NEW PRODUCT INTRODUCTION STRATEGY: THE CASE OF FUSION ANDMULTI-FUNCTION PRODUCTS

By

YUWEN CHEN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

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c© 2008 Yuwen Chen

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To everyone who helped me.

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ACKNOWLEDGMENTS

I want to thank my wonderful parents. They provided me the greatest education in

my life and generously let me pursue my dreams with unconditional love and support.

I want to say ”thank you” to my friend, Eric. His life and business experiences helped

me to revise my resume and show my full potential. When I was less than positive in my

outlook, Eric patiently listened to my frustration, shared with me his life stories, guided

me to see life and view the world with a different lens, and he also taught me how to

windsurf. From him and my personal experience, I also have learned that “in life, things

happen for a reason.”

For my dissertation, I especially appreciate Dr. Vakharia’s fully supporting me

to pursue this interesting topic. He believed in me, encouraged me to keep going, and

provided opportunities to be free of financial worries. I want to thank Dr. Carrillo for

patiently reading through my numerous editions and providing me with helpful comments.

A special thank-you to Dr. Alptekinoglu. His rigorous critiques and contribution resulted

in a successful publication of our paper. Many thanks go to all of the professors, who have

assisted me to finish this dissertation. I want to thank all the staff and schoolmates of

ISOM, and friends in Gainesville, my Tango friends, my online and overseas friends. It was

they who gave me help, laughs, entertainment, and understanding when I need them.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Fusion Products: An Overview . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Market Impact of Fusion Products . . . . . . . . . . . . . . . . . . . . . . 151.3 Portfolio Decisions of Fusion and Single-Function Products . . . . . . . . . 181.4 Designing Fusion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Diffusion of Fusion and Single-Function Products . . . . . . . . . . . . . . 211.6 Focus of This Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 OPTIMAL PRODUCT PORTFOLIO STRATEGY: TWO SINGLE FUNCTIONPRODUCTS AND ONE MULTIFUNCTION PRODUCT . . . . . . . . . . . . 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Fusion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Product Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.3 Vertical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Optimal Product Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . 382.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6 Implications and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 FUSION PRODUCT DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Literature: Product Variety and Product Line Selection . . . . . . . . . . . 523.3 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.2 Dominant Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 One All-in-One Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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4 PRODUCT DIFFUSION MODEL FOR SINGLE-FUNCTION AND FUSIONPRODUCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 Diffusion of Single-Function Products . . . . . . . . . . . . . . . . . . . . . 804.4 After the Availability of the Fusion Product . . . . . . . . . . . . . . . . . 834.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5.1 Market Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.2 Role of the Fusion Product Supplier . . . . . . . . . . . . . . . . . . 954.5.3 Synergy of the FP: Faster Diffusion Speed, Higher Margin, and Market

Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.4 Maturity of the Fusion Technology and the Development Cost . . . 994.5.5 Time Related Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.6 Market Size and Substitution . . . . . . . . . . . . . . . . . . . . . . 1014.5.7 Market Overlap: From Two SPs to One FP . . . . . . . . . . . . . . 1014.5.8 Summary of the Optimal Launch Time . . . . . . . . . . . . . . . . 1024.5.9 Impacts on the Total Profit . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Key Results in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Key Results in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3 Key Results in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.4 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

APPENDIX

A CONCAVITY OF THE PROFIT FUNCTION . . . . . . . . . . . . . . . . . . . 115

B PROOF FOR THEOREM 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C HESSIAN MATRIX FOR CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . 120

D THE OPTIMAL QUANTITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

E QUANTITY AND PROFIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

F PROOF FOR THEOREM 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

G PROOF SUPPLEMENT FOR THEOREM 3.1 . . . . . . . . . . . . . . . . . . 124

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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LIST OF TABLES

Table page

1-1 Examples of Single-Function Products and Corresponding Fusion Products . . . 12

2-1 Results of Theorem 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-2 Dominant Product Portfolio Strategies. . . . . . . . . . . . . . . . . . . . . . . . 40

2-3 Comparative Statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3-1 Comparison of Product Line Selection Models . . . . . . . . . . . . . . . . . . . 55

3-2 Variable Notation for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3-3 Congruent Substitution Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3-4 Incongruent Substitution Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3-5 Optimal Product Portfolio Reacting to Parameter Value Change . . . . . . . . . 70

4-1 Notation and Acronyms for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 82

4-2 Optimal Launch Time in Various Cases . . . . . . . . . . . . . . . . . . . . . . . 93

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LIST OF FIGURES

Figure page

2-1 r13: Price versus Market Effects at Different Levels . . . . . . . . . . . . . . . . 45

2-2 r23: Price versus Market Effects at Different Levels . . . . . . . . . . . . . . . . 45

2-3 r13 and r23: Cost versus Market Effects at Different Level . . . . . . . . . . . . . 46

2-4 r13 and d3: Dominant Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2-5 r23 and d3: Dominant Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2-6 r13 and r23: Dominant Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4-1 Development Cost at Different Launch Time of FP . . . . . . . . . . . . . . . . 84

4-2 Examples of Cumulative Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4-3 Optimal Launch Time of the Fusion Product versus Various Factors . . . . . . . 97

4-4 Impact of Parameter Change on The Profit . . . . . . . . . . . . . . . . . . . . 104

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of DOCTOR OF PHILOSOPHY

NEW PRODUCT INTRODUCTION STRATEGY: THE CASE OF FUSION ANDMULTI-FUNCTION PRODUCTS

By

Yuwen Chen

May 2008

Chair: Asoo J. VakhariaCochair: Janice E. CarrilloMajor: Business Administration

Devices that integrate multiple functions together are popular in consumer electronic

markets. Examples include the cellular phone that takes digital pictures and plays

MP3’s, the PDA with cell phone, and multifunction office machines. We describe

these multifunction devices as fusion products since they fuse together products which

traditionally stand alone in the marketplace.

The first contribution of this study is the guidelines of the optimal product portfolio

for a firm who offers two single-function and one multi-function products. We utilize a

linear price-demand function which captures the substitution effects between products.

We identify the possible product portfolios and the corresponding condition which favor

each of them. The fusion product with higher profit margin and less cannibalization of

the single-function products is more likely to be included in the optimal portfolio. The

single-function products that have low profit margins and can be easily substituted are

often replaced by the fusion products. This provides a potential explanation for the

proliferation of multi-function products in the market coupled with the simultaneous

reduction of single-function products offered.

The second contribution of this study provides insights of what fusion products should

be designed and offered and what product portfolio should be chosen when there are more

than two functions can be integrated into any combination of fusion product. Even though

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the number of possible portfolios can be extremely large, we propose an algorithm which

efficiently identifies the optimal solution. The optimal product portfolio is often a small

subset of all product variants and offering all-in-one product is not always optimal.

The third contribution of this study is building a product diffusion model which

considers the switching from single-function products to fusion product. This model

assumes two single-function products are already in the market and the firm tries to

find the optimal launch time for the fusion product. Based on Bass diffusion model, we

conclude that the firm should consider the diffusions of all related products. We identify

some factors that are unique to the whole diffusion of single-function and fusion products:

the competitive role of the supplier, the development cost, the switching rates from

single-function products to the fusion product, and how many users who will eventually

buy one fusion product instead of two single-function products.

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CHAPTER 1INTRODUCTION

1.1 Fusion Products: An Overview

New technology advances have blurred the boundaries among various products.

Previously, printers, scanners, copiers, fax machines, PDAs, and cell phones were all

independent stand-alone products with little, if any, overlap in terms of functionality.

In recent years, however, we have been introduced to many multi-function products

(MFPs, we refer to multi-function products as fusion products.1 ) such as printers with an

integrated scanner, copiers with fax capability, PDAs that can also be used as cell phones,

and cell phones that can take digital photographs, record digital video, and play digital

music. Although fusion products are not an entirely new phenomenon (e.g., the Swiss

Army knife, the clock-radio, and the cassette player and radio tuner), it has been argued

that technology advances in digital electronics are the primary drivers for the more recent

proliferation of such products in the current market place [36].

We call integrating and consolidating all functions into an all-in-one device fusion

technology. Fusion products prospers when the fusion technology becomes mature and is

acceptable to the end users. In the digital electronics market, the emergence of standards

and protocols in networking, interface and wireless technology makes product-fusion

possible and attractive [36]. Fusion products capture the strong demand of electronic

users: elminating the need for multiple devices by consolidating seamlessly all the different

functions; computing, organizing, communication, input/output transforming, and

transmitting, data storage, and entertaining, into a single device. Purchasing a fusion

product normally costs the buyer less than the total cost of several single-function

1 In the dissertation, we use “multi-function product” and “fusion product”interchangeablely depending on the context and the convenience of acronyms. For similarreasons, the hyphen in ‘single-function’ is sometimes omitted. In Chapter 2, we usemulti-function product (MFP) to contrast with single function product (SFP). In Chapters3 and 4, we use fusion product (FP) to contrast with single-function product (SP).

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Table 1-1. Examples of single-function products and corresponding fusion products.

single-function products Examples of fusion productsCopy machine, printer,

scanner and fax machineDifferent combinations ofall-in-one office machines

CD player and radio CD and radio stereoCell phone, digital camera, digitalmusic player, PDA, hand-held PC

Different combinations ofmulti-function personal devices

TV and DVD player TV/DVD combo setSunglasses and digital music player Digital music player embedded sunglasses

Refrigerator, TV, Computer Refrigerator with built in TV & internet access

products. In some cases, when the buyer evaluates function consolidation with a high

premium, the price of a fusion product may be greater than the sum of single-function

products, such as PalmOne’s Treo 700.

We refer to these new multi-function devices as fusion products for several reasons.

First, “multi-function product” (sometimes called “multi-function peripheral” and both

are abbreviated as MFP) has been commonly described in business and technology

journals ([16], [25], [3], [4], [36]) to indicate a multi-function office machine which puts

a copier, a printer, a fax machine, and a scanner into an all-in-one device. Second,

‘Integration’ is widely used in new product development and design. In many new product

development papers ([20], [23], [35]), the integration focuses on the development process

or cross-function cooperation, which centers on product development operations. Better

integration of multi-function design team and faster product development are the major

concerns of these papers. However, ‘integration’ in new product development is a very

different concept from the ‘integration’ of functionalities into a product. We avoid using

‘integration’ to designate multi-function devices. Third, according to the Merriam-Webster

dictionary, ‘fusion’ means “a merging of diverse, distinct, or separate elements into a

unified whole.” We think ‘fusion’ more properly specifies the harmonizing, integrating, and

consolidating of a variety of functions in an all-in-one multi-function device.

In Table 1-1, we list some examples of fusion products and their corresponding

single-function products. One interesting fusion product is the Oakley Trump Polarized

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Sunglasses, which adds a digital music player on a pair of sunglasses. The website (http:

//oakley.com/o/o2213d) stated:

Flip-up lenses with contours that maximize protection against sun, wind andside impact. Patented XYZ Opticsr for razor sharp clarity at all angles ofvision, even at lens periphery. Plutoniter lens material to filter out 100% ofall UVA, UVB, UVC and harmful blue light. Durable O Matterr frame withless than 1.8-ounce total weight for all-day comfort.... Absolute music freedomwith no wires or cords to dangle or tangle. [It] stores up to 120 songs on the512 MB version or up to 60 songs on the 256 MB version Solid-state NANDflash memory

Right after the launch in 2004, wearing such a pair of sunglasses and enjoying digital

music is associated with a high premium. In November 2004, one pair of Oakley Trump

Polarized Sunglasses was priced at approximately $500 USD, which is costlier than a

pair of premium sunglasses, approximately $150 USD, plus a 512 MB iPod Shuffle music

player, which costs approximately $100 USD.

Many market analysts predict that a multi-function personal gadget will eventually

replace many personal electronic products. In Gartner’s report2 , 53% of all PDA’s have

some sort of integrated cellular capability3 . The PDA market is driven toward this

direction from wireless service carriers who subsidize substantially for the device, and they

drive the PDA phone price down in exchange of monthly data service fee 4 5 Based on

Gartner’s Latin America market report, in the first quarter of 2003, the single-function

2 In Gartner’s report, the definition of PDA includes single-function PDA andcellular-equipped PDA and PDA is seen as a data-centric (voice second) device. Thesmartphone, which offers all the attributes of PDA, is a voice-centric (data second) device.

3 Study: smart phone, PDA shipment up 57 percent, by S. Ferguson, October 10, 2006,www.eweek.com.

4 Gartner says PDA shipments grew 32 percent in Q3, November 17, 2006, www.cellular-news.com/story/20464.php

5 Worldwide PDA and smartphone shipments market grew 57 percent, October 11,2006, www.cellular-news.com/story/19802.php..

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inkjet printer sales declined by 5% compared to the same period during 2002, while

the sales of multi-function inkjet printer increased 176% during the same period. Many

fusion products cannot be easily categorized into one product market because they

simultaneously perform multiple functions across a product category spectrum.

Rust et al. [80] point out several drivers for of the proliferation of fusion products.

First, adding a new feature or an extra function costs little or even nothing, such that

“engineers can’t resist the temptation to equip existing electronic components with more

functions.” Second, the manufacturer is “aiming to hit two birds with one stone.” Third,

the marketers believe that “more is better” and adding additional features will make the

product more appealing. From the demand side, Thompson et al. [82] find that most

consumers6 do perceive that “more is better” before they really buy and use the fusion

products. It was observed that adding too many features decreases the usability of the

fusion product to the consumer and they experience buyer’s regret. Thompson et al.

[82] mention that the product design engineers do not foresee this problem in the early

developing stage of fusion product design.

Note that each major function integrated in a fusion product typically has an

associated single-function device available in the market. The fusion product, as a result of

coexistence with the single-function product, substitutes the demand of its corresponding

single-function component products to some degree. From the view point of aggregate

demand, some buyers of fusion products will not buy some (or all) single-function

products that have been integrated into the fusion products. Conversely, some buyers

who have certain single-function products may not purchase the fusion products, which

consolidate those single-function products, because those single-function products have

6 In this dissertation, we use ‘buyer’, ‘consumer’, and ‘customer’ exchangeablly toindicate entity who purchases or uses the product. We aslo use ‘firm’, ‘supplier’ and‘manufacturer’ exchangeablly to represent the entity who manufactures, produces, or sellsproducts.

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satisfied their needs. The complicated demand substitution makes the product portfolio

strategy issue diffcult to analyze. The factors affecting the demand of one product is not

limited by just itself but also by others products that are beyond the previous product

boundaries. The next section will discuss the market effect of fusion products.

1.2 Market Impact of Fusion Products

When the traditional single-function products overlap with these new fusion products,

some kind of competition and substitution is unavoidable. Consider a first time buyer

of either a single- or a multi-function product. The customer faces options between

single-function printers and multi-function printers. If the customer determines he would

never use the other functions, then a single-function printer is a better choice since it

is normally less expensive than a counterpart of multi-function printer. If the customer

believes that a scanner, a copier, or a fax machine may be useful, the customer may decide

to purchase the multi-function printer if the extra functions are justified by the additional

cost above the single-function printer. Therefore, in purchasing a fusion product, the

customer has to evaluate his multi-dimension needs, multi-quality levels of the fusion

products, and the total cost. The assessment process of the fusion product’s purchasing

decision is more complicated. The consumer may be willing to compromise if one or two

minor functions fall short of his requirements when overall the fusion product has the best

value.

There is more evidence showing the market expansion and growth of the fusion

products. A business mobility survey conducted by NOP World Technology in 2005 shows

that sixty percent of large organizations (1000-plus employees) have used PDA phones

and more than one-third of large companies use smart phones. Approximately two-thirds

of large organizations use or plan to use wireless technology to access company databases

and files. Email, calendaring, and scheduling are seen as the most important functions for

wireless data communication in these firms [29]. A market research report released by IDC

in November 2004 projects that the average growth rate of multi-function peripherals from

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2003 to 2008 will be 21.8%. The worldwide unit shipments will increase from 29.25 million

units in 2003 to 78.41 million units in 20087 .

It is not surprising that fusion products enter into the market(s) as high-end devices

since they put so many functions together. Therefore, initially the price of the fusion

product is typically at a premium to its corresponding single function products. The

fusion product may go through several generations of upgrades and developments before

it evolves into a fully acceptable fusion product. Early versions of Treo and Blackberry

had many defects which were removed in new versions of the same products. While a

PDA-phone and a multi-function copier-printer have become popular in the market, other

fusion products are still struggling for market penetration. Smart phones, which emerged

later than the PDA phone, try to consolidate pocket-PC function into a cellular-phone,

but typing on such a small device as a laptop substitute is difficult; hence, smart phones

are not as popular as PDA phones [47]. Some fusion products have been launched for

years but they are not accepted by the market (i.e., Internet TV) or stay in niche market

(i.e., DVD-TV combo set.)

As the component and production costs decrease, the prices of fusion products

decrease and become more competitive with the single-function products. In terms of

unit variable cost, a multi-function printer cost is approximately 25% greater than a

quality-comparable single-function printer. The moderate variable cost increase of the

fusion product comes from sharing the same interface among multiple functions. However,

the firm has to invest a substantial amount of money in fusion product development.

After several generations of improvement and cost reductions in multi-function-related

components, the prices of the fusion products are competitive compared with the

7 Worldwide multifunction peripheral 2004-2008 forecast and analysis, IDC, November2004, www.mindbranch.com/listing/product/R104-17717.html

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single-function products, and the quality and usability of the fusion products are

improved.

There are fewer and fewer single-function products, and more and more fusion

products in the office machine and cellular phone markets. One evidence of this is the

shelf display of the retail stores. Now, buying a pure cell phone without the additional

functions leads to limited choices. Buyers are still able to find a single-function document

scanner or radio tuner in the retail stores, but the choices available are fewer than in the

past. Fusion products have gradually become a mainstream choice in many markets.

To resist the encroachment of the fusion products, the single-function products have

to improve their quality, which will validate their value and prolong the life cycles of the

single-function products. One example is the digital camera. Cell phones which contain

a digital camera function are predicted to out sell the non-camera phone in 2006 world

wide [86]. Camera phones are now equipped with 3-megapixel resolution or above (i.e.,

Sony Ericsson K800 and Nokia N80, N93), flash light and digital zooming. The market

of low-end single-function digital cameras has declined. Samsung B600’s cellular phone

containing a 10-megapixel resolution surpasses many mid-level single-function digital

cameras. To combat the camera phone’s encroachment into their market, the makers of

digital cameras have not only increased the resolution, the optical zooming, and the LCD

screen size, but also added more features such as image stabilization, high-sensitivity

mode, and consecutive shooting. However, some industry forecasters still predict the

fusion product will eventually cause the demise of single-function digital camera [8].

Single-function PDA’s shipments also suffer from the growth of PDA phones and

smart phones. According to IDC’s survey, single-function PDA shipments were down

22.3% in the first quarter of 2006 compared to the same quarter a year ago and that was

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the ninth consecutive quarter of year-over-year decline8 . Despite the continued decline,

Palm still launched their latest PDA -‘LiveDrive’ and they continue to position it as a

personal mobile manager9 which aims to satisfy consumer’s demand for more powerful

handheld function: color screen, infrared port, bluetooth, Wi-Fi, memory slot expansion,

voice recording, MP3 playback function, micro hard drive, video viewer, hand-held

Windows Office software.

1.3 Portfolio Decisions of Fusion and Single-Function Products

Imagine a situation that the buyer’s demand can be satisfied by single-function

products or fusion products. For the potential buyer who can choose between these two

products, the choice might depend on several factors. Is the extra function justified by

the extra cost? Does each function satisfy the minimum need of each aspect? Is the fusion

product well integrated? Is the interface of the product easy to use? If the functions

between two options differ from each other, what is the trade-off between them?

The following excerpt from Purchasing magazine [4] points out the desirability of

fusion products and their impacts on the consumers and the market:

...An MFP (multifunction product) combines copiers, fax machines andprinters into one unit, which means less hardware and lower cost per copy(CPC) for buyers. So, a company can save money by combining all functionsinto one device....Customers are becoming more comfortable with MFPs so wesee demand declining for single-function fax machines and copiers... A reporton the MFP market published recently by Gartner shows that while demand isrising, it’s more robust in some segments than others. Much of the growth insales of MFPs for the office is for printer-based units, which are taking volumeaway from single-function printers.

8 PDA Market shrinks as mobiles take customers, April 28, 2006, www.cellular-news.com/story/17163.php

9 PalmOne introduces lifedrive, its first mobile manager product, PalmOne, Inc.November 2004, www.palm.com/us/company/pr/2005/051805a.html

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The perception and utilization of consumers for a single-function product and for

a fusion product are different. All buyers of the fax machine will use it to receive or

send documents through the phone line. There is not much variation in a consumers’

perception and utilization in a single-function product. In contrast to the simplicity of

single-function products, the multi-function capabilities of a fusion product complicate the

consumers’ perception and utilization. The buyers of a fusion product may not utilize (or

not understand how to use, or not even realize some functions exist) all functions in the

device. Also, though buying the same fusion product, different consumers may emphasize

a different set of functionalities of the fusion product.

As a result, for different individuals, the utility of a fusion product might not be

the sum of the utilities from all its component products. Some consumers are willing to

pay premium for the PDA phone because they really need well-integrated PDA phones

that can send emails, integrate address book, and access databases. Some consumers just

utilize the basics of the extra functions because they use them infrequently. In contrast,

some consumers are just more adventurous and they are early adapters for new products.

Fusion products give these consumers new experiences, trendiness, and convenience. Other

consumers are more suspicious about new products and they are imitators in the product

diffusion process. Hence, some consumers are willing to pay a high-premium for a Treo 650

but others are not. The consumer’s utility heterogeneity with regards to fusion products

generates more complexity in demand analysis.

1.4 Designing Fusion Products

The manufacturers of single-function products need to consider the following

questions. Does it have the capability to integrate several functions together in a

single product? Are other product manufacturers also thinking about offering fusion

products? How difficult is the product-fusion for each manufacturer? How long does it

take for the fusion product to be available on the market? What is the cost to develop

a fusion product? How much is the extra variable cost for each extra function? What

19

level of functionality in each aspect of function should be integrated together? If the

manufacturers produce several single-function products, what are the impacts of fusion

products on the demand of these single-function products?

The first three questions for the manufacturers are technology level related. The

most important criterion of availability of fusion products, especially in electronics, is

the technology advances. These technology advances need certain investment in product

design and development, which normally is assumed to be a fixed cost. Similar to any

new product development effort, fusion products normally need some time to go through

several generations before being widely accepted in the market. If the fusion product is

just at the initial stage, then severe functionality discount often comes from immaturity

of the new combined technology. Normally, each generation of fusion products has

improved consolidation in functionalities that make that fusion product more attractive

and competitive. However, the diffculty of integration for all involved manufacturers might

not be symmetrical. The digital camera can easily be added to the cellular phone and

this idea has become very popular in the market. But, embedding a cellular phone into

a digital camera is still rare10 and awaiting acceptance from the market. A copier-based

multi-function machine consolidates fax, printer, and scanner functions without too much

trouble. In contrast, a fax-based multi-function machine normally can be used to copy a

single sheet of document but cannot work as an office copier machine.

In addition, the fixed development cost and the variable cost to design a fusion

product vary and depends on the degree of functionality of the additional functions and

how well these functions are integrated together. The cost of adding a fax function to a

printer is less expensive, because the extra variable cost might not be much more than a

control integrated circuit (IC). Size is another issue regarding portable fusion products.

Adding so many functionalities into one small device is a challenge in product design. As

10 The device that ate everything? The Economist 374 (8417) (2005) 16

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electronic devices shrink in size, the current components used in single-function products

may not be put together in a fusion product. In general, fusion products have space

limitations and need smaller components that are normally more expensive. For this

reason, the variable cost of fusion products can be more than the sum of total variables

costs of its component products.

The question about what level of functionality in each function should be integrated is

also critical to the manufacturers. Normally, the added function will not work better than

the single-function product. For example, the scanning function of a multi-function printer

might have a smaller scanning size or lower resolution, and a fax-machine-based fusion

product cannot copy a book. The manufacturers may also have different fusion options

to integrate the fusion products. Each fusion product model has different levels of extra

functions and attracts consumer demand in a variety of ways.

