that these switc hes o ccur quite often. The u nderlying sales dy n amics foun d b et we en b oth p erio d s nicely matc hed a m ultiplicativ e pro cess. Ho wev er, ev en though the m ultiplicativ e term in the p ro cess d ispla ys a size- dep end en t relationship when a steady lognormal distribution holds, it s ho ws a siz e-indep endent relationship when th e p ow er-la w distribution holds. Th is d ifference in the un derlying dynamics is resp onsib le for the difference in the t wo observ ed distributions. Keyw ords: pow er-la w distribution, lognormal distribution, m ultiplicativ e pro cess, sales distrib u- tion, sales dynamics P ACS num b ers : 89.65.Gh; 05.40.-a ; 02 .50.-r ; 02 .50.Ng ∗ Corresp o nding author: Ryohei Hisano Address: 2-1, Nak a, Kunitachi C ity , T okyo 186-8 603, Ja pan T el: +81-42-58 0 -835 7 F ax: +8 1-42 -580- 8357 E-mail a ddress: em07201 0 @yaho o .co.jp 1 I. INTR ODUCTIO N Since P areto p oin ted out in 1896 that the distribution of income exhibits a hea vy-tailed structure [1 ], ma ny pap ers has argued that suc h distributions can b e found in a wide range of empirical data that describ e not o nly economic phenomena but also biological, ph ysical, ecological, sociolog ical, and v arious man-made phenomena [2]. The list of the measuremen t s of quan tities whose distributions hav e b een conjectured to ob ey suc h distributions includes firm sizes [3], cit y p o pula t io ns [4], frequency of unique w ords in a giv en no v el [5-6 ], the biological genera b y num b er of sp ecies [7], scien tists by n umber of published pap ers [8], w eb files transmitted ov er the in ternet [9], bo ok sales [10], and pr o duct marke t shares [11]. Alo ng with these rep orts the argumen t ov er the exact distribution, whether these heavy -tailed distributions ob ey a lognormal distribution or a p ow er-la w distribution, has b een rep eated o v er man y y ears as w ell [2]. In this pap er w e use the statistical tech niques dev elop ed in this literature to clarify the sales distribution of consumer electronics. T o illustrate the heavy -tailed distribution’s app earance, r a ndom growth pro cesses are widely used as the appro ximation of it s underlying dy namics. Gibrat, who built up on Kapteyn and Uv en’s w ork, w as the first to prop ose the simplest form of this t yp e of mo del, usually know n as the m ultiplicativ e pro cess, to describ e the app earance of heavy -tailed distributions in firm size distributions [12]. His work is significan t in mark et structure literature [13]. Ev en 70 y ears after G ibrat’s b o ok, more and more me asures of quantities are found that a r e conjectured to ob ey this type of pro cess. Among rece n t w orks, F u et al. [14] is crucial b ecause it w as p erhaps the first w ork to consider the hierarc hical structure of institutions. No one den ies that firms gro w in size a nd scop e and that suc h grow th is heav ily influenced by the succe ssful launc h of new pro ducts. F u et al. mo deled pro ducts as eleme n tary sales units assuming tha t they ev olve based on a random multiplicativ e pro cess. They ex tended the usual model of prop ortional gro wth to illustrate the size v ariance relationship f ound in g r o wth distributions at differen t lev els of aggregation in the econom y by considering hierarc hical structure. Man y studies hav e analyze d pro duct sales. Sornette et al. [10] used a b o o k sales databa se from Amazon.com and perfo rmed a time series analysis of b o ok sales by classifying endoge- nous and exogenous sho c ks. With a database of newspap er and magazine circulation, Picolli et al. [15] used the m ultiplicativ e pro cess to illustrate the link b et w een ten t-shap ed log 2 gro wth distributions and the p ow er-law distributions found in the gro wth of newspap er and magazine sales. Ho we v er the exact dynamics o f pro duct sales remains an op en question. In this pap er w e clarify the distribution and the dynamics of the sales of consumer elec- tronics using a unique database of pro duct sales from the Japanese consumer electronics mark et. T he data w ere recorded daily , making it p ossible to t r a c k the actual sales v olume of eac h pro duct in a more detailed manner and to mo