Never-buyers of consumer non-durables

Fi e , C. A icle Ne e -buye s o consume non-du ables Sou h A ican Jou nal o Business Managemen P o ided in Coope a ion wi h: Uni e si y o S ellenbosch Business School (USB), Bell ille, Sou h A ica Sugges ed Ci a ion: Fi e , C. (1984) : Ne e -buye s o consume non-du ables, Sou h A ican Jou nal o Business Managemen , ISSN 2078-5976, A ican Online Scien i ic In o ma ion Sys ems (AOSIS), Cape Town, Vol. 15, Iss. 3, pp. 155-161, h ps://doi.o g/10.4102/sajbm. 15i3.1121 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217866 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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The adi ional Nega i e Binomial Dis ibu- ion app oach o Eh enbe g o he ques ion is p esen ed. P e ious- ly unpublished wo k ca ied ou a he G adua e School o Business Adminis a ion, Uni e si y o he Wi wa e s and, Is e iewed and hypo heses a e pu o wa d ha he obse ed la ge ze o cell in he pu chase equency dis ibu ions may be caused by he exis ence o a g oup o ne e -buye s o he p oduc , o by he supe imposi ion o a leas wo dis inc buying popula ions, p e iously iden i ied as b and-loyal and mul ib and/b and-swi chlng households. The esul s o he esea ch aimed a es ing he i s hypo hesis a e p esen ed he e. Two ca e ully moni o ed da a se s we e modelled using ze o-augmen ed Nega i e Binomial and Sichel dis ibu ions. The da a we e p e iously shown o exhibi he necessa y mean households pu chase/consump ion s a iona i y. In- di idual b ands in one da a se (pu chases o oile soap) we e shown o ollow he p edic ions o he adi ional heo y - he p opo ion o non-buye s dec easing wi h ime. In he second da a se (consump ion o packaged soup) he p opo ion o non- consume s o he b ands ell owa ds ze o as he leng h o he ime pe iod s udied was inc eased, bu a a a e as e han ha p edic ed by he heo y. The hypo hesis o he exis ence o ne e - buye s/use s o indi idual b ands in hese wo p oduc classes was he e o e ejec ed. S. A . J. Bus. Mgm . 1984, 15: 155- 161 In die a ikel wo d die konsep an nooi -kope s an nle-duu same e b uike sgoede e besp eek. Eh enbe gse adisionele Nega ie · binomiaaldis ibusie-benade ing in e band me die aag is oo ges el. Vo ige ongepublisee de we k ui ge oe by die Nag aadsebes uu skool, Uni e si ei an die Wi wa e s and is besp eek, en hipo eses wo d oo ges el da die waa genome g oo nul-sel in die e koops- ekwensiedis ibusies miskien 'n ge olg mag wees an die bes aan an 'n g oep nooi -kope s an die p o- duk, o as 'n ge olg an die opeenplasing an en mins e wee a - sonde like kope populasies oo heen geiden i isee as handels- naam-lojaal en handelsnaam-wlsselende gesinne. Die esul a e an die na o sing wa gedoen is om die ee s e hipo ese e oe s, wo d gegee. Twee so g uldig gekon olee de s elsels da a is gemodellee me geb uik an die nul e mee de de Nega ie - binomiaal- en Sichel-dis ibusies. lndi iduele handelsname in een da as esel (aankope an oile seep) olg die oo spelling an die a- disionele eo ie -die p opo sie an nie- e b uike s neem me e yd a . In die weede da as elsel ( e b uik an e pak e sop) he die p opo sie an nie- e b uike s an die handelsname na nul gedaal me e lenging an die ydpe k wa bes udee is, maa een 'n snelheid innlge as wa oo spel was deu die eo le. Die hipo ese da daa nool -kope s/geb uike s an indi iduele handelsname in die wee p odukklasse bes aan he , is dus e we p. S.