Technology advances and competition make product-fusion an invertable trend.

If a firm is not a leader in offering fusion products, it might be encroached by other

competitors that offer fusion products. As a result, manufacturing fusion products might

be a survival strategy for every firm. All firms have to evaluate the penetration of fusion

products, the impact on the demand of single-function products and a market transition

process from single-function to multi-function. Of course, not all single-function products

will be replaced by fusion products. However, what type of single-function product can

survive alone will remain a constant question. As technology advances, more firms may

choose to become fusion product providers realizing that product-fusion might be a

continuous and unbounded process.

1.5 Diffusion of Fusion and Single-Function Products

In section 1.2 we presented evidence of shipment declines of some single-function

products after the launch of the fusion products. There is also plenty of evidence

indicating many electronic products are converging to fusion products. If we analyze

21

the overall market from product life cycle’s point of view, we find that the changes of

market transition from single-function products to a fusion products occurs gradually.

One example of a fusion product which has almost taken over its component

single-function products is the portable radio tuner/CD (radio/cassette) stereo. Now it is

difficult to buy a single-function portable radio tuner, single-function portable cassette,

and single-function portable CD player. It shows that the single-function to multi-function

transition has almost finished in the portable radio tuner/CD (radio/cassette) market.

However, not every fusion product will eventually cause the demise of its component

single-function products. The TV-DVD combo set is an example of a fusion product that

only has a niche market11 . There is no empirical research explaining why a TV-DVD

combo fails to captures the market especially when integrating these two functions

together makes sense. One possible reason might be the price of a TV-DVD combo is

not sufficiently attractive to the consumers. At the price comparison website Nextag.com

(12/13/2006), an RCA DVD player and an RCA 20” CRT TV costs around $50 USD

and $150 USD, respectively, but an RCA 20” CRT TV equipped with DVD player costs

above $200 USD. The purchase of a TV-DVD combo does not provide any savings for the

consumer. Another possible reason is that both products are still evolving in technology.

The DVD technology is going from traditional DVD to Blue-ray DVD (a standard of high

definition DVD) and the TV market is transitioning from heavy-weight CRT technology to

thin-and-light LCD or Plasma technology.

The diffusion and substitution process that an FP gradually replaces its component

products is predicted but has not been investigated. Many questions related to fusion

product diffusion are in need of investigation. These questions include: when should a firm

introduce the fusion product and in what conditions? How does the availability of the FP

affect the diffusion of these SPs as well as the FP? What other factors are influential to

11 The pioneer of TV-DVD, TV-VCR, also failed to become a mainstream product.

22

the launch timing decision? How does the diffusion processes impact the current products?

How does the maturity factor of SPs influence the diffusion of all products?

1.6 Focus of This Research

The dynamics brought by the fusion products to the market are enormous. There are

also many questions worth investigating. From the consumer’s point of view, Thompson

et al. [82] have shown that too many features make the fusion product less attractive.

Marketing oriented research can start from consumer’s utility analysis to discuss fusion

products and single-function products. From product design’s point of view, how many

functions should be consolidated into a size limited all-in-one device is an important

question. After integrating several functions, how to design a more user friendly interface

to increase the usability of the fusion product is also a challenging work. The 60 Minutes

show on CBS (Get Me The Geeks!, Jan. 28, 2007) reports that new high-tech electronics

(HDTV or cable networking) have become so complicated, that even high tech geeks have

difficulties to set up a new product. Product design oriented research is another direction

of fusion product research.

However, this dissertation investigates the fusion product phenomena from the point

of view of the manufacturer.

First, we analyze the product portfolio decision for a manufacturer produces two

distinct single-function products and one two-function (multi-function) product in Chapter

2. We characterize the demands in a price-linear form such that the price function of a

product depends on how much quantity the product is offered as well as its substitutes.

The substitution effects between two products depends on the functionalities in them. We

identify five possible product portfolios and characterize the conditions which favor each of

them.

Second, we focus on the design decision for fusion products. As more and more

functions can be consolidated into an all-in-one device, the number of possible products

grows exponentially because different fusion products can be designed with several

23

combinations of single functions. We extend the stylized analytical model in Chapter

2 to a situation where n functions can now be integrated into any combination of

multi-function products. We develop structural results which identify a dominance

relationship between the parent and their children portfolios, and we provide a search

algorithm that can quickly find candidates of optimal product portfolios from an enormous

number of possible portfolios.

Finally, the launch of the fusion product will change the diffusion process for

single-function products. If a fusion product is launched early, the single-function products

may complete their life cycles early. However, launching a new fusion product earlier may

require more R&D investment. It should also be noted that substitution effects, relative

profitability and potential market size may also play a role in the firm’s launch decision.

We propose a product diffusion model that reviews two single-function products and one

fusion product and incorporates the factors above into the model. We investigate factors

that impact the launch time decision.

24

CHAPTER 2OPTIMAL PRODUCT PORTFOLIO STRATEGY: TWO SINGLE FUNCTION

PRODUCTS AND ONE MULTIFUNCTION PRODUCT

2.1 Introduction

In this section, we investigate the product portfolio strategy for a single firm which

can offer two single function products and one multi-function product. Some firms like

office machine makers have been in a similar situation. While the fusion technology

can integrate the printer and the scanner together, HP should re-evaluate their product

portfolio among printers, scanners and multi-function office machines.

We observe that the buyers of multi-function office machines often use the all-in-one

devices to replace some or all single function products of the multiple functions. For

example, most consumers and SOHO (Small office/home office) buy just all-in-one offices

machines to save space and the total purchase cost. For these buyers, the scanning and fax

functions on the all-in-one offices machines, even though not so good, are acceptable for

their low volume of usage.

But for a larger business or the consumers with high volume of usage on the scanning

and fax functions, the all-in-one offices machines fall short of their needs. The add-on

functions on all-in-one office machines are often discounted. For example, the scanning

function often has a lower resolution and the fax function is lack of auto-redial and

multiple receivers feature. As a result, these high-volume users may still purchase

single-function products for each function because single-function products have complete

and high-end features.

From the analysis above, the firm which offers single function and multi-function

products must evaluate the substitution effects between a single function and a multi-function

products. Note that the degrees of substitution effects vary in functions and that may

depend on the design of the multi-function product and the function itself. In the

all-in-one office machine example, the printing function between a single function printer

and a multi-function office machine are pretty similar; but the scanning function between

25

a single function scanner and a multi-function office machine demonstrate some quality

gap.

Due to the substitution effects, the demand of a single function product and a

multi-function product overlaps. While the multi-function product grabs the demand

from its single-function substitutes, it also expands the market. Some consumers only

purchase when the multi-function product is available. But some buyers of single-function

products choose to ‘upgrade’ to the multi-function product. This has been observed in

the multi-function cell phone market and Apple’s ‘iPhone’ is an example. As a result, the

demand of each product shows some degree of overlapping with other substitutes but it

has its own characteristics.

In terms of the variable cost, a single function product and a multi-function product

may share some common components. Flat-bed style scanners use the same flat-bed

skeleton, glass and and some optical components in all single function and multi-function

scanners. But the multi-function scanner needs a more complex control card to execute

other functions. As we discuss in Chapter 1, the variable cost of a multi-function product

does not equal to the sum of total cost of its component functions.

Now, let us discuss some literature related to multi-function products.

2.2 Literature

2.2.1 Fusion Products

In recent years, business and technology journals have continuously conducted reports

on fusion products. The reports focus on issues from “how to select the right fusion

product” [3], to “strategy of the manufacturer” [4], to “new market trend” [22] [31], to

“product design critique” [25], and to “fusion product review” [28] [37] [13].

Fusion products such as office machines and cellular phones are becoming more

mature in markets. Several articles [4] [28] [41] report the demand of the fusion products

increased when the quality of fusion products became more acceptable. Schonfeld [41]

points out that Blackberry evolved from resembled pagers with miniature keyboards in the

26

first version to “a mobile phone for thumbing-out e-mail messages” in the latest version.

Magid [28] implies earlier multifunction devices before then had “too many compromises

in at least one of the functions.” After overcoming initial compromises, fusion products

have now shown that they are strong substitutes of single function products.

Here, we summarize these business and technology reports on fusion products.

First, as the prices of fusion products decline, the fusion products are more affordable

to consumers. Having a fusion product means “less machine count” (total number of

single function devices) and lower cost for consumers [4]. Second, there are almost

limitless functionalities which can be integrated into one unit [4] [48]. As technology

advances continue, product-fusion will emerge and prosper in more product fields. Third,

fusion products normally work well in one or two functions but not all of them and the

consumers should evaluate what level of performance they require and how much they are

willing to pay [25] [28].

There are some disadvantages associated with fusion products. First, Magid [28]

mentions that “If your stand-alone scanner or fax machine breaks, you can still print.

But if any component goes down on a multifunction machine, the whole system may be

unusable until the machine is fixed.” Second, it is difficult to find out how every function

works in a multifunction device and the learning curve can be steep [13]. Third, Breeden

and Soto [13] point out some fusion products cannot multitask, which is different from

using several single function machines. Even if some fusion products can handle multitasks

simultaneously, the length of time to complete the task is greater due to shared memory.

Product design integration is another topic in product-fusion that has been

investigated. Though integration of design is one aspect of product-fusion, current

literature about integration of product design focuses on maximizing the performance.

The research from [6] , [5], [45] discuss the integration issue of IC and component. Allawi

[6] studies the integration of analog-digital IP blocks. The focus of the paper is on

the performance and design issue of circuit routing. Allan [5] discusses the superiority

27

of multifunction modules over single function modules. Toker et al. [45] show that

a multifunction filter can perform three basic functions simultaneously. However,

these papers concentrate on the performance or design issue of the product instead of

fusion products’ impact on the market. Also, the products discuss in these models are

components of other final products and do not relate to the final market demand directly.

2.2.2 Product Bundling

Bundling is the concept most similar to product-fusion and has been investigated

well in theoretical models. A bundle is a package of several products that is sold as a

single unit by the firm in the market. In most bundling models, the supplier decides

whether they should bundle their products and whether the bundle should coexist with

their components. The consumers, depending on their utility functions and the prices of

products, purchase either the bundled package or some (or all) of the single component

products. The firm normally provides a bundled package with a lower (than the sum of

prices of component products) price. We expand on the difference between product-fusion

and bundling in the summary section.

In general, bundling models originate from economics theory. All models assume

each consumer has a reservation value vector for all products and the firm knows all the

reservation value vectors. Rational consumers choose combinations of single components

or a bundled package by maximizing their utility. Most models except those presented

by Hanson and Martin [24], Dansby and Conrad [18], and Bakos and Brynjolfsson [9]

assume product costs and reservation values of consumers are additive. For the bundled

product to be strictly competitive, the bundled price has to be less than the sum of all

component prices. All models assume that the resale of components from the consumers

is not allowed. It is a strong assumption because in the real world consumers often take

advantage of bundling or promotions by aggregating demand together.

Product bundling was first suggested by Stigler [44], who viewed it as a strategy for

a firm to accomplish price discrimination when heterogeneous consumers have different

28

willingness to pay (reservation values). Bundling has been investigated extensively in the

economics literature and more recently in the information goods area.

Early research investigated the issues of the optimal strategy of sellers, consumer

surplus and the effects on competition (e.g., [1], [18], [39], [40] [30], and [24]).

Adams and Yellen [1] first define two bundling strategies for a monopoly producer.

Pure bundling strategy means the firm only sells goods in package and mixed bundling

means the firm sells the components separately as well as in packages. They show that

bundling can be profitable without the assumption of production cost saving. They

conclude that, in most circumstances, bundling can be more lucrative than simple

monopoly pricing.

Dansby and Conrad [18] consider two other possibilities: subadditive and superadditive.

Subadditive (superadditive) means the total cost of the bundle is less (more) than the sum

of the costs of all components. They also allow the consumers to buy two component

products, called self-bundle, instead of bundle one when mixed bundle is adapted. If all

buyers are utility-additive, offered-bundle and self-bundle will not exist simultaneously.

They find when the utility of consumers is subadditive the bundle price under mixed

bundle might be higher than that under pure bundle.

Schmalensee [39] analyzes a bundling model that a monopolist combines a single

product with another product in a competing market. Schmalensee also finds if two

products are negatively correlated the monopolist can profit from price discrimination

when the buyers reveal their reservation values. Schmalensee [40] finds that while the

average willingness to pay is high and the correlation coefficient is positive, pure bundling

dominates pure component strategy in a symmetric normal case because it increases profit

by extracting more consumer surplus.

McAfee et al. [30] find that, in a monopoly situation, the pure component will

never be optimal if the monopolist can monitor a purchase made by consumers. Offering

mixed bundle allows the monopolist to segment groups of customers. If the monopolist

29

cannot monitor the purchase, the optimal conditions for mixed bundling can be found.

In a duopoly situation, if the distribution of two products is independent, then the pure

component will never be in Nash equilibrium.

Hanson and Martin [24] develop a generalized bundling analysis when the monopolist

sells many separate components in the market. They build a complicated linear

programming model and show that if the consumers are rational and wish to maximize

their utility, the consumers in each market segment (segmented by their reservation values

and market prices of all components) will purchase exactly one bundle or will not make a

purchase. This property can be formulated as a disjunctive constraint.

Salinger [38] analyzes the impact of the correlation of reservation values on optimal

bundling strategy. When the reservation values of two products are negatively correlated,

high unit costs make bundling unattractive compared to components’ aggregation. When

the reservation values of two products are positively correlated, then bundling needs larger

cost saving to be the optimal strategy. When the reservation values of two products are

perfectly negatively correlated, which reduces the variation of reservation values, the firm

can use bundling to extract all consumer surpluses.

Chen [14] investigates the bundling strategy when the primary product is in a

perfectly competing duopoly market. Each duopolist can bundle with one or more of the

goods. He finds that bundling works as a product differentiation device for the firms.

Some marketing oriented research focuses on retail or information goods bundling

(e.g., [34] [10]. Even though knowing that the consumer’s utility function is a widely

accepted assumption in bundling analysis, how consumers judge, and how they perceive

and evaluate the bundle deal are not studied till recently (e.g., [51], [26], and [42]).

Mulhern and Leone [34] point out in some sense promotion is a form of price

bundling. Discount promotions for some items are commonly used to drive the demands

of other non-discount products in the retailers. They call this approach implicit price

30

bundling. No matter whether the relationship between two products is substitute or

complementary, the price on one item will affect the demand for another item.

Bakos and Brynjolfsson [9] study the bundling of information goods. Due to very low

marginal cost, they find bundling a large number of information goods makes it easier to

predict the evaluation of consumers. The multiproduct monopolist can extract more profit

by selling bundles of information goods. Bakos and Brynjolfsson [10] extend their previous

bundling of information goods to a competitive setting. When the marginal costs are low,

they conclude that there exist economies of aggregation even without network externality

or economies of scale or scope. By adding new information contents to the current one,

the firm can employ economies of aggregation.

Yadav and Monroe [51] conduct an experiment to explore how buyers perceive the

savings in a bundling price. Their outcomes show the seller’s pricing strategy significantly

influences the buyers’ perception on two savings and their purchasing behavior. Two

savings are both important to the buyers although the bundle saving is more influential to

the total transaction value.

Simonin and Ruth [42] use a quasi-experiment to investigate bundling strategy used

in the introduction of the new product and its impact on consumer price perception.

Bundling a new product with the current brand (or product) reduces the risk and is

strongly related with brand extension. Bundling a new product with a well-linked brand

generates a favorable attitude toward the bundle. The brand of the new product has

greater influence on the attitude toward the bundle than the brand of the tie-in product.

Kaicker et al. [26] investigate the choice of the buyers’ component versus bundling

purchase by comparing the real price and the price expectation. They find that subjects

prefer a bundle when faced with mixed gains and low net mixed losses, and subjects prefer

a components purchase when they perceived multiple gain and high net mixed losses. One

major constraint of their research is to not consider a mixed bundling strategy.

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2.2.3 Vertical Differentiation

Another track of research related to product-fusion is product vertical differentiation.

Most vertical differentiation models focus on one base product market that has several

variants based on quality which varies on one dimension (for two dimensions of quality

differentiation see Baumol [11], Vandenbosch and Weinberg [46]). The models assume

the consumers have the same ranking of the variants of this product. If all variants have

the same price, all consumers will choose the best quality product.The heterogeneity of

consumers can be on income budget distribution (e.g., [11], [21]) or on taste distribution.

Most models assuming difference in taste also assume the distribution is uniform (e.g.,

[32], [46], [50]). Ansari et al. [7] relax the taste distribution to a generalized beta

distribution that shows the results are very different from uniform distribution. Two

papers investigated vertical differentiation over time, which means that intertemporal

competition exists ([33] and [19]).

Baumol [11] analyzes the optimal posture of a new product or a new store along

two dimensions of product characteristics or retailer stance. The results show that the

new product (shop) must exist above the boundary of a convex region formed by current

products (shops). All the consumers during a certain range of attribute slope buy the

same product or shop in the same store.

Gabszewicz and Thisse [21] investigate a non-cooperative duopoly game facing a large

number of consumers with identical ranking of products but different incomes. Higher

average income helps the profitability of both firms; however, up to a certain point, the

lower quality firm will be out of the market since all consumers can afford high quality

products. If the higher quality firm improves its product quality, both equilibrium prices

go up. Surprisingly, when the lower quality firm improves its product quality, it might

reduce its equilibrium price which makes itself worse-off.

Moorthy [32] relaxes a strong assumption of Pigouvian third-degree price discrimination

consumer-isolation and investigated the impact of consumer selection on supplier’s market

32

segmentation strategy. Under the self-selection of consumers, the monopolist cannot

determine the optimal quality levels and prices for all segments separately; thus, social

welfare decreases. The supplier may reduce the total number of products to reduce

cannibalization.

Bolton and Bonanno [12] explore the vertical differentiation two-layer supply chain

structure. When the supply chain is not vertically integrated, a linear-price contract

between the manufacturers and retailers gives rise to vertical inefficiency. They derive an

optimal contract by restricting the set of retail prices and price-dependent franchise fees.

They find when the willingness to pay for quality does not vary much with income, there

are fewer incentives for retailers to use price discrimination.

Waterson [49] discusses oligopoly models of vertical differentiation in quality. On the

producer side, under certain condition of income range, duopoly equilibrium holds. Larger

market size leads to higher quality products and the higher quality producer earns more

revenue. He finds the lower quality producer is in an underdog situation; however, change

of income distribution might have a different conclusion.

Choi et al. [15] investigate the Nash equilibrium of the duopoly problem while

the individual consumer’s purchasing behavior is heterogeneous and in logit model

form. Concave profit function of single firm is a necessary condition of Nash equilibrium

while low price sensitivity of consumers is a sufficient condition. They find that higher

quality will increase higher equilibrium price, but that makes the competitor’s price

indeterminant.

Cremer and Thisse [17] study two types of product differentiation: horizontal and

vertical. They prove that a large class of horizontal differentiation models is a special case

of vertical product differentiation model, but not vice versa. The research also finds that

in vertical differentiation the top and bottom quality firms earn the same profit that is

higher than the firm’s profit by providing medium quality.

33

Moorthy and Png [33] examine the timing of product introduction while there are

two segments in the market. If the seller introduces two products simultaneously, it should

lower the quality of the low-end model and reduce the high-end price. However, the seller

can also commit to the market about its launching schedule if the seller introduces two

products sequentially.

Ansari et al. [7] investigate the one-dimension product differentiation in a non-uniformly

distributed consumer market. They analyze the positioning and pricing game when the

market has two, three, and four brands and the firms make the decision simultaneously or

sequentially.

Vandenbosch and Weinberg [46] analyze two-dimensional vertical differentiation

in a two-firm situation. They determine that there are three types of results. The first

result is that the two firms will maximize differentiation in one dimension and minimize

differentiation in another dimension, which is called MaxMin product differentiation.

The second result is that one firm takes a max-max quality position and another takes a

min-min quality position. The third result shows two firms maximally differentiated in

one dimension and partially differentiated in another one. MaxMin differentiation is the

normal case.

Wauthy [50] assumes that two firms differ in level of quality and the consumers differ

in their tastes of quality. He concludes that the Nash equilibrium prices are functions

of the degree of population heterogeneity and the degree of product differentiation.

One major contribution of this paper is it includes covered and uncovered market

configurations.

2.2.4 Summary

Even though the study of product bundling and vertical differentiation gives us many

insights into the product-fusion problem, product fusion has its own characteristics that

differ from these two well-studied topics.

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Product-fusion has several properties which differ from product bundling. First,

production cost of fusion products can be greater or less than the sum of all costs of

its single component products and it also a function on integration difficulty. Second,

consumers’ reservation values for fusion products do not necessarily equal the sum of

reservation values of its component products. Some consumers might need the fusion

product greatly for simplifying its function, such as universal remote controller, or for

saving space, such as an integrated printer-copier-scanner-fax machine. Third, limitation

on interface integration, which sacrifices user friendliness for additional functions, makes

the fusion product not a total substitute of the single-function product. Finally, the

supplier of a bundled product can freely choose component products and sell them

together so they can reach a high-level of price discrimination and extract additional

consumer surplus. The extra expense of bundling might be some repackaging cost.

However, the manufacturers of fusion products cannot easily provide many varieties to

satisfy different types of consumers. Each fusion product needs a certain investment for

product development and design.

A fusion product may be seen as a high-end version of its single component products.

However, viewing the fusion product as a high-end substitute of the original product will

ignore the impact of the fusion product on the other single-function product market. It

is likely that every unit sale of Treo 650 displaces a potential buyer of a cellular phone,

a PDA, and a low-end digital camera. The dynamic brought by fusion products is much

more complex. We think the problem of product-fusion will not be sufficiently explained

by vertical differentiation and should be analyzed separately.

2.3 Preliminaries

A single firm has the capability of offering three products in the market (i = 1, 2, 3).

Products 1 and 2 are assumed to be single function products while product 3 is a

multi-function product which is designed such that product 1 is the base product and

35

product 2 is the non-base product.1 An example of such a multi-function product is

the camera phone such as the Sony Ericsson S710a. This product provides almost all

the functionality of a stand-alone cellular phone (the base product) but a lower level

of functionality as compared to a digital camera (in terms of attributes such as picture

quality, flash, zooming capabilities, and mode adjustment).

We assume a linear, downward-sloping demand curve as a function of price. As an

approach to model price-demand relationships, linear demand curves are quite common in

the literature ([43], [74] [27] [58] [77] [66] ). Besides analytical tractability, linear demand

curves also possess several attractive features. First, the price elasticity of demand is

increasing in price, i.e., the higher the price, the more sensitive the demand is to changes

in price (this is not true for other popular forms of price-demand functions such as

Cobb-Douglas). Second, the linear demand function has a utility based explanation

from first principles [72]. For a complete discussion of when linear demand curves are

appropriate, the reader is referred to LaFrance [63]. The specific inverse demand functions

we assume for each product are as follows:

p1 = a1 − q1 − r13q3 (2–1)

p2 = a2 − q2 − r23q3 (2–2)

p3 = a3 − q3 − r13q1 − r23q2 (2–3)

where ai > 0 ∀i can be interpreted as the maximum price for product i; qi is the quantity

of product i sold by the firm; ri3 (0 < ri3 < 1; i = 1, 2) is the substitutability in-

dex of the multi-function product in relation to the existing single function product i.

Consistent with the multi-function product incorporating product 1 as the base product

1 Product 3 can also be designed such that product 2 is the base product and product 1is the non-base product and the remainder of the analysis holds by symmetry.

36

and product 2 as the non-base product, we assume that r13 > r23.2 Finally, note that

from a design perspective, we assume that the multi-function product has almost the

complete functionality of base product 1. However, the substitutability index r13 is

assumed to be strictly less than 1 to incorporate the feature that product 3 is not a

perfect substitute for product 1 from a demand perspective. For example, a PDA phone

could be designed such that it incorporates the complete functionality of a stand-alone

PDA but from a market demand perspective, the PDA phone might not be a perfect

substitute for the PDA.