del the dynamics from a more empirically driv en approa ch. W e n umerically analyzed more than 1200 sales distributions recorded on a daily basis from Octob er 1 2004 to F ebruary 29 200 8. Using the uniform most p ow erful test, w e statistically sho w that sale distributions differ among differen t p erio ds and o ccasionally exhibit a p ow er-law behavior. W e also sho w with the m ultiplicativ e pro cess that the under- lying ingredien ts o f sto c hastic growth itself are differen t among these p erio ds. Moreo ver our findings are compatible with the mathematical results rep orted b y Ishik aw a et al. [16]. The pap er is org a nized as follo ws. Section 2 pro vides an ov erview of our data set. Se c- tion 3 in tro duces the sales distribution of consumer electronics, and Section 4 illustrates our statistical tec hnique regarding the v erification of a true p ow er-law distribution. Using this statistical tec hnique, in Section 5, w e show wh y the p o w er-law b eha vior found in Section 3 can b e considered a gen uine pow er-law b ehav ior. Section 6 reports ho w sales distribution c ha nges ov er time. Sales distribution exhibits b oth p ow er-law and lognormal distributions. Section 7 fo cuses on the underlying dynamics of sales, providing ano ther source o f evidence that the dynamics of sales differs among differen t p erio ds. Section 8 pro vides further dis- cussion and a conclusion. I I. SALES D A T A OF CO NSUMER ELE CTR ONIC S Consumer electronics c ha ins sell pro ducts suc h as TVs, p ersonal computers, a udio devices, refrigerators, digital cameras, air conditioners, and DVD recorders. Their ann ual rev enue amoun ts to 5.9 trillion yen in Japan. In this pap er we in ves tigate distribution using the sales data of digital cameras from 23 different consumer electronics c hains in Japan collected by a priv ate compan y called BCN Inc. This dataset co ve rs ab out 45% of all consumer electronics c ha ins in Japan including o v er 1,400 retail stores [17]. The data w ere recorded daily cov ering the perio d from October 1 2004 to April 30 2008. 3 I I I. SALES DISTRIBUTIO N OF C ONSUMER E LECTRONICS W e fo cused on the top selling pro ducts using cum ulativ e distribution P > ( S ) defined as P > ( x ) := P r [ X ≥ x ] = Z ∞ x f ( x ′ ) dx ′ (1) where f ( x ) des crib es the probability densit y func tion. The cum ulativ e distribution of the sales volume of digita l cameras on April 1 2005 is show n on a double logarithmic scale (Fig. 1). It exhibits a heav y-tailed structure. T o in v estigate the exact characteristic of its distribution, w e also depict the maxim um lik eliho o d estimate of a logno rmal distribu- tion, assum ing that all v alues a b ov e 1 o b ey ed a lognormal distribution and the maxim um lik eliho o d estimate of a pow er-law distribution, ass uming that all v alues ab o v e 16 ob ey ed a p ow er-law distribution. A lognormal distribution fits nicely for almost a ll p oints except the last three. F or the p oin ts ab ov e 16 including these last three p oints, at first glance it seems that a p ow er-law distribution fits b etter. In this pap er w e num erically j udge whether a simple lognorma l distribution or a lognormal distribution with a p o w er-law t a il displa ys a b etter fit using the statistical tec hnique dev elop ed b y Malev ergne et al. [1 8]. The imp or- tance of distinguishing betw een these tw o distributions lies in the fact that not only does the tail describ e the top selling pro ducts but these pro ducts which seems to exist in the p ow er-la w region a ccoun t f o r a b out 80% of total sales ; identifying the dynamics of these top selling pro ducts is imp ortant. IV. TESTING A PO WER-LA W DISTRIBUTIO N A T T HE T AIL T o judge whether a p ow er-la w distribution o r a lognormal distribution displa ys a b etter fit for v a lues o v er a threshold, one natural w ay is using a mo del selection tec hnique betw een a p ow er-law distribution and a singly truncated lognormal distribution t ha t puts the trun- cation p oin t iden tically as the lo w er b ound o f a pow er-law distribution (for instance , see Clauset et al. [19]). The basic c hange of the v ariables sho ws that a loga rithm of a random v ariable, whic h obeys a pow er-law distribution, is an exp onen tial distribution, but a loga- rithm of a singly tr uncated lognormal distribution is a singly truncated normal distribution. Hence the test of a p o w er law against a singly truncated lognormal is equiv alent to testing an exp onential distribution against a singly truncated normal distribution in the log-size distribution of the orig inal measure of quan tit y . 