-A . Tydsk . Bed y sl. 1984, 15: 155-161 C. Fi e G adua e School o Business Adminis a ion, Uni e si y o he Wi - wa e s and, P.O. Box 31170, B aam on ein 2017, Republic o Sou h A ica Accep ed May 1984 In oduc ion In he ield o consume non-du ables, an in e es ing segmen o he ma ke , known as 'ne e -buye s', has caused some dis- cussion in he li e a u e. Eh enbe g (1972) claimed ha , gi en a su icien ly long pe iod o ime, e e y consume will e en ually pu chase a p o- duc a leas once, and so he ne e -buye does no exis . I a s udy shows a numbe o non-buye s in a pe iod o ime, Eh enbe g belie es ha , had he ime o he s udy been longe , hese non-buye s would no ha e been ound. In ac , his Nega i e Binomial Dis ibu ion (NBD) heo y which p oposes ha , p o ided he ma ke is s a iona y equen ly pu chased consume non-du ables can be modelled using he NBD, (Equa ion Al) p edic s his ou come. As ime inc eases, he p opo ion o buye s o he p oduc as p edic ed by he NBD ends o uni y (Equa ion A2). (A b ie desc ip ion o he s a is ical dis ibu ions men ioned in he ex is gi en in he Appendix.) Pu ely om a logical poin o iew one may seek o aise issue wi h Eh enbe g on he ques ion o ne e -buye s. Non- smoke s p obably ne e buy ciga e es, non-d i e s who do no own ehicles a e unlikely o pu chase pe ol. P oduc classes can he e o e be iden i ied in which, by logical conside- a ions alone, one would expec o ind some ne e -buye s. Ne e heless, one should concede o Eh enbe g ha such examples p obably ep esen a small p opo ion o he con- sume non-du ables ma ke . Eh enbe g also claimed ha his model is as applicable o an indi idual b and as i is o he p oduc class as a whole. Howe e , wi h espec o indi idual b ands he si ua ion may be qui e di e en . Eh enbe g and o he o e seas w i e s ha e, wi h ew published excep ions, only es ed he NBD model in i s uni a ia e o m. Whe e hey ound depa u es om he model, hey asc ibed hem o e ec s which hey labelled ' he shel ing phenomenon' o ' he a iance disc epency'. They belie ed his e ec -an obse ed discon inui y in he equen- cy o hea ie pu chases a o close o he numbe o uni s, equal o he numbe o weeks o pe iod in he ime pe iod o he s udy- o be caused by a small sec o o he ma ke pu - chasing he p oduc egula ly. Sichel (1975) in es iga ed his phenomenon using obse a- ions on packaged-soup consump ion. By hei e y na u e, being consu.-np ion-based, such da a, acco ding o Eh enbe g's heo y, ough no o exhibi he e ec desc ibed abo e. Howe e , 'shel ing' was obse ed in he da a. She also ound ha o he s a is ical es s e ealed signi i- can depa u es om Eh enbe g's NBD model. As was subse- 1S6 quen ly demons a ed by Zachos (l 977), al hough uni a ia e dis ibu ions p edic ed by NBD heo y may be accep ed by he chi-squa e es , p edic ions in he bi a ia e onn o he NBD o en p o ide e y unsa is ac o y models. As a esul , bo h o he own wo k and an ea lie s udy by Tucke (1973), Sichel (1975) sugges ed ha he ue phen~ menon o epea -consump ion/buying may be ep esen ed by wo dis inc ly di e en popula ions: (a) A ha d-co e o s able, b and-loyal households which con- sume egula ly in acco dance wi h Eh enbe g's o iginal model.' These he e o e would be expec ed o ollow he co ela ed NBD model (Equa ion A3). (b) 'A ai numbe o b and-swi ching, mul ib and consum- ing households.' This g oup is s a is ically cha ac e ized by he unco ela ed bi a ia e NBD (Equa ion AS) in suc- cessi e ime pe iods. She pu o wa d he hypo hesis ha he obse ed bi a ia e dis ibu ion a ose om he supe imposi ion o hese wo popula ions. This hypo hesis was suppo ed by he wo k on he a ia ion o ma ke pene a ion, numbe o new and epea consume s and co ela ion coe icien s wi h inc easing leng hs o he obse a ional pe iod. She ound ha he obse a ions usually lay be ween he p edic ions o he co ela ed