Niche Market or Mass Market?

Transcription

1 Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value. We characerize properies of he demand funcion ha deermine his ranking wihou calculaion of he variables. JEL classi caion: D4, L Keywords: Markes, monopoly, price Inroducion The main goal of his work is o invesigae he ranking of wo economic variables he monopoly price and he mean value of a produc, ha is, he produc valuaion by he buyer wih an average ase. If he mean value is no less han he monopoly price, he marke is considered as a mass marke, in which he monopolis sells he produc o a relaively large fracion of consumers. Oherwise, he marke is considered as a niche marke, in which he produc is supplied only o consumers wih high valuaions. The di erence beween he wo ypes of he marke can be also seen in he conex of buyer s informaion. If a buyer wihou any prior informaion eners he monopolisic marke, hen she values he produc a he expeced valuaion. Thus, his buyer is willing o purchase he produc a he (saic) monopoly price if he marke is a mass one, bu declines o do so if he marke is a niche one. In he lieraure, he answer o he quesion abou which variable is bigger is crucial, since i may qualiaively a ec resuls and heir implicaions. For example, Bergemann and Välimäki (2006) show ha in a dynamic model of experience goods pricing, all markes can be classi ed as eiher mass markes or niche markes wih qualiaively di eren equilibrium paerns for prices and quaniies. In he mass marke, he dynamic equilibrium prices decline over ime and uninformed buyers purchase in all periods. In I am very graeful o Kalyan Chaerjee, Vijay Krishna, and Alexander Tarasov for very helpful commens and discussion. All misakes are my own. y Deparmen of Economics, McMaser Universiy, Kenneh Taylor Hall, 280 Main Sree Wes, Hamilon, ON, Canada L8S 4M4. [email protected]. Phone: (905) x Fax: (905)

2 conras, in a niche marke, he opimal prices are iniially low followed by higher prices ha exrac surplus from he buyers wih a high willingness o pay. Anoher relaed work is by Johnson and Mya (2006), who invesigae he response of a monopolis o changing dispersion of he disribuion of consumer s ases. They show ha in conras o a mass monopolis, a niche monopolis prefers he higher dispersion of consumer s values. However, i remains unclear wha properies of he disribuion of consumer s valuaions deermine he marke ype. The di culy comes from he naure of he variables. While he monopoly price depends on he local properies of he demand funcion, which deermine he marginal revenue, he mean value aggregaes informaion abou he whole demand. In his work, we provide a simple crierion for he demand funcion, which deermines he marke ype as a niche or a mass marke. The main feaure of his crierion is ha i does no involve he calculaion of hese endogenous variables. The crierion requires only he knowledge of he properies of he hazard rae funcion and he relaionship beween he consumer s lowes value and he uni cos of producion. In addiion, we analyze he response of he marke ype o some common modi caions of he disribuion of consumers values. The model In our analysis, we use he basic model of he monopolis, who sells he produc o he buyer wih a privaely known valuaion. The valuaion is a random draw from a disribuion F (), which is suppored on [; ] ( ; ], has a coninuous posiive densiy f () ; 2 (; ), and he mean value e = E [] <. Tha is, he demand funcion is F (). The producion coss are deermined by he uni cos c 2 [0; ). Le M be a monopoly price, i.e., i solves he pro -maximizaion problem max () = max ( F ()) ( c) : () Noe ha he imposed condiions on F () and c imply ha here exiss a monopoly price M < (Van den Berg, 2007). Given he hazard rae funcion () = f () F () ; he su cien condiions for each ype of he marke are deermined by he following heorem, which is he main resul of he paper. All proofs are colleced in he Appendix. Theorem If he reciprocal hazard rae = () is convex (concave) and is above (below) c, hen here exiss a unique M, which is below (above) e. Noe ha due he speci c feaures of seings, Bergemann and Välimäki (2006) use a slighly di eren de niion for mass and niche markes. They compare he monopoly price o he produc s expeced value, which is, due o informaional bene s, is no less han he mean value. If he informaional bene s are no high, hen he expeced value is likely o be near he mean value. Thus, he main idea behind heir de niion is he same as ours: o deermine he siuaion, in which he buyer wihou any prior informaion (ha is, she evaluaes he produc a he expeced value) is ready o pay he saic monopoly price. Similarly, Johnson and Mya (2006) compare he monopoly price o he roaion poin of an ordered family of disribuions. Thus, heir de niion canno be deermined for a single disribuion. However, in many cases, e.g., for heir ruh-or-noise ordering (see also Lewis and Sappingon, 994), he roaion poin coincides wih he mean value. 2