The variable manufacturing cost for product i is ci and to rule out trivial cases, we

assume ci < ai. For the multi-function product, we do not make any assumption on the

magnitude of this cost in relation to the total cost of single function products 1 and 2 (i.e.,

c3 can be larger or smaller than c1 + c2), since the extent of technological difficulties and

synergies for manufacturing an integrated product, which combines the functionalities of

the existing products into a single product, makes either of these two cases possible.

Given this setting, the key questions for the firm are as follows. Is it beneficial

to include the multi-function product in the firm’s optimal product portfolio set? If

so, should it be included as an alternative to the single function products or should it

complement one or both of these products? Are there a set of dominant product portfolio

strategies in this setting? What are the key parameters which define the choice of one

strategy over another? In order to address these and related questions, we now analyze the

optimal product portfolio decisions for the profit maximizing firm.

2 We also assume that r213 + r2

23 < 1 in order to ensure that demand for products 1 and 2is an increasing function of the price of the substitute product 3 and vice versa [73].

37

2.4 Optimal Product Portfolio Strategies

Given our three product scenario, the firm’s product portfolio selection problem is

equivalent to the following problem of setting quantities:

Maximize Π = q1(d1 − q1 − r13q3) + q2(d2 − q2 − r23q3) +

+q3(d3 − q3 − r13q1 − r23q2) (2–4)

subject to:

qi ≥ 0 for i = 1, 2, 3 (2–5)

where di = ai − ci ∀i and can be interpreted as the maximum profit margin for

product i. It is relatively straightforward to show that Π is strictly and jointly concave in

the decision variables (see Appendix A) and thus, the first-order conditions are necessary

and sufficient to identify an optimal solution (q∗1, q∗2, q

∗3) to our problem. However, there

is no guarantee that this solution is feasible (i.e., q∗i ≥ 0 ∀i). Hence, there are several

potential product portfolio strategies that emerge as being optimal for the firm. These

strategies are:

• All-Product Strategy (APS): This corresponds to the strategy of offering all 3products to the market and the associated product portfolio set is {1,2,3}.

• No MFP Strategy (NMFPS): This corresponds to the strategy of simply offering thetwo single function products to the market and the associated product portfolio set is{1,2}.

• Partial MFP Strategy 1 (PMFPS1): This corresponds to the strategy of offering themulti-function product along with the base single function product and the associatedproduct portfolio set is {1,3}.

• Partial MFP Strategy 2 (PMFPS2): This corresponds to the strategy of offeringthe multi-function product along with the non-base single function product and theassociated product portfolio set is {2,3}.

• Single MFP Strategy (SMFPS): This corresponds to the strategy where only themulti-function product is offered and the associated product portfolio set is {3}.

38

In order to identify when each of these strategies is optimal, we first define:

α1 = r13d1 + r23d2

α2 = min{1− r223

r13

d1 + r23d2,1− r2

13

r23

d2 + r13d1}

α3 = max{ d1

r13

,d2

r23

}

Using these definitions, Theorem 2.1 below provides parametric guidelines for identifying

the optimal product portfolio strategies.

Theorem 2.1: The optimal product portfolio strategy for the firm can be identified as

follows.

1. If d3 ∈ (0, α1], the optimal product portfolio strategy is NMFPS;

2. If d3 ∈ (α1, α2], the optimal strategy is APS.

3. If d3 ∈ (α2, α3), and• If d2

r23≤ d1

r13, then the optimal strategy is PMFPS1; and

• If d2

r23> d1

r13, then the optimal strategy is PMFPS2.

4. If d3 ∈ [α3,∞), then the optimal strategy is SMFPS.

Proof: See Appendix B.

Table 2-1 summarizes the optimal product portfolio strategies based on the results

stated in Theorem 2.1. For each of these strategies, details on the product quantity

offerings, resulting firm profits, and corresponding prices for each product for each are

shown in Table 2-2.

The results of Theorem 2.1 are intuitively appealing. Recall that d3 is the maximum

profit margin for the multi-function product. It is only when this maximum profit margin

is less than or equal to the weighted average (where the “weights” are the substitutability

parameters in the inverse demand functions) maximum profit margin of both single

function products, that the firm should not include the multi-function product in its

product portfolio. However, if the maximum margin on the multi-function is greater than

this weighted average, the multi-function product is always part of the product portfolio

39

Table 2-1. Results of Theorem 2.1.

ParameterSetting

Optimal ProductPortfolio Strategy

Corresponding ProductPortfolio

d3 ∈ (0, α1] NMFPS {1,2}d3 ∈ (α1, α2] APS {1,2,3}

d3 ∈ (α2, α3)PMFPS1 (when d2

r23≤ d1

r13)

PMFPS2 (when d2

r23> d1

r13)

{2,3}{2,3}

d3 ∈ [α3,∞) SMFPS {3}where α1 = r13d1 + r23d2; α2 = min {1−r2

23r13

d1 + r23d2,1−r2

13r23

d2 + r13d1}; α3 = max { d1r13

, d2r23} .

Table 2-2. Dominant Product Portfolio Strategies.

APS NMFPS SMFPS PMFPS1 PMFPS2q∗1 0.5xβ1 0.5d1 NA 0.5y(d1 − d3r13) NA

q∗2 0.5xβ2 0.5d2 NA NA 0.5z(d2 − d3r23)

q∗3 0.5xβ3 NA 0.5d3 0.5y(d3 − d1r13) 0.5z(d3 − d2r23)

Π∑3

i=1[0.5q∗i di] 0.25(d2

1 + d22) 0.25(d2

3) 0.25yβ4 0.25zβ5

p1 0.5(a1 + c1) 0.5(a1 + c1) NA 0.5(a1 + c1) NA

p2 0.5(a2 + c2) 0.5(a2 + c2) NA NA 0.5(a2 + c2)

p3 0.5(a3 + c3) NA 0.5(a3 + c3) 0.5(a3 + c3) 0.5(a3 + c3)

where β1 = d1(1 − r223) + d2r13r23 − d3r13; β2 = d1r13r23 + d2(1 − r2

13) − d3r23; β3 = −d1r13 − d2r23 + d3;β4 = d2

1+d23−2r13d1d3; β5 = d2

2+d23−2r23d2d3 x = (1− r2

13 − r223)

−1; y = (1− r213)

−1; and z = (1− r223)

−1.

for the firm. Another key result is that in a specific range for the parameter d3, one of

the partial multi-function strategies would dominate. The choice between these partial

multi-function strategies is a function of the maximum profit margin (adjusted by the

substitutability parameters) for the two single function products. When the maximum

profit margin for the base product adjusted by the substitution parameter r13 dominates

the maximum profit margin for the non-base product adjusted by the substitution

parameter r23, then the base product and the multi-function product should be part of

the firm’s product portfolio. Of course, if the reverse is true, then the non-base product

and the multi-function product should comprise the product portfolio for the firm. Finally,

note that only the multi-function product is included in the firm’s product portfolio when

40

it’s maximum profit margin dominates the maximum of the adjusted profit margins of

both single function products.

An analysis of the impact of each parameter on the changes in optimal quantity

offerings and changes in the corresponding profit for each product portfolio strategy is

shown in Table 2-3. Examining this table, we obtain the following additional insights:

• As expected, an increase in the maximum profit margin for a product leads to anincrease in the optimal quantity offering of that product. Of course, this increase inthe quantity is complemented by a decrease in the optimal quantity offering of thesubstitute product. For example, for PMFPS1, an increase in the maximum profitmargin for product 1 (i.e., d1) leads to an expected increase in the optimal quantityoffering for product 1 (q∗1) and a simultaneous decrease in the optimal quantityoffering for product 3 (q∗3) given that these products are market substitutes. Similareffects are observed across all the product portfolio strategies.

• For APS, there is one effect for changes in parameters d1 and d2 which needs someclarification. For example, when d1 increases, there is an increase in the optimalquantity offering for product 1 and a decrease in the optimal quantity offering ofproduct 3 since these products are substitutes. However, an increase in d1 alsoincreases the quantity offering for product 2 even though products 1 and 2 areindependent in terms of functionality and demand substitution effects. The reasonthis occurs is that under this strategy, all three products are included in the productportfolio and hence, a decrease in the quantity offering of product 3 tends to drivean increase in the quantity offering for product 2 - in essence, there is a spilloverquantity effect when all three products are included in the firm’s optimal productportfolio. If we compare this result to that obtained under NMFPS, then thesespillover effects do not occur since multi-function product 3 is not included in theoptimal product portfolio. Thus, under this strategy, increases in d1 will increasethe quantity offering of product 1 but there is no change in the quantity offerings forproduct 2.

• In terms of changes in the demand substitution parameters, the effects on optimalquantity offerings under each strategy cannot be easily determined. Note that theseparameters are only relevant when we have an optimal product portfolio whichincludes the multi-function product and one or more of the single function products(i.e., APS, PMFPS1, and PMFPS2). In the case of all such strategies, we find thatdepending upon the other parameters, increases in a substitution effect parameter,could result in either an increase or decrease in the optimal quantity offering of eachproduct included in the product portfolio. However, there is a symmetry to theresults for PMFPS1 and PMFPS2. For example, for PMFPS1, when within a certainrange of the parameter d3, increases in r13 lead to decreases in the quantity offeringfor product 1 and increases in the quantity offering for product 3; while outside this

41

Table 2-3. Comparative Statics.

APS NMFPS SMFPS PMFPS1 PMFPS2

q∗1 q∗2 q∗3 Π q∗1 q∗2 Π q∗3 Π q∗1 q∗3 Π q∗2 q∗3 Π

d1 ↑ ↑ ↑ ↓ ↑ ↑ NC ↑ NA NA ↑ ↓ ↑ NA NA NA

d2 ↑ ↑ ↑ ↓ ↑ NC ↑ ↑ NA NA NA NA NA ↑ ↓ ↑

d3 ↑ ↓ ↓ ↑ ↑ NA NA NA ↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑

r13 ↑ ↓†, ↑‡ ↓/, ↑. ↑/, ↓. ↓ NA NA NA NA NA ↓?, ↑◦ ↑?, ↓◦ ↓ NA NA NA

r23 ↑ ↓σ, ↑τ ↓•, ↑ρ ↑σ, ↓τ ↓ NA NA NA NA NA NA NA NA ↓4, ↑5 ↑4, ↓5 ↓where:↑ (↓) indicates an increase (decrease); NC indicates no change; and NA indicates not applicable.† applies when d3 > d2r23 + (1 + r2

13 − r223)

−12d1r13(1− r23). ‡ applies when d3 < d2r23 + (1 + r213 − r2

23)−12d1r13(1− r23).

/ applies when d3 > d2r23 + (2r13)−1d1(1 + r213 − r2

23). . applies when d3 < d2r23 + (2r13)−1d1(1 + r213 − r2

23).• applies when d3 > d1r13 + (1 + r2

23 − r213)

−12d2r23(1− r13). ρ applies when d3 < d1r13 + (1 + r223 − r2

13)−12d2r23(1− r13).

σ applies when d3 > d1r13 + (2r23)−1d2(1 + r223 − r2

13). τ applies when d3 < d1r13 + (2r23)−1d2(1 + r223 − r2

13)? applies when d3 − d2r23 > r23(d2 − r23d3). ◦ applies when d3 − d2r23 < r23(d2 − r23d3).4 applies when d3 − d1r13 > r13(d1 − r13d3). 5 applies when d3 − d1r13 < r13(d1 − r13d3).

range for the parameter d3, the reverse holds. For APS a similar symmetry does existbut it is also complemented by an additional result. For example, in a certain rangeof values defined by other parameters, increases in r13 could actually increase theoptimal quantity offerings for both products 1 and 3 but there would be also be adecrease in the quantity offering for product 2 through spillover effects.

• In the context of firm profits, results for increases in the maximum profit parameters(di) are in line with expectations (i.e., the firm profits increase as these parametersincrease). For the demand substitution parameters, the impact on profits is relevantonly for APS, PMFPS1 and PMFPS2. In all cases we find that as the substitutioneffects increase, the profits under any of these strategies decline. This impliesthat although increases in these parameters could lead to increases (decreases)of the quantity offerings, overall the firm profits are adversely impacted by thesesubstitution effects parameters.

The analysis of the optimal product portfolio strategies so far has been focused on

providing parametric guidelines for the firm to choose a specific strategy. Further insights

into the changes in the quantity offerings and firm level profits with changes in key

parameters have also been discussed. One issue which has not been explored is the impact

of simultaneous changes in these key parameters on choices between the set of portfolio

strategies. In order to do this, the next section presents a numerical analysis based on

secondary data for two single function products and one multi-function product currently

available in the market.

42

2.5 Numerical Analysis

Over and above examining the impact of simultaneous parameter changes on the

optimal product portfolio, the purpose of the numerical analysis is to also address the

following managerial questions:

1. Maximum Profit Margins versus Market Effects: The maximum profit margin forthe MFP is represented by the parameter d3 and is determined as a function of themaximum price (a3) and cost (c3). Thus, key questions in this context would be:• What is the impact of the maximum price for the MFP (i.e., a3) as compared to

the maximum prices of the single function products (i.e., a1 and a2) in identifyingthe optimal strategies and how are these choices moderated by the demandsubstitutability parameters? This could be potentially useful to evaluate the trade-offbetween pricing decisions and market effects (represented by the substitutabilityparameters).• What is the impact of the relative cost for the MFP (i.e., c3) as compared to

the costs for the single function products (i.e., c1, and c2) in identifying the optimalstrategies and how are these choices moderated by the substitutability parameters?In essence, we are interested in understanding the trade-off between efficiency (asrepresented by the costs) and market effects (represented by the substitutabilityparameters).

2. Strategy Regions and Market Effects: How are the optimal strategy regionsmoderated by the substitutability parameters? This could provide insights intohow market effects impact the optimal product portfolio.

In order to address these questions, we use office machines (specifically printers and

scanners) as our products for analysis. There is assumed to be a single manufacturer (e.g.,

HP) who has the capability of offering a printer (base product 1), a scanner (non-base

product 2), and a printer-scanner (MFP - product 3) to a single market. Price quotes from

www.Officedepot.com and www.Nextag.com and the gross margin data from HP’s financial

report are used to set the maximum prices, unit costs and the maximum profit margins for

two single function products (products 1 and 2). These parameters values are set as:

Product 1 (Printer): a1 = $680, and c1 = $240 which implies that d1 = $440; and

Product 2 (Scanner): a2 = $580, and c2 = $280 which implies that d2 = $300.

Since profit margins are a function of maximum prices and costs, we generate three

figures. In Figures 2-1 and 2-2, we set cost of the MFP (c3) to be equal to $ 400 and then

43

vary a3 in the range $ [600, 1200]; while in Figure 3, we set the maximum price of the

MFP (a3) to be equal to $ 900 and then vary c3 in the range $ [200,600]. This essentially

implies that the Figures 2-1 and 2-2 illustrate the impact of maximum price of the MFP

while Figure 2-3 illustrates the impact of MFP cost. Summary observations based on these

figures are:

• Based on Figures 2-1 and 2-2, it is obvious that a larger maximum price of the MFP(a3) as compared to the maximum prices for the two-single function products (a1 anda2) is necessary for the MFP to be included in a product portfolio.

• Based on Figure 2-3, it appears that the MFP is included in an optimal portfoliowhen the cost of the MFP (c3) is lower than the sum of individual costs for the twosingle function products (i.e., c1 + c2).

• A larger maximum price for the MFP (a3) is required to include the MFP in anoptimal product portfolio when substitution effects (i.e., r13 or r23) increase. Forexample, in Figure 2-1, we see that the MFP is included in an optimal productportfolio when a3 ≥ 740 for r13 = 0.5 while this is the case only when a3 ≥ 872 forr13 = 0.8. Similarly, Figure 2-2 shows that the MFP is included in an optimal productportfolio when a3 ≥ 718 for r23 = 0.2 while this is the case only when a3 ≥ 814for r23 = 0.5. Managerially, this indicates that when the market effects as reflectedby the substitutability indices increase, then this must be accompanied by a highermaximum price for the MFP for it to be included in the firm’s product portfolio.

• Once an MFP is included in the optimal product portfolio, then it is also interestingto note that the transition between strategies APS and PFMPS1/2 occurs with alower maximum price for the MFP when substitution effects are larger. For example,Figure 2-1 shows that the transition between APS and PMFPS1 occurs whena3 ≥ $1182 given r13 = 0.5 while this same transition between APS and PMFPS2occurs when a3 ≥ $982 given r13 = 0.8. Similarly Figure 2-2 shows that the transitionbetween APS and PMFPS1 occurs when a3 ≥ $1048 given r23 = 0.5 while this sametransition between APS and PMFPS2 occurs when a3 ≥ $1164 given r23 = 0.2. Thus,from a managerial perspective, the move towards to a portfolio strategy where theMFP is included with one of the two single function products occurs with a lowermaximum price for the MFP when the substitution effects are higher.

• If the substitution effects are low, then the MFP is more likely to be included in theoptimal portfolio even if the cost is higher as compared to when substitution effectsare higher. For example, when r13 = 0.8 and r23 = 0.4, the MFP is included in anoptimal portfolio when c3 ≤ $428. On the other hand, when r13 = 0.6 and r23 = 0.4,the MFP is included in the optimal portfolio provided c3 ≤ $516. Managerially, thispoints to the fact that if the market perceives that the MFP is more of a substitute

44

Figure 2-1. Price versus Market Effects: Optimal strategy when a1=680, c1=240, a2=580,c2=280, c3=400, and r23=0.4.

Figure 2-2. Price versus Market Effects: Optimal strategy when a1=680, c1=240, a2=580,c2=280, c3=400, and r13=0.6.

for the single function products, then internal cost controls need to be more stringentto ensure that the MFP is included in the optimal portfolio.

• As with the case of maximum prices, it can also be seen that the transition betweenstrategies APS and PMFPS2 occurs quicker when the substitution effects are large.For example, the transition APS to PMFPS2 occurs when c3 ≤ 329 when r13 = 0.8and r23 = 0.4 while this transition does not occur at all with a lower value of eitherr13 and/or r23 within the range of values for c3 explored in Figure 2-3.

45

Figure 2-3. Cost versus Market Effects: Optimal strategy when a1=680, c1=240, a2=580,c2=280, a3=900.

In sum, these results indicate that when substitution effects are higher, the MFP

needs to command a higher maximum price to be included in the product portfolio.

However, once this is achieved, higher substitution effects actually lead to the MFP

included in the product portfolio either with one of the two single function products or by

itself with lesser required increases in the maximum price. In a similar vein, we also see

that when substitution effects are higher, the base cost for the MFP can be lower for it to

be included in the product portfolio. However, once this is achieved, higher substitution

effects actually lead to the MFP being included in the product portfolio with one or more

of the single function products with smaller reductions in MFP cost.

In order to illustrate these regions, we start by setting d1 = $440 and d2 = $300. In

addition, Figure 2-4 is generated by fixing r23 = 0.4 and simultaneously varying r13 in the

range [0.5,0.95] and d3 in the range $ [380,800]; Figure 2-5 is generated by fixing r13 = 0.6

and simultaneously varying r23 in the range [0.1,0.5] and d3 in the range $ [380,800]; and

Figure 2-6 is generated by fixing d3 = $600 and simultaneously varying r13 in the range

(0,1) and r23 in the range (0,1). Figures 2-4 and 2-5 illustrate the following key results:

• In Figure 2-4, when d3 ≤ $523, the optimal choice is between NMFPS and APS andfor the latter strategy to dominate the former in this range, a larger maximum profit

46

margin for the MFP (i.e., d3) is required as the substitution effect increases. Onced3 ≥ 523, we note that the MFP is always included in the product portfolio regardlessof the substitution effects parameter r13. However, the manner in which the MFP isincluded in an optimal portfolio is moderated by both the maximum profit marginand substitution effect parameter.

• In Figure 2-5, we observe a similar result to Figure 2-4. Hence, when d3 ≤ $415, theoptimal choice is between NMFPS and APS and for the latter strategy to dominatethe former, a larger maximum profit margin for the MFP (i.e., d3) is required asthe substitution effect increases. Based on the parameter settings for this figure,we note that the MFP is always included in the product portfolio regardless of thesubstitution effects parameter r13 when d3 ≥ $415.

Figure 2-4. Dominant regions with d1=440, d2=300, and r23=0.4

Managerially the results in Figures 2-4 and 2-5 can be interpreted to formulate the

following general guidelines for strategy choices:

1. APS is more likely to be the optimal choice when demand substitution effects aresmall and the profit margin of the MFP is not too large or too small. The profitmargin of the MPF must be least equal to the linear combination of the profitmargins and the substitution indices of the SFPs;

2. PMFPS1 and/or PMFPS2 are more likely to be the optimal choice when either thedemand substitution effects are large or the maximum profit margin for the MFP islarge; and

47

Figure 2-5. Dominant regions with d1=440, d2=300, and r13=0.6

3. SMFPS is more likely to be optimal when both the demand substitution effects arelarge and the maximum profit margin for the MFP is large.

The final set of results in Figure 2-6 offer a different perspective on the strategy

regions which all include the MFP in the optimal product portfolio. Given our assumptions

that r13 > r23 and r213 + r2

23 < 1, the dark dashed lines in this figure indicate the bounds

for the feasible region for strategy choices. Given the parameter settings of the maximum

profit margins for the three products, managerial guidelines based on these results are:

Figure 2-6. Dominant regions with d1=440, d2=300, and d3=600

48

• If the substitution effect between product 1 and the MFP is significantly higher thanthe substitution effects between product 2 and the MFP, then the MFP should beincluded with product 2 in the optimal portfolio (PMFPS2);

• If the substitution effect between product 2 and the MFP is significantly large andnot too different from the substitution effect from the substitution effect betweenproduct 1 and the MFP, then the MFP should be included with product 1 in theoptimal portfolio (PMFPS1); and

• The MFP should be included by itself (SMFPS) only when both substitution effectsare significantly large (regardless of the difference between them).

2.6 Implications and Summary

As new technology advances make more and more MFPs available, a key decision

for firms is whether to include these products in their optimal product portfolios.

By integrating demand substitution effects, costs and prices associated with single

function and multi-function products, we examine this decision for a single firm. Using a

stylistic model that incorporates several features unique to MFPs, we are able to provide

normative guidelines on dominant product portfolio strategies. These dominant strategies

are: No MFP Strategy (NMFPS); All Product Strategy (APS); Partial MFP Strategies

(PMFPS1 or PMFPS2); and the Single MFP Strategy (SMFPS).

We are able to identify the key parameters driving the choice between these dominant

strategies. To start with, it is necessary for the firm to understand and parameterize the

demand substitution effect between each single function product and the MFP. Assuming

this information is available, we are able to show that the firm’s optimal choice of a

product portfolio strategy is driven primarily by the maximum profit margin associated

with the MFP. At one extreme, if this maximum profit margin for the MFP is less than

or equal to the weighted average profit margin for the two single function products,

then the MFP should not be included in the optimal product portfolio (i.e., NMFPS is

optimal). On the other hand, the other extreme case is that if the maximum profit margin

for the MFP dominates the adjusted maximum profit margins for both single function

products, then it is optimal for the firm to include only the MFP in its product portfolio

49

(i.e., strategy SMFPS is optimal). Within these extremes, the other strategies which

include one or more single function products and the MFP are optimal. These results

also provide some insights into how the firm could potentially influence the choice of a

portfolio strategy. Since the maximum profit margin for the MFP is a function of the cost

associated with the MFP (lower the cost, higher the maximum profit margin), this could

be viewed as an incentive to lower the manufacturing costs associated with the MFP so

that it could be included in its optimal product portfolio choice.

An analysis of the demand substitution effects also leads to some interesting insights.

If these effects are higher (for both the base and non-base product in relation to the

MFP), then the likelihood of the MFP being included in the optimal product portfolio is

lower. Thus, there is a necessity to focus on both quantity and margin effects in making

the decision to include an MFP in the optimal product portfolio. Further, there are

interactions between the demand substitution effects and the maximum profit margin of

the MFP which moderate the choice of the partial MFP strategies. In essence, smaller

(larger) values of the demand substitution effect between the base (non-base) product and

the MFP tend to increase the possibility of the base product and the MFP being included

in an optimal product portfolio. On the other hand, larger (smaller) values of the demand

substitution effect between the base (non-base) product and the MFP tend to increase the

possibility of the non-base and MFP being included in an optimal product portfolio.