4 Next, as show n b y Castillo [20], an exp onen tial distribution and a singly t r uncated normal distribution ha v e the follo wing relat io nship: f S T N ( x ; α, β , A ) → f exp ( x ; λ )1 x > A as ( α, β ) → ( λ, 0) , (2) where A denotes the truncation p oint and α and β are the parameters of a singly truncated normal distribution with the follow ing relationship: α := − µ − A σ 2 β := 1 2 σ 2 (3) where µ is t he usual mean, σ is the standard dev iation. This implie s that an exp onential distribution is in the b oundary line of a singly truncated norma l distribution. This rela- tionship illustrates wh y a singly truncated normal distribution (singly truncated lognormal distribution) so closely resem bles an exp onen tial distribution (p ow er-law distribution) if β b ecomes incresaingly close t o 0. Fig. 2 show s the maxim um lik eliho o d es timate assum ing an exp o nen t ial distribution and a singly t r uncated no rmal distribution for the log-size distri- bution of digit a l camera sales on April 1 20 0 5, setting the truncation p oint as A = log (16). Observ e from the maxim um lik eliho o d estimate that a singly truncated normal distribution with suffic ien tly small closely resem bles an expo nen t ial distribution. Considering this relationship, a natural test to distinguis h an exp o nen tial distribution from a singly truncated normal alternat ive is to test the departure from the exp onen tial form (n ull hypothesis) against the singly truncated normal alternativ e (alt ernativ e h yp othesis) using the like liho o d ratio test that ev aluates statistic W = 2 ( L ( ˆ θ ) − L ( ˜ θ )) (4) where L denotes log-likelihoo d function ˆ θ = ( ˆ α, ˆ β ), whic h is the maxim um likelih o o d estimate under the full mo del, and ˜ θ = ( ˜ λ, 0) = ( 1 ¯ x , 0) in its expo nen t ia l form. Castillo and Puig [21] sho w ed the following: 1) the lik eliho o d ra t io t est is the unifor m most p ow erful un bia sed t est in this case; 2) the lik eliho o d ratio test could easily b e p erformed using the clipp ed co efficien t of v ariation (i.e. c = min { 1 , ¯ c } , where ¯ c is the empirical co efficien t of v ariation); and 3) the c ritical region of the tes t could be appro ximated with a high degree of accuracy e v en for small samples using saddle p oint appro ximation. Malev ergne et al. [18], who discussed whether a lognormal suffices or a p ow er-la w distribution sh ow s a better fit for the upp er 5 tail o f the size distribution of US cit y size dat a , concluded tha t the upp er tail of the size distribution of US cities is in fa ct a p ow er la w. Figure 3 sho ws the test using the sales distribution found in April 1 2005. Starting from the top 10 pro ducts w e recursiv ely calculated the p-v alue of the test using Castillo and Puig’s metho d. W e then calculated the p oin t where the p-v alue of the test first falls within the critical region (in this pap er, α = 0 . 1) min us 1. F or the sales distribution found in April 1 2005 this point is 68. This implie s that for the 68 p oin ts ab ov e this threshold the p o w er-law distribution is not rejected and sho ws that the upp er ta il of the distribution of sales found in April 1 2005 w a s w ell fit b y a p o w er-law distribution. V. DISTRIBUTION ANAL YSI S OF SALES In a small sample data set, we often observ e ”p ow er-law b ehavior” (straigh t line in the cum ulative distribution depicted on a double logarithmic scale) ev en if it we re actually sampled from a theoretical lognorma l distribution. Fig. 4 sho ws tw o cases t ha t plot the cum ulative distribution o f syn t hetic dat a sets randomly sampled from a theoretical lo g normal distribution with the same parameters a s the sales distribution o f digita l cameras on April 1 2005 (Fig. 1). In the one case depicted in t he left panel, note