and un- co ela ed models. Fi e & Ba ne (l 979) ound simila esul s in he oile soap ma ke . Joannides (l 976) onnula ed a bina y model by combining he co ela ed and unco ela ed bi a ia e NBDs. He showed ha his model could p oduce many o he obse ed shapes, pa e ns (including a shelO, and alues o pa ame e s which he o iginal NBD heo y could no . In p ac ice, pu chase/ consump ion equency dis ibu ions can be obse ed in which he size o he z.e o cell appea s exces- si ely la ge when compa ed o he dis ibu ion as a whole. This sugges s ha he dis ibu ion is no d awn om a single homogeneous popula ion, bu ha a leas wo dis inc popula- ions a e in ol ed. Joannides (1976) iden i ied hese popula ions as b and-loyal and mul ib and, b and-swi ching households. Howe e , he same dis ibu ion shape could a ise om he supe imposi ion o a buying popula ion (which may, o a gi en ime pe iod con ain non-buye s who will buy in a u u e pe iod) and a g oup o ne e -buye s (whose dis ibu ion would consis o a ce ain equency in he ze o cell only). The objec i e o he esea ch epo ed in his a icle was o es hese hypo heses by in es iga ing he p opo ion o ne e - buye s in wo equen ly pu chased Sou h A ican consume non-du ables p oduc classes. Da a The wo da a se s analysed we e: Soap The da a e e ed o as 'soap' o igina e om he oile soap p oduc class. In Sou h A ica his p oduc class comp ised mo e han hi y- i e di e en b ands, bu was domina ed by he wo leading b ands, Lux and Palmoli e. The da a base used consis ed o a sample o 614 Whi e households' bimon hly pu chases o oile soap, o e he pe iod Ap il o No embe 1978, de ails o which we e collec ed by Ma ke Resea ch A ica using he consume panel me hod. Goodman (1979) es ed he da a and concluded ha pu - chasing a es o e he eigh -mon h pe iod we e s a iona y. The obse a ions had one peculia i y wo hy o men ion. When S.-A . Tydsk . Bed y sl. 1984, 15(3) hey we e plo ed in he onn o equency dis ibu ions, e en- numbe ed pu chase equencies appea ed high ela i e o he odd-numbe ed pu chase equencies, gi ing a zig-z.ag shaped dis ibu ion (Figu e 1). This was a ibu ed by Goodman o a consume p e e ence o pu chasing he p oduc in e en- numbe ed lo s. 200 100 oL----~~~~~~:::::::c.-...:A::.. 0 2 4 6 8 10 12 14 No. OF OOTS PURCHASED Figu e 1 Obse ed equency dis ibu ion o Palmoli e soap - eigh mon hs (N = 614) Fo he pu pose o es ing a dis ibu ion i o his da a i was he e o e necessa y o g oup he cells. The z.e o cell was allowed o s and alone, he one-plus- wo pu chase equency cells, he h ee-plus- ou cells, e c., we e g ouped oge he . Soup In he ea ly 1970s he packaged-soup ma ke in Sou h A ica was domina ed by he leading b and (he e designa ed b and A) ollowed a some dis ance by he only o he b and wi h a easonable ma ke sha e (designa ed b and B). Nine o he b ands made up he es o he ma ke . All b ands we e p esen ed in uni onn packagings o ou o six se ings each. The da a e e ed o as 'soup' in his pape we e collec ed by Sichel (1975) as pa o a Sou h A ican Na ional Dus bin Panel Su ey. Housewi es we e p o ided wi h a bin in o which hey disca ded emp y packe s o soup, he con en s o which had been consumed by hei amilies. E e y wen y-eigh days (i.e. a s anda d mon hly in e als) he housewi es we e isi ed by in e iewe s who eco ded he con en s o hei bins. In onna ion ela ing o 512 households' consump ion o six een wen y-eigh day pe iods was colla ed. Soup consump ion displayed a seasonal pa e n. I d opped du ing sp ing and ose in au umn. Th ee sepa a e s a iona y pe iods could hus be iden i ied: Pe iods one h ough ou ( he i s s a iona y win e pe iod): -pe iods se en h ough en ( he s a iona y summe pe iod); -pe iods hi een h ough six een ( he second s a iona y win e pe iod). Simila mean consump ion le els we e ound o he wo win e pe iods. This se o da a di e s undamen ally om he oile soap obse a ions in ha i con ains consump ion and no pu chase da a. Thus idosync asies o indi idual households such as he S. A . J. Bus. Mgm . 