12 PDMA VISIONS OCTOBER 2002 VOL. XXVI NO. 4

Technical Report

Want to know how diffusion speed varies across countriesand products? Try using a Bass modelBy Christophe Van den Bulte, Assistant Professor of Marketing, The Wharton School of the University of Pennsylvania(vdbulte@wharton.upenn.edu)

For global product managers, one big question is how fast a new product is likely to sellin different countries. This question is daunting. But slowly research and data are buildingup that can answer that question. In a meta-analysis of diffusion speed research,Christophe Van den Bulte takes the mystery out of diffusion speedand gives productmanagers a useful quantitative tool for measuring it in certain product categories incertain regions of the world.

all published studies and look for patterns.This piggy-back strategy where one doesnot collect new raw data but instead ana-lyzes other peoples analyses has long beenaccepted in medicine and psychology. It iscalled meta-analysis. This is the researchstrategy I will follow. But how does onequantify diffusion speed?

Using the Bass modelFor over 30 years, marketing scientists

have been using a simple mathematicalmodel to study the diffusion of innovations.It is often referred to as the Bass model,after Professor Frank M. Bass who first ap-plied it to marketing problems. Here is howthe model works.

There is a market consisting of m con-sumers who will ultimately adopt. (I use theword consumers, but the model can be

Christophe Van den BulteUniversity of Pennsylvania

How does the speed at which newproducts get adopted and diffusethrough the market vary acrosscountries and products? Oneway to try to answer that question is bycollecting data on the diffusion of a largenumber of products in a large number ofcountries, and analyzing them to identifysystematic patterns.

However, building such a database wouldbe an enormous task. Fortunately, thereis an alternative. For over four decades,sociologists and marketing academics havebeen studying the diffusion of new products.So, the alternative is to pool the results of

applied to business markets as well.) Letscall N(t-1) the number of people who havealready adopted before time t. The modelassumes that the probability that someoneadopts given that he or she has not adoptedyet consists of two factors. First, there is afixed factor p that reflects peoples intrin-sic tendency to adopt the new product.Second, there is a factor that reflects wordof mouth or social contagion, such thatpeople are more likely to adopt the largerthe proportion of the market that has

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I n 1980, the U.S. department of En-ergy used the Bass model to forecastthe adoption of solar batteries. Itperformed a survey of home buildersto assess their perceptions of the market-place to obtain reasonable values for pand q. Feeding them into the model, theresearchers came to the conclusion thatthe technology was not far enough alongto generate positive word-of-mouth. As aconsequence, they decided to postponewide scale introduction until the tech-nology had improved enough so thatusers would be quite satisfied with it,resulting in a higher q and hence fastersales growth after launch.

In the early 1990s, DirecTV plannedthe launch of its subscription satellitetelevision service. It wanted to obtainprelaunch forecasts over a five-yearhorizon. The forecast were based on the

Bass model, and the values for its para-meters were obtained from a survey ofstated intentions combined with thehistory of analogous products. Theforecasts obtained in 1992 proved to bequite good in comparison with actualsubscriptions over the five-year periodfrom 1994 to 1999.

Several firms have reported using theBass model to their satisfaction. Often,they end up extending the simple modelto further aid their decision making. In themid-1980s, RCA used an extension of theBass model to forecast the sales of musicCDs as a function of the sales of CD play-ers. The model was quite accurate in itspredications. Some movie exhibitors inEurope now use an extension of the Bassmodel to forecast box office revenues formovies, and to decide on how manyscreens to show a movie.

Copyright 2002 Product Development & Management Association. All rights reserved. No part of this publication may be reproduced byany means, nor transmitted, nor translated into machine language, without the written permission of the publisher, PDMA.

13PDMA VISIONS OCTOBER 2002 VOL. XXVI NO. 4

already adopted. Since that proportion issimply N(t-1) divided by m, the rate at whichnew people adopt is p + qN(t-1)/m, where qcaptures the influence of word of mouth.So, since the number of people who havenot adopted yet before time t is m - N(t-1),and the rate at which these people turninto new adopters is p + qN(t-1)/m, onecan express the number of adoptions occur-ring at time t as:

N(t) - N(t-1) = [p + qN(t-1)/m] x [m - N(t-1)]

The model has three unknown parameters:the market size m, the coefficient of innova-tion p, and the coefficient of imitation q. Theparameters can be estimated from real datausing standard statistical software or evenusing the Solver tool embedded withinMicrosoft Excel.