that the tail follo ws a lognormal distribution. Ho w eve r, in the other, ev en if w e used the statistical t ec hnique explained in Section 4, the lo wer b ound estimated f rom the pro cedure returns a v alue of 77 for the distribution denoted in circles, confirming a p ow er-law behavior at the tail. Hence to judge whether distributions found in a ce rtain perio d are w ell desc rib ed b y a p ow er-la w distribution w e m ust consider all the distributions found during that p erio d. The left panel of Fig. 5 sho ws the estimated low er b ound f rom the 107 dates during Jan uary 22 2005 to May 8 2005 and the righ t panel sho ws the estimated lo w er bound of the first 107 syn thetic data sets rando mly sampling from a theoretical log normal distribution w ith the same parameter as the sales distribution found on April 1 2005. The low er b ound from the real data is cle arly quite stable, which prov es that the p ow er-law b eha vior found in the sales distribution of April 1 2005 reflected a generating pro cess that pro duces a gen uine p ow er-law b eha vior and not the result of a process that generates a logno r ma l distribution. Fig. 6 also sho ws their probability densit y , confirming that the b ehaviors found in Fig. 5 ar e also quite statistically differen t. Fig. 7 sho ws the time ev olution of the p o w er-law e xp onent during 6 Jan uary 22 20 0 5 to Ma y 8 2005. The p ow er-law exp o nen t is stable and fl uctuates ar o und v alue µ = 1 . 3 ± 0 . 1 whic h is quite close to the p ow er la w exp o nen t found for city size [18] and we alth [22]. The p erio d when the p o w er-la w behavior b ecomes stable w a s rep eatedly found and sho ws that the lognormal distribution do es not sufficien tly desc rib e the sales distribution of digital cameras on a da ily ba sis. VI. HO W DISTRIBUTION CHANGES O VER TIME Next w e fo cus on all the other dat es in o ur data set. Fig. 8 sho ws their estimated rank thresholds from Octob er 1 2004 to F ebruary 29 2008. The p erio d at whic h w e successiv ely observ ed high estimated rank thresholds is not only Jan uary 22 2005 to Ma y 8 2005 but is a lso found in other par t s o f the data. Ho w ev er, there is a perio d when the estimated rank threshold do es not behav e as if a p ow er-la w b eha vior really exists at its distribution tail: p erio d January 16 2006 to August 8 2006 (Fig . 8). Fig. 9 shows a t ypical cum ulative distribution observ ed during this p erio d. The lognormal distribution a dequately explains the sales distribution for all p oints. Fig. 10 also show s the histogram of the estimated rank thres hold of this p erio d. F or this perio d a simple lognormal distribution adequately describes the distribution of sales . Therefore w e conclude that sales distribution reflects when they we re observ ed. VI I. SALES D YNAMIC S O F CONSUMER ELECTRONICS It is w ell kno wn that prop ortional growth principle applies to firm gro wth [23]. F u et al sho w ed that not o nly do es this principle apply to firms, but it also applies to differen t aggre- gation of t he econom y f rom coun t ries, industry sector, and to pro ducts sho wing theoretically that this stems from its elemen tary sale unit (i.e. sales of pro ducts) ev olving accro ding to a random m ultiplicativ e pro cess [14,24]. Sa k a i and W atanab e inv estigated further this issue confirming that dynamics of pro ducts determines firms growth b y analyzing sales of pro ducts sold at g ro cery stores in Japan [25]. Picoli et al used the m ultiplicativ e pro cess to mo del the dynamics of circulation of newspap ers and magazines [15]. Motiv ated by these literature w e w o uld use S ( t + 1) = | b ( t ) S ( t ) + ǫ ( t ) | ǫ ( t ) ∼ Gaussian (0 , σ ) (5) 7 to describ e the underlying dynamics of sales. This assumes a preferential lik e model for sales whic h requires age and av erage sales of pro ducts to correlate in a exp onen tial fashion if the distribution of lifetime is exponential. It is rep o rted that the lifetime distribution often follow s exp o nen t ia l functions in comp etitiv e mar kets [26]. Although this relatio nship could not b e easily v erified directly with pr o ducts itself b ecause lifetime o f digital camera is short (usually 