1984, 15(3) equency o shopping and he numbe o uni s pu chased on each shopping occasion a e emo ed. Consump ion should 1so he e o e p oduce a mo e uni o mly dis ibu ed se o da a. Ne e -Buye s Ne e -buye s o he p oduc ield In he oile soap p oduc ield, all o he 614 households in he sample had pu chased some soap du ing he eigh -mon h pe iod o he su ey. Thus no ne e -buye s o oile soap ex- is ed in he sample. Howe e , in he packaged-soup ma ke , a e wo win e pe iods, 20 o he 512 households had no ye consumed. Sichel (1975) concluded ha , because he e we e so ew non-consu- me s, hese 20 households did no p o ide su icien e idence o he exis ence o ne e -consume s. This conclusion is accep ed by he au ho and he analysis below is hus de o ed o indi i- dual b ands. Ne e -buye s o indi idual b ands: A s udy o he soap and soup da a e eals ha e en a e eigh and six een mon hs espec i ely, subs an ial p opo ions o he households in he samples had no pu chased/ consumed he leading b ands (Table 1). Table 1 Non-buye s o leading b ands P opo ions o non-buye s B and o e pe iod s udied Lux 0,450 Palmoli e 0,492 Sunligh Mild 0,590 B eeze 0,651 Shield 0,694 Soup B and A 0,203 Soup B and B 0,309 A isual inspec ion o he b and pu chase equency dis ibu- ions e ealed ce ain in e es ing phenomena. Fi s , in almos all cases a kink o a la ening ou , and some imes e en a sligh maximum, occu ed a abou he second o hi d cell. In he case o he oile soap dis ibu ions, he p oblem o he con- sume s' a ou ing o e en-numbe ed pu chases (discussed abo e) may ha e been con ibu ing owa ds his e ec . Second- ly, in isually sweeping a smoo h cu e upwa ds h ough he da a poin s, one would ha e expec ed such a cu e o in e cep he equency axis a a much lowe poin han ha ac ually eco ded. An example om eacq da a se is shown in Figu es 1 and 2. The hypo hesis pu o wa d is ha hese wo e ec s a e caus- ed by an 'excess' o ze os in he i s cell o he dis ibu ions, and ha his excess in ac comp ises ne e -buye s/consume s o he b and. I was an icipa ed ha he obse a ions could bes be model- led using a ze o-augmen ed dis ibu ion (See (c) in Appendix). Resul s Analysis o each o he da a se s is discussed sepa a ely. Analysis o he Soap Da a Goodman (1979) ound ha he i o he Sichel dis ibu ion (Sichel, 1971) wi h -y = -2,5 ((b) in Appendix) was gene ally supe io o he NBD wi h espec o he oile soap p oduc class (all b ands included). As a esul o his inding, she ex- 100 so 6 8 10 12 14 No. O OOTS Coo.HO Figu e 2 Obse ed equency dis ibu ion o combined b ands A and B o soup - i s ou mon hs (N = 614) amined only he Sichel dis ibu ion when analysing he indi i- dual b ands wi hin he p oduc ield. Acco ding o Sichel (1980), indi idual b ands need no neces- sa ily ollow he same dis ibu ion as ha pos ula ed o a o al p oduc ield, and he e o e Goodman's (1979) omission in no s udying he ze o-augmen ed NBD as a po en ial model o he indi idual b ands is open o some c i icism. This s udy he e o e began by a emp ing o i a ze o- augmen ed NBD model o he indi idual b and da a o Lux and Palmoli e soaps. Ea lie esea ch (Fi e & Ba ne , 1979) had shown ha o all ime pe iods s udied, he chi-squa e es ejec ed he hypo hesis o an o dina y NBD model o he da a g ouped