Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. Yi-Nung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322-353 April 3, 995 (curren version Feb, 996) JEL codes: Keywords: L2, L86 nework exernaliies, sofware, upgrade Absrac: There are many producs which have lile or no value in isolaion, bu if combined wih oher producs or services, hey generae more value. The uiliy ha a given user derives from hese producs depends upon he number of oher users who consume he same producs. This kind of posiive consumpion exernaliy is defined as a nework exernaliy. Nework exernaliies are pervasive in he compuer in he compuer sofware marke. While sofware vendors consider pricing sraegies, hey mus also ake ino accoun he impac of nework exernaliies on heir sales. Our main ineres in his paper is o describe he firm's sraegies and behavior in he presence of nework exernaliies. Based on he wo-sage model in his paper, we find ha he sofware firm will charge a lower price o arac more users in he firs sage, hen charge a higher price as i does in monopoly as radiional economic heory indicaes. And our dynamic model shows ha he opimal upgrade ime occurs when gross profi of he firs ediion equals he gross profi of he second ediion of he sofware. This implies ha eiher oo earlier or lae promoion of he new sofware version will causes profi loss. *Graduae suden. Phone: (8) 755-946. E-mail: slmk@cc.usu.edu. WWW: hp://cc.usu.edu/~slmk/. Address: USU P.O. Box 69, Logan UT 84322. would like o hank Terry Glover, Chris. Fawson, and Chris Barre for heir paience o read and considerable commens on an earlier draf of his paper.
. nroducion There are many producs for which he uiliy derived by a consumer depends on he number of he agens consuming he good. n oher words, posiive consumpion exernaliies arise. Kaz and Shapiro (985) defined and illusraed hree ypes of nework exernaliy effecs. () Direc physical effecs; generaed hrough he direc physical effec of he number of agens purchasing he produc. Examples include fax machines, elephone and daa neworks, ec. (2) ndirec effecs; or wha migh be called he hardware-sofware paradigm. For example, an agen buying a PC will no merely ake he number of PCs sold ino accoun, he will also consider sofware availabiliy. Similar examples are video games and video players. (3) Pospurchase service nework effecs; for which vivid case is he auomobile marke. An agen will ake ino consideraion he pospurchase nework service size in addiion o he car's qualiy. As Kaz and Shapiro (994) poined ou hese kinds of nework exernaliy effecs will rouble economic heory boh in marke equilibrium and in he marke performance. Tha is, if he marke presens such nework exernaliies, an equilibrium migh no exis, or muliple equilibria migh exis. And he fundamenal heorems of welfare economics may no apply in his siuaion. Los of ineresing problems arise in markes wih such nework exernaliies. The personal compuer indusry is perhaps he bes case of nework exernaliies, boh for hardware and sofware. The hardware compeiion beween Apple and BM compaible PCs has prevailed for years. The operaing sysem war beween Microsof and BM has become a ho issue recenly. The new upgrade informaion abou some imporan sofware such as Lous -2-3 spreadshee is usually paid aenion by he people as well as sock marke. The issues in which we are ineresed in he sofware indusry are wha he pricing and promoion sraegies are given he nework exernaliies in he marke. --
n his paper, a wo-sage saic model and a simple dynamic model of sofware monopoly will be inroduced. examine he monopolisic firm's pricing sraegies boh for sofware inroducion and upgrading. The dynamic model indicaes he opimal upgrade ime of he sofware. The paper is organized as follows: secion discusses consumers' decision behavior, based on consumers' behavior, he wo-sage saic model in secion demonsraes he firm's pricing policy in secion. hen apply opimal conrol heory o analyze he firm's dynamic behavior in he presence of nework exernaliies in secion V. This model also proposes he opimal upgrade ime for he sofware monopolis. Some conclusions are drawn in secion V.. Choices of Consumers and Marke Equilibrium Assume he sofware, x, exhibis nework exernaliies, i.e., uiliy comes no only from consuming sofware x iself, bu also from he number of oher consumers who adop he same producs. There are k poenial consumers and one firm in his