6 to 12 mon th) due to pro duct turno ve r, this could b e roughly v erified when w e observ e the a ve rage daily mark et share of brands during there lifetime with their a ge (fig.11). If m ultiplicativ e noise b is indep endent of the for mer size of S , then S leads to a steady p ow er-la w distribution [27]. Ho wev er if it is size dependent, S departs fr o m a p ow er-la w distribution [28]. In this section, w e sho w that sales dynamics follo w this m ultiplicativ e pro cess and use it to reexamine the differences in the sales distribution found in Section 6 from the usually assumed elemen ts o f a sto c hastic grow th pro cess. When BCN Inc. collected this data set, they made new con tracts with other stores to offer sales data and g enerated an apparen t artificial c ha ng e of pro duct sales along time. T o cope with this problem, w e in tro duce normalized sales, ¯ S i ( t ) = S i ( t ) / 1 n P n i =1 S i ( t ), in- stead of actual sales to compare t w o distan t p erio ds. Here, n is num b er of pro ducts in the mark et. All t he results in this section can also b e repro duced using mar ket share, ˆ S i ( t ) = S i ( t ) / P n i =1 S i ( t ), as we ll. T o begin our empirical inv estigation w e cut the scatter plots of b oth p erio ds into equal logarithmic bins: 0 . 7 ≤ ¯ S low < 2 . 1 7 5 ≤ ¯ S mid < 6 . 7 6 ≤ ¯ S hig h < 21 (Fig. 12). Th e basic idea is to observ e whether t he dis tribution of sales gro wth f o r one w eek, ¯ S i ( t +1) ¯ S i ( t ) , dep ends on ¯ S i ( t ). W e sa w in Section 6 that the tail prop ert y of the sales distribution follows a p o w er la w for Jan uary 22 2005 to Ma y 8 20 0 5 and a lognormal for January 16 2006 t o August 8 200 6 (Fig. 8). Hence, as shown in Fig . 13, w e compare the distribution of log sales gr o wth for 1 w eek, log ¯ S i ( t +1) ¯ S i ( t ) , observ ed during p erio ds Jan uary 22 20 05 to Ma y 8 200 5 and Jan uary 16 2006 to August 8 2006. Note that while the p ositiv e v alues of the middle and high areas are quite differen t during Jan uary 16 20 06 to August 8 20 0 6, they seem to coincide for the lo g gro wth distributions observ ed during January 22 2005 to M ay 8 2 005. Note also that log gro wth distribution could be w ell de scrib ed as a double exp onen tial distribution a nd that for the negativ e log growth rates, the probabilit y densit y coincides. This suggests that while the m ultiplicativ e t erm for the high and middle areas is size indep enden t during Jan uary 8 22 2005 to Ma y 8 2 005, it is size dep enden t during Jan uary 16 2006 to August 8 2006 and sho ws differen t b eha viors. The same observ ation can also b e made using the t wo sample Kormogorv- Smirnov tests. T able 1 sho ws the p-v alue from the test for tw o pairs, ”high vs middle” and ”middle vs lo w”, for t w o p erio ds, Jan uary 22 2005 to Ma y 8 20 05 and Jan uary 16 2006 t o August 8, resp ectiv ely . The only pair for whic h the test do es not reject the n ull h yp othesis is the ”high and middle” pair found in Jan uar y 22 2005 to May 8 2005. F rom these observ ations, the sales dynamics can b e described by the m ultiplicativ e pro- cess: ¯ S ( t + 1 ) = | b ( ¯ S ( t )) ¯ S ( t ) + ǫ ( t ) | b ( ¯ S ( t )) =    b low ( ¯ S ( t )) if ¯ S ( t ) < 2 . 175 b mid ( ¯ S ( t )) = b hig h ( ¯ S ( t )) if ¯ S ( t ) ≥ 2 . 175 (6) where t describ es the time during Jan uary 22 2005 to Ma y 8 2 005 and ¯ S ( t + 1) = | b ( ¯ S ( t )) ¯ S ( t ) + ǫ ( t ) | b ( ¯ S ( t )) =          b low ( ¯ S ( t )) if ¯ S ( t ) < 2 . 175 b mid ( ¯ S ( t )) if 2 . 175 ≤ ¯ S ( t ) < 6 . 76 b hig h ( ¯ S ( t )) if 6 . 