as sugges ed abo e. Pa ame e es ima es o he model we e made on ung ouped da a. As can be seen om Table 2, he ung ouped da a ' ailed' he chi-squa e es . A de ailed s udy o he es s e ealed ha he 'odd-e en' pu chase phenomenon was esponsible o a subs an ial p opo ion o he o al chi-squa e. The es s we e hus epea ed using g ouped da a. The ze o-augmen ed Sichel dis ibu ion (-y = - 0,5) was hen i ed o he same g ouped da a using he same ail-end g ouping o cells o expec ed e- quency less han 5,0. F om Table 2 i can be seen ha , con a y o Goodman's expec a ions, he be e i was p o ided by he ze o-augmen ed NBD model. An example o he i s ob ained is shown in Figu e 3. Sys ema ic de ia ions o he wo models we e ound. The Sichel dis ibu ion was i s oo low, hen oo high, hen oo low again, ela i e o he NBD. Acco ding o Sichel (1980) he na u e o he de ia ions led one o he conclusion ha dec eas- ing he alue o 'Y om -0,5 o -1,5 o -2,5 would no signi ican ly imp o e he i . I was he e o e concluded ha , in he oile soap ield, in- di idual b and pu chase equencies a e bes modelled by he uni a ia e ze o-augmen ed NBD. One o he implica ions o he ze o-augmen ed model is ha es ima es ob ained o q, he p opo ion o ne e -buye s in he sample, should be independen o ime. Table 3 shows he esul s ob ained om he i ing o bo h he NBD and Sichel ze o-augmen ed models. These es ima es a e ce ainly no cons an . 1S8 S.-A . Tydsk . Bed y sl. 1984, 15(3) Table 2 Soap: Resul s o chi-squa e es s o ze o-augmen ed dis ibu ions NBD NBD Sichel ('y = - 0,5) Ung ouped G ouped G ouped Lax 1s 2 mon hs P(x2~ 1,531 I 2)< =0,465 • a 1s 4 mon hs P(x2 !l;: 16,379 I 5) < = 0,050 P(x2 ~ 2,023 I 2) = 0,364 P(x2~ 2,548 I 2)=0,275 6 mon hs P(x2 !l;:29,982 I 8)< =0,050 P(x2~5.747 I 3)=0,124 P(x2~ 7,791 I 3)=0,051 8 mon hs P(x2~ 19,5(,6 I 11)=0.052 P(;x2~7,385 I 5)=0,194 P(x2~ 11,422 I 5)<0,050 Palmoli e 1s 2 mon hs P(x2~ 1,482 I 2)=0,477 a a 1s 4 mon hs P(x2 !l;:25,474 I 5)<0,050 P(x2 !l;:2,536 I 1)=0,lll P(x2~ 2,916 I 1)=0,088 6 mon hs P(x2 !l;:28,420 I 7)<0,050 P(x2 !l;:5,991 I 3)=0,112 P(x2 !l;: 8,398 I 3) < 0,050 8 mon hs P(x2~21,031 I 9)<0.050 P(x2~9,181 I 4)=0,057 P(;x2!l;: 12,931 I 4)<0,050 'No deg ees o l llom le o chi-squa es s l10 240 21) 11kl ; 150 z a ~120 a OOSERVED o ZEl{)-AUH:NTED NBD • ZERO-~NTED smn (y=-0.5) 'JO ZERO-AUMNTED NIil : 9=0.709 q=0.371 k=168 P(JCH 3855) =0.194 60 ZER>-AUMNTED S10£L: 8=0.814 q:O 406 p:1.87 (y=-0.5 a PRICH) P b11.422 l5l<O 05 J) 0 0 2 4 6 8 10 12 14 16 Nn CF L lTS PLR:HASED Fl&a e 3 Obse ed and i ed ze o-augmen ed dis ibu ions o Lux soap -eigh mon hs (N = 614) Table 3 Soap: Es ima es o p opo ion o ne e - buye s q de i ed om ze o-augmen ed models Lux 1s 2 mon hs 1s 4 mon hs 6 mon hs 8 mon hs Palmoli e 1s 2 mon hs 1s 4 mon hs 6 mon hs 8 mon hs P opo ion o ne e -buye s NBD Siebel 0,634 0,635 0,498 0,515 0,410 0,440 0,37) 0,407 0,686 0,686 0,474 0,503 0,450 0,485 0,376 0,430 I ne e -buye s do exis , one would expec he obse ed p o- po ion o non-buye s o a p oduc o e en ually le el ou a he p opo ion o ne e -buye s in he sample, whe eas he p o- po ion o non-buye s as p edic ed by Eh enbe g's heo y would end o ze o wi h leng hening ime pe iods (Eh enbe g, 1972). Figu e 4 is an example o he obse ed and expec ed p o- po ions o non-buye s o Lux soap. The expec ed p opo - ions we e de i ed using Equa ion A4. The pa ame e es ima es we e made by he me hod o maximum likelihood using he obse a ions o he i s wo-mon h pe iod. In no case did he obse ed p opo ion lie abo e ha p e- dic ed by Eh enbe g's heo y. As a esul o hese indings, he hypo hesis o he exis ence o ne e -buye s o indi idual b ands in he oile soap ma ke was ejec ed. Analysis o he Soup Da a In his analysis he non-consume o e he en i e six een-mon h pe iod o he su ey we e conside ed, despi e he ac ha he da a did no exhibi s a iona i y o e he pe iod s udied. Al hough he use o he six een-mon h pe iod is heo e ically no co ec , i does a leas gi e some guidance as o he p o- po ion o non-consume s o e a leng hy pe iod. As will be seen om Figu e S, he obse ed poin plo ed a six een S. A . J. Bus. Mgm . 