marke. Wihou aking he nework exernaliies ino accoun iniially, he sand-alone willingness o pay for he sofware x is R i (h), i =, 2,..., k. R i 's are ranked by he willingness o pay, i denoes consumer's rank order, and h denoes an individual's preference parameer, which can be viewed as a echnology index. Superior echnology enables consumers o enjoy more wih he new echnology, so consumers are willing o pay more for i, i.e., R ( h ) / i h >. is assumed o be consan here for presen discussion in his paper and we will suppress R(h) in following analysis. Assume every poenial consumer demands eiher one uni of produc or none, each makes a discree choice. n he absence of nework exernaliies, he consumers' demand schedule D for his produc could be drawn in he order of he willingness o pay. (See, Figure ) -2-
price Po x j } x Excess willingness o pay for j =R j + V j ()-p D x Figure. Following Kaz and Shapiro's (985) assumpions, suppose he nework exernaliies funcion for ih individual is v i (x), where 2 vi vi >, <, v 2 i ( ) =, and lim v'( x) =. x x x So, in he presence of nework exernaliies, he willingness o pay for x is R ( h) + v ( x). n sage, firs he firm chooses p, and xo consumers whose willingness o pay is greaer han or equal o he price, i.e., R ( i h ) p, will purchase in he firs period. Noice ha nework exernaliies do no ye influence consumer choice since he produc is no ye esablished. However, sage consumpion knocks he marke ou of equilibrium, for nework exernaliies now emerge. Marginal consumers who originally didn' ener he marke now wan o join he nework because hey perceive posiive nework exernaliies. So x consumers, where x = x x >, ener he nework in sage. Those whose willingness o pay are greaer han or equal o p will ener he nework, i.e., R ( h) + v ( x ) p, i = x +, x + 2,..., k. D denoes demand in he i i second sage. By dynamic adjusmen he nework will grow a a declining rae unil he effec of nework exernaliies vanishes. Finally, in sage, he nework reaches equilibrium wih demand denoed by D* as shown in Figure 2. Suppose he firm se an iniial price P. Then a corresponding nework growh funcion, G( vx ( ), p ), associaed wih his pricing sraegy P. emerges (Figure 2). A ime =, he nework size is X. Because prior period purchase increases he presen willingness o pay for poenial users, some of hose marginal users who didn' ener he nework previously will now join he -3- i i
nework. Therefore, a ime =, he nework grows o X. For he same reasons, users join he nework coninuously unil he excess willingness o pay vanishes. The nework will follow along a concave growh funcion and reach he equilibrium X * * = X. The * equilibrium se is ( X, p ). f he firm se a differen price, P, insead of P iniially, a corresponding nework growh funcion, Gvx ( ( ), p) equilibrium, X would lead o a differen = X * *, he equilibrium se is ( X, p ). The firm's differen iniial pricing 2 sraegies, Pj, collecively map ou he differen long run equilibrium nework ses * {( X j, pj )}, based on he nework growh funcion, G( vx ( ), p j ), yielding a he nework demand funcion, D*, as indicaed in figure 2. Thus, he long-run marke equilibrium condiion can be expressed as, * R *( h) + v *( x ) p = x x j price p (x*,p ) (x*,p ) p x D D D2 D * x x 2 x * x x + G(v(x),p ) x 3 G(v(x),p ) x 2 x x x x... 2 x * x Figure 2. Dynamic adjusmen of nework growh.. Two-Sage Saic Model -4-
Following he consumer behavior and long-run marke equilibrium condiion presened in secion, now consider a wo-sage saic model of he sofware monopolis's behavior. Assume he monopolis provides wo successive versions of a sofware package. n he firs sage he offers an inroducory version, x ; in he second sage he offers an upgrade version, x. We assume ha he consan marginal coss of hese wo versions are c and c 2 respecively. The monopolis looks for prices p and p 2 (he prices for x and x ) ha maximize he presen value of he profi sream from he wo successive versions of he sofware, given he exisence of nework exernaliies. The effec of nework exernaliies is assumed o be lagged. Tha is, consumers' decisions o purchase he inroducion version does no expec he nework exernaliies for he presen version. When he upgrade version becomes available, consumers consider he exising nework size for he inroducory version in deermining wheher o upgrade or no. Hence, he monopolis's profi funcion is, Π= π + π = ( p c ) x ( p ) + δ ( p c ) x x ( p ), p () ( ( )) 2 2 where δ = /( + r ), a discouned facor and r is he discoun rae. The firs order condiions require,, Π x x x = ( p c) + x ( p) + δ( p2 c2) = (2) x Π 2 x = δ p2 c2 + x p2 ( ) ( ) 2 =, (3) Sofware vendors usually provide wo prices for upgrades; one for owners of he firs version, and a higher price for new users. This is convenional price discriminaion. is easy o show here are similar resuls under his siuaion if imposed by price discriminaion. However, we don' examine he price discriminaion policy here. -5-