3 The proof is based on he fac ha he condiions on = () play a dual role. Firs, hey guaranee ha () ( c) is sricly pseudo-monoone ha resuls in he uniqueness of M. 2 Given his propery, i can be shown ha e M if and only if ( e ) ( e c). These preliminaries imply ha e M if and only if e = E [= ()] + = ( e ) + c: Finally, because of Jensen s inequaliy, he condiions in he heorem are, in fac, he su cien condiions for each ype of he marke. Two commens are in order. Firs, he resul above demonsraes ha he marke ype depends on wo facors: he convex properies of = () and he ranking of c and. For a xed disribuion, a high c favors a niche marke, since i increases M while keeping e he same. If c becomes su cienly high, hen he mass marke ineviably swiches o he niche marke. 3 The converse is no necessarily rue: if 0 and = () is concave, hen even c = 0 does no change he ranking. Second, he reciprocal hazard rae is relaed o he virual value funcion () = = () ; which represens he marginal revenue and is inensively used in he lieraure on aucions and mechanism design. 4 Thus, he convexiy and he concaviy of = () imply he concaviy and convexiy of (), respecively. In general, classes of he disribuions ha have eiher a convex or a concave reciprocal hazard rae are quie rich. For example, for gamma, Weibull, power, log-logisic, and Burr families of disribuions parameerized by he shape parameer k, = () is convex if k and concave oherwise. 5 Moreover, even he class of disribuions wih he linear = () = s + ; conains several sandard families. For his class, he characerizaion of he marke ype is simple: e M if and only if c. This class is deermined by he disribuions, which are he soluions o he di erenial equaion: df () d = F () ; (2) s + wih he iniial condiion F () = 0. All soluions o (2) are parameerized by he riple (s; ; ) and have he form of F (js; ; ) = =s s + ; s + where s;, and deermine he shape, he scale, and he lower bound of he suppor, respecively. For example, riples (0; ; 0) ; ( ; ; 0) ; ( =2; ; 0) ; and 2 A funcion ' () is sricly pseudo-monoone if for every and 0 6= ; ' () ( 0 ) 0 implies ' ( 0 ) ( 0 ) < 0 (Hadjisavvas e al., 2005). Equivalenly, ' () 0 implies ' ( 0 ) < 0 for all 0 >. 3 For example, if c e, hen M > c e. 4 See, for example, Krishna (2002). 5 For he Burr family F () = + k c ; k > 0; c > 0, = () is convex if k, and concave if k. 3

