1KKU Res. J.(be) 2012; 11(1)

Generalized Network Externality Function

Arnut Paothong 1* and G.S. Ladde

1 Department of Mathematics and Statistics, University of South Florida 4202 East Fowler Avenue, PHY 114, Tampa Florida, 33620-5700 USA* Correspondent author: apaothon@mail.usf.edu

Abstract

With regards to thenetworkexternality function, thefirstderivative is commonlyconsidered tobe

positive.However,thesignofitssecondderivativeisbasedontheassumptionaboutthemarginalnetworkvalue

whenthenetworkgrows.Threekeyassumptionsareconstant,decreasing,orincreasingwhichgenerateadifferent

functionalform,linear,concave,convexfunction,respectively.Thereisanefforttomixthesestrictassumptions

by combining themwhich generate theS-shaped function such as logistic,Gompertz or Sigmoid function.

Unfortunately,todoso,theS-shapedfunctioncausesaninappropriatedomain,negativenetworksize.Inthis

work,wedevelopadynamicmodeltogeneralizeafunctionalformofnetworkexternalityfunctionwhichnot

onlykeepsallgoodpropertiesoftheS-shapedfunction,butalsohasadesireddomain.

Keywords: Networkexternality,S-shapedfunction,Sigmoidfunction

JEL Classification: C51,D01,D11

KKU Res. J.(be) 2012; 11(1): 1-9http : //resjournal.kku.ac.th

2 KKU Res. J.(be) 2012; 11(1)

1. Introduction

The network externality function (Katz

andShapiro, 1985) is the function that describes the

relationshipbetweennetworkvalueanditscorresponding

size.Thisfunctionissuccessfullyutilizedinthestudyof

economicsofnetworkindustries(Shy,2001).Weremark

thattheconceptofnetworkexternalitywasintroduced

byBell’semployee,N.Lytkins(1917).Historically,the

developmentofthefunctioniscenteredontheassumption

ofproperties.LetNbenetworksize, bethe

networkexternalityfunctionandbe thenetwork

externalityvalue.

Becauseofthedefinitionofnetworkexternality

(Economides, 1996), network externality value is

commonly assumed to increasewhen network size

increases;thatis,thefirstderivativeofis positive.

Hence,

(1.1)

Lately,thediminishingconcept,inadditionto

thepositiveslope,isalsonormallyassumed

(1.2)

The properties in (1.2) supports that the

networkexternalityistheconcavefunction.However,

in general,Hans-WernerGottinger (2003, p.17) said

threekeyassumptionsabout therelationshipbetween

network size and network externality value relate to

linear, logarithmic and exponential functional form.

Thelinearfunctionpostulatesthat,asnetworksgrow,

themarginalvalueisconstant.Thelogarithmicfunction

postulates that, as a network grows, themarginal

valuediminishes.Networkexternalitiesatthelimitin

this formulationmustbeeithernegativeorzero.The

exponentialfunctionpostulatesthat,asanetworkgrows,

themarginal value increases,which in the popular

business and technology press, has been named

‘Metcalfe’sLaw,RobertMetcalfe (1995).Moreover,

anS-shapedfunction,inadditiontotheseassumptions,

isamixtureofanexponentialfunctionandalogarithm

function;thatis,theearlyadditionstothenetworkadd

exponentially,yetlateradditionsadddiminishinglyin

networkexternalityvalue.TheS-shapedfunctionisan

increasingfunctionandhastwohorizontalasymptotes

which are the appropriate properties of a network

externalityfunction.InadditiontotheS-shapedfunction,

anN-shapedfunctionisanothermixture;thatis,early

additionsadddiminishingproperty,yetlateradditions

addexponentialproperty.Insummary,theshapeofa

networkexternalityfunctionisbasedontheassumptions

ofparticulargoods.SeeTable1.1.