Assume he second order condiions hold. Define price elasiciy as ε i xi pi =,, i = 2,. x i i Equaion (3) can be rewrien as p * ( 2 + ) = c2, (4) ε 2 where p 2 * is opimal upgrade price. This is he sandard monopolisic pricing policy; i means marginal revenue equals marginal cos. f here is only one upgrade, he monopoly follows he pricing principle of MR = MC. Furhermore, of paricular ineress is o look furher a equaion (2). Afer muliplying boh sides wih / x and rearranging i, we can obain * * x p ( + ) = c δ( p2 c2) ε x. (5) where p * * is opimal upgrade price. Since ( p c ) is nonnegaive 2, x / x, be he 2 nework exernaliies, should be posiive. Therefore, we find ha in he presence of nework exernaliies, a profi-maximizing monopolis charges a lower price for he inroducory version of he sofware han wihou he nework exernaliies (where x / x = ). is inuiive ha a monopolis ses a lower inroducory price o arac more users in order o generae greaer demand in nex round since he firm knows here exis nework exernaliies. Equaion (5) indicaes ha he monopolis equalizes presen marginal revenue and he presen marginal cos plus a (negaive) presen opporuniy cos caused by fuure nework exernaliies effecs. V. The Dynamic Model and The Decision of Opimal Upgrade Time 2 * * Equaion (3) can be rearranged as x ( p2) = ( p2 c2) x / p. Since 2 x (.), and x / p2 <, * ( p c ). -6-
The firm is assumed o be able o provide enough amoun of he nework good o saisfy fully demand over he produc life cycle. Based on he assumpions given in earlier secions, we form he dynamic adjusmen of he nework size which can be expressed by he differenial equaion, dx = Gvx ( ( ), p), (6) d G(.) is he nework growh funcion which is a funcion of nework exernaliies and price. G(.) is defined as marginal growh of nework effec. As v(.) increases, he posiive v growh of he nework, G(.), gradually vanishes. i.e., 2 Gv ( ) 2 <. v The monopoly firm chooses an opimal pricing sraegy o maximize discouned presen value of he profi sream, aking ino accoun he impacs of he differen nework growh pahs of he nework associaed wih differen prices. Sales in ime are represened as y, which is he change of he nework size in he dx infiniesimal ime inerval, i.e., y = = G( ). Hence, he oal sales from ime o d T ime T is given by xt ( ) = yd. For simpliciy, assume he firm ses a fixed price over he his produc's lifeime. The firm's problem becomes T r Max e ( p c) yd s.. dx = Gvx ( ( ), p), d x(o)=. (7) where r is he discoun rae. The Hamilonian is r Η= e ( p c) y + φ G( v( x ), p) (8) Define λ r = e φ by he curren value ofφ, hen we have curren value Hamilonian H = ( p c) G(.) + λg(.) (9) = ( p c+ λ ) G(.) Assume ha here exiss inerior soluion. The necessary condiions for his maximizaion problem are, -7-
H λ Gvx p = ( p c+ ( ( ), ) ) + Gvx ( ( ), p) = () dλ H = rλ d x G v rλ p c λ = ( + ) v x () dx d = y = G( v( x ), p), and x()=. (2) where "^" means he opimal soluions for each variable. These necessary condiions yield he opimal ime pahs for λ and x, and hen y = G( v( x ), p) and opimal p. λ is convenionally inerpreed as he shadow value or shadow price of nework growh, G(.). The ransversaliy condiion ells us ha λ, λ GT T ( ) =, where T is he erminal ime. Rearrange equaion () and subsiue y for G(.) o ge y p ( c + ) y = λ (3) p y Define he elasiciy of demand by ε =, and rewrie equaion (3), y p p ( c + ) = λ (4) ε This is jus radiional heory of monopoly pricing; he monopolis produces where marginal revenue equals marginal cos, i.e., MR = p + = c = MC. n he presence ε of nework exernaliies, he monopolis chooses a price lower han c because he value of marginal nework growh, λ, also affecs marginal cos. As Figure 3 indicaes graphically, he monopolis lowers his price (P n ) below ha which prevails in he absence of nework exernaliies (P m ) o induce more presen demand because his ulimaely leads o larger sales volume and profis. -8-