50

CHAPTER 3FUSION PRODUCT DESIGN

3.1 Introduction

In the previous chapter, we investigated the optimal product portfolio decision for a

stylized setting. In this chapter, we extend our analysis to a situation where n functions

can now be integrated into any combination of fusion products. The objective is to gain

insights into which combinations of functions should be integrated in designing fusion

products.

Fusion products with multiple functions are thriving in the market. Multi-function

office machines are often equipped with three or four functions. A multi-function cell

phone can now integrate functions of cell phone, digital camera, PDA, music player, and

GPS all into a single device. Apple’s iPhone is advertised to replace four single-function

devices. There are several factors that contribute the popularity of fusion products. A

multi-function portable device has the advantage of space saving and is free of integration

hassles between functions. The multi-function office machines are normally priced just

slightly higher than single-function products since they have been in the market for a long

time. But a well-integrated multi-function cell phone can be priced with a high premium.

The availability of the fusion technology increases the product variety and the

complexity of product portfolio decisions for the firm. For example, when a fusion

technology which integrates five functions together is available, there are in fact 31

different function combinations of products that are technologically achievable. In terms

of possible product portfolio, there are totally 231 − 1 portfolios to choose from. However

we do not observe all sort of function combinations in the personal portable device market

if we only consider these five functions: cell phone, digital camera, PDA, music player,

and GPS. For example, the combinations of digital camera with music player, PDA with

GPS, or GPS with digital camera are not available in the market. There is no explanation

about why these function combinations are not attractive to the manufacturers of personal

51

portable devices. Some odd multi-function devices (i.e., a digital camera with cell phone

function added) have been introduced into the market. But these products receive

negatives reviews and low acceptance from the consumers, then are soon phased out from

the market.

From a technology development, product design, and product introduction perspective,

we addresses the second decision. Thus, we explicitly assume that the firm has access to

(through self-development or external sources) the technology required to offer an FP.

Given that this is the case, the product design decision is addressed using a product

portfolio perspective and thus, the general approach we propose would also help a firm

make the strategic decision on the optimal combination of SPs and FPs (which defines a

product portfolio) to introduce to the marketplace. The following literature reviews the

research of product line selection and product portfolio as product variety increases.

3.2 Literature: Product Variety and Product Line Selection

In Lancaster [64], product variety in most papers refers to the number of variants

within a specific product group, but the view of product variety and the questions

being asked varies a lot. There are four different views concerning product variety: the

individual consumer, the individual firm, market equilibrium, and the social optimum.

Sorenson [71] uses an evolutionary perspective to test the product variety strategy in

the computer workstation industry. The author examines 179 firms and 1,276 products

from 1980 to 1996. The results illustrate that product variety becomes less valuable

when the total number of products in the market increases. Product culling can remove

badly-performed products but also decreases the ability of the firm reacting to the change

of consumers’ preference.

Ramdas and Sawhney [67] develop measurement procedures for life-cycle sales

volume and cost for line extension of assembled products. Sharing components can bring

a cost benefit because of demand pooling for component. However, similarity among

assembled products decreases the revenue because these products substitute each other’s

52

demand. The authors decompose the life-cycle sales volume and develop several measure

procedures, and then an optimization model is used to identify a subset of line extensions

that has maximum incremental profits.

Ramdas et al. [68] identify three organizational approaches for component sharing,

which achieve product variety with lower component variety and cost for assembled

products. Three conceptual constraints - component-to-product feasibility, system-to-product

feasibility, and component interactivity - are also discussed.

Loch and Kavadias ([62]) recognize the inherent combinatorial complexity of

optimally determining a project portfolio when analyzing this decision from the technology

development and R&D investment level. They focus on the dynamic allocation of

resources over a fixed planning horizon which can guide managers in the development

of a new product with several associated product lines.

Other papers have addressed the product line and pricing problem. Dobson and

Kalish [57] formulate a mathematical program which assumes that the customers in

certain market segments are homogeneous and can be aggregated. They find that, due

to fixed cost of assortment and the cannibalization effects, having too many variants may

decrease the total profit. Despite a different modeling approach, our numerical examples

also suggest that the optimal product portfolio is normally a small subset of all possible

products. Chen and Hausman [56] propose a parsimonious choice-base conjoint analysis

which leads to an efficient algorithm to solve the product line/price problem. Different

from ranking/rating based conjoint analysis that often depends on enumeration or

heuristic procedures to select the product line, their method can be applied to a problem

with realistic size of product line and solved by commercial mathematical programs.

Hopp and Xu [61] investigate product line selection while the firm uses modularity to

reduce the introduction cost of new product. Assuming existence of economy of scope

without economy of scale, they find that modularity offers more variety in the product

line, which also benefits a higher market share and a price premium. It was observed, in

53

a multiple-segment market, a risk-adverse firm may sometimes reduce product variety to

save production cost.

One track of product assortment models utilizes multinomial logit (MNL) to model

the consumer’s utilities and purchasing choices. Cachon et al. [55] investigate the retailer’s

assortment problem when consumer search is possible. They differentiate three models of

consumer search: no search, independent assortment search and operlapping independent

assortment. In some cases, the optimal assortment is within a defined popular set of

products for no search and independent search models. Our model confirms this result

utilizing a different demand based approach. While including the cost of assortment,

they find that the optimal assortment can be found by full enumeration. In contrast, we

provide structural results to aid in the identification of the optimal product portfolio.

Other papers also use MNL choice models to investigate the product line and stocking

problem. Aydin and Ryan [53] analyze the product line and pricing decisions in three

situations. They find that when the shelf space is limited, the optimal product line

consists of a number of models with the highest average margins. Smith and Agrawal [70]

develop a multi-item inventory system to meet the individual item’s service level. Van

Ryzin and Mahajan [75] analyze an inventory model in which a sequence of consumer

arrivals purchases only their first choice. Mahajan and van Ryzin [65] is similar to van

Ryzin and Mahajan [75] except that the consumers substitute among product variants if

their first choice is not available.

The comparison of product line selection models above with our model is listed in

Table 3-1.

We can differentiate our fusion product problem by contrasting to the product line

selection models in terms of the timing of decision. From product development to product

display in the retail shop, this first stage of FP decision making concerns itself with

such technology development. The research is exemplified by Loch and Kavadias ([62]).

Afterward, the firm makes a strategic decision at the market level concerning both (a)

54

Table 3-1. Comparison of product line selection models.

Model StructureDemand MNL Substitution Fixed cost Inventory

CharacteristicsDobson and Kalish Aggregate No No substitution Yes No(1998) Demand

Van Ryzin and Individual Yes No substitution No YesMahajan (1999) discrete choice

Aydin and Ryan Individual Yes Substitution based No Yes(2000) discrete choice on availability

Smith and Individual No Substitution based Yes YesAgrawal (2000) discrete choice on availability

Chen and Hausman Aggregate Yes No substitution No No(2000) demand

Mahajan and van Individual Yes Substitution based No YesRyzin (2001) discrete choice on availability

Cachon, Terwiesch Individual Yes Substitution based Yes Yesand Xu (2005) discrete choice on availability

This model Aggregate No Substitution based No Nodemand on functionality

which FPs to bring to market and (b) how much of each product variant to produce based

on aggregate demand projections. This is the focus of the model introduced here. At last,

the firm decides the production and inventory stocking issues, which are investigated as in

[55], [70], [75], and [65].

For a more thorough literature view of product variety, we refer the readers to

Lancaster [64], Ho and Tang [60], and Ramdas [69]. Regarding applying financial portfolio

theory to product portfolio, see Cardozo and Smith [76] and Devinney and Stewart [78].

3.3 General Model

3.3.1 Preliminaries

Assume there exists a product-fusion technology that can integrate any combination

of functions 1 to n into FPs. As a result, the firm has the capability to offer m = 2n − 1

different products which includes n products each with a single functionality and m − n

55

FPs. Instead of directly addressing the problem of how many functionalities should

be combined when designing FPs, we approach this issue indirectly using a product

portfolio perspective. In essence, we focus on identifying an optimal product portfolio

(which includes at most m products) and note that by examining the components of such

a portfolio, the firm can identify which functionalities should be incorperated in each

product. Our contention is that such an approach is more comprehensive since it provides

input into the design decision (for FPs) by integrating product substitution effects and

market demands.

When the fusion-technology makes FPs possible, then the demands for all products

are more dynamic due to substitution effects. Consequently, the optimal quantity decision

for each product is influenced by the substitute product. Because no two products are

exactly identical to each other, we assume each product k has its own market potential ak

and variable cost ck. We denote a and c as the m × 1 market potential vector and m × 1

variable cost vector of m products, respectively. The variable notations used in this paper

are summarized in Table 3-2.

If any two of these m products possess similar functionality, then there exists

some degree of substitution between the two markets for these products. Using the

manufacturer of office machines as an example, single-function products which could

potentially be offered are the fax machine, copier, printer, and scanner (i.e., n = 4).

Given no overlap in functionalities between each of these products, they are not

considered substitutes. With the availability of fusion-technology, the manufacturer

now has the capability of offering 11 FPs (i.e., 24 − (4 + 1)) and depending upon the

functionalities included in each of them, these could be considered substitute products.

For example, assume that the manufacturer introduces an FP which integrates the

functionalities associated with a printer and a copier. In this case, this product would be a

substitute for the single function printer, the single function copier, and other FPs which

56

Table 3-2. Variable Notation for Chapter 3

SP Single-function productFP Fusion productpk Price of product kqk Quantity of product k offered by the firm (decision variable)ak Market potential (the maximum amount of willingness-to-pay) of product kck Unit variable cost of product kdk Scaled profit margin of product k, dk = ak−ck

2

S The optimal product portfolio with s distinct productss The cardinality of the optimal product portfolio SFk Function krk,j Substitution index representing one unit of product j on the price of product kγk,j Average substitution index between products k and jΠS Profit function of portfolio SK A subset of single-function products associate with product portfolio SλK Linear combination of profit margins from all SPs in KθK Concavity index composed of the substitution indices of all SPs in K

NFPS No fusion product strategyAPS All product strategyPFPS Partial fusion product strategySFPS Single fusion product strategy

incorporate the functionality of a printer and/or the scanner (e.g., printer/fax; copier/fax;

printer/copier/fax).

To differentiate various levels of combinations of function, in this paper, a fusion

device equipped with all functions, is called all-in-one, and a fusion device with only

some functions is called some-in-one. For example, consider a four-function set containing

functions of copying, scanning, printing and fax. Then, a copier/printer and a printer/copier/fax

are examples of some-in-one, and a device with four functions is an example of an

all-in-one product. Due to the complexity of the model, we assume that the fusion

technology is exogenously given.

Before specifying the inverse demand function, we first describe substitution effects.

The substitution among m products can be represented by a m × m asymmetric

57

substitution matrix r:

r =

1 r1,2 r1,3 · · · r1,m

r2,1 1 r2,3 · · · r2,m

r3,1 r3,2 1 · · · r3,m

......

.... . .

...

rm,1 rm,2 rm,3 ... 1

,

where 0 ≤ rk,j < 1 (k 6= j). A small (large) rk,j is associated with weak (strong)

substitution effect of product k substituted by product j. If there is no function

overlapping between two product k and j, rk,j should be zero; otherwise, rk,j should

be a number less than 1. Different from Chapter 2, the substitution between any two

products in this model can be asymmetric ([81], [52]). Normally, a high-end product

has a stronger substitution effect to its low-end substitute than vice-versa. For example,

an all-in-one printer has stronger substitution effect on the single function printer than

the converse. Managers can utilize the estimation techniques shown in Ben-Akiva and

Gershenfeld [54] and Hendel [59] for assessing the substitution matrix for their firm. Also,

a technique similar to that described in Chen and Hausman [56] for choice-base conjoint

analysis can be adapted to derive aggregate demand level parameters.

Let a be the market potential vector, q be m × 1 quantity vector and r be the

substitution matrix. The inverse demand functions are as follows.

p = a− rq.

Let c be the m× 1 variable cost vector and d be the m× 1 profit margin vector, such that

dk = ak−ck

2is the k-th element of d (k = 1, 2, . . . , m). Then the profit function of the firm

is

(GP ) : Π = qT (p− c) = qT (a− c− rq) = qT (2d− rq)

s.t. q ≥ 0.

58

We do not include the fixed cost in the objective function since we assume the

investment in technology has already been made and thus, there is no additional fixed cost

of function combination selection.

Because of asymmetric substitution effects among all products, offering all m products

may not be an optimal strategy for the firm. Using office machine products as an example,

even though a printer with faxing function is a technologically achievable product,

consumers cannot find such a product in the market. This implies that offering all FPs

might not be an optimal strategy. Hence, the objective of the firm is to find the optimal

product portfolio and the quantities for product variants that maximize profit. Facing

m(= 2n − 1) technologically achievable products, the firm has a total of 22n−1 − 1 product

portfolios to choose from.

Let us first start by evaluating whether the objective function to model GP is strictly

concave in the decision variables. In order to do this, we note that the Hessian (see

Appendix C) is defined as:

H =

1 γ1,2 γ1,3 · · · γ1,m

γ1,2 1 γ2,3 · · · γ2,m

γ1,3 γ2,3 1 · · · γ3,m

......

.... . .

...

γ1,m γ2,m γ3,m · · · 1

= (−2)γ,

where γk,j = 12(rk,j +rj,k) (k, j ∈ {1, 2, ..., m} and k 6= j) represents the average substitution

effect between products k and j. In order to establish concavity, we need to show that the

principal minors of H alternate in sign. Although this can be easily shown for the case

of n = 2 functionalities with some additional restrictions on the substitution effects (see

Chapter 2), it is analytically difficult to reach this conclusion when FPs can be designed

with n ≥ 3 functionalities. Also note that, in some cases, concavity of the objective

function does not guarantee that by simultaneously solving the FOC, we can determine

59

the optimal quantities since such an interior solution might violate the non-negativity

constraints on the these decision variables.

Based on this, we start by formulating the Lagrangian for our profit maximizing

model as follows:

(GL) : ΠGL = qT (p− c) = qT (2d− rq) + qT ν,

where νk is the lagrangian multiplier of quantity qk (k = 1, . . . , m) and ν is the lagrangian

multiplier vector. The FOC for this model (which identify necessary conditions for

optimality) lead to the following solution for the quantity offering and lagrangian

multiplier vectors (see Appendix D):

q = [γ]−1(d +1

2ν) (3–1)

ν ≥ 0 (3–2)

and, of course, for each product offering k, qkνk = 0∀k. This leads to some interesting

observations. For some product offering k, if qk > 0, then we have νk = 0; otherwise, when

νk = 2(∑m

j=1,j 6=k γk,jqj − dk) > 0, qk = 0. It is also possible that both the the quantity

offering and the lagrangian multiplier are zero simultaneously, which occurs when there is

a boundary solution. The content of νk implies that a product k with a “relatively small”

profit margin is more likely to have a positive lagrangian multiplier and hence, not be part

of the product portfolio while the product with “relatively large” profit margin is more

likely to be included in the product portfolio. However, a high profit-margin product may

not be selected if the substitution effects with other products are too strong. This ‘rule

of thumb’ with asymmetric substitution effects resonates with some of the results in [93]

when they focused only on symmetric substitution effects.

Now that we have some insights into which product offerings will probably be

included in a solution to our problem, let us turn our attention to identifying the optimal

set of product offerings. In order to do this, we first define a product portfolio S as one

which consists of specific non-zero quantity offerings for each product included in the

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portfolio and also has an associated profit function which is concave. For such a given

portfolio, let dS be the maximum scaled profit margin vector, qS be the quantity offering

vector, rS be the substitution effects vector, and γS the average substitution effects vector.

Then, it is easy to show that the non-zero quantity offering vector qS and the associated

profit for a given portfolio S are (see Appendix E):

qS = [γS]−1dS (3–3)

ΠS = qTS rT

S qS

=dT

S Adj [γS]dS

|γS| (3–4)

In case of symmetric substitution effects, the non-zero quantity offering vector qS and the

associated profit can be restated as:

qS = [rS]−1dS (3–5)

ΠS =dT

S Adj r[S]dS

|rS| (3–6)

when we have symmetric substitution effects.

3.3.2 Dominant Portfolios

With m potential products, there are theoretically 2m − 1 distinct portfolios (e.g.,

with 3 functionalities, m = 23 − 1 = 7 products and thus, 27 − 1 = 127 potential product

portfolios) of this type. Given that the number of distinct portfolios is substantially

large, we develop analytical results which can help to pare down the number of potential

portfolios which could be potentially optimal. Let S be a product portfolio of s products.

If S ′ = S ∪ {j} and j /∈ S, then we call S ′(S) the parent (child) portfolio of S(S ′).

Constructing the hierarchy of product portfolios, we know that a portfolio consisting of all

m products is at the highest level of parent portfolios since it contains all possible product

variants in a single portfolio. In contrast, one-product portfolios are the lowest level of

child portfolios. A parent portfolio with i products contains i direct children portfolios,

such that each child portfolio has one product less than its direct parent. For example,

61

if S ′ = {1, 2, 3} then this portfolio has three direct children portfolios: {1,2}, {1,3} and

{2,3}.If the profit function associated with a parent portfolio S is concave, this is an

important factor in determining the optimality of that portfolio. We can assess the

concavity of the associated profit function, we can do this simply by ensuring that all

principal minors of γS have positive determinants. As a result, if a product portfolio S has

an associated profit function which is concave, the the following theorem establishes an

important dominance relationship between parent and child portfolios.

Theorem 3.1: Assume |γS| > 0 and qS is a positive optimal quantity vector for portfolio

S with concave profit function. Let S ′ be a product portfolio created by adding another

product j (j /∈ S) into portfolio S. If qS′ is also a positive optimal quantity vector for

portfolio S ′ and γS′ is invertible, then

1. If |γS′| > 0, then S ′ dominates S; else

2. S dominates S ′.

Proof. See Appendix F.

The key implication of this result is as follows. A parent portfolio S ′ dominates a

child portfolio S if and only if S ′ has an associated profit function which is concave and

the quantity offerings for all products included in S are all positive. In essence, this also

implies that S ′ dominates all its child portfolios. This result can be used to reduce the the

number of potential product portfolios which need to be evaluated so that the firm can

identify an optimal portfolio of products.

We use the dominance relationship established through Theorem 3.1 to propose a

search algorithm for finding the optimal product portfolio. More specifically, by starting

with smaller children portfolios and adding single product variants to these portfolios,

all potential portfolios for evaluation can be identified. The determinant of the average

substitution matrix γ for each children portfolio is the building block of each potential

parent portfolios since these have already been evaluated and computed. Note that a

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portfolio with all product variants is unlikely to be concave, especially if the substitution

effects are high between the possible product variants. While the dominance result alone

does not determine the optimal portfolio, it can be used as a basis to identify good

candidate portfolios which can then be exhaustively evaluated to identify the optimal

portfolio. The proposed algorithm to determine such an the optimal portfolio is as follows.

1. Set i = 0 at iteration 0. Maintain a list that contains a null portfolio S = {∅}.

2. For each concave portfolio S in the list, add another product j such that j /∈ S.Hence every S ∪ {j} is an (i + 1)-product portfolio.

3. If, at iteration i + 1, there is no (i + 1)-product portfolio with concave profit function,Go to Step 6; otherwise, continue to the next step.

4. For each (i + 1)-product portfolio with concave profit function, examine the optimalquantities by solving the FOCs of the profit function. For each portfolio, if allquantities are positive, add this portfolio to a list of concave feasible parent portfolioand remove all children portfolios of this newly-added portfolio.

5. Set i = i + 1, Go to Step 2.

6. Compare the profits of the portfolios in the list of concave feasible parent portfolio,and the optimal product portfolio is the one with highest profit.

Each iteration in the search algorithm saves computation time through two

mechanisms. First, it is easy to check the concavity of the profit function corresponding to

newly composed portfolios through an evaluation of the principal minors of the associated

γ matrix. Thus, if this is not the case, we save computation time by not proceeding

to compute the quantity offerings of all the products in a portfolio through the FOC.

Second, even if the objective function for a given portfolio is concave, it is possible that

the quantity offerings for all products included in this portfolio are not positive. In this

case again, we do not include such a portfolio as a possible one to evaluate for identifying

the optimal portfolio.

A reasonable question following the results above concerns the specific properties

of the optimal portfolio. Unfortunately, due to the complexity of the general model and

63

the matrix form, analytical results are difficult to derive. In Section 3.4, we investigate

a special case of FP design where a firm currently offering n SPs would like to obtain

insights as to whether to also design a single all-in-one FP and offer it to the market.

3.4 One All-in-One Model

Given this setting, the potential product set for the firm is n SPs and one all-in-one

fusion product f that integrates all the functions of the n SPs. This is obviously a special

case of our general model and our focus is to gain insights into the composition of the

optimal product portfolio. Thus, we primarily focus on assessing whether the FP is

included in the optimal portfolio. To start with, since substitution only exists between

each SP and the all-in-one, the substitution matrix r is a relatively sparse (n + 1)× (n + 1)

matrix as follows:

r =

1 0 · · · 0 r1,f

0 1 · · · 0 r2,f

......

. . ....

...

0 0 · · · 1 rn,f

rf,1 rf,2 · · · rf,n 1

Based on this, the profit function for the firm is:

ΠAIOM =n∑

k=1

qk(ak − ck − qk − rk,fqf ) + qf (af − cf − qf −n∑

k=1

rf,kqk) (3–7)

To start with, we define several product portfolios: (a) APS (All Product Strategy)

which consists of all SPs and the FP (i.e., APS = {1, . . . , n, f}); (b) NFPS (No Fusion

Product Strategy) which consists of all SPs (i.e., NFPS = {1, . . . , n}); (c) SFPS (Single

Fusion Product Strategy) which consists of only the fusion product (i.e., SFPS = {f});and (d) PFPS (Partial fusion product strategy) which consists of some SPs and the

fusion product.For this special setting, the process to identify an optimal portfolio is relatively

straight forward and we proceed as follows:

1. Evaluate portfolio APS as follows.

64

• Check whether the associated profit function for this portfolio is concave. This canbe done by simply ensuring that (1 − ∑n

k=1 γk,f ) ≥ 0. If this is not the case, thenAPS cannot be the optimal portfolio and goto 2; otherwise, continue.

• Compute qf =df−

∑nk=1 γk,f dk

1−∑nk=1 γk,f

and qk = dk − γk,fdf (for k = 1, . . . , n).

• If qf and all qk are positive, then APS is the optimal portfolio and STOP else, goto2.

2. Evaluate portfolio NFPS. In this case, it is trivial to show that for this portfolioqk = dk ∀k, qf = 0, and ΠSFP =

∑ni=1 d2

k.

3. Evaluate portfolio SFPS. In this case, it is trivial to show that for this portfolioqf = df , qk = 0 ∀k, and ΠF = d2

f .

4. Evaluate all possible portfolios PFPS. Using the dominance relationship establishedthrough Theorem 1, compare all possible parent portfolios in this set to identify the“best” PFP portfolio (defined as one which provides the maximum profit). If such aportfolio exists, goto step 6, else goto step 5.

5. Compare portfolios NFPS and SFPS and the one with the higher profit is optimaland Stop.

6. Compare the “best” PFPS portfolio in step 4 to the NFPS portfolio and the onewith the higher profit is optimal.

As is obvious, Step 4 is this process is computationally intensive. However, given

the sparsity of γ, it is much easier to implement the process to search among all PFPS

portfolios. Recall that each potential PFPS portfolio contains some combination of

the SPs and the fusion product f . Define K as the set of SPs included in a specific

PFPS - call this portfolio PFPSK . For this portfolio, define λK =∑

k∈K dkγk,f and

θK = 1 − ∑k∈K γ2

k,f . Then if θK ≥ 0, the profit function associated with portfolio

PFPSK is concave. Further, the optimal quantity offerings for all products included

in portfolio PFPSK , can easily be computed as qf =df−λK

θKand qk = dk − γk,fqf for

all k ∈ K. Of course, if the profit function is concave and all these quantity offerings

are positive, then PFPSK is a candidate portfolio for evaluation in step 4. In terms of

the resulting profit for PFPSK , this can also be determined quite easily as ΠPFPSK=

∑k∈K d2

k + θK−1(df − λK)2.