76 ≤ ¯ S ( t ) (7) where t des crib es the time during Jan uary 16 2006 to August 8 2006. Here, ǫ ( t ) ∼ Gaussian (0 , σ ) . Where ¯ S i ( t ) is la r g e, t he log gr o wth distribution displa ys a size-indep enden t relationship with ¯ S i ( t ) during January 22 2005 to May 8 2005, but it displa ys a size- dep enden t relationship with ¯ S i ( t ) during Jan uary 16 2006 to August 8 2006. As sho wn b y Ishik aw a et al. [16], if t he log growth distribution is w ell described by a double ex p o- nen tia l distribution and the probabilit y densit y coincides for negative log growth rates but exhibits a size dep enden t relationship for positiv e v alues, then the multiplicativ e pro cess de- scrib ed in Eq. (7 ) will generate a steady lognormal distribution. Recall that during p erio d Jan uary 16 2006 to August 8 2006 this condition is satisfied. On the other hand, as sho wn b y T ak a y asu et al. [27], Eq. (5) theoretically generates a p ow er-la w distribution when the gro wth distribution is indep enden t of ¯ S i ( t ). Therefore while Eq. (6) g enerates a distribution with a p o w er-law tail, Eq. (7) generates a simple lognormal distribution that explains the difference in the underlying dynamics for the tw o perio ds in whic h w e observ ed differe n t distributions. 9 VI I I. CONCLUSION AND FUR T HER DISC USSIONS This pap er sho w ed how the sales distributions o f pro ducts ev olv es when w e observ ed them daily . W e show ed that the distribution of the top ranking pro ducts switc hes b et w een lognormal and p ow er-la w distributions dep ending on the timing, suggesting that the un- derlying dynamics among these p erio ds differs. This structural difference in t he underlying dynamics was well established from the usually assumed ingredien t s o f the growth pro cess as w ell providing ano t her source o f evidence that the dynamics b et w een these t w o p erio ds differ. Our r esult is mathematically compatible with Ishik a w a et al. [16], who illustrated the app earance o f b ot h pow er-law and lognormal distributions under a m ultiplicativ e pro cess. W e only in v estigated digital cameras in this pap er; ho w ev er suc h findings can b e established with man y other pro ducts in consumer electronics markets as w ell. An interesting question to p onder is wh y the switc h b eha vior found in Section 6 o ccurred. In our case the main source of the switc h can probably b e explained b y pro duct turno v er. In pro duct markets suc h as the digital camera market product life cycle is short taking only ab out 6 to 12 mon th for a particular brand to c hange from a n old pro duct to new one due to pro duct comp etition. Those pro duct turno v er usually ta kes place on F ebruary and August b efor e the agg regate demand fo r digital cameras starts to rise. F ig. 14 s how s the time evolution of t he n um b er of pro ducts and the low er b ound. As denoted in fig. 14, the p erio ds coincide when w e observ ed a steady p o w er-la w behavior and a rapid increase in the n um b er o f pro ducts (i.e. when rapid pro duct turnov er take place), explaining the switc h b eha vior from a lo gnormal to a p ow er la w. When a lar ge n um b er of new pro ducts ar e b orn sim ultaneously , sales distributions are generated b y a mixture of old and new pro ducts making sales dynamics to b e more accurately described as Gibrat’s la w (i.e. size indep enden t gro wth rate). An empirical study taking these pro duct turno v er effect is future w ork. Since researc hers are equipped with more detailed data from actual mark ets w e can in v estigate actual mark et co ordination in a more detailed sense. These studies are imp ortant not only for economics literature but also for phys ics b ecause suc h social sys tems as t he mark et are one natural la b orato r y for inv estigating co ordination under complex sy stems. W e hop e this line of studies con tinues to b e fruitful for bo th ph ysics and economics. 10 Ac knowledgmen ts The authors w an t t o thank D. So rnette, A. Ishik aw a, and S. F ujimoto for their helpful suggestions concerning this w ork. Without them this w ork w ould nev er ha v e reached its curren t lev el. W e w o uld also lik e to thank BCN Inc. for prov iding its data. Man y thanks go es to the tw o anon ymous r eferees who ga ve us helpful commen ts as w ell. This researc h is a part of a pro ject en titled: Understanding Inflatio n D ynamics of the Japanese Econom y , funded by JSPS Grant-in-Aid for Creativ e Scien tific Researc h (18 GS0101). T ak a yuki Mizuno w a s supp orted b y funding from the Kamp o F oundation 2009. [1] P areto,V., ”Cours D’Economie P olitique”, Rouge, Lausanne et P aris, (1896). [2] Mitzenmac her , M., ”A Brief History of Generativ e Mo dels for Po wer La w and Lognormal Distributions”, Inte rnet Mat hematics, V ol. 1, No. 2, 226-2 51, (2004). [3] Gibrat, R., ”Les In egalites Economiques”, L ib rairie du Recueil Sirey , P aris, (1931). [4] Auerbac h, F., ”Das Gesetz d er Bv olk eru ngsk onzen tra yion. Pe termann’s Geo graphiche Mit- terilungen”, 59, 73-76. , (1913) . [5] Estoup. 