p Pm D* Pn ym yn MR c c- λ y Figure 3. Monopolisic pricing in he presence of nework exernaliies. Opimal Upgrade Time Now exend he above one-generaion model o a wo-generaion seing problem and ry o find he opimal sofware upgrade ime. Assume he firm will offer only one upgrade version of he sofware. Given he nework exernaliies, firs we assume ha he second ediion, x is perfecly backward compaible wih he firs ediion, x. means x ( ) = x ( ) where denoes he erminal ime for he firs ediion. There won' be any firs ediion of he sofware available afer he inroducion of he second ediion. Under such assumpions, he opimal conrol problem becomes Maximizing r r e ( p c ) y d + e ( p c ) y d T 2 (5) s.. y = G( v( x ), p), and x ( ) =, y = G( v ( x ), p ), and x ( ) = x ( ) 2 Subsiue y = dx / d and y2 = dx / d ino equaion (5), i can be rewrien as dx T r r dx e ( p c ) d + e p c d d ( ) d r r = e ( p c ) dx + e ( p c ) dx T -9-
T r r ( ) ( ) r rt r ( ) ( ) = ( ) r p c e dx + ( p c ) r e dx = ( p c ) e x () + ( p c ) e x () = ( p c ) e x ( ) + ( p c ) e x ( T) e x ( ) (6) n order o maximize he sum of he wo generaions' profis over ime. The firs order condiion is o ake he derivaive of equaion (6) wih respec o and se i o be zero. Then we obain ( p c) e dx ( ) + ( p2 c2) e d dx ( ) = d r r (7) Muliplying boh sides wih e r and rearranging i, Equaion (7) can be rewrien as dx dx / d / d y( ) G(.) = = = y ( ) G (.) 2 p p c, or (8) c ( p c ) G (.) = ( p c ) G (.) (9) 2 According o equaion (9), he opimal upgrade ime occurs when gross profi of he firs ediion, ( p c ) G (.), equals he gross profi, ( p c ) G (.), of he second ediion of he 2 sofware, or a he ime when he raio of he nework growh of wo generaions is equal o he reciprocal of he gross profi raio. The opimal upgrade ime and he firm's profi flow are shown in figure 4. $ (p-c) G (.) (p-c) G (.) 2 x *( ) x (T) x (T) x,x Figure 4. The opimal upgrade ime and he firm's revenue. --
$ (p-c) G (.) Profi Loss (p-c) G (.) 2 x '( ) x (T) x,x Figure 5a. A case of laer promoion. $ (p-c) G (.) Profi Loss (p-c) G (.) 2 x "( ) x (T) x,x Figure 5b. A case of earlier promoion. Figure 5a and 5b show he wo cases in which sofware promoion occurs oo early or oo lae shown in figure 5a and 5b. We imagine ha someimes when sofware vendors announce a delay in inroducing nex version of sofware, he reasons o delay usually is ha he new version has no been well developed ye. However, according o our model, anoher ineresing explanaion migh be ha he presen version is sill relaively profiable in he mean ime. V. Conclusions According o our wo-sage saic model, we found ha he monopolis will charge a lower price for he inroducion version of sofware han a monopolis usually does. makes sense ha a monopolisic firm ses a lower price o induce more users o ge ino --
he nework in he firs sage in order o exrac more profi from he nework effecs in he second sage. The firm's sraegy is nohing bu seing he price a he poin where marginal revenue equals marginal cos which is he sum of marginal cos a presen period and discouned opporuniy cos in he fuure. n our dynamic model, he similar pricing sraegies of he monopolisic firm are also observed a which marginal revenue is equal o presen marginal cos minus shadow value of he nework growh. And he firm's opimal upgrade ime is shown in our wogeneraion dynamic model. We can see ha he opimal upgrade ime is a he ime when gross profi of he firs version is equal o he gross profi of he second version. There are a lo of ways in which he models could be generalized and exended. For example, i would be more general o assume here is monopolisic compeiion in he sofware marke. f we exend he models o more han one firm in compeing he nework marke, i will be much ineresing o see wha he firms' behaviors and pricing sraegies including he cooperaion and predaion. The models can also be exended o link he hardware and sofware since here mus be some nework-effec relaionship beween hem. We already saw ha Novell, a big nework sofware provider, is going o incorporae hardware producion. nel, a bigges 8x86 CPU supplier, has enered he sofware marke o ry o ge more business based one he nework effec of i CPU produc. As we see, he linkage beween hardware-sofware paradigm will become an ineresing issue and i deserves more researches. -2-
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