65

In the final steps of the process outlined above, steps 5 and 6 require some explanation.

Note that if there exists even one PFPS which is a candidate for an optimal solution,

then according to the process described above portfolio SFPS is never in contention as

an optimal portfolio since SFPS is always a “child” portfolio for any potential PFPS.

This justifies skipping step 5 provided there is at least one PFPS which is identified

as a candidate in step 4. It follows that the comparison in Step 6 (between the “best”

PFPS and NFPS) is also quite straightforward. Assume that K∗ represents the set of

SPs in the ‘best’ PFPS identified in step 4. Then if θ−1K∗(df − λK∗)2 >

∑k/∈K∗ d2

k, PFPS

dominates NFPS and vice versa. Of course, if there is no PFPS which is feasible (which

is quite unlikely), step 5 simplifies the search process for the optimal portfolio by simply

comparing portfolios NFPS and SFPS.

The results for this special setting (where a firm can offer a single FP incorporating

the functionalities of n distinct SPs) indicate that it is highly likely that the FP will be

included in the firm’s optimal product portfolio (since it is included in SFPS, APS, and

all possible PFPS). From an FP design perspective, this implies that a firm should make

an attempt to design an FP which integrates the n functionalities included in each SP.

In the next section, we focus on the general case where the firm can design FPs with any

combination of n functions. Given that this problem is analytically complex, we resort to

a numerical analysis based on secondary data.

3.5 Numerical Examples

The analysis in Section 3.4 is related to the product design decision when a firm can

offer an all-in-one device which integrates n functions together. Since the firm has the

technology to fuse n SPs together, it is likely that technology is available to fuse subsets

of the component products. However, because of the complexity of this more general

problem, analytic results are difficult to obtain. In this section, we perform numerical

analysis to gain further insight into this problem. Specifically, the dynamics of changes in

66

the substitution indices and the profit margins are investigated for the complete model as

discussed in Section 3.3.

For the numerical examples, we use Sony to illustrate a firm who offers fusion

products based on digital camera, MP3 music player, and cell phone functionalities.

Sony is a manufacturer of digital cameras and MP3 players, and it also maintains a

joint-venture with Ericsson to produce cell phones, which adopts Sony’s technology

to provide multi-function cell phone [83]. Sony owns 50% share of Sony Ericsson (SE

hereafter); hence, Sony has strong influence in the joint-venture’s strategy. Moreover,

recent articles have discussed how SE has adjusted the number of product variants that

it is providing to the market [85]. The president of the corporation has commented that,

“We are confident that the remainder of the year will see us further capitalize on this new

broader portfolio,” which includes cell phone, camera and MP3 product variants.

We index the cell phone, the digital camera, and the MP3 player as (single-function)

products 1, 2, and 3, respectively. In terms of SPs, product models SE T105, CyberShot,

DSC-S700, and Walkman NWZ-A816 are examples of products 1, 2, and 3, respectively.

Since all cell phones offered by SE in 2007 have extra functions, we use an older model

(T105) as an example of a single function cell phone. Products 4, 5, and 6 are some-in-one

products which combine two of the single-function products together. SE K550i is an

example of product 4 that integrates functions of digital camera and cell phone. Sony

does not provide products 5 and 6 to the market that combine a cell phone and an MP3

player or a digital camera with an MP3 player. For the three-function all-in-one product,

SE W810i is an example of product 7 in our model. Note that since the camera phones

normally adopt low image resolution, we choose a low-end digital camera model and only

analyze the impact of fusion products on the low-end market.

Regarding profit margin estimations, we use data from Sony and SE ’s annual report.

The average gross profit margin rate of SE in the last three year is 28%, while Sony’s

annual report shows that the company-wide gross profit margin is 37%. There is no

67

available profit margin data from any of Sony’s specific product categories. Therefore, we

utilize industry data from Sony’s competitors to estimate the gross profit margins for the

SPs. According to news reports, Nokia’s cell phone [88], Canon’s digital camera [87], and

Creative’s MP3 player1 have gross profit margins at 15 %, 23 % and 23 %, respectively.

Based on the market price data, the prices of products 1, 2, 3, 4, and 7 are $120, $150,

$150, $200 and $240. We extrapolate the prices for products 5 and 6, since these are

not currently offered by Sony. The profit margin is then calculated using both the unit

price and the gross profit margin. According to this estimation method, the scaled profit

margins (dk) of products 1 to 7 used in our analysis are set as $24, $38, $38, $50, $53, $57

and $67, respectively.

To characterize the impact of the substitution matrix on the optimal solution, we

actually consider two different sample matrices A and B as shown in Table 4 and Table 5.

These two matrices allow us to capture some effects of the landscape of substitutability

indices on the optimal solutions. The values shown in matrix A are more realistic for

Sony’s three product market, in that there are relatively high substitution indices between

the products which contain similar functions. These single-function products under

consideration are fairly congruent, in that it is easy to fuse them into a single product

and the newly fused product is serving a similar market as the original single-function

products. In contrast, the values shown in matrix B reflect those associated with a more

incongruent set of products. The substitution indices are lower, as the combination

products seem to create a new market with less overlap with the original markets for the

single-function products.

We consider symmetric substitution matrices for the seven products as shown in

Tables 4 and 5. Note that this analysis also applies to any asymmetric substitution

matrices which can be ‘averaged’ to find these two matrices. A zero in the matrix denotes

1 Nokia predicts lower profit margins in coming years, 2006, www.teleclick.ca

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the fact that there are no substitution effects for the corresponding product pair. For

example, because products 1, 2, and 3 are single-function products, there is no functional

overlap between these different markets. Consequently, the values for γAi,j and γB

i,j between

these three products are equal to zero.

The results of six different numerical examples are shown in Table 6. A summary of

the input parameters as well as the corresponding optimal portfolio and objective values

are shown for each example. The first three cases correspond to the cell-phone, digital

camera, and MP3 player markets and use matrix A for the substitution matrix. Case 1

reflects the initial scenario using parameters described in the previous paragraphs where

a single fusion product is the optimal solution. In this situation, it appears that offering

a single all-in-one product captures the most profit for the firm. Cases 2 and 3 show the

impact of changes in specific profit margin parameters on the optimal portfolio. For case

2, an increase in the profit margin for the first single-function product (i.e., the cell phone)

has no effect on the optimal portfolio.

For case 3, an increase in the profit margin for the third single-function product (i.e.

the MP3 player) changes the optimal portfolio slightly. In this case, it is now optimal

to sell the single-function digital camera, single-function MP3 player and two-function

camera phone to the market. This result is supported in the press by anecdotal evidence

which points to the popularity of camera phone. The research report released by ABI

Research in 2005 [86] projects that the shipment of camera phone is predicted to surpass

the shipment of single-function cell phone. The low-end digital camera sales also shrink

significantly by the encroachment of the high-end digital camera and the camera phone.

The remaining examples utilize the matrix B which reflects a more incongruent

product set with lower substitution indices. In general, the optimal product portfolio

for these examples includes more product variants and is more sensitive to parameter

changes than those shown for matrix A. In comparing case 1 to case 4, the optimal

product portfolio includes both the all-in-one fusion product (i.e. product 7) and also a

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Table 3-3. Matrix A

γAi,j 1 2 3 4 5 6 7

1 1 0 0 0.8 0.8 0 0.82 0 1 0 0.6 0 0.6 0.63 0 0 1 0 0.9 0.9 0.94 0.8 0.6 0 1 0.85 0.85 0.85 0.8 0 0.9 0.85 1 0.9 0.956 0 0.6 0.9 0.85 0.9 1 0.957 0.8 0.6 0.9 0.8 0.95 0.95 1

Table 3-4. Matrix B

γBi,j 1 2 3 4 5 6 7

1 1 0 0 0.5 0.5 0 0.52 0 1 0 0.6 0 0.6 0.63 0 0 1 0 0.9 0.9 0.94 0.5 0.6 0 1 0.6 0.7 0.75 0.5 0 0.9 0.6 1 0.9 0.956 0 0.6 0.9 0.7 0.9 1 0.957 0.5 0.6 0.9 0.7 0.95 0.95 1

Table 3-5. Changing the profit margins (d1 and d2) and the results.

Case γ d1 d2 d3 d4 d5 d6 d7 Opt. Portfolio q∗ Profit1 γA

i,j 24 38 38 50 53 58 67 {7} {67} $ 44892 γA

i,j 30 38 38 50 53 58 67 {7} {67} $ 44893 γA

i,j 24 38 44 50 53 58 67 {2, 3, 4} {12.5, 44, 42.5} $ 45364 γB

i,j 24 38 38 50 53 58 67 {4, 7} {6, 62.7} $ 45075 γB

i,j 24 44 38 50 53 58 67 {2, 5} {44, 53} $ 47456 γB

i,j 24 38 44 50 53 58 67 {1, 2, 3, 4} {4.5, 14.6, 44, 39} $ 4548

some-in-one product (i.e. product 4). Specifically, those customers in the market for only

the first or second single-function products (i.e. those contained in product 1 or product

2) will choose between the two different products (i.e. product 4 or product 7). Customers

in the market for the third single-function product (i.e. product 3) will buy the all-in-one

fusion product (i.e. product 7).

In cases 5 and 6, the profit margin parameters are varied for SPs 2 and 3 and the

optimal product portfolio for these cases changes significantly. In case 5, the profit margin

for product 2 is increased. As a consequence, the optimal product portfolio now includes

the single-function product 2 and the some-in-one product 5, which indicates the firm

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should offer digital camera (for function 2) and MP3 phone (for functions 1 and 3) to

the aggregate market. Similarly, in case 6, the profit margin for product 3 is increased.

Consequently, the optimal product portfolio now includes all single-function products 1,

2 and 3, and the some-in-one product 4. Cases 5 and 6 implies that, ceteris paribus, the

all-in-one should be withdrawn when one of the SPs becomes more profitable. This may

happen when adding three functions significantly decreases the usability such that there is

a lack of synergy between the single-function products. Interestingly, the result from these

cases implies that when a single function product is associated with a relatively high profit

margin, the firm should not combine this function with others to sell it as part of a fusion

product.

Some additional managerial insights based on this analysis are as follows. First, when

the substitution effects are relatively high, a portfolio containing a smaller number of

products is likely to be optimal. If a single all-in-one fusion product has high margins,

then this model dominates the product portfolio. However, when a stand alone single

function product has relatively high profit margins, then it is less likely that a fusion

product containing this function should be offered. Instead, the firm should sell the

single function product independently and combine other lower margin functions into a

some-in-one product. Lastly, small changes in parameter values can cause large changes

in the optimal portfolio. When the set of products under consideration is somewhat

incongruent (i.e. the substitution effects are low), then the product portfolio in general is

somewhat larger and more sensitive to small changes in the profit margins.

3.6 Conclusion

If technology makes it possible to integrate many functions into one device, firms

might be contemplating introducing fusion products into the market. However, this

may lead to product proliferation, excessive self cannibalization, and consumer “feature

fatigue.” Manufacturers must decide how to intelligently fuse these technologies into

different product variants so as to design an appropriate fusion product for the market.

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We have analyzed a normative model to gain insights into this design decision. Even

though the number of possible product portfolio configurations is large, we develop an

algorithm which can exploit parent-child portfolio relationships with a simple check on

concavity properties of the objective function.

There are several managerial insights generated from the analysis of the model. In

general, the optimal portfolio and hence the ‘best’ product designs are a function of two

important parameters: profit margin and substitution effects. A product with higher profit

margin and smaller substitution effects with other products is more likely to be included

in an optimal portfolio and thus, this product design should be initiated. However, facing

any possible combination of function integrations, substitution and cannibalization cannot

be avoided if the firm intends to offer many different products. Since the firm’s objective

is profit maximization, a careful investigation and evaluation of all (single-function,

some-in-one, all-in-one) possible products is the best way to achieve optimality while

taking into account the cannibalization effects.

In general, the firm should not manufacture too many different fusion products

(FPs) simultaneously. Strong cannibalization effects among these FPs imply selecting

the right fusion product is important. In essence, the firm should not complement the

FP with too many component or other FPs when the substitutability indices are high.

It is also interesting to note that our general-form model can also be applied to the

problem of product variety in a certain market segment. Kraft Foods, Inc.[84] found they

have launched too many similar products in one market segment. Introducing too many

products induces strong cannibalization among their own products.

In contrast, the product portfolio and corresponding product design is more difficult

to determine when cannibalization effects are small. This situation can occur when the set

of single-function products (SPs) under consideration are incongruent, or when the fusion

products (FPs) create a significantly different market than the original SPs. Numerical

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results show that the optimal portfolio in this situation generally contains a wider variety

of products and is more sensitive to changes in the profit and cannibalization parameters.

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CHAPTER 4PRODUCT DIFFUSION MODEL FOR SINGLE-FUNCTION AND FUSION

PRODUCTS

4.1 Introduction

The diffusion of a product is described as the projection of the sales rate and

cumulative sales throughout the product’s life cycle. According to the interpersonal

communication theory in a social network, the diffusion of an innovation first reaches

the early adopters. Gradually these adoptions reach the rest of a social system through

interpersonal communication with early adopters. Assuming the adoption pattern is a

normal distribution, Roger [114] categorizes the adopters into five groups based on the

timing of their purchase: innovators, early adopters, early majorities, late majorities, and

laggards.

So far, product diffusion research focuses on the diffusion within a product category.

As we point out in Chapter 1, the current diffusion models do not capture the transition

from single-function products to fusion products. In this chapter, we construct a new

product diffusion model that is as parsimonious as possible to capture the diffusion process

of the FP and their interactions with the diffusion processes for the SPs. After the launch

of the FP, the FP starts encroaching on the demand of the SPs and changes their diffusion

paths. We are interested in the optimal launch time decision for the fusion product and

the factors that influence the decision.

We observe that the introduction of the FP influences the diffusion of SPs in many

ways. To illustrate, radio and CD player functions have been integrated soon after the

compact disc was used as a media of music, and the radio and CD combo has become a

mainstream product. Shipments of the single-function PDA also suffer from the growth of

the PDA phones and the smart phones. According to IDC’s survey, single-function PDA

shipments were down 22.3% in the first quarter of 2006 compared to the same quarter a

year ago and that was the ninth consecutive quarter of year-over-year decline. A combined

product of TV and DVD has been introduced in the market for a while but it fails to have

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strong impact on the sales of either the TV or the DVD player. Though we do not intend

to explain why some FPs soon replace the SPs and others do not, these examples illustrate

the dynamics brought by the launch of the FP.

The diffusion and substitution process where a fusion product gradually replaces

its single-function products has not been investigated in the literature. The following

questions related to product diffusion still need to be answered. (a) How long will it take

to penetrate to a certain level of market acceptance? (b) When should the firm introduce

the fusion product? (c) How does the manufacturer speed up the penetration of a new

fusion product?

The phenomenon above is similar to the diffusion process of several generations of

products, but there is a major difference: the research on the diffusion process of several

generations only focuses on one product category, but the fusion products influence the

product diffusion across several product categories. For example, the camera phone

has changed the market landscape in the cellular phone and the digital camera. The

diffusion model which treats the FP as a new generation of an SP might not be sufficient

to capture all of the dynamics brought by the FP. In addition, how well the fusion product

consolidates its component products has a strong influence on the adoption of the FP.

Thus, using a multi-generation diffusion model, the diffusion paths derived from two SPs

might be very different.

Moreover, if there is synergy among some SPs then fusing them together into an FP

is appealing to the market. For the consumer, the synergy includes space saving, lower

total cost, and function synchronization. The PDA phones can share the contact list

without the hassles of synchronization and allow the users to carry only one device. For

the firm, the synergy of launching FPs is manufacturing cost saving. A printer, a copier,

a scanner and a fax machine can all share the same platform and interface which makes a

multifunction office machine cost only at a small premium compared to a single-function

printer. The cost savings can make it possible for the FP be priced lower than the sum

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of all SPs. The early generations of multifunction office machines had high profit margins

due to manufacturing cost savings and strong demand.

Finally, we are interested in the role of the FP’s manufacturer. The FP’s manufacturer

can be an new entrant or an incumbent of one or two current SPs. In the office machine

market, the manufacturer of the scanner and the printer is also the manufacturer of the

all-in-one office machine. But in the personal portable device market, the first PDA phone

was introduced by Palm, which is a PDA manufacturer, while the successful iPhone was

introduced by Apple, which is the leader in the digital music player market. We will also

analyze the impact of the competitive role of the FP’s manufacturer on the FP’s launch

decision and on the diffusion processes.

In next section, we will review the literature related to the topic of product diffusion

and highlight those most related to our model.

4.2 Literature

Bass [89] proposes an S-shape diffusion pattern as a new product is introduced to

the market then diffuses gradually to the whole market. The S-shape is defined as the

cumulative sales of the product, and the sales rate over time is usually a bell shape, which

experiences several stages from birth, growth, peak, to decline. The diffusion pattern

has several important components: total market potential (m), innovation coefficient

(φ) and imitation coefficient (ψ) [102]. As a new product is launched, it first reaches

some innovative adopters in the whole market. As time goes by, the product’s prevalence

spreads from the ‘neighborhood’ of the innovative adopters as the neighbors imitate and

become the followers. Based on the time of adoption, the buyers can be categorized to one

of five groups: innovators, early adopters, early majority, late majority, and laggards [102].

The Bass model is known for its robustness in characterizing the diffusion process without

incorporating decision variables.

Mahajan et al. [102] point out that new product diffusion models have two different

strategic decisions: prelaunch/launch decisions vs. postlaunch decisions. Bayus [91]

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investigates the firms’ new technology investment decision in high-definition TV when the

various forecasts of consumer acceptance of the new technology have a wide gap between

optimistic and pessimistic results. Other than the demand forecast, prelaunch samplings

for “target adopters” are also critical to the diffusion path. Jain, Mahajan and Muller [97]

compare the word-of-mouth effects on durables and nondurables in the first adoption.

In postlaunch diffusion research, Mahajan et al. [102] illustrate six applications:

timing of successive generations, capacity decisions with product diffusion ([97] and [96]

), determining the market value on anticipated penetration ([99]), market saturation

assessment and expansion opportunity for the retailers ([104]), estimation of lost sales due

to pirated sales ([95]), and lost sales due to patent infringements ([103]).

There are many papers related to product or technology diffusion. For a summary of

the existing literature, we refer the reader to Majahan et al. [100] and the book edited by

Majahan et al. [102].

Here, we review several papers that are most relevant to our fusion-product diffusion

problem and related to the diffusion and substitution between several generations

of technology or product. Note that most of the literature in this area is limited to

addressing successive generations of products in a product category.

Norton and Bass [105] construct a diffusion model for repeat-purchase products where

the new technology brings successive generations of innovation. Different from previous

market-share substitution models (e.g. [94]), their diffusion model has the ability to

estimate a market potential and forecast the demand trajectory. Assuming that a firm

plans for several generations of innovation, where each innovation increases the market

potential, their model captures the substitution of sales from earlier to later generations.

By fitting their model into the DRAM’s and SRAM’s diffusion, they found that once

the new generation is introduced, the sales of the older generation begins to decline.

Compared to Fisher and Pry’s [94] market share substitution, Norton and Bass assert that

their model generates more accurate results in terms of estimation of the market share.

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Wilson and Norton [112] investigate the optimal time to introduce new generations

of a durable product, and they find that several factors primarily influence the result:

the substitution effect and diffusion between two products, the relative margins, and

the planning horizon of the firm. They assume that the first product has higher margin

but slow sales growth, and the new extension has lower margin but expands the market

potentials. The results illustrate that the firm should introduce the line extension as soon

as it is available or should never introduce it at all. However, they point out at certain

value of parameter, a static analysis and a dynamic analysis may get different conclusion

because the rate of growth is not considered.

Mahajan and Muller [101] extend the Bass diffusion model that captures the

substitution of successive generations in the technology innovation1 . Under some

assumptions that simplify the complexity of their diffusion model, they find that the

new generation should be either introduced as soon as it is available or at the maturity

stage of the previous generation product. This “now or at-peak” decision depends on the

gross profit margins, the diffusion and substitution parameters, the relative size of market

potentials, and the discount factor. The discount factor plays an important role on their

“now or at-peak” conclusion while Wilson and Norton’s [112] “now-or-never” rules do

not consider the discount factor. They apply their model in IBM’s mainframe’s diffusion

process and find out IBM introduced two successive generations to the market too late.

Putsis [107] applies the diffusion model into the diffusion of stand alone TVs and

VCRs by considering both cross-product demand and supply issues. Their empirical data

supports the hypothesis that product differentiation and new product introduction are

more likely to occur when the existing products start to slow at saturation levels.

1 Note that Mahajan and Muller [101] characterize the demand function in a differentmanner than Wilson and Norton [112]. Our model is more similar to Mahajan and Muller[101].

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Van den Bulte [110] investigates the diffusion speed of 31 electrical household durables

in the United States from 1923 to 1996. The results show a statistically significant

acceleration on the diffusion speed. Other factors affecting the diffusion speed are:

economic conditions, demographic change, whether there is a competing standard early in

the life cycle, and the amount of innovation investment required. The author emphasizes

that management should pay careful consideration to the probability, timing of takeoff

and speed after the takeoff, especially for the product which requires a large up-front

investment, and has a high uncertain pay-off and longer take-off time.

Talukdar et al. [108] investigate the diffusion of six products in 31 countries, which

covers 60% of world population. They find that the diffusion of a product introduced

earlier in one country is useful to explain the diffusion coefficients in other countries, while

the past experience of one product is more useful to estimate the penetration level of

another product in the same country.

Several papers address the topic of parameter estimation for diffusion models. Norton

and Bass [105] assume that the innovation and imitation parameters over generations are

the same. They admit that this is a strong assumption that leads to reasonable fit to the

empirical data. Mahajan and Muller [101] assume the same innovative parameter but a

different imitation parameter. Sohn and Ahn [2] discuss the diffusion of several generations

of technology from cost to benefit analysis. They use Monte Carlo simulation to find the

initial factors on the diffusion for new information technology. Also, Pae and Lehmann

[106] investigate the correlation between the inter-generation time and the diffusion data

fitting. They report that the later generation seems to have smaller initial sales (smaller

innovation parameter) but a faster rate of growth (larger imitation parameter) based on

their 30 pairs of two-generation products. They reach this conclusion from their ordinary

least squares regression analysis which obtains the changes in parameter values (φ and

ψ) between generations. However, Van den Bulte [111] cautions about that Pae and

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Lehmann’s [106] finding may be a methods artifact. Van den Bulte quotes that several

simulation studies ([109], [92]) have the same phenomenon observed.

Kim et al. [113] conduct an empirical model of the diffusion of telecommunication

products in Hong Kong and South Korea. Their dynamic market growth model captures

the technology substitution as well as the interaction of inter-product category effects.

Their diffusion fitting results show that their model can make better prediction than

the general Bass model. The market potential of one category or one generation is

significantly affected by others products and by the overall structure of a geographic

market. However, they also point out that their empirical model cannot answer strategic

questions such as the optimal launch time of a new product.

A key question influencing the firm’s decision to introduce an FP concerns its current

product offerings of SPs in the marketplace. Early years of the literature, incumbent

firms are believed to have advantage in the competition [118]. Three primary sources

of advantages of the incumbents are: (1) technological leadership, (2) preemption of

assets, and (3) buyer switching cost. However, in recent years, many late entrants

leapfrogged into the market-leader positions [119]. These leapfrogging new entrants

include Amazon.com, Apple, Blackberry, and Palm. Shankar et al. [120] investigate how

late entrants outsell pioneers in the pharmaceutical industry. They find that a innovative

late entrant can create a sustainable advantage by enjoying a higher market potential

and hurt the product diffusion of other brands. In this chapter, we do not investigate

the advantages/disadvantages of the incumbents or the new entrant. Instead, we allow

the manufacturer of the FP to be either a new entrant or the incumbent of one or two

SPs and understand the impact of the competitive role of the FP’s manufacturer on the

diffusions of all products.