Attractive propertiesThe model has several attractive prop-

erties. When q is larger than p, the cumu-lative number of adopters N(t) follows thetype of S-curve often observed for reallynew product categories. When q is smallerthan p, the cumulative number of adopt-ers follows an inverse J-curve often ob-served for less risky innovations such asnew grocery items, movies, and musicCDs. Exhibit 1 on this page shows thesepatterns, where I rescaled the curves tobe cumulative penetration curves by divid-ing N(t) by m. Exhibit 2 on this page showsthe corresponding patterns for the propor-tion of new adoptions occurring at time t,i.e. [N(t) - N(t-1)]/m. Note that an S-curveas shown in Exhibit 1 for the cumulativeproportion corresponds to the familiarbell-shaped curve for the non-cumulativeproportion of adopters (Exhibit 2) de-scribed in Geoffrey Moores popular bookslike Crossing the Chasm.

Diffusion speed is captured by p and qThe parameters p and q provide us infor-

mation about the speed of diffusion. A highvalue for p indicates that the diffusion hasa quick start but also tapers off quickly. Ahigh value of q indicates that the diffu-sion is slow at first but accelerates af-ter a while. Of course, the number of newadoptions must start to decline at somepoint in time, since the number of peoplewho have not adopted yet [m - N(t-1)]becomes smaller and smaller. Interest-ingly, once one knows p and q, one cancalculate the time at which the peak num-ber of adoptions occurs as:

t* = ln (q/p) / (p + q)

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Exhibit 1: The Bass model fits the classic S-shaped cumulative diffusion curves

Exhibit 2: The Bass model fits the classic bell-shaped diffusion curves

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14 PDMA VISIONS OCTOBER 2002 VOL. XXVI NO. 4

How p and q varyHaving found a way to quantify diffu-

sion speed in terms of p and q and havingchosen to perform a meta-analysis, I cantranslate the broad question I startedwith into something feasible: Taking allpublished applications of the Bass diffusionmodel, what can one say about the speedof diffusion of innovations as capturedin the parameters p and q? Specifically,how do p and q vary across products andcountries?

Constructing the databaseI constructed a database containing

1586 sets of p and q parameters, from 113papers published between January 1969and May 2000. Note that some of these 1586observations include multiple p and q val-ues for the same diffusion process. Forinstance, many studies have investigatedthe diffusion of color television in the U.S.I drastically reduced this database. First,I deleted all entries relating to filmsand drugs, because these products dif-fused much faster than average. Next, Ikept only those 40 countries for which Ihave data about the extent of collectiv-ism and risk avoidance of their nationalculture. Furthermore, I only kept data re-lating to the period between 1950 and1992, since this is the only window forwhich high-quality international data onpurchasing power per capita are avail-able. Finally, I collapsed multiple obser-vations of the same product in the samecountry into a single average value. Theseseveral steps reduced the data set to 188unique sets of p and q.

Averages are better than nothing, butmarket analysts and managers want morefine-grained information. Say you have anew product that perhaps is not evenlaunched yet. Based on the diffusion history

of previous product categories, what kindof diffusion curve can you expect for yournew product? That is, what values for pand q should you use in the last equationgiven above to forecast your adoptioncurve? The answers from my statisticalanalyses are given in Exhibits 3 and 4.As a baseline, I took consumer durableslaunched in the US in 1976. The exhibits

Exhibit 4: Whats your best guess for q?

Best guess 90% confidence intervalBaseline case:US, consumer, durable, launch in 76 0.409 0.355 0.471

For other cases, multiply by the following factors

Cellular telephone 0.635 0.465 0.868

Non durable product 0.931 0.713 1.216

Industrial 1.149 0.909 1.451

Non commercial innovation 2.406 1.488 3.891

Western Europe 0.949 0.748 1.203

Asia 0.743 0.571 0.966

Other regions 0.699 0.429 1.137

For each year after 1976, multiply by 1.028 1.018 1.039

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The Bass model is a handytool to look at diffusion patterns.

In short, the Bass model is ahandy model that one can use toquantify the speed of diffusion.Better still: assuming that westart at the time of launch (where

t = 0 and N(0) = 0), and that wehave values for m, p and q thatwe feel comfortable about, themodel can also be used to fore-cast future adoptions using thefollowing formula:

N(t) = m x [1 exp{-(p+q)t}] /[1 + (q/p) exp{-(p+q)t}]

Exhibit 3: Whats your best guess for p?