4 (s; 0; ) ; s 2 (0; ) ; > 0 correspond o he exponenial, he uniform, he asymmeric riangular, and he Pareo disribuions, respecively. In addiion, () > 0; 2 (; ) requires s + > 0. Finally, e < if s <. Transformaions of demand Given he above resuls, we invesigae he response of he ranking o changes on he demand side. For insance, acquiring informaion by he buyer spreads he disribuion of buyer s poserior valuaions and may change he ype of he marke (Johnson and Mya, 2006). Similarly, replacing he produc wih a newer version or appearance of a subsiue produc also a ecs he disribuion of buyers valuaions. An implicaion of Theorem is ha he marke ype is no a eced by any ransformaion of he disribuion F () = T (F ()), which preserves he convexiy (or he concaviy) of = () and he ranking of and c. Hereafer, we refer o ransformaions ha do no a ec he convex properies of = () as he convexiy preserving ransformaions. Typical examples of such ransformaions are a runcaion of he original disribuion from below, a linear ransformaion of he value, or he minimum of i.i.d. variables. For such ransformaions, a swich of he marke ype is possible only via a change in he ranking of and c. Corollary If = () is convex, a convexiy preserving ransformaion F () swiches he mass marke o he niche marke only if < c. If = () is concave, hen a convexiy preserving ransformaion swiches he niche marke o he mass marke only if > c. Tha is, if = () is convex and c, hen a swich from he mass marke o he niche one happens only if he ransformaion generaes a posiive mass of consumer s ypes, which value he produc below c. Similarly, if = () is concave, he reverse swich happens only if all consumer s values are bounded away from he cos of producion. We use he observaion abou he convexiy preserving ransformaions o invesigae he response of he marke ype o linear ransformaions of values. These ransformaions have many useful applicaions, since hey cover such cases as a shif and a srech in he disribuion, and a change in he variance. Consider a family of disribuions F (), which are obained from he original disribuion by linear ransformaions = + ( ) + ; where > 0 re ecs he spread of he disribuion around poin, and re ecs he shif. Tha is, ( ) F () = F ; where F (:) is a coninuous disribuion wih he uni variance and a sricly posiive densiy on (; ). In addiion, i is assumed ha () ( c) is sricly pseudo-monoone for max f ; cg. 6 Lemma characerizes he parameer se, which separaes he wo ypes of he marke. 6 This condiion holds if, for example, () is sricly increasing. 4

5 Lemma For a family of disribuions F (), he marke is a mass marke if and only if (( ) + c) = = ( e ) e. The lemma above allows o see how he marke ype is a eced by some sandard ransformaions. For example, if he supplier replaces he produc by a new one, which shifs he buyer s value by > 0, hen such a replacemen favors he mass marke. Tha is, here exiss he cu-o level = = ( e ) e + c, such ha he marke is a mass marke if and only if. If he iniial produc in he mass marke is replaced by a new one, which increases he buyers valuaions by he facor >, his does no change he marke ype. In conras, inroducing new informaion abou he produc s characerisics may increase he variance of consumers values while while keeping he same mean. This favors a niche marke. 7 Conclusion While he relaionship beween he monopoly price and he average consumer ase can make a signi can impac on predicions of models of markes, he basic properies of he demand funcion, which deermine his relaionship, were no explored. We ll his gap by characerizing simple su cien condiions on he disribuion of consumer s values, which deermine he ranking of he wo variables. Appendix Claim M 0 = max fc; g. Also, M = 0 only if > c. Proof Firs, c implies () 0 < ( F ( 0 )) ( 0 c) = ( 0 ), where 0 2 (c; ). Second, if > c, hen 0 =. For any <, we have () = c < c = (). Tha is, M max fc; g, and M = only if > c. and The rs-order condiions for M are F ( M ) f ( M ) ( M c) = ( F ( M )) ( ( M ) ( M c)) = 0 if M > 0 ; Hence, i follows ha ( F ( M )) ( ( M ) ( M c)) 0 if M = > c: ( M ) ( M c) = if M > 0, and (3) ( M ) ( M c) if M = > c: Lemma 2 If () ( c) is sricly pseudo-monoone for 0, hen M is unique, and e M if and only if ( e ) ( e c). 7 For he riple ( ; ; 0 ) = (; 0; e ), he family F () is a special case of he variance-ordered family of disribuions wih he roaion poin e (see Johnson and Mya, 2006). I follows from Proposiion in Johnson and Mya (2006) ha once he dispersion is su cienly high, he marke becomes he niche one. Lemma complemens heir resul by idenifying he criical dispersion = ( e ) ( e c) ha changes he marke ype. 5