1984, 1S(3) 1,0 --o-- OBSERVED - EXPECTED 0.4 +-----,--"'T"""----,.----- --- 0 2 4 6 8 LE {iTH OF OBSERVATIONAL PERIOD I 11'.JNTHS I Figu e 4 P opo ion o non-buye s o Lux soap as a unc ion o leng h o obse a ional pe iod (N = 614) mon hs is well wi hin he gene al sweep o he cu e. The consump ion equency dis ibu ions o he indi idual b ands we e no modelled excep o he case o he combined b ands A and B o e he eigh s a iona y win e mon hs. The esul s o he chi-squa e es s on he wo ze o-augmen ed models we e: NBD Sichel P<x2 ~ 2S, 732 I 27) = 0,531 P(x 2~33,304 I 26)=0,166 The NBD model, which p o ided he be e i , ga e an es ima e o ze o o he p opo ion o ne e -consume s. Because o his esul , he i ing o models o he indi idual b ands was omi ed, and he g aphs o obse ed and expec ed ( ia Eh enbe g's NBD heo y) p opo ions o non-consume s o b and A, b and B and b ands A and B combined we e im- media ely s udied. The pa ame e es ima es o b ands A and B we e based on a e ages o each o he i s ou mon hs' es ima es, calcula ed using he me hod o momen s. Fo he combined b ands he maximum likelihood pa ame e es ima es o he i s mon h we e used. I was ound ha he obse ed p o- po ions o non-consume s app oached ze o mo e apidly han he expec ed p opo ions as p edic ed by he Eh enbe g heo y. An example o he obse ed and expec ed p opo ions o non- consume s is shown in Figu e 5. On he basis o his e idence, one mus he e o e also e- jec he hypo hesis o he exis ence o ne e -consume s o in- di idual b ands in he wo da a se s s udied. Concluslons Fo he wo da a samples a ailable, no e idence could be ound o suppo he hypo hesis ha he appa en excessi e size o he ze o cell in he pu chase/ consump ion equency dis ibu- ion was due o he supe imposi ion o a buying/consuming popula ion and a g oup o ne e -buye s/consume s. The q pa ame e s o he z.e o-augmen ed NBD (which i was hoped would p o ide an es ima e o he p opo ion o ne e - buye s o oile soap) we e no cons an o e he a ious ime pe iods s udied. In addi ion, wi h leng hening ime pe iods, he obse ed p opo ion o non-buye s dec eased a almos ex- ac ly he same a e as ha p edic ed by he Eh enbe g NBD 1,0 ---o--- OBSERVED ......,.__ ~XPECTED 159 0,0 +--,---.,..--.....----,--,--,---,---,--- 0 2 4 6 8 1l 12 14 16 LE «iTH O OBSER ATIONAL PERIOO !l'llNTHSl Figu e 5 P opo ion o non-consume s o b and A soup as a unc ion o leng h o obse a ional pe iod (N = S 12) heo y. This heo y does no p o ide o he exis ence o ne e - buye s. A simila calcula ion was made o he indi idual b ands in he soup p oduc ield. The e o e, on he basis o he a ailable e idence in wo p o- duc ields, he ne e -buye /consume hypo hesis was ejec ed. The Bina y Model o Joannides (1976) should hus be con- side ed as he explana ion o he obse ed dis ibu ion shapes. He showed ha he ma ginal dis ibu ions ob ained by sum- ming a co ela ed NBD, ep esen ing loyal epea -buye s, and an unco ela ed NBD, ep esen ing b and-swi ching/mul i- b and buye s, displayed high ze o cell equencies as well as poin s o in lec ion o seconda y modes close o he ze o cell. Thus he esul s ob ained, oge he wi h a conside a