Best guess 90% confidence intervalBaseline case:US, consumer, durable, launch in 76 0.016 0.012 0.021

For other cases, multiply by the following factors

Cellular telephone 0.226 0.125 0.409

Non durable product 0.689 0.415 1.143

Industrial 1.058 0.679 1.650

Non commercial innovation 0.365 0.146 0.910

Western Europe 0.464 0.296 0.729

Asia 0.595 0.360 0.981

Other regions 0.796 0.315 2.008

For each year after 1976, multiply by 1.021 1.002 1.041

15PDMA VISIONS OCTOBER 2002 VOL. XXVI NO. 4

give values for p and q for such cases,and also indicate how you should changethose values if your case does not fit thatbaseline. The exhibits also report 90percent confidence intervals. Those areintervals or windows of values such thatone would expect the (unknown) truevalue to fall within that interval with a90 percent probability. As such, the in-tervals reflect uncertainty.

When applying the values in Exhibits 3and 4, shown on this page, to your owncase, it would be a good idea to do so forseveral values within the confidence inter-vals. This will allow you to see how sensi-tive the predicted curve is to the level ofuncertainty in my analysis. Also, as a gen-eral rule in forecasting, you should not useonly forecasts from the Bass model, orfrom any other model or technique. Itwould be better to obtain several estimatesof the level of adoptions using differentmethods, and then to average them.

The results shown in Exhibit 3 andExhibit 4 describe differences acrossproduct categories and countries. I alsoperformed several additional analyses toexplain why such differences occur. Spaceconstraints do not allow me to presentdetailed results, so I limit myself to themain conclusions.

First, in countries with a collectivisticrather than individualistic culture (e.g.,Japan vs. US), q is higher. This makes alot of sense, since people in collectivisticcultures care more about others opin-ions. The effect, however, is not verystrong. Second, in countries with higherpurchasing power per capita, p is higher.That makes perfect sense as well, sincehaving a higher purchasing budget makesit easier to adopt new products immedi-ately. Finally, products and technologiesthat exhibit network effects (like the VCRand the fax machine) or require heavyinvestments in complementary infra-structure by suppliers (like television orcellular telephone) have a higher q. Thisis consistent with prior research indicat-ing that for such categories, people tendto wait until enough other people haveadopted and, when there are competingstandards, until it has become clear what

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technology will survive. Interestingly, palso tends to be lower for products withnetwork effects, making the S-shape evenmore pronounced.

ConclusionsThere are systematic differences in

diffusion of global products which emergeusing a Bass model analysis. Here are myconclusions:

The Bass model is a handy tool to lookat diffusion patterns. Moreover, thequantitative best guesses from meta-analysis can be useful for predictingfuture adoptions, even when the producthas not been launched yet.

There are systematic regional differencesin diffusion patterns.

The average coefficient of innovation p(speed of take-off) in Europe and Asia isroughly half of that in the U.S.

The average coef-ficient of imitationq (speed of lategrowth) in Asiais roughly a quar-ter less than thatin the U.S. andEurope.

Also, economic dif-ferences explainnational variationsin speed betterthan cultural dif-ferences do.

There are system-atic product dif-ferences in diffu-sion patterns. Forinstance, take-offis slower for non-d u r a b l e s a n dp ro d u c t s w i t hc ompeting stan-dards that requireheavy investmentsin infrastructure,while late growth

There are systematic productdifferences in diffusion patterns.

is faster for industrial products andproducts with competing standardswhich require heavy investments ininfrastructure. w

Endnotes: Armstrong, J. Scott , Editor, Principles

of Forecasting: A Handbook for Research-ers and Practitioners. Boston: KluwerAcademic Publishers, 2001. [See also

Professor Armstrongs website: market-

ing. wharton.upenn.edu/forecast.]

Bass, Frank M., A New Product Growth

Model for Consumer Durables, Manage-ment Science, 15 (January 1969), pp.215-227. [Website: basseconomics.com.]

Lilien, Gary L. and Arvind Rangaswamy,

Marketing Engineering: Computer-As-sisted Marketing Analysis and Planning,2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2003. [Website: www.mktgeng.com.]

Lilien, Gary L., Arvind Rangaswamy, and

Christophe Van den Bulte, Diffusion Mod-

els: Managerial Applications and Soft-

ware, Pp. 295-336 in New-Product Dif-fusion Models, edited by Vijay Mahajan,Eitan Muller and Jerry Wind. Boston, MA:

Kluwer Academic Publishers, 2000.

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