4.3 Diffusion of Single-Function Products

For our model, we focus on the situation where two SPs are currently in the market,

and one FP has become technologically achievable and ready for launch. The notation

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used in this chapter is listed in Table 4-1. We assume that the market currently has two

distinct SPs, denoted as 1 and 2, respectively. Let mi be the total adopters (or market

size) for product i (i = 1, 2). We assume that, without other interference, the demand

trajectory for SP i is exogenous and influenced by two factors: the independent innovation

coefficient φi and the word-of-mouth (imitative) coefficient ψi. Note that the production

constraints and uncertainty of demand are not modeled in this paper. As a result, the firm

always produces exactly the amount of the demand; that is, the sales rate at a particular

time equals the demand. According to the Bass Model [89], the sales rate 2 for product i

at time t can be formulated as follows.

si(t) =mi(1 + ai)bie

bit

(ai + ebit)2i = 1, 2, (4–1)

where ai = ψi

φi

3 and bi = ψi + φi. The general Bass model in Equation 4–1 is known for its

robustness in characterizing the diffusion process of a new innovation by assessing proper

parameters without incorporating decision variables. Note that the equation assumes the

current time is at t = 0. However, when we analyze the sales trajectories for the existing

SPs, the products have normally been in the market for some time. We assume when

the fusion technology is achievable, the SPs 1’s and 2’s ages are t1 and t2 periods old,

respectively. Thus, we allow for the possibility that the two SPs are at different stages in

their respective life cycles. For example, the cellular phone for the consumer market in the

U.S. was launched in the early 80’s, but the first successful PDA for the consumer market

was launched in 1997 by Palm.4

2 Refer to [97] and [96] for diffusion models which explicitly consider production.

3 Note that the variables ai and di (i=1,2,3) used in this chapter have differentmeanings from previous chapters.

4 Earlier versions of PDA were not successful in the consumer market. For example,Apple launched its PDA, Newton, in 1993. Newton had many defects and the productdemand never took off. Apple discontinued Newton in 1998.

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Table 4-1. Notation and Acronyms.

m1,m2 Total adopters for SPs 1 and 2m3 Total adopters that can be only captured by the FPai A denominator coefficient in Φi, ai = ψi/φi, i = 1, 2, 3bi An exponent coefficient in Φi, bi = φi + ψi, i = 1, 2, 3φi The innovative coefficient for product i, i = 1, 2, 3ψi The imitative coefficient for product i, i = 1, 2, 3

si(t) Sales (demand) rate of product i before FP’s launch at time t, i = 1, 2sm33 (t) Sales (demand) rate of FP 3 from m3

sni (t) Sales (demand) rate of product i after FP’s launch at time t, i = 1, 2, 3

ti The age of SP i at t = 0, i = 1, 2.tm Technology cost maturityτ Launch time of FP 3, decision variable

T3 End of the planning horizondi Unit net profit for product i, i = 1, 2, 3α The convex development cost parameter

Ri ∈ [0, 1], the aggregate proportion of product i’s demand switching to the FPv0 The development cost of “launch now” policyw The proportion of the switched adopters from SPs 1 and 2 that is not overlappedκ The multiplier related to the diffusion speed of the FP

Πi Total profit of scenario i, i ∈ {I, II.1, II.2, III}FP Fusion productSP Single-function product

Given that the SPs have been in the market for some time, the demand at time t for

product i should be revised as:

si(t) =mi(1 + ai)bie

bi(t+ti)

(ai + ebi(t+ti))2i = 1, 2 (4–2)

Let t = 0 be the present time. For t > 0, si(t) represents the sales rate of product i

at time t. The diffusions of two SPs are independent and deterministic when there is no

interference from other factors. However, the fusion technology changes the dynamics of

the product diffusion.

At present time (t = 0), the manufacturer of the FP owns the fusion technology and

considers entering into the market. From the discussion in the literature, the manufacturer

of the FP can either be a new entrant or an incumbent. This model investigates the

optimal launch time of the fusion product when the fusion product manufacturer is in one

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of the scenarios: (I) a total new entrant, (II) a manufacturer of one SP5 , and (III) the

manufacturer of two SPs.

4.4 After the Availability of the Fusion Product

The launch of the FP disturbs the diffusions of SPs 1 and 2. We assume that at

t = 0, the manufacturer of the FP can choose to launch a fusion product with a

fixed development cost v0, or launch the fusion product later. A classic trade-off that

many firms face concerns the interplay between time-to-market, product quality and

development cost [91]. An early-launched new product will grab the market sooner,

but this strategy also contains the risks of high development cost, lower quality from

poor design, and failure of product transition. A U-shaped development cost function is

proposed by Smith and Reinertsen [115], Gupta et al.[116], Murmann [117] and Bayus [91].

If the firm wants to launch the FP immediately, the firm needs to invest more in product

development. If a new product development project is delayed, then the development cost

can be trimmed by reducing the development time. Let tm to be the time with minimal

development cost of this fusion technology; hence, tm represent the technology cost matu-

rity. Similar to Bayus [91], we assume the development cost v of the FP launched at time

τ follows a U-shape as described below.

v(τ) = v0 − ατtm + ατ 2/2 = v0 − αt2m/2 + α(τ − tm)2/2 τ ≥ 0

v =dv

dt= α(t− tm) for t < τ (4–3)

The development cost curve and the first derivative of development cost with respect

to time are shown in Figure 4-1. The first derivative of the development cost is decreasing

(increasing) if the FP is launched before (after) tm, and the minimal total development

5 In this section, our analysis for scenario II assumes that the firm is the manufacturerof both SP 1 and FP. The scenario that the firm is the manufacturer of both SP 2 and FPis symmetric to scenario II and is thus omitted. However, in the numerical analysis, weseparate the former and the latter situations as II.1 and II.2 scenarios, respectively.

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cost (v0−αt2m/2), occurring when the FP is launched at tm, is assumed to be positive. The

coefficient α indicates the convex increasing rate of the development cost when the launch

time deviates from tm. The second part of Figure 4-1 shows that v is increasing in time.

Note that this is a major difference from the model in Mahajan and Muller [101]. They do

not consider the impact of development cost, which may vary with the launch time of the

new generation.

Figure 4-1. Development Cost at Different Launch time of FP.

The substitution between the FP and the SPs is similar to the substitution between

products of different generations. To make the model tractable, we assume that there is

only one generation of FP considered. The FP made available by the fusion technology

is indexed as product 3 and is associated with additional adopters of population size m3.

The composition of m3 includes (1) the consumers who are not included in m1 or m2, and

(2) the consumers who had purchased one or two SPs before the launch of the FP and will

‘upgrade’ to purchase the FP. Van den Bulte and Lilien [109] point out the m3 can come

from population growth.

Let τ be a decision variable which is the launch time of the FP. The launch time

is critical to the firm because it often impacts on many aspects of diffusion. First, to

achieve an earlier launch time, the firm often needs to allocate more investment in product

development. Second, the earlier the launch of the new product, the more potential

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buyers of the old products may switch to buy the new product. Third, the firm’s strategic

concerns for the launch time may depend on the role of the FP’s manufacturer (scenario I,

II or III) due to different objective functions which are considered.

Many factors may have an impact on the launch timing decision: the relative

profitability of all products, the diffusion speed, the development cost of a new product,

the number of new adopters, and the switching rate of the future buyers from the old

products to the new product. In the real world, a converse causal relationship may exist

where the launch time influences these factors. To simplify the model’s complexity, we

assume only the development cost of product introduction is affected by the launch time

decision. The total adopters m3, the profit margin of FP, the switching rate, and the

diffusion parameters are independent of the launch time. Similar to existing literature

on multi-generation diffusion models, we assume that the launch time of the FP will not

impact the values of other parameters.

However, the launch of FP 3 provides the future adopters in m1 and m2 an option.

Norton and Bass [105] and Mahajan and Muller [101] both assume a new generation

product expands the market and the substitution effect between the new and old

generations of the product is captured by some proportion of consumers for the old

generation product that will switch to a new version of the product. We have a similar

assumption and we allow different switching rates for the two SPs. Normally, the SP with

a higher switching rate has a higher similarity to the FP and we call it the base SP of

the FP, and the SP with smaller substitution is called the non-base SP. For example,

an all-in-one printer works pretty well as a printer (base SP) and less well as a scanner

(non-base SP). For the future buyers of the the single-function printer, the consumers

normally buy either a single-function printer or a multi-function printer. Specifically, it is

more likely that consumers for the printer will switch to the fusion product. Conversely,

the future buyers of a single-function scanner are less likely to purchase the fusion product

since the scanning in the fusion product is the secondary feature.

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Note that many consumers are often simultaneously included as future buyers for

multiple distinct SPs. For example, many consumers have requirements for both a printer

and a scanner. When an FP is launched and attracts these future buyers of SPs, some

buyers may eventually “buy one FP instead of two distinct SPs”. As a result, in our

model, we have to avoid repeatedly counting the switch of SP future buyers to the FP. It

is very difficult to determine exactly how many future buyers from SPs switch to buy the

FP. To simplify the estimation of the switch, we assume that a fraction w (0 ≤ w ≤ 1) of

the total switches from both SPs is contributing to increasing the demand for FP 3.

Now we construct the sales trajectories after the launch of the FP. To simplify the

notation, we drop the time argument in related variables when possible. Let sni be the

sales of product i considering the consumer switch after the launch of the FP, then

sni = si(1−Ri) for i = 1, 2, (4–4)

sn3 = sm3

3 + w(s1R1 + s2R2) (4–5)

where 0 ≤ Ri < 1 (i = 1, 2) represents the switching rate of the demand of SP i switching

to the FP, and sm33 = m3(1+a3)b3eb3(t−τ)

(a3+eb3(t−τ))2indicates the future adopters of FP 3 from m3 [89].

Note that Mahajan and Muller [101] use a similar form to capture the switching from the

old to the new generation. The expression Risi (i = 1, 2) represents the future adopters

that switch from SP i. However, due to overlapping customer needs, only w(s1R1 + s2R2)

will purchase the FP.

Let d1, d2 and d3 be the unit profit of products 1, 2, and 3, respectively. We do not

consider the discount factor in our objective function for two reasons. First, it is difficult

to identify when and how much the development cost occurs at any point of time since

this cost occurs before the launch time. Second, omitting the discount factor allows us

to identify other factors influencing the optimal solution. The firm’s goal is to find the

optimal launch time which maximizes their profit.

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We previously mentioned the manufacturer of the FP can be either an incumbent

or a new entrant. In this model, the FP’s manufacturer is will be one of three following

scenarios (I) a total new entrant, (II) a manufacturer of one SP6 , and (III) the manufacturer

of two SPs.

The objective functions of the firm under the three scenarios are as follows.

maxτΠI =

∫ T3

τ

[d3sn3 ]dt− v(τ)

maxτΠII =

∫ τ

0

[d1s1]dt +

∫ T3

τ

[d1sn1 + d3s

n3 ]dt− v(τ)

maxτΠIII =

∫ τ

0

[d1s1 + d2s2]dt +

∫ T3

τ

[d1sn1 + d2s

n2 + d3s

n3 ]dt− v(τ)

The firm in scenario I is a new entrant and is concerned chiefly with the profit of

the FP after the launch. The firm in scenario II needs to evaluate the total profit from

SP 1 before the FP’s launch and the total profit of SP 1 and FP 3 after the launch. The

objective function of the firm in scenario III includes the profit of SPs 1 and 2 before

the FP’s launch and the total profit of all products after the FP’s launch. Take the first

derivative of the profit functions with respect to τ , we obtain

dΠI

dτ= −d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α(τ − tm)

dΠII

dτ= d1R1s1(τ)− d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α(τ − tm)

dΠIII

dτ= d1R1s1(τ) + d2R2s2(τ)− d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α(τ − tm)

6 In this section, our analysis for scenario II assumes that the firm is the manufacturerof both SP 1 and FP. The scenario that the firm is the manufacturer of both SP 2 and FPis symmetric to scenario II and is thus omitted. However, in the numerical analysis, weseparate the former and the latter situations as II.1 and II.2 scenarios, respectively.

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By setting the first order conditions zero, we obtain:

− α(τ − tm) = d3[sm33 (T3) + w(R1s1(τ) + R2s2(τ))] (4–6)

d1R1s1(τ)− α(τ − tm) = d3[sm33 (T3) + w(R1s1(τ) + R2s2(τ))] (4–7)

d1R1s1(τ) + d2R2s2(τ)− α(τ − tm) = d3[sm33 (T3) + w(R1s1(τ) + R2s2(τ))] (4–8)

Unfortunately, the optimal launch times solved by the first order conditions above

are in a complicated form7 , such that no closed-form solution can be obtained. However,

we can still get some insights from these equations. The right hand sides of the FOCs

represent the marginal profit from the FP, and the left hand sides of the FOCs equal

the marginal development cost plus the marginal substitution loss profit from the

SP(s). For example, in scenario I, the optimal launch time only occurs between now

and the technology cost maturity (tm) since the RHS is always positive. The managerial

interpretation is that the firm should launch the FP at the moment that the marginal

profit of the FP equals to the marginal cost from the development. The marginal profit

has an interesting composition: it is a product of FP’s profit margin times the sum of the

sales of FP at the end of planning horizon and the switch demand from SPs at the launch

time. Note that the sales of FP at the end of planning horizon comes into the equation.

Given a certain planning horizon, the launch time of the FP will decide the cumulative

sales level of the FP. The sales of the FP from m3 always starts from zero; as a result, the

sales of the FP from m3 at the ending becomes part of the marginal profit. Note that this

result implies that the length of planning horizon will impact on the decision of launch

7 Wilson and Norton [112] mention the optimal launch time solution can be foundby taking the derivative of the profit function with respect to the launch time decisionand find the value of the launch time when the derivative vanishes. However, they alsopoint out that the value of the optimal launch time is “sufficiently complicated that it ischallenging to get insight about its implication.” Mahajan and Muller [101] use optimalcontrol theory to investigate the optimality condition, but do not obtain a closed formsolution.

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time. Similarly, in scenario II (III), the optimal launch time occurs when the marginal

profit of the FP equals to the sum of the marginal cost from the development and the lost

profit(s) from SP(s) 1 (and 2).

The second derivatives of the profit functions with respect to τ are presented below.

Since s1, s2, and sm33 can be either positive or negative depending on the increasing or

decreasing sales rates, the global concavity of the profit function cannot be guaranteed.

d2ΠI

dτ 2= d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α

d2ΠII

dτ 2= −d1R1s1(τ) + d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α

d2ΠIII

dτ 2= −d1R1s1(τ)− d2R2s2(τ) + d3[s

m33 (T3) + w(R1s1(τ) + R2s2(τ))]− α

Given that the first derivatives of sales rate are positive before the sales peak and

negative after the sales peak, the second order conditions show that the profit function

is generally concave when τ is small and the sales from the SPs are still increasing. But

in some cases, at some point after the sales peaks of the SP, the profit function starts to

become convex decreasing as τ increases. This case is more likely to occur in scenario I

than scenarios II and III since the magnitude of negative terms in scenario I is smaller. In

our numerical examples, some cases show the profit functions are strictly concave in the

planning horizon.

Even though the FOC is in complicated form and the profit function can be neither

convex or concave, we can still obtain the candidates of the optimal launch time by setting

the FOC zero. Note that the FOC’s being zero is just a necessary condition of the optimal

solution.

From the composition of dΠI

dτ, dΠII

dτand dΠIII

dτ, we notice that dΠIII

dτ> dΠII

dτ> dΠI

dτ

when all parameter values remain the same for the three scenarios and the concavity of

the second order condition holds. From the structure of the FOCs in three scenarios, we

can conclude that the new entrant (scenario I) should choose a launch time which is earlier

than the firm in scenario II, which again should choose a launch time earlier than the firm

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in scenario III. This is a reasonable outcome since a new entrant has no concern about

the cannibalization of the FP to the SPs, and the firm in scenario III must evaluate the

cannibalization of the FP to two SPs. When the firm of the FP is also the supplier of SPs,

extra profit from the FP sales comes with some sales loss from the SPs. As a result, the

opportunity cost of an earlier launch is higher for the firm in scenario III than the firms in

I and II when other conditions remain the same.

When dΠi

dτ> 0 for i ∈ {I, II, III} and for any τ ∈ [0, T3], this indicates that the

FP is very profitable compared to the development cost and the profit loss from SP(s),

and the firm of scenario i should introduce the FP immediately. Similarly, when dΠi

dτ< 0

for τ ∈ [0, T3], the the firm of scenario i should never introduce the FP (τ ∗ = T3) since

the possible lost profits from SP plus the development cost is higher than the profit from

the FP. Note that the magnitude of α, which represents the convex increasing rate of

total development cost, is included in the necessary conditions. When α is large, the total

development cost will increase significantly when the launch time of the FP deviates from

tm. In this circumstance, dΠi

dτ> 0 or dΠi

dτ< 0 for i ∈ {I, II, III} is less likely to occur and

the optimal launch time of the FP will be more ‘centered’ to tm. If the development cost

over time is a flatter curve, which has a smaller α, then the optimal launch time of the FP

will be earlier (later) compared to the case with a steeper development cost curve. That is,

the optimal launch time of the FP is more likely to deviate from tm when the development

cost over time is a flatter curve.

From the analysis above, we know that no closed-form analytical solution can be

obtained. Due to this constraint, we also cannot obtain the sensitivity analysis on the

optimal solution. In the next section, we exemplify many sets of numerical examples to

study the optimal FP launch time.

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4.5 Numerical Analysis

In this section, we use different sets of parameter values to investigate the diffusion

problem of two SPs and one FP8 . We investigate the impact on the the optimal solution

of the examples by changing one parameter value for all cases. However, across all

comparisons, we select one base case that is shown in every comparison. The parameters

we use for the comparison can be categorized into five groups. The first group is the

synergy of the fusion product: the diffusion speed, the relative profit margin, and the new

future adopters of the fusion product. The second group is related to the technology and

the development cost: development cost curve and the technology cost maturity. The

third group contains time related factors: the age of the SP and the planning horizon. The

fourth group concerns the SP related parameters: their relative size and the substitution

effects of the FP on the SPs. The last group is about the overlap degree in the future

adopters of two SPs.

In the real world, the SPs may have different market sizes and profit margins.

To reflect this situation, in our numerical examples, we let SP 2 have a larger market

size (m2 > m1) and a higher unit profit (d2 > d1)9 . In scenario II, we create two

sub-scenarios: (II.1) the supplier of the FP is the supplier of SP 1; (II.2) the supplier

of the FP is the supplier of SP 2. For each scenario, we conduct ten comparisons as

mentioned above and evaluate the influence on the optimal launch time of the FP and the

total profit.

8 There is no reliable market data available for fitting the diffusion model with SPs andFPs. First, it is lack of clear-cut boundaries between FPs and SPs. Second, the marketsurvey firms use various definitions of categorization of products (see section 1.5). Forexample, a PDA phone which is categorized in the PDA shipment by a market researchfirm can be included in cell phone sales in another firm. These two reasons make theindustrial data unreliable for fitting the product diffusions.

9 Later, in one comparison, we will change the ration of m2 : m1 from 4:1 to 1:1, but therelative profitability of d2 and d1 remains the same in all cases.

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The parameter values of the base case are: m1 = 100,m2 = 200, m3 = 200, d1 =

1.5, d2 = 2.5, d3 = 4, φ1 = φ2 = 0.0049, ψ1 = ψ2 = 0.64410 ,φ3 = κφ1, ψ3 = κψ1, t1 =

2, t2 = 4, T3 = 20, tm = 4, R1 = 0.5, R2 = 0.3, w = 0.7, κ = 1, α = 10, v0 = 28011 .

The base case has the following properties: (1) the FP has a unit profit which equals the

sum of the two SPs’ unit profits, (2) the two SPs and the FP have the same diffusion

parameters, (3) the two SPs are still ‘young’ and the fusion technology will be mature

soon (in four periods). We consider multifunction office machines as one example. The

printer can be seen as SP 2 and the scanner can be seen as SP 1. Since the fusion product

normally integrates closely related functions and both the printer and the scanner are

PC peripherals, setting the same innovation and imitation parameters for two SPs is

reasonable. The innovation and imitation parameter values used in our examples are based

on the diffusion parameters from [90]. Note that, κ indicates the diffusion speed multiplier

of the FP with respect to the diffusion speed of the SPs 1 and 2. When κ is greater than

one, the FP has a faster diffusion speed than the SPs. In the base case, we assume that

the diffusion speed of the FP is the same as the diffusion speed of the SPs.

Based on the scenarios we describe above, the fusion product manufacturer can be

one of four different roles (I) a total new entrant, (II.1) a manufacturer of SP 1, (II.2) a

manufacturer of SP 2, and (III) a manufacturer of two SPs. Table 4-2 lists the optimal

solutions from all cases and all four scenarios conducted in this paper. In section 4.5.1 we

contrast the impact of different launch times on the diffusion processes. In section 4.5.2,

we compare the results among four scenarios. The impacts of parameter changes will be

discussed in sections 4.5.3 to 4.5.8.

10 For our optimization problem, the magnitude of φ’s and ψ’s impact on the planninghorizon. Smaller φ’s and ψ’s will need a longer planning horizon to capture the wholediffusion, and the timing influence would be ‘diluted’. To present a more dramatic change,we select larger values of φ’s and ψ’s in our examples.

11 This implies the minimized development cost at τ = tm is 200.

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Table 4-2. Optimal launch time of FP and the profit under four scenarios and 31 cases.

Case Difference τ∗I Πτ∗I τ∗II.1 Πτ∗

II.1 τ∗II.2 Πτ∗II.2 τ∗III Πτ∗

III

0 Base case 1.5167 832.76 1.6857 909.74 2.6024 1180.33 3.0166 1262.861 κ=1.1 1.5306 833.22 1.7027 910.25 2.6375 1181.24 3.0706 1264.032 κ=1.2 1.5353 833.36 1.7085 910.41 2.6504 1181.53 3.0915 1264.443 κ=1.3 1.5369 833.40 1.7104 910.46 2.6551 1181.63 3.0993 1264.574 d3 = 3 1.8617 567.94 2.1009 646.31 3.2495 925.74 3.8687 1013.655 d3 = 5 1.2435 1099.51 1.3730 1175.59 2.1058 1440.16 2.3808 1519.676 d3 = 6 1.0182 1367.62 1.1225 1443.08 1.7240 1703.42 1.9190 1781.177 m3 = 150 1.5219 632.93 1.6921 709.92 2.6159 980.66 3.0375 1063.308 m3 = 250 1.3060 866.50 1.4449 942.77 2.5607 1297.56 2.9640 1397.809 m3 = 300 1.1223 901.07 1.2393 976.81 2.5194 1414.81 2.9121 1496.7810 α = 5 0.6603 853.37 0.8157 928.09 1.9407 1187.49 2.4337 1266.7011 α = 15 1.9995 820.35 2.1738 899.17 2.9313 1176.60 3.2809 1261.0912 α = 20 2.3239 811.97 2.4972 892.30 3.1328 1174.29 3.4329 1260.0713 tm = 2 0.4781 872.58 0.5802 946.90 1.2698 1201.29 1.4914 1277.5714 tm = 3 1.0151 855.08 1.1464 930.57 1.9604 1192.48 2.2696 1271.4115 tm = 5 1.9993 805.35 2.2208 884.27 3.2127 1164.43 3.7502 1251.7016 t2 = 3 1.8422 848.99 2.0447 927.21 2.6424 1203.86 3.0541 1286.6317 t2 = 5 1.1855 812.03 1.3358 887.98 2.5867 1141.84 3.0165 1224.3118 t2 = 6 0.9114 787.24 1.0676 862.50 2.6085 1084.31 3.0580 1166.9819 T3 = 10 0 810.84 0 884.37 0 1130.87 0 1204.3920 T3 = 15 1.1249 818.52 1.2334 894.10 1.8191 1155.81 1.9999 1233.7521 T3 = 25 1.5368 833.41 1.7104 910.47 2.6554 1181.65 3.1002 1264.6022 m1 = 50 1.6735 767.36 1.7698 806.06 2.9839 1120.04 3.2542 1162.3423 m1 = 150 1.3819 898.60 1.6087 1013.55 2.3167 1242.36 2.8187 1364.0124 m1 = 200 1.2632 964.78 1.5375 1117.46 2.0886 1305.55 2.6498 1465.6625 R2 = 0.2 1.8519 789.47 2.0818 867.79 2.6589 1176.46 3.0874 1259.3826 R3 = 0.4 1.2421 877.93 1.3729 954.02 2.5469 1184.25 2.9467 1266.4027 R4 = 0.5 1.0121 924.50 1.1176 999.95 2.4923 1188.22 2.8778 1270.0128 w = 0.6 1.6992 795.33 1.9019 873.01 2.9427 1148.11 3.4630 1233.2529 w = 0.8 1.3562 870.79 1.5007 947.23 2.3081 1214.17 2.6361 1294.8030 w = 0.9 1.2132 909.32 1.3390 985.31 2.0546 1249.28 2.3184 1328.53

* The parameter values of base case are: m1 = 100,m2 = 200,m3 = 200, d1 = 1.5, d2 = 2.5, d3 = 4,φ1 = φ2 = 0.0049, ψ1 = ψ2 = 0.644, φ3 = κφ1, ψ3 = κψ1, κ = 1, t1 = 2, t2 = 4, T3 = 20,12

4.5.1 Market Penetration

The launch of the fusion product often changes the dynamics of related single-function

products. In this subsection, we exemplify the cumulative sales of the SPs and the

FP, which are shown in Figure 4-2. The original cumulative sales of SPs without the

competition of the FP are represented by thin solid lines, where the SP 1’s (2’s) diffusion

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has lower (higher) cumulative sales curve with t1 = 2 and m1 = 100 (t2 = 4 and

m2 = 200). The diffusion of the FP is represented by a thick solid line and the long dashed

line approaching 200 represents the cumulative sales of the FP solely from m3. The short

dashed lines represent the cumulative sales of SPs 1 and 2 after the launch of the FP.