Remark 1:

TheexampleofanN-shapednetworkexternality

function is the product that develops itselfwhen the

networksizereachesacertainlevel.Thesmartphones,for

example,hasfewapplicationsinthebeginninginwhich

diminishing concept inmarginal network externality

isapplied.Later,whennetworksizeincreases,thereare

more developers creatingmanynewapplications.At

thispoint,themarginalnetworkexternalityisnolonger

diminishing,butexponentiallyincreasing.

3KKU Res. J.(be) 2012; 11(1)

Table 1.1 Thecharacteristicofnetworkexternalityfunctionundervariousassumptions

Assumption ( ) Shape

I Zero line

II Negative logarithm

III Positive exponential

IVpositivefortheearlygrowth,

negativeforlategrowthS-shape

Vnegativefortheearlygrowth,

positiveforlategrowthN-shape

Nextsection,wedevelopadynamicprocess

togeneralizeafunctionalformofanetworkexternality

functionwhichmatchesallassumptionsandkeepsall

appropriateproperties.Thedevelopmentofthisfunction

ismotivated by Sigmoid function also called the

Sigmoidalcurve(vonSeggern,2007).

2. Network Externality Process

The linear, logarithm, exponential and

S-shaped function can be considered the solution

of corresponding differential equations. Thewell

knownS-shapedfunctionisSigmoidfunction,logistic

function,Gompertzfunctionandacumulativedistribution

function. Sigmoid function is the solution of the

differentialequation.

(2.1)

where Thesolutionof(2.1)is

(2.2)

where The equation (2.2)

isincreasinganddifferentiablefunctionoverdomain,

, and has two horizontal asymptotes,

and This

functionmeets the commonproperties of a network

externalityfunction(KatzandShapiro,1985),

and However, i ts domain,

representingthenetworksize,isinappropriatebecause

the network size cannot be negative. See Figure

2.1a. In this section, we introduce a differential

equation in which its solution is the generalized

networkexternalityfunctionthatkeepsallpropertiesof

anS-shapedfunctionandeliminatestheinappropriate

domain.Letusdefinesometerms.

Definition 2.1 Lower Limit of Network

ExternalityValueisthegreatestlowerbound or

theinfimumoftherangeofagivennetworkexternality

function, .

Definition 2.2 Upper Limit of Network

ExternalityValueistheleastupperbound or

thesupremumoftherangeofthenetworkexternality

function, .

Thus,byitself,thenetworkexternalityvalue

canbeexplainedbytheSigmoiddifferentialequation

(2.3)

for

Definition 2.3LowerLimitofNetworkSizeis

thegreatestlowerboundortheinfimumofthedomain

4 KKU Res. J.(be) 2012; 11(1)

ofagivennetworkexternalityfunction, .

Definition 2.4UpperLimitofNetworkSize

istheleastupperboundorthesupremumofthedomain

ofthenetworkexternalityfunction, .

Likeabove,thenetworksizecanbeexplained

bytheSigmoiddifferentialequation

(2.4)

for

Now,wearereadytointroducetheconceptof

networkexternalityprocess.Thedifferentialequation

described by the ratio of the Sigmoid differential

equationofnetworkvalue(2.3)totheSigmoiddifferential

equation of network size (2.4) is called the network

externalitydifferential equation.All goods are called

networkgoods if theirmarketability is influencedby

thenetworkexternalityprocess.Hence,

(2.5)

where i s the

network externality differential equation and its

solutionis

(2.6)

where .Theequation (2.6) iscalled the

generalized network externality function. For further

graphicaldetails,seeFigure2.1b.

Remark 2:

Withoutlossofgenerality,therateofchange,

,ofadifferentialequation(2.5)canberelaxedtohave

a negative signwhich implies the negative network

externality.

Remark 3:

If we assume , , and set

and , then the generalized

networkexternalityfunctionfulfillsthepropertiesofa

cumulativedistributionfunction(CDF).Ingeneral,we

cansaythat isacumulativedistribution

functionofacontinuousrandomvariablebetweenthe

lowerlimitofnetworksizeandtheupperlimitofnetwork

size.

Figure2.1a Figure2.1b

Figure 2.1TheSigmoidandGeneralizedNetworkExternalityFunction

5KKU Res. J.(be) 2012; 11(1)

Beforewediscussmoreabouttheproperties

and advantages of generalized network externality

function,letfirstusdefinemoretechnicalterms.