6 Proof The sric pseudo-monooniciy of () ( c) implies he sric pseudo-monooniciy of 0 () = ( F ()) ( () ( c)) ; 2 [ 0 ; ). By Proposiion 2.5 in Hadjisavvas e al. (2005), () is sricly pseudo-concave on he convex se [ 0 ; ). Hence, () is sricly quasi-concave on [ 0 ; ], so ha i has a unique maximizer M. Then, we show ha ( e ) ( e c) implies e M. If M =, hen e > = M. If M > 0, hen ( M ) ( M c) = from (3). Since ( e ) ( e c), we have e > c, so ha e > 0. Also, i follows from he sric pseudo-monooniciy of () ( c) ; 0 ha () ( c) > ; > e. Hence, M e. Finally, we show ha ( e ) ( e c) implies e M. From (3), we have ( M ) ( M c). Also, he sric pseudo-monooniciy of () ( c) ; 0 implies () ( c) > ; > M. Hence, e M. Proof of Theorem. Since e < and f () > 0; 2 (; ), hen e = d ( F ()) = ( F ()) j + F () d = F () d + = F () f () d + = f () f () d + = E () () Firs, suppose ha = () is convex and c. If <, hen = () is sricly decreasing. By conradicion, le = (y) = (x) for some x and y > x. Noe ha = () = 0, so ha y = resuls in he conradicion = (y) = (x) > 0. Also, y 2 (x; ) can be expressed as y = x+( ) for some 2 (0; ). Then, he convexiy of = () resuls in he conradicion = (y) = (x)+( ) = () = = (x) < = (x). Thus, () is sricly increasing. Tha is, () ( c) is sricly decreasing and, hence, sricly pseudo-monoone. If =, we show ha () ( c) is sricly pseudo-monoone for 0, ha is, () ( c) implies ( 0 ) ( 0 c) > ; 0 >. By conradicion, suppose ha (x) (x c) and (y) (y c) for some x and y > x. Noe ha he convexiy of = () implies ha () ( c) for all y. (Oherwise, if (z) (z c) > for some z > y, hen y 2 (x; z) can be expressed as y = x + ( ) z for some 2 (0; ). Then, (x) (x c) and (z) (z c) > imply = (x) x c and = (z) < z c. The convexiy of = () resuls in +. = (y) = (x) + ( ) = (z) < (x c) + ( ) (z c) = y c; which leads o he conradicion (y) (y c) >.) Hence, () = ( c) ; y. This leads o F () = e R (u)du e R y (u)du R y u c du y c = c e yr (u)du, and 6

7 e = F () d F () d e y yr = (y c) e (u)du ln ( c) j y =, yr (u)du y y c c d which conradics he fac ha e <. Thus, () ( c) is sricly pseudo-monoone for 0. Moreover, by Jensen s inequaliy, we have e = E [= ()] + = ( e ) + = ( e ) + c; which gives ( e ) ( e c). By Lemma 2, M is unique and e M. Second, suppose ha = () is concave and c. By Claim, we have M > 0 = c, so ha ( M ) ( M c) =. We show now ha () ( c) is sricly pseudo-monoone for 0. By conradicion, suppose ha (x) (x c) and (y) (y c) for some x > c and y > x. Since x 2 (c; y), i can be expressed as x = c + ( ) y for some 2 (0; ). Then, (y) (y c) implies = (y) y c. The concaviy of = () and = (c) > 0 resul in = (x) = (c) + ( ) = (y) > ( ) (y c) = x c; which leads o he conradicion (x) (x c) <. Applying Jensen s inequaliy leads o ( e ) ( e and e M. c). By Lemma 2, M is unique Proof of Lemma. By Lemma 2, e () M () if and only if ( e ()) ( e () c). Since e () = e + ( ) + ; and () = ( ) ; i follows ha ( e ()) ( e () c) = (e ) ( e + ( ) + c) = ( e ) e + ( ) + c ; if and only if ( ) + c ( e ) e : References Bergemann, D. and J. Välimäki, 2006, Dynamic pricing of new experience goods, Journal of Poliical Economy 4, Hadjisavvas N., S. Komlòsi and S. Schaible, 2005, Handbook of Generalized Convexiy and Generalized Monooniciy, Nonconvex Opimizaion and is Applicaions, (Springer, 7

8 Boson). Johnson, J. and D. Mya, 2006, On he simple economics of adverising, markeing, and produc design, American Economic Review 93, Krishna, V., 2002, Aucion Theory, (Academic Press, San Diego). Lewis, T. and D. Sappingon, 994, Supplying informaion o faciliae price discriminaion, Inernaional Economic Review 35, Van den Berg, G.J., 2007, On he uniqueness of opimal prices se by monopolisic sellers, Journal of Economerics 4,