Because the influence of the FP’s launch is the same, the cumulative sales for all scenarios

will not be any different.

Figure 4-2. Examples of Cumulative Sales.

Figures 4-2.1 and 4-2.2 are intended to illustrate the impact of alternate fusion

product launch times on the diffusion for all products. Figure 4-2.1 shows the market

penetration if the FP is launched at τ = 1. As the FP is launched, the cumulative sales

for SPs 1 and 2 increase by smaller rates as some potential adopters switch to buy the

FP. Eventually, the cumulative sales of SPs 1 and 2 drops from 100 to 52.1, and from 200

to 149.4 after the launch of the fusion product. Conversely, the cumulative sales of FP 3

increases from 200 to 241.4. The total units of SPs drop 98.5 but it only gets 41.1 units

of FP increases. The difference is due to the overlap of consumers. Some consumers who

have needs in two SPs eventually switch to the FP and just buy one FP instead of two.

Figure 4-2.2 shows the market penetration if the FP is launched at τ = 3. If the FP

is launched later, such as in Figure 4-2.2 τ = 3, then the impacts of the FP to the SPs

are moderated. The cumulative sales of SPs remain intact before the FP is launched. The

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cumulative sales of SPs 1 and 2 are 57.8 and 164.7, which are higher than those in Figure

4-2.1, and the cumulative sales of FP is 231.6.

Next, we investigate the factors which influence the firm’s optimal launch time

decision for the fusion product.

4.5.2 Role of the Fusion Product Supplier

First, we investigate the role of the FP manufacturer in four scenarios. The results

in Table 4-2 and ten sub-figures in Figure 4-3 show that the more substitution that the

FP has with its current SP(s), the greater the cannibalization concerns for the firm,

and the later the firm should introduce the FP. The cannibalization effect is highest in

Scenario III, then Scenario II.2, then Scenario II.1, then Scenario I, which has no concern

of cannibalization.

The results are not surprising because a new entrant has nothing to lose and a firm

who offers SPs and FP must be concerned with (1) the increased sales of the FP which

comes with the decreased sales of SPs; (2) some future adopters choose to “buy one

FP instead of two SPs,” which is reflected by parameter w. We should note that the

cannibalization is captured by many factors, including the relative magnitudes of total

future adopters, unit profits, the switching rate, and the degree of the overlap. Different

scenarios also reflect different cannibalization effects depending on the firm’s mix of

current product offerings. For example, in Figure 4-3.8, the firm in scenario II.1 acts more

closely to the firm in scenario I when the total market is relatively lower (m1 = 50). When

m1 = 200, the firm in scenario II.1 has more cannibalization concern since m1 is larger;

as a result, the gap in the optimal launch time between scenarios II.1 and I is larger. The

same parameter change may have different degrees of impact on the firms in different

scenarios. In Figure 4-3.9, the gap of optimal launch times between scenarios II.1 and II.2

is wider when the potential cannibalization is larger. A larger R2 for the firm in scenario

II.1 is beneficial since more consumers from m2 will switch to buy the FP; however, to the

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firm in scenario II.2, a larger R2 means more cannibalization and lost sales due to“from

two SPs to one FP.”

4.5.3 Synergy of the FP: Faster Diffusion Speed, Higher Margin, and MarketExpansion

In section 1.5, we mention that the synergy of the FP may include space saving,

lower total purchasing cost, and function synchronization. There are several ways that

the synergy of the FP is reflected in the model. First, when two functions have greater

synergy, the diffusion of the FP may be faster. Second, when the FP has greater synergy

to integrate two functions together, it normally has a higher unit profit. Third, the FP

might has a larger number of future adopters when it has greater synergy.

Norton and Bass [105] use the same innovation and imitation parameters to fit the

diffusion data of different generations of DRAM. However, van den Bulte and Lilien

[109] point out that Bass models does not account the declines of real prices, improving

performances and increasing distribution penetration. These factors increase the total

adopters and result in a downward pressure on the estimates (i.e., ψ and φ). Van den

Bulte [110] investigates the changes of the diffusion speed of 31 electrical durables in the

United States from 1923 to 1996. He defines diffusion speed as the time to reach a certain

market penetration level and measures it by the slope coefficient of the logistic diffusion

model. The variance of diffusion speed is primarily explained by the purchasing power, the

demographics, and the maturity of the products. Excluding the variance, this study finds

that the diffusion speed is increasing with statistical significance.

Based on the above research, we use a multiplier κ to represent whether the FP has

an equal or larger diffusion speed than its component SPs. In the base case we set the

fusion product’s innovation and imitation parameters of the FP at the same level as the

SPs (κ = 1) . In cases 1 to 3, we set κ at 1.1, 1.2, and 1.3, respectively. Table 4-2 shows

that, in all scenarios, the diffusion speed has no significant impact on the optimal launch

time and the total profit when other parameters remain unchanged. Even though the

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Figure 4-3. Optimal launch time of the fusion product versus various factors.

97

diffusion speed is important to forecast when the demand of FP will grow faster, our

results shows the optimal launch times are very similar despite the different diffusion

speed.

If the synergy of the two functions is reflected on the FP’s unit profit, then the FP’s

optimal launch time is strongly influenced. Cases 4, 5, and 6 are comparable situations

to the base case with alternate values for the FP’s unit profit. The results in Figure 4-3.2

show that when the FP is less profitable, the launch time should be postponed to more

closely match the time of technology maturity. The development cost structure in our

model makes our optimal launch time different from Wilson and Norton’s [112] “now or

never” and from Mahajan and Muller’s [101] “now or at-peak” conclusions. When there is

a higher unit profit for the FP, the launch time should be earlier in order to capture more

demand from the SPs. It should be noted that the FP’s optimal launch time should be no

later than the technology cost maturity, which occurs when the FP has a relatively high

unit profit compared with the two SPs together. If the FP is less profitable, it is possible

that the optimal launch time can be postponed after tm under a certain circumstance.

When the FP’s profit is only slightly larger than SP 2, the low d3 will cause some profit

loss due to the switching “from two SPs to one FP” thus deterring the launch of the FP.

Kurawarwala and Matsuo [98] find the innovative and imitative parameters are easy

to assess, but the magnitude of the total adopters has high uncertainty. Cases 7, 8, 9 and

the base case show that the FP’s optimal launch time is very insensitive to various market

sizes of the FP (Figure 4-3.3) when the cannibalization is strong. The results mean that if

the higher synergy of the FP is represented by more future adopters while other conditions

remain the same, then the FP’s optimal launch time is not much different from the case

with a lower synergy FP for firms with existing products. For the firms in scenarios I and

II.1, FP’s market size has stronger influence to the optimal launch time.

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4.5.4 Maturity of the Fusion Technology and the Development Cost

When the initial fusion technology which combines two functions is available, the

technology might not be mature immediately. If the fusion technology has a longer

development time, speeding up the launch of the FP normally requires a higher level of

product development investment. The product development cost might have different

increasing rates in different circumstances. For some FPs like office machines, the convex

development cost might have a lower rate of increase. For other FPs, such as PDA phone,

the convex development cost may increase at a steeper rate since the integration is more

challenging. In this section, we analyze the optimal launch time of the FP under various

times to technology maturity and various development cost curves.

Cases 10, 11, 12 and the base case compare different slopes of the development cost

curves, which is represented by α. Figure 4-3.4 shows that a development cost curve

with a steep slope, which has a larger α, leads to a later optimal launch time of the FP.

Furthermore, the impact on the optimal time to market is sizable. As time-to-market

becomes more costly, the firm should wait till the fusion technology is more mature.

Cases 13, 14, 15 and the base case compare various values for the technology cost

maturity. We find that the FP’s optimal launch time should be early when the fusion

technology’s maturity is early. When a fusion technology will be mature immediately, it

implies the integration of function is not difficult. The technology break-through of chip

design in recent years shows the sign of a mature fusion technology, therefore many fusion

products are available soon after a new function (e.g., MP3 player, digital camera) is

invented and offered in the market. If the fusion technology’s maturity is late, the convex

cost curve hinders the firm from an early launch of the FP.

4.5.5 Time Related Factors

So far, our numerical examples are all based on two SPs that are in their early

stages of life cycles: SPs 1 and 2 were just launched for 2 and 4 periods, respectively. As

technology advances at an increasing pace, the fusion technology is often available soon

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after a new function is invented. However, the gap between two SPs’ launch times might

be several years or more than a decade. For example, the time gap between the first cell

phone and first PDA launched in the U.S. market is 17 years.

In this section, we investigate the impact of the ages of the SPs. Since the fusion

technology appears very soon after the launch of the later launched SP, we investigate the

impact of various values for t213 . Cases 16, 17, 18 and the base case show that the firm in

scenarios I or II should pursue an early launch of the FP when SP 2 is currently ‘older’.

However, the impact of the SP’s age on the optimal launch time of scenarios II.2 and III

is vaired. For these two scenarios, the firm should pursue a later launch of the FP when

SP 2’s age is “young” or “close to mature”; in some immediate age of SP 2, the firm in

scenarios II.2 and III should launch the FP slightly earlier.

Evaluating the impact of SP age on optimal launch time, we find that the cannibalization

is a possible explanation for the difference. When the cannibalization is weak, as a result,

as SP 2 ages and its demand is closer to peak, the firm should launch the FP sooner to

capture the demand switch. But if the firm has strong degree of cannibalization concern,

the firm postpones the launch of FP when SP 2 is young since the switch from SP 2 is also

small at this moment. If t = 0 is about the flexion point of SP 2’s sales rate, then the sales

rate of SP 2 is at its maximum increasing rate. In this case, due to a larger switch of from

SP 2 to the FP, the firm should launch the FP earlier. The decreasing then increasing

optimal launch time along with t2 only occurs when the cannibalization is strong (i.e.,

scenarios II.2 and III). Even though the impacts on the optimal time in scenarios II.2 and

III are not influential, in some settings the impact can be stronger.

Cases 19, 20, 21 and the base case compare the optimal launch time of the FP for

different values of the length of the planning horizon. Similar to Wilson and Norton [112],

we find that the planning horizon is a decisive factor for the optimal launch time when

13 The impact of t1 is symmetric to t2 hence we omit it.

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it is short. If the planning horizon is long enough, then it does not influence the optimal

launch time much. When T3 is only ten periods, the optimal launch time for the FP is

‘now’; however, when T3 is 25 periods, the optimal launch time is as similar as when

T3 is 20 periods. We find that a short planning horizon will make the diffusion speed

(i.e., κ > 1) of the FP become influential. If the planning horizon is short, then a faster

diffusion speed will shorten the diffusion process; as a result, it induces the firm to launch

the FP early.

4.5.6 Market Size and Substitution

Figure 4-3.8 shows the changes of m1 to investigate the impact of SP’s relative size

of total future adopters. Changing m1 from 50 to 200, the total future adopters ratio

between SPs 1 and 2 varies from 1:4 to 1:1. For all scenarios, smaller m1 leads to a later

optimal launch time of the FP. However, the impact is moderate, especially for the firm

in scenario II.1. This is because less future adopters of SP 1 causes less future adopters to

switch from SP 1 to FP 3. Note that when two SPs have a similar size of future adopters,

the gap of the optimal launch times between scenarios II.1 and II.2 is also smaller, which

means the two scenarios have more similar cannibalization concerns.

To address the substitution effects, we investigate the impact of different degrees

of switching rate R2. A larger value of R2 means that a higher proportion of future

adopters may switch to the FP. The higher the R2 is, the higher proportion in demand

of SP 2 will be substituted for the FP. Cases 22, 23, 24 and the base case compare the

impact of different degrees of R2. The results in Figure 4-3.9 shows that when the FP’s

substitution rate is higher, the firm should launch the FP early. Note that the impact of

R1 is symmetric to R2.

4.5.7 Market Overlap: From Two SPs to One FP

To avoid repeatedly counting the switches from two SPs, we use a parameter w

(w ≤ 1) to adjust the total switches from two SPs. When w is larger (smaller), there is

fewer (more) overlapped future adopters who would “buy one FP instead of two SPs.”

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A lower bound of w should be the minimum of switches from SP 1 and SP 2. The last

comparison is to investigate the overlap degree of the future adopters from two SP

markets.

Finally we change the value of w to 0.6, 0.8, and 0.9 in cases 28, 29, and 30,

respectively, in contrast to the base case with w = 0.7. Figure 4-3.10 shows that when

there is more overlap between the two SPs’ future adopters, the firm should introduce the

FP later because the buyers from the SPs will eventually purchase just one FP rather than

two SPs. This result implies that when the firm intends to launch the FP to “hit two birds

with one stone,” they need to carefully estimate how many future buyers for the markets

of the two SPs will eventually buy just one FP.

4.5.8 Summary of the Optimal Launch Time

Wilson and Norton [112] find that the substitution effects, the diffusion between two

generations of similar products, and the relative profitability are major factors influencing

the optimal launch time of the new product. Our numerical examples support their

conclusion. Mahajan and Muller [101] demonstrate that the relative market sizes between

generations are also influential to the introduction time. In our product diffusion models,

the numerical analyses show that the market size of the FP matters for the firm with

less cannibalization concern. Besides, there are additional factors impacting the launch

decision of the fusion product: (1) the shape of the development cost curve for a early

launch of FP, (2) the technology cost maturity, (3) the age of the earlier launch SP (4)

the overlap degree from two SP markets, and the most important (5) the competitive role

of the fusion product supplier. The influence of the FP’s market size, the diffusion speed

of the FP, and the planning horizon on the optimal launch time are not notable for these

numerical studies.

In terms of the demand model structure, our model is similar to [101]. However, while

the firm also considers the convex development cost, the optimal launch time is often

between ‘now’ and the technology cost maturity when the FP is relatively lucrative. In

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some extreme cases (i.e., short planning horizon or an FP with very low profit margin),

the optimal launch time is either “now” or “at peak (of the SP).”

Here we summarize the insights from the numerical examples.

1. The greater the cannibalization concern of the firm, the later the FP should belaunched.

2. The optimal launch time is pushed back as the following factors increase: thesteepness of the development cost curve for an early launch of the FP, the technologycost maturity, the planning horizon, and the overlap degree from two SP markets.

3. The optimal launch time should be set earlier as the following factors increase: theFP’s profit margins and the switching rate from two SPs to a single FP.

4. If the FP is profitable enough to offset the lost sales of SPs, the FP’s optimal launchtime will be between ‘now’ and the technology cost maturity. When the FP is moreprofitable or the early-launch is less costly, the firm should introduce the FP earlier.

5. When the planning horizon is relatively long, the market size and the diffusion speedof the FP do not have influential impact on the optimal launch time of the FP.

6. The ages of the SPs show strong non-linearity on the optimal launch time whenthe firm has increased cannibalization concerns. The optimal launch time is convexdecreasing then increasing as the ages of the SPs increases from young to mature.

4.5.9 Impacts on the Total Profit

The analysis above focuses on the optimal launch time of the FP. Now we evaluate

the financial performance of the launch timing decision. Figure 4-4 demonstrates the

impact of parameter change on the profit. For each sub-figure and each scenario, the

profit from the base case is set as the reference. The total profits of other cases due to

some parameter changes are show as ratios relative to the profit of the base case. In

all sub-figures except in Figures 4-4.2, 4-4.3, 4-4.8, and 4-4.9, the ratios are shown in a

moderate range of [0.9, 1.1]. In other sub-figures ratios are shown in a large range which

can properly demonstrate their total variation.

We find that the diffusion speed is not influential to the total profit. We can easily

tell that the diffusion speed (Figure 4-4.1) given a relatively long planning horizon

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Figure 4-4. This figure shows the impact of parameter change on the profit. The profit ofthe base case is standardized as 1 and contrasted by the profits of other cases.

104

(T3 = 20) dos not impact the profit for all scenarios. The FP’s market size is not

influential to the optimal launch time in scenario III, however, the market size (like the

unit profit margin) has a large impact on its total profit in Figure 4-4.3.

We find that the interactions between some factors and the firm’s competitive role

has a joint impact on the total profit. For example, in Figure 4-4.2 the profit margin of

the FP has strong impact on the profit of the firm with less cannibalization concern. In

Figure 4-4.3, a larger market size of the FP has larger impact on the profit of the firm

with more cannibalization concern. But, if the market size of the FP is small, then it has

larger impact on the profit to the firm with less cannibalization concern. The development

cost (Figure 4-4.4) and the technology cost maturity (Figure 4-4.5) have moderate impact

to the total profit, which varies less than five percent of the profit of the base case. Even

though we find a short planning horizon will induce the firm to launch the FP much

earlier, its impact on the total profit is minor. This is because that launching the FP

much earlier will make the diffusion of the FP finish as much as it can in a short planning

horizon, as a result, the overall loss in total profit is moderated.

From Figure 4-4, we can obtain the following insights.

1. If the synergy of the FP is reflected by a higher profit margin or a larger market size,the synergy has a strong impacts on the total profit in all scenarios.

2. The diffusion speed has no influential impact on the total profit when the planninghorizon is long enough.

3. A short planning horizon and the switching rates of two SPs to FP have moderateimpact on the total profit, especially when the firm offers three products and has agreater cannibalization concern.

4. When the firm has the maximum cannibalization concern as in scenario III, theimpact on the total profit from the factors (except the market sizes and profitmargins) is more moderated than in other scenarios.

5. When the firm has a minimum cannibalization concern as in scenario I, the optimallaunch decision has a greater impact on the total profit than a firm which has highercannibalization. This result implies that a new entrant should be more careful whenit decides the launch time of the FP.

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4.6 Conclusion

As product-fusion penetrates and expands in the electronic markets, the projection

of product diffusion becomes more complex compared to the diffusion process within one

product category. This is the first product diffusion model that investigates the dynamics

of two SPs and one FP simultaneously, as a stylized product diffusion model is constructed

that is as parsimonious and complete as possible. Incorporating the development cost and

reflecting the phenomenon where consumers may transition “from two SPs to one FP”,

the model intends to answer the optimal launch time of the FP. Due to the complexity

and enormous dynamics of the model, no closed form analytical solution of optimal launch

time is available. However, we conduct several sets of numerical comparisons showing the

impacts of parameter change on FP’s optimal launch time and total profit.

We find that the competitive role of the FP’s manufacturer is influential to the launch

decision of the FP. When the supplier of the FP is also the supplier of the SP(s), the

higher cannibalization results in a later launch time of the FP. This finding suggests the

firms involved should evaluate who owns the fusion technology and assess the possible

actions of the FP introduction.

As the launch time of the FP is the decision variable, our numerical examples

demonstrate that several factors are influential to the optimal launch time of the FP.

When the FP has a higher profit margin, the firm should launch the FP earlier. The

diffusion speed or the market size of the FP matters only when the planning horizon is

short. A flatter development cost curve, early maturity of the fusion technology, and a

higher switching rate from SP to FP also imply an earlier launch. However, if there are

more consumers switching “from two SP to one FP,” then the firm should launch the

FP later. The age of the SP has a more complex influence on the optimal launch time

decision. For the firm with more cannibalization concern, the firm should launch the

FP later when the SP is ‘young’ or ‘old,’ and launch the FP earlier when the SP is at

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a fast-growing stage. If the cannibalization concern is low, then an older SP implies an

earlier launch of the FP.

Because we incorporate many realistic functions for the development cost and the

maturity time of the fusion technology, the optimal launch time policy is different from the

conclusions of Wilson and Norton’s [112] “now or never” and from Mahajan and Muller’s

[101] “now or at-peak.” When the FP is relatively profitable (normally it is), the optimal

launch time of the FP is between ‘now’ and the maturity of the fusion technology. In

general, the more profitable the FP is, the earlier the FP should be launched.

The numerical examples demonstrate the impact of parameter changes on the optimal

launch time and the total profit. The diffusion speed is not influential to the optimal

launch time as well as the total profit when the planning horizon is relatively long. The

unit profit of the FP are both influential to the total profit and the optimal launch

time. Other parameter values are more influential to the optimal launch time, but their

impacts on the total profit is moderated, which depends on the role of the firm and the

cannibalization effect. The results provide guidelines to firms when they assess the values

of the diffusion parameters.

However, our model constructs a valid skeleton for empirical and numerical

applications. Many important variables are simplified in our model. While applying

to numerical data fitting and forecasting, more detail of the correlation between many

variables can be constructed in the data fitting. For example, the diffusion speeds and

substitution effects might depend on the launch time, and the product price may decrease

over time. Numerical data fitting has more flexibility to incorporate the dynamics among

many factors.

We want to mention that, to analyze the diffusion of the FP, a complete and reliable

industrial survey is strongly needed, especially in the market of the personal mobile

device, the PC, home entertainment devices, and office machines. As we discuss above,

a well-categorized statistics for the sales of related products is the first step for this

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research. The FP diffusion analysis would be biased without proper categorization. This

is especially challenging because many FPs integrate more than four or five functions in a

single device. The interactions among all correlated products are difficult to capture due

to various function combinations of fusion products. The current market research shows

a deficiency of categorization in their market reports. A potential limitation to gathering

data from high technology industry is that different firms may categorize fusion products

in incongruent ways. We suggest the categorization of different FPs should be made as

complete as possible.

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CHAPTER 5SUMMARY

5.1 Key Results in Chapter 2

We first investigate the product offering strategy for a firm which provides two

distinct single-function products and one multi-function product. Using a stylistic model

that incorporates several features unique (demand substitution effects, costs and prices)

among these products, we identify five dominant product portfolio strategies. These

dominant strategies are: No MFP Strategy (NMFPS); All Product Strategy (APS);

Partial MFP Strategies (PMFPS1 or PMFPS2); and the Single MFP Strategy (SMFPS).

The firm should understand and parameterize the demand substitution effect between

each single function product and the MFP. Assuming this information is available, we

demonstrate that the maximum profit margin associated with the MFP and the degree

of the substitution effects are key factors to the firm’s optimal choice. To offer the MFP,

the MFP’s profit margin should be no less than the weighted average profit margin for

the two single function products, which is the lower bound of offering the MFP. As the

MFP becomes more profitable, the firm should first discontinue the single-function product

which has lower adjusted profit margin. Offering only the MFP is optimal when the MFP

is very profitable and it indicates the firm should use the MFP to replace two distinct

single-function products.

These results also provide some insights into how the firm could potentially influence

the choice of a portfolio strategy. Since the maximum profit margin for the MFP is a

function of the cost associated with the MFP (lower the cost, higher the maximum profit

margin), this could be viewed as an incentive to lower the manufacturing costs associated

with the MFP so that it could be included in its optimal product portfolio choice.