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[27] T ak a ya su, H., S ato, H., and T ak a y asu, M., ”Stable Infinite V ariance Fluctuatio ns in Ran- domly Amplified Langevine Sys tems”, P h ysical Review Letters, V ol. 79, No. 6, (1997) . [28] Mizuno, T., T ak a yasu, M., and T ak a yasu, H, ”The mean-field app ro ximation mo del of compan y’s income growth”, Ph ys ica A, 332 , 403 -411, (200 4). 13 FIG. 1: Cum ulativ e d istribution of sales v olume of d igital cameras sold on April 1 200 5. Slashed line sho ws fitted maxim um lik eliho o d estimate assumin g all p oin ts ab o v e 1 ob ey ed a lognormal distribution, and co nti nuous line sho ws fitted maxim um lik eliho o d estimate assuming all p oin ts ab o v e 16 obeyed a p o w er-la w distribution. P arameters of b oth distributions are depicted as well. 14 FIG. 2: Log-size d istribution of sales distribution found on April 1 2005 for v alues o v er A = l og (16). Maxim u m lik eliho o d estimates of b oth exp onenti al and singly truncated normal distributions are depicted along with their p arameters. 15 FIG. 3: Right panel d epicts p-v alue of test of n ull h yp othesis w here distribu tion’s up p er tail is p ow er against alternativ e singly truncated lognormal distribution as a fu n ction of rank threshold. 16 FIG. 4: Two examples of rand omly sampling from a lognormal d istribution with parameters µ = 1 . 97 , σ = 1 . 36. Th ere are 250 p oin ts in b oth distributions. Note the p o we r-la w behavior at the distribution’s tail denoted by circles. Estimated lo wer b ound for crosses is 9 and 77 for circles. 17 FIG. 5: Left panel sh o ws esti mated rank threshold from distribution of sales v olume from January 22 2005 to Ma y 9 2005. Right panel sh ows estimated rank th reshold for first 107 synthetic data sets ( LN (1) , LN (2) , ..., LN (107)) obtained in exp erimen t. Estimated rank threshold o f 7t h and 22nd data sets denoted as A and B are used to depict Fig. 4. 18 FIG. 6: Histogram comparison of estimate d r ank threshold. Circles d enote h istogram fr om syn - thetic data sets, and diamonds d enote histogram from real data. 19 FIG. 7: Time evolutio n of pow er-la w exp onent found du ring perio d Jan u ary 22 200 5 to Ma y 8 2005. 20 FIG. 8: T im e ev olution of estimated rank threshold for en tire p erio d (Octob er 1 2004 to F ebruary 29 2008) . 21 FIG. 9: Cumulativ e d istribution of s ales v olume of digital cameras sold on Marc h 27 2006. Con- tin uous line sho ws fitted maximum lik eliho o d estima te assuming that all p oints ab o ve 1 ob ey ed a lognormal distribution. Maxim um lik eliho o d estimate of parameters is written inside parentheses. 22 FIG. 10: Histogram o f estimated r ank thresh old fr om real data (January 16 20 06 to Au gust 22 2006) . 23 FIG. 11: Averag e daily mark et share of brands v ersus its age in semilog sc ale. Con tin uous line sho ws the least squares fi t for the exp onential h yp othesis. 24 FIG. 12: Cutting scatter p lots int o equ al logarithmic b ins. Left p anel depicts sales of successiv e w eeks from p erio d January 22 2005 to Ma y 8 2005, and right panel depicts sales of su ccessiv e wee ks from p erio d Jan uary 16 2006 to August 8 2006. 25 FIG. 13: Left panel shows distribu tion of log gro wth for one w eek for p erio d Jan u ary 22 2005 to Ma y 8 2005 . Right panel sh o ws distribution for January 16 2006 to August 8 2006. Although righ t panel clea rly seems size dep endent , b et w een the middle and h igh areas, the left seems to coincide. 26 FIG. 14 : Estimated rank threshold for all dates w ith time evo lution of n umber of prod ucts on mark et. 27 KS Test Jan 22 – May 8 2005 Jan 16 – Aug 8 2006 Low v s Mid 2.10E-13 2.58E-14 Mid vs High 0.324 1.00E-05 T ABLE I : Result of t wo sample Kormogorv-Smirno v test. Numb ers inside sho w p -v alues. Ro ws represent s pairs and columns denote p erio d. Null h yp othesis states that tw o distributions are iden tical, and the alternative states that they are different. 28