Definition 2.5 The Lower-Left Terminal

Point (LLTP) is a pair of the greatest lower bound

of the domain and range of the function.Hence, the

LLTPof generalized network externality function is

Definition 2.6 TheUpper-Right Terminal

Point (URTP) is a pair of the least upper bound

of the domain and range of function. Hence, the

URTPof generalizednetwork externality function is

Let be the line that connects these two

points.Thustheequationof is

.LetbetheratiooftheslopeofLLTPandthe

initialpointtotheslopeofURTPandtheinitialpoint

ofthegeneralizednetworkexternalityfunction.

Proposition 2.1.(seeFigure2.2)For

Proof i. If locates on the same line as

two terminalpoints, theslopeofany

two points are equal. Thus, which

implies

Proof ii. Let andlocatesonline ,.Hence,

locatesunderline and

Proof iii. Let andlocatesonline

,.Hence, locatesabove

line and

Figure 2.2 Theillustrationforproposition3.1

3. Properties of the Generalized

Network Externality Function

Inthissection,wecomparesomeproperties

of the network externality function under various

assumptions to the generalized network externality

function(2.6)andalsoitsadvantages.FromTable1.1,

five assumptions aboutmarginal network externality

valuewhennetworkgrowsarelisted.Letusassigna

particular function for thefirst four assumptions.To

makethemcomparable,wealsosetthesamedomain

andrangeforallfunctions(ifpossible).

AssumptionI:

AssumptionII:

AssumptionIII:

AssumptionIV: ,Sigmoidfunction(2.2)

3.1 Domain

In economics, the network size, domain of

networkexternalityfunction,isconsiderednonnegative

value.For ,and ,wecanadjustthem

to get an appropriate domain by shifting, rescaling,

and constraining the domain. The domain of is

inappropriate,includingthenegativevalue,eventhough

theshapeoftheSigmoidfunctionismoreappropriate.

Ifweshiftorrescaleitsdomaintoadesiredinterval,we

willdroptheconvergentpropertyattheendbehavior.

The domain of the generalized network externality

6 KKU Res. J.(be) 2012; 11(1)

functionis .Thisisanadvantage,itnotonly

hasthedesireddomainbutalsosavesallpropertiesof

Sigmoidfunction,especiallytheendbehaviorofboth

sides, and .

3.2 Range

In economics, the network value, range of

network externality function, is also considered

nonnegative value. All function, and

generalizednetworkexternalityfunctionhavethesame

range,

3.3 Monotonicity

Fromthedefinitionofnetworkexternality,with

lowerlimitandupperlimit,thepositivemarginalvalue

ofnetworkvaluein(1.1)isexpected.Again,allfunctions

havethisproperty.However,regardingtodiminishing

concept,theSigmoidandgeneralizednetworkexternality

functionhavethisproperty.Ifweadjustthedomainof

Sigmoid function,wewill lose this property.This is

anotheradvantageofthegeneralizednetworkexternality

function;ithasthediminishingpropertyforbothsides,

3.4 Inflection Point

Ineconomics,theinflectionpointofnetwork

externalityaffects the locationofcriticalmassof the

network.The functions, and , have no

criticalpoint.ThesecondderivativeofSigmoidfunction

(2.2)is

(3.1)

Thus, the Sigmoid function has only one

criticalpoint , and the secondderivativeof

generalizednetworkexternalityfunction(2.6)is

(3.2)

Italsohasonlyonecriticalpoint which

satisfiesthecondition.

(3.3)

3.5 Concavity

The concavity of the network externality

function is varying, based on the assumption about

marginalnetworkvalueofparticulargoods.Thefunctions,

,havezero,negativeandpositiveconcavity,

respectively. The Sigmoid function has positive

concavityintheearlygrowthandnegativeinthelater

growth.Thisisanotheradvantageofgeneralizednetwork

externalityfunction,moreflexibleinconcavity.Itcan

providelinear,concave,convex,S-shapedandN-shaped

functionbycontrollingitsparameters.Thefollowingare

threescenarioswhichgivesvariousshapes.