An analysis of the demand substitution effects also leads to some interesting insights.

When the MFP has strong substitution effects with its base and non-base product, offering

the MFP is only justified by a higher profit margin of the MFP. In essence, smaller

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(larger) values of the demand substitution effect between the base (non-base) product and

the MFP tend to increase the possibility of the base product and the MFP being included

in an optimal product portfolio. On the other hand, larger (smaller) values of the demand

substitution effect between the base (non-base) product and the MFP tend to increase

the possibility of the non-base and MFP being included in an optimal product portfolio.

We also find that the demand independence assumption between single-function products

does not hold once the MFP is included in the portfolio. In the APS, two single-function

products interact more like complimentary products.

5.2 Key Results in Chapter 3

When product-fusion technology makes integrating many functions into one device

possible, a multiple product manufacturer is facing a difficult product portfolio decision

as well as the decision of what function to fuse. We construct a normative model that

tries to help the business to solve these two problems while it has too many possible

options to choose. Under our assumptions of inverse demand functions and substitution

matrix relationship, we provide an efficient algorithm that can quickly find out the optimal

portfolio when the complexity of problem is huge.

There are several managerial insights generated from the analysis of the model. In

general, the optimal portfolio can be easily found, but its composition depends on two

major parameters: profit margins and substitution matrix. An optimal portfolio often

contains just a small subset of all product variants, especially when the substitution

effects are strong. In contrast, the product portfolio is more difficult to determine when

cannibalization effects are small but existent between the different product variants. This

situation can occur when the set of single-function products (SPs) under consideration is

somewhat incongruent, or when the fusion products (FPs) create a significantly different

market than the original SPs.

Second, all-in-one fusion product is not always included in the optimal portfolio. This

case may occur due to high variable cost of the all-in-one, design conflict between some of

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the functions, and lower demand from over-design and ‘feature fatigue.’ The firm should

resist the temptation of integrating all functions into a single device when one of above

exists. The concavity condition in the objective function implies that the firm should not

complement the FP with too many SPs or other FPs when the substitutability indices are

high. Introducing too many products induces strong cannibalization among their products

rather than grabbing market share from their competitors.

Finally, numerical results show that the optimal portfolio in this situation generally

contains a wider variety of products and is more sensitive to changes in the profit and

cannibalization parameters. Note that the objective of a firm should be maximizing its

profit rather than provide any product that is technologically achievable.

5.3 Key Results in Chapter 4

In Chapter 4, a stylized product diffusion model is constructed to investigate the

interaction on the diffusion of two single-function products after the launch of a fusion

product. Incorporating the development cost and reflecting the phenomenon of “from two

SPs to one FP”, the model provides some guidelines for the optimal launch time decision

of the FP. We conduct numerical comparisons and show the impacts of parameter change

on FP’s optimal launch time and total profit.

We find that the competitive role of the FP manufacturer is important to the launch

decision of the FP. An FP manufacturer which is a new entrant should introduce the FP

to the market earlier than the FP manufacturer who is also the manufacturer of one or

two SPs. The higher cannibalization concern of the firm results in a later launch time

of the FP. This finding suggests the firms involved should evaluate who owns the fusion

technology and assess the possible actions of the FP introduction.

Our numerical examples demonstrate several factors are influential to the optimal

launch time of the FP. When the FP has a higher profit margin, the firm should launch

the FP earlier. The diffusion speed or the market size of the FP matters only when the

planning horizon is short. A flatter the development cost, early maturity of the fusion

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technology, and a higher switching rate from SP to FP also imply an earlier launch.

However, if there are more consumers switching “from two SP to one FP,” then the

firm should launch the FP later. The age of the SP has a more complex influence to the

optimal launch time decision. For the firm with more cannibalization concern, the firm

should launch the FP later when the SP is ‘young’ or ‘old,’ and launch the FP earlier

when the SP is at a fast-growing stage. If the cannibalization concern is low, then an older

SP implies an earlier launch of the FP.

Note that the factors which are influential to the optimal launch time of the FP do

not always have strong impact on the total profit, and the impacts on the total profit also

differs across scenarios. The results provide guidelines to firms when they assess the values

of the diffusion parameters.

5.4 Future Research

There are several interesting extensions for future research. First, it may be

worthwhile to incorporate competition into the multi-function problem, and the

two-function context of Chapter 2 might be a good starting point. Suppose two firms

are monopolists in two distinct product markets and each firm can offer an MFP in either

of the two markets. Under such a setting, it might be interesting to analyze issues such as:

(a) what are the optimal product portfolio strategies for both firms? (b) does there exist

any Nash equilibrium for this duopoly game? (c) under what conditions will one of the

firms retreat from the market?

A second avenue of future research is to focus on developing a dynamic model

incorporating learning effects and market growth into the MFP problem. As technology

advances and process improvement, the MFPs become more acceptable in terms of the

price and the quality. Mahajan and Muller [101] investigate several generations of IBM’s

mainframe server and find two generations of IBM servers are launched to late. In the

MFP problem, the joint consideration of quality improvement, unit cost down and the

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demand growth makes the introduction of the MFP crucial to the firm’s success and

survival.

Third, when the manufacturer tries to fuse many functions into an all-in-one device,

the product space, platform and human interface are confined in a limited scale. How

should the manufacturer design the fusion product? From the outside the product

dimensions, what is the best human interface design? Which function should the fusion

function used as the platform base? When the product dimensions is too small to put all

together, how should a firm to segment different FPs?

Fourth, an empirical fitting for the diffusion of the FP is strongly needed in many

product markets, especially in personal mobile device, PC, home entertaining and office

machines. As previously mentioned, a well-categorized statistics for the sales of related

products are the stepping stones for this research. Without reasonable categorization,

any FP diffusion fitting would be biased. This is especially challenging because many FP

integrate more than four or five functions in one device. Chapter 3 shows the complexity

of product portfolio with correlated n functions. With different function combinations of

FP, the interactions among all correlated products are difficult to capture. The current

market research shows deficiency of categorization in their market reports. We suggest the

categorization of the FP should be as complete as possible. As the result, this statistics

can be easily re-categorized to apply to different product diffusion analysis when it is

needed. Several questions are needed to be answered: Whether the diffusion speed of

the FP is significantly different from its component SPs? Do two products have similar

diffusion pattern when they has higher similarity and substitution effects? Whether a new

FP is launched at a right time? Does a new FP capture most of its demand from other

SPs or FPs? Or a new FP generates a significant amount of new potential adopters? An

empirical research to answer these questions would have much contribution to the business

when they choose to introduce new product or enter a market.

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Finally, a survey and comparison research from consumer utility’s point of view will

help to answer the following questions. How do the consumers perceive the FP and which

component SP is more similar (or substitutable) to the FP? What component SPs will

be replaced if the consumer buy one FP? Why do some FPs replace its component SPs

very quickly, but other do not? Why does the demand of one component SP shrink very

quickly after the launch of the FP, but another component SP is resistent to the FP?

We recommend that the conjoint analysis might be applicable to the FP and SP market

research.

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APPENDIX ACONCAVITY OF THE PROFIT FUNCTION

Claim: Π is strictly and jointly concave in q1, q2, q3.

Proof: In order to show that Π is strictly and jointly concave in q1, q2, q3, it is

necessary to show that the determinants of the hessian (defined below) alternate in sign.

Now given that:

Π = q1(d1 − q1 − r13q3) + q2(d2 − q2 − r23q3) +

+q3(d3 − q3 − r13q1 − r23q2)

the hessian and its determinants are:

H =

∂2Π∂q2

1

∂2Π∂q1∂q2

∂2Π∂q1∂q3

∂2Π∂q2∂q1

∂2Π∂q2

2

∂2Π∂q2∂q3

∂2Π∂q3∂q1

∂2Π∂q3∂q2

∂2Π∂q2

3

=

−2 0 −2r13

0 −2 −2r23

−2r13 −2r23 −2

|H11 | = |H1

2 | = |H13 | = −2 < 0

|H212| = 4 > 0, |H2

13| = 4(1− r213) > 0, |H2

23| = 4(1− r223) > 0

|H3123| = −8(1− r2

13 − r223) < 0 by assumption.

Since the determinants of the hessian alternate in sign, we conclude that Π is strictly

and jointly concave in q1, q2, q3.

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APPENDIX BPROOF FOR THEOREM 2.1

On examining Table 2.1, we note that each strategy provides a feasible solution (i.e.,

q∗i ≥ 0) in terms of the parameter d3 when:

• 0 < d3 < ∞ ⇒ NMFPS and SMFPS are both feasible.

• r13d1 + r23d2 ≤ d3 ≤ min {r−113 d1(1 − r2

23) + d2r23, r−123 d2(1 − r2

13) + d1r13} ⇒ APS is

feasible.

• r13d1 < d3 < r−113 d1 ⇒ PMFPS1 is feasible.

• r23d2 < d3 < r−123 d2 ⇒ PMFPS2 is feasible.

The remainder of this proof is provided depending upon the range of values for the

parameter d3 in the Theorem.

Case 1: d3 ∈ (0, α1] or 0 < d3 ≤ r13d1 + r23d2

To start with, it is obvious that since r13d1 + r23d2 > r13d1 and r13d1 + r23d2 > r23d2,

in the range 0 < d3 < r13d1 + r23d2, the potentially feasible strategies are NMFPS, SMFPS,

PMFPS1, and PMFPS2. Keeping in mind our technical assumption of r213 + r2

13 < 1 which

implies that 1 − r213 > r2

23 and 1 − r223 > r2

13, let us examine the differences in profits

between the feasible strategies.

ΠNMFPS − ΠPMFPS1 = 0.25[(d21 + d2

2 − y(d21 + d2

3 − 2r13d1d3)]

= 0.25y[d22(1− r2

13)− (d1r13 − d3)2]

> 0.25y[d22r

223 − (d1r13 − d3)

2] since 1− r213 > r2

23

= 0.25y[(d2r23 − d1r13 + d3)(d2r23 + d1r13 − d3)

≥ 0

This last statement is true since: (a) d3 − d1r13 ≥ 0 which is a feasibility condition for

PMFPS1; and (b) d2r23 + d1r13 − d3 ≥ 0 which is the range for the parameter d3 we

are investigating. Hence, we can conclude that NMFPS is preferred over PMFPS1. In a

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similar manner it is possible to show that ΠNMFPS − ΠPMFPS2 > 0 and thus, NMFPS is

also preferred over PMFPS2.

Now in the range 0 < d3 ≤ r13d1 + r23d2, we know that ΠSMFPS = d23 is monotonically

increasing. Thus, it achieves its maximum when d3 = r13d1 + r23d2 and hence, let us

consider the following:

ΠSMFPS(d3 = r13d1 + r23d2)− ΠNMFPS

= (d1r13 + d2r23)2 − (d2

1 + d22)

= d21r

213 + d2

2r223 + 2d1d2r13r23 − (d2

1 + d22)

= −(d21 + d2

2)(1− r213 − r2

23) + (2d1d2r13r23 − d21r

223 − d2

2r213)

= −(d21 + d2

2)(1− r213 − r2

23)− (d1r23 − d2r13)2

< 0

As a result, when 0 < d3 < r13d1 + r23d2 we know that the profits under NMFPS dominate

the profits under SMFPS, PMFPS1, and PMFPS2. Hence, in this range, the preferred

strategy is NMFPS.

Case 2: d3 ∈ (α1, α2] or

r13d1 + r23d2 < d3 ≤ min {r−113 d1(1− r2

23) + d2r23, r−123 d2(1− r2

13) + d1r13}In this range, the solution provided by APS is feasible. Given that this solution

is globally optimal for our problem (since Π is strictly concave - see Appendix 1), it is

obvious that APS would dominate all other potentially feasible strategies for this range.

Case 3: d3 ∈ (α2, α3) or

min {r−113 d1(1− r2

23) + d2r23, r−123 d2(1− r2

13) + d1r13} < d3 < max{r−113 d1, r

−123 d2}

In general, PMFPS1, PMFPS2, NMFPS and SMFPS are all feasible strategies in this

range. We consider two separate sub-cases to identify the dominant strategy.

Case 3A: r−113 d1 ≤ r−1

23 d2

117

In this case,

r−113 d1(1− r2

23) + d2r23 − (r−123 d2(1− r2

13) + d1r13) = (1− r213 − r2

23)[r−113 d1 − r−1

23 d2] < 0

This implies that the range specified in Case 3, can be restated as r−113 d1(1− r2

23) + d2r23 <

d3 < r−123 d2. In this range, PMFPS1 is infeasible since r−1

13 d1 − (r−113 d1(1 − r2

23) + d2r23) =

r223(r

−113 d1−r−1

23 d2) < 0. Thus, under Case 3A, the feasible strategies are PMFPS2, NMFPS,

and SMFPS. Comparing profits for these strategies:

ΠPMFPS2 − ΠSMFPS = 0.25z[(d2 − r23d3)2] > 0

Now it is easy to show that ΠPMFPS2 is monotonically increasing in the range for d3 given

by Case 3A. Thus, the profits under PMFPS2 are minimum when d3 = r−113 d1(1 − r2

23) +

d2r23 + ε where ε is set to be sufficiently small. Say ε ≈ 0, then consider:

ΠPMFPS2(d3 = r−113 d1(1− r2

23) + d2r23)− ΠNMFPS

= d22 + (1− r2

23)−1(d3 − r23d2)

2 − (d21 + d2

2)

= (1− r223)

−1(r−113 d1(1− r2

23))2 − d2

1

= r−213 d2

1(1− r223)− d2

1 > 0 since 1− r223 > r2

13 ⇒ r−213 (1− r2

23) > 1

Given these results, we can conclude that PMFPS2 is the dominant strategy for Case 3A.

Case 3B: r−113 d1 > r−1

23 d2

In this case,

r−113 d1(1− r2

23) + d2r23 − (r−123 d2(1− r2

13) + d1r13) = (1− r213 − r2

23)[r−113 d1 − r−1

23 d2] > 0

This implies that the range specified in Case 3, can be restated as r−123 d2(1− r2

13) + d1r13 <

d3 < r−113 d1. In this range, PMFPS2 is infeasible since r−1

23 d2 − (r−123 d2(1 − r2

13) + d1r13) =

r213(r

−123 d2−r−1

13 d1) < 0. Thus, under Case 3B, the feasible strategies are PMFPS1, NMFPS,

118

and SMFPS. Comparing profits for these strategies:

ΠPMFPS1 − ΠSMFPS = 0.25y[(d1 − r13d3)2] > 0

Now it is easy to show that ΠPMFPS1 is monotonically increasing in the range for d3 given

by Case 3B. Thus, the profits under PMFPS1 are minimum when d3 = r−123 d2(1 − r2

13) +

d1r13 + ε where ε is set to be sufficiently small. Say ε ≈ 0, then as with Case 3A, it can be

shown that:

ΠPMFPS1(d3 = r−123 d2(1− r2

13) + d1r13)− ΠNMFPS > 0

Given these results, we can conclude that PMFPS1 is the dominant strategy for Case 3B.

Case 4: d3 ∈ [α3,∞) or max{ r−113 d1, r

−123 d2} ≤ d3

When r−113 d1 ≤ r−1

23 d2

ΠSMFPS(d3 = r−123 d2)− ΠNMFPS

= r−223 d2 − (d2

1 + d22)

= r−223 d2(1− r2

23)− d21

> r−223 d2r

213 − d2

1 since 1− r223 > r2

13

> 0 since r−113 d1 < r−1

23 d2 ⇒ r−123 d2r13 > d1

Similarly, when r−113 d1 > r−1

23 d2 ΠSMFPS(d3 = r−113 d1) − ΠNMFPS > 0. Let A = max

{r−113 d1, r

−123 d2}, it is obvious that ΠSMFPS(d3 = x) > ΠSMFPS(d3 = A) for ∀x > A. Since

PMFPS1 and PMFPS2 are infeasible in this region, SMFPS is the only dominant strategy

when max {r−113 d1, r

−123 d2} ≤ d3.

119

APPENDIX CHESSIAN MATRIX FOR CHAPTER 3

Taking the first and second derivatives of the objective function for problem (GP), we

obtain∂Π

∂qk

= 2dk − 2qk −m∑

j=1,j 6=k

(rk,j + rj,k)qj, k = 1, 2, . . . , m,

∂2Π

∂q2k

= −2 k = 1, 2, . . . , m

∂2Π

∂qk∂qj

= −(rk,j + rj,k), k, j = 1, 2, . . . , m, k 6= j.

Based on this:

H =

−2 −(r1,2+r2,1) −(r1,3+r3,1) ··· −(r1,m+rm,1)−(r1,2+r2,1) −2 −(r2,3+r3,2) ··· −(r2,m+rm,2)−(r1,3+r3,1) −(r2,3+r3,2) −2 ··· −(r3,m+rm,3)

......

......

...−(r1,m+rm,1) −(r2,m+rm,2) −(r3,m+rm,3) ··· −2

= (−2)

1 γ1,2 γ1,3 · · · γ1,m

γ1,2 1 γ2,3 · · · γ2,m

γ1,3 γ2,3 1 · · · γ3,m

......

. . ....

...

γ1,m γ2,m γ3,m · · · 1

= (−2)γ,

where γk,j = 12(rk,j + rj,k) (k, j ∈ {1, 2, ..., n} and k 6= j) represents the average substitution

effect between products k and j.

120

APPENDIX DTHE OPTIMAL QUANTITIES

The equation system for the FOCs is shown below. Solving the optimal quantity

vector by Cramer’s rule, we obtain the following:

γT q =

1 γ1,2 γ1,3 · · · γ1,m

γ1,2 1 γ2,3 · · · γ2,m

γ1,3 γ2,3 1 · · · γ3,m

......

. . ....

...

γ1,m γ2,m γ3,m · · · 1

q1

q2

q3

...

qm

=

(a1 − c1 + ν1)/2

(a2 − c2 + ν2)/2

(a3 − c3 + ν3)/2

...

(am − cm + νm)/2

=

d1 + ν1

2

d2 + ν2

2

d3 + ν3

2

...

dm + νm

2

= d +ν

2

q∗ = [γT ]−1(d +1

2ν) = [γ]−1(d +

1

2ν)

ν ≥ 0

qkνk = 0,∀k.

121

APPENDIX EQUANTITY AND PROFIT

Let S be any portfolio with all positive-quantity products, that is qk > 0,∀k ∈ S, then

the optimal quantities and the profit are shown in E–1 and E–2, respectively.

qS = [γST ]−1dS = [γS]−1dS (E–1)

ΠS = qST (pS − cS) = qS

T (2dS − rSqS) = 2qST dS − qS

T rSqS

=2dS

T Adj[γS]dS

|γS| − dST [γS]−1rSAdj[γS]dS

|γS|

=dS

T{2Is − [γS]−1rS}Adj[γS]dS

|γS|

=dS

T{2Is − [γS]−1[2γS − rST ]}Adj[γS]dS

|γS|

=dS

T{2Is − 2[γS]−1γS + [γS]−1rST ]}Adj[γS]dS

|γ|= dS

T [γS]−1rST [γS]−1dS = [[γS]−1dS]T rS

T [γS]−1dS

= qTS rS

T qS

(E–2)

122

APPENDIX FPROOF FOR THEOREM 3.1

The profit difference between the parent S ′ and the child S portfolios is

ΠS′ − ΠS =|γS|dS′

T Adj [γS′ ]dS′ − |γS′ |dST Adj [γS]dS

|γS||γS′|=|γS′||γS| (qS′j)

2,

where the last equality follows from Corollary 1 in Appendix G. Because |γS| > 0, we

obtain

ΠS′ − ΠS

> 0, if |γS′| > 0;

< 0, if |γS′| < 0.

123

APPENDIX GPROOF SUPPLEMENT FOR THEOREM 3.1

Let A be an (n × n) matrix and let B be the (n − 1) × (n − 1) submatrix obtained

by deleting the last row and last column of A. Let Adj(A) and Adj(B) be the adjugate

matrices of A and B. Let x′ = (x1, . . . , xn)T , x = (x1, . . . , xn−1)T , y′ = (y1, . . . , yn)T and

y = (x1, . . . , yn−1)T .

Theorem 1. Let R be the matrix obtained from A by replacing the last row by x′T and let

C be the matrix obtained from A by replacing the last column by y′. Then

|B|x′T Adj(A)y′ − |A|xT Adj(B)y = |R||C|.

Proof. First, we recall the standard notation Ai,j for the the (n − 1) × (n − 1) submatrix

obtained from A by deleting row i and column j. Also, the (i, j) cofactor of A is

(−1)i+j|Ai,j|. Then Adj(A) is the matrix whose (i, j) entry is the (j, i) cofactor of A.

To prove this theorem, we must reformulate it a little. Let (Adj(B))+ be the n× n matrix

obtained from Adj(B) by adding a last row and column of zeros and let F be defined as:

F := |B|Adj(A)− |A|(Adj(B))+. (G–1)

Then equation (G–1) is equivalent to

x′T Fy′ = |R||C|. (G–2)

Let r be the column vector whose i-th entry is the (n, i) cofactor (−1)i+n|An,i| of A.

Similarly, let c be the column vector whose i-th entry is the (i, n) cofactor of A. (Thus

r = c if A = AT .) Then by the cofactor expansion of determinants, we have

|R| = x′T r and |C| = cTy′ (G–3)

so we can rewrite (G–2) as

x′T Fy′ = x′T rcTy′.

124

This equation expresses the equality of two bilinear forms. Since two bilinear forms are

equal if and only if they are represented by the same matrices, the theorem is equivalent

to

F = rcT . (G–4)

For 1 ≤ i, j ≤ n, let fij denote the (i, j) entry of F . Then (G–4) is equivalent to the

equations

fij = (−1)i+j|An,i||Aj,n|, (1 ≤ i, j ≤ n). (G–5)

To prove (G–5) we now examine each entry of F , using the definition (G–1). If either the

row index or the column index is equal to n, then the entry is simply that of |B|Adj(A).

Thus,

fnn = |An,n||B|, and for 1 ≤ i ≤ n− 1,

fin = (−1)i+n|An,i||B|,

fni = (−1)n+i|Ai,n||B|.

Since B = An,n, we see that (G–5) holds whenever i or j is equal to n.

It remains to check fij for 1 ≤ i, j ≤ n− 1. From (G–1), we see that

fij = (−1)i+j|Aj,i||B| − (−1)i+j|Bj,i||A|. (G–6)

For these values of i and j the equation (G–5) follows immediately by applying

Lemma 1 below to (G–6). This completes the proof of the theorem.

Lemma 1. Let 1 ≤ i, j ≤ n− 1. Then

|A||Bj,i| = |Aj,i||B| − |An,i||Aj,n|.

Proof. Lemma 1 is a classical formula of Jacobi (1833), sometimes called the Dodgson

Condensation Formula.

125

To apply this general theorem in the proofs of Theorem 3.1 we set S ′ = S ∪ {j},A = γS′ , B = γS, x′ = y′ = dS′ and x = y = dS. Since γS′ is symmetric, we have |R| = |C|in this case. Furthermore, if we set qS′ = (γS′)

−1dS′ , then by Cramer’s Rule, we have

qS′j = |R||γS′ | . Therefore we obtain the following.

Corollary 1. With the notation above,

|γS|dS′T Adj[γS′ ]dS′ − |γS′|dS

T Adj[γS]dS = (|γS′|qS′j)2.

For the proof of Theorem 3.1 the corollary yields

ΠS′ − ΠS =|γS|dS′

T Adj [γS′ ]dS′ − |γS′|dST Adj [γS]dS

|γS||γS′| =|γS′ ||γS| (qS′j)

2.

126

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BIOGRAPHICAL SKETCH

Yuwen Chen is a doctoral candidate in Operations Management at the University

of Florida. His research focuses on new product strategy related to multi-function

products. Yuwen is also interested in problems on technology management, operations

and supply chain management. Yuwen received his bachelor degrees in Accounting from

National Taiwan University in Taipei, Taiwan in 1993. Yuwen had three years of working

experiences in high-tech industry in Taiwan and California. Yuwen plans to graduate in

May 2008 and pursue a career in an academia environment that has a balance of teaching,

research and the connection with real business.

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