ion o he Joannides model, lead one o he conclusion ha pe haps b and-disloyal buye s/ consume s a e be e ep esen ed by a dis ibu ion o equencies a he han he subse he eo (ze o cell equencies only) implied by he ne e -buye hypo hesis. These esul s ha e impo an implica ions o he ma ke e s o consume non-du ables. Ins ead o de o ing ime and e - o in pe suading ne e -buye s o use hei p oduc s (in a gene ic sense), hey would be s ongly ad ised o unde ake ad e ising campaigns aimed a wooing exis ing use s o o he b ands o he p oduc in o ying hei pa icula b and. Finally, his s udy highligh ed he need o model he ield o consume non-du able pu chase pa e ns om a mul ib and pe spec i e, a he han by looking a indi idual b ands in isola- ion. Resea ch in his a ea has been unde aken and will be epo ed on in he u u e. Re e ences Eh enbe g, A.S.C. 1972. Repea buying: heo y and applica ions. Ams e dam: No h Holland Publishing Co. Fi e , C. 1980. The de elopmen o new ma hema ical models o mul ib and buying. Unpublished M.B.A. disse a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Fi e , C. & Ba ne , A.E. 1979. Tes ing he NBD epea buying model o exposu e pe iods o a ying leng hs. Unpublished Repo . Goodman, M.A. 1979. Mul ib and pu chasing beha iou - heo y and applica ions. Unpublished M.B.A. disse a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Joannides, J. 1976. A new dis ibu ion o epea consump ion. Un- published M.B.A. disse a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Mo ison, D.G. 1969. Condi ional end analysis: a model ha allows 160 o non-use s. J. Ma ke . Res., VI, 342-346. Sichel, H.S. 1971. On a amily o disc e e dis ibu ions pa icula ly sui ed o ep esen long- ailed equency da a. In P oceedings o he Thi d Symposium on Ma hema ical S a is ics (N.F. Laubsche , ed.). S.A. C.S.l.R. (P e o ia), Sl-97. Sichel, H.S. 1973. S a is ical alua ion o diamondi e ous deposi s . Jou nal o he Sou h A ican Ins i u e o Mining and Me alle gy, 23S-239. Sichel, H.S. 1974. On a dis ibu ion ep esen ing sen ence leng h in w i en p ose. Jou nal o he Royal S a is ical Socie y, 137(1), 2S-34. Sichel, H.S. 1980. P i a e Communica ion. Sichel, N. 197S. P oblem a eas in epea consump ion models. Un- published M.B.A. disse a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Tucke , A.E. 1973. The epea pu chase pa e ns o non-du able con- sume goods in selec ed households. Unpublished M.B.A. disse - a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Zachos, G. 1977. Non-du able consump ion pa e ns o indi idual households. Unpublished M.B.A. disse a ion, Uni e si y o he Wi wa e s and, Johannesbu g. Appendix A b ie desc ip ion o he s a is ical dis ibu ions men ioned in he ex is gi en he e. (a) The NBD Repea -buying Model (i) Uni a ia e Case The model assumes ha an indi idual household's pu chases a e cha ac e ized by a Poisson dis ibu ion. In o de o s udy ma ke beha iou as a whole, he indi idual household's dis ibu ions a c compounded using a 'Y dis ibu ion o he indi idual a e age pu chase equencies pe uni ime (Sichel, 1971). The esul is he Nega i e Binomial Dis ibu ion (NBD): Ax I J = [kl(k + m)lk ex + k) l ml(k + m)]X (Al) (k) (x+ I) whe e xis he numbe o pu chases in ime , 0< m < co ,0< k < oo , and x = 0,1,2, ... The p opo ion o buye s o he p oduc in ime is gi en by b1 = 1 - (I + mlkF" (A2) whe e a buye is de ined as one who buys a leas one uni du ing ime pe iod . (ii) The Bi a ia e Co ela ed NBD The bi a ia e model is ob ained by mixing he p oduc o wo uni a ia e Poisson dis ibu ions wi h he 'Y dis ibu ion, o o m he bi a ia e co ela ed NBD model. In i s symme ic