Scenario I:

The rate of change of (2.5) equals to the

reciprocal of the slopeof .With this condition, the

equation(2.6)isreducedto

(3.4)

Proposition 3.1. For

Proof Let Thus,Theequation(3.5)can

berewrittenas

(3.5)

(3.6)

From(3.2),thesecondderivativeofgeneralizednetwork

externalityfunctioncanberewrittenas

(3.7)

7KKU Res. J.(be) 2012; 11(1)

Plug(3.6)into(3.7),wewillget

(3.8)

Hence,from(3.6)and(3.8),

Fromproposition3.2,when

theequation(2.6)isreducedtolinearfunction.Inother

words, isthespecialcaseofthegeneralizednetwork

externalityfunction.When thegeneralized

network externality functionmeets the increasing

marginal network externality value assumption

(assumptionIII)andwhen itmeets

the decreasingmarginal network value assumption

(assumptionII).SeeTable3.1.

Scenario II:

For this scenario, the generalized network

externalityfunctionisS-shapedandalsopreservesall

propertiesoftheSigmoidfunction.SeeTable3.1.

Scenario III:

For this scenario, the generalized network

externalityfunctionisN-shaped.SeeTable3.1.

Insummary,thefollowingsaretheadvantages

ofthegeneralizednetworkexternalityfunction

i. Ithasanappropriatedomain.

ii. ItpreservesallgoodpropertiesoftheSigmoidfunction.

iii. ItmeetsallassumptioninTable1.1.

Table 3.1Thegeneralizednetworkexternalityfunctioninsomeconditionofparameters

8 KKU Res. J.(be) 2012; 11(1)

iv.

v. Figure 3.1Theillustrationofnetworkexternalityfunction

4. Conclusion

Including the combination of assumptions,

therearefiveassumptionsaboutthemarginalnetwork

externality valuewhennetwork size increases.They

generate different functional forms.Unfortunately,

eachfunctionhassomedisadvantages.Thefunctions,

do not have the diminishing concept

andtheshapeoffunctionistoostrict.Eventhoughthe

Sigmoidfunction, ,hasmoreflexibleshapeandmeets

commonpropertiesofnetworkexternalityfunction,it

hasaninappropriatedomain.Ifwefixitsdomaintothe

desiredinterval,wewilllosesomegoodproperties.By

implying concept of differential equations, thiswork

introduces the new function, called the generalized

networkexternalityfunction,whichnotonlypreserves

all properties of Sigmoid function but also fix the

inappropriatedomain to theadjusteddesireddomain.

Moreover,allnetworkfunctionsunderfiveassumptions

canbereplacedbythisgeneralizednetworkexternality

function.Inthestudyofitsproperties,therateofchange

andtheinitialpointofgeneralizednetworkdifferential

equationplaytheimportantroleincontrollingtheshape

ofgeneralizednetworkexternalityfunction.

5. Reference

Economides, N. 1996. Network externalities,

complementarities, and invitations to enter.

European Journal of Political Economy.12(2):

211–233.

9KKU Res. J.(be) 2012; 11(1)

Gottinger. H.W. 2003. Economies of Network

Industries.London,UK:Routledge.

Katz, M.L., and Shapiro, C. 1985. Network

Externalities,Competition, andCompatibility,

American Economic Review.75:424-440.

Metcalfe,R.1995.Metcalfe’sLaw:NetworkBecomes

more Valuable as it ReachesMore Users.

InfoWorld.17(40):53.

Paothong,A.2011:DynamicEquilibriumforNetwork

Goods,Ph.D.thesis.UniversityofSouthFlorida,

USA(inpreparation).

vonSeggern,D.2007.CRCStandardCurvesandSurfac-

eswithMathematics.Florida,USA:CRCPress.

Shy,O.2001.TheEconomicsofNetworkIndustries.

Cambridge,UK:CambridgeUniversityPress.