o m (whe e successi e pu chase pe iods a e o equal leng h) i can be w i en J x I J = (l+Umlk)-1c (x+y+k) [ ml(k+Um) +" (A3) ,Y (k) (x+ 1) e,,+ I) whe e k>O, m >0, k( J = k, m( )= m, x=0,1,2, ... , y=0,1,2, ... The ma ke pene a ion (p opo ion o buye s o he p oduc in ime J is gi en by c" = l -(1 + mlk)-k (A4) (iii) The Unco ela ed Bi a ia e NBD Sichel (1980) de i ed he unco ela ed bi a ia e NBD in an a emp o model he beha iou o p oduc -disloyal consume s (b and- swi che s and mul ib and buye s) whose consump ions a e unco - ela ed be ween wo ime pe iods. The unco ela ed NBD is ob ain- ed om he p oduc o wo uni a ia e NBDs. I can be w i en in he o m I l ______________________ _ (b) (c) S.-A . Tydsk . Bed y sl. 1984, 1S(3) k>O, m>O, k( )= k, m( )= m, x=0,1,2, ... , y=0,1,2, ... In he case o his dis ibu ion, he p opo ion o buye s is ,/' = I - (1 + ml k) - '" The Sichel dis ibu ion (A6) This dis ibu ion was also de i ed by Sichel in 1971. I can be w i - en in he o m _ ( =e)l (a0/2)' K (a) 1/J(. ) -K (a( ' I - 0)) ! • +} l = 0,1,2, ... (A7) I is de i ed by mixing he Poisson dis ibu ion wi h he lexible mixing dis ibu ion >.) = j2 '{ =e)/ a0P • >. ~ -1 • exp! _ ( _!_) >. _ a2 e (AS) 2K (a 'l -0) 0 4 } ' In his dis ibu ion - oo < -y < oo , O < 0 < I and a >0 a e he h ee pa ame e s and K_ (.) is he modi ied Bessel unc ion o he second kind o o de (.). , , ( ) ep esen s a amily o disc e e dis ibu ions. Many o he be - e known disc e e dis ibu ions such as he NBD, Poisson, geome ic, e c., a e special o limi ing o ms o 1/J(. ) (Sichel, 1974). Making gamma nega i e allows one o gene a e an en i ely new se o disc e e dis ibu ions. The dis ibu ions a e ma hema ically di - icul o handle unless 'Y is a hal -in ege . -y could he e o e be se , a p io i, o - O,S and alues o - l ,S o -2,S would be used in o de o y and imp o e he i i 'Y = - O,S is no sa is ac o y. Two pa ame e s, a and 0, ha e he e o e o be es ima ed. I he p opo ion o obse ed equenci~ in he ze o cell is la ge (gene al- ly abo e S00 o), he me hod o equa ing mean and ze o cell p opo - ions is ai ly e icien (Sichel, 1973). I his is no he case, a maxi- mum likelihood me hod is a ailable (Sichel, 1971). Howe e , his me hod is e y complica ed, so when he i.e o cell is small, he me hod o momen s is gene ally used (Sichel, 1974). Ze o-augmen ed dis ibu ions A la ge numbe o obse ed pu chase/consump ion equency dis ibu ions ha e an (wha appea s o he eye), excessi ely la ge mo cell. I espec i e o he ue cause o his e ec , one equi es o be able o model such dis ibu ions. In o de o do so, i is assumed ha his cell is made up o wo sepa a e g oups. The si ua ion is illus a ed g aphically in Figu e Al· An implica ion o his model is ha , i a ha d co e o ne e -buyen does exis , he es ima e o IJ should be independen o ime. The alues o IJ o di e en b ands wi hin a p oduc ield need no , o cou se, be simila . ~ ~q 1i / / / / / .... .... .... ( 1-q )do (0) -- 0 1 2 3 i. S -.,_ OF I..HITS PIR:HASEO / CON9.J £0 la11n Al A g aphical ep esen a ion o ne e -buye s S. A . J. Bus. Mgm . 1984, 15(3) Mo ison (1969) showed ha he exis ence o ne e -buye s would in oduce a sys ema ic bias in o he NBD model. He he e o e p o- posed a model o he o m gi en abo e. In his model h ee pa ame e s equi e es ima ion: he k and m pa ame e s o he NBD and lj, he p opo ion o ne e -buye s. 161 Sichel (1980) p oposed a simila modd based on he Sichel dis nbu- ion. He e, oo, h ee pa ame e s equi e es ima ion: a and 0 o he Sichel, and lj he p opo ion o ne e -buye s. The echniques o es ima ion o he pa ame e s o bo h dis ibu- ions a e discussed by Fi e (1980).