THE IMPACT OF DESIGN QUALITY-BASED PRODUCT DIFFERENTIATION ON THE COMPETITIVE DYNAMICS IN AN OLIGOPOLY

Ramesh G. Kini Department of Information and Systems Management

School of Business and Management The Hong Kong University of Science and Technology

Clear Water Bay, Kowloon, Hong Kong

Panagiotis Kouvelis1 John M. Olin School of Business

Washington University in St. Louis Campus Box 1133, One Brookings Drive

St. Louis, MO 63130 (email: kouvelis@olin.wustl.edu)

Samar K. Mukhopadhyay School of Business Administration

University of Wisconsin Milwaukee, WI 53201

October, 2001

1 Any correspondence may be addressed to the second author, listed alphabetically.

Abstract

In this paper we examine the relationship between the firm's investments in design quality

improvement and its ability to thrive or even survive in an oligopolistic setting. We start by

providing the contextual background for our approach, and then proceed to formulate a differen-

tial game theory model with six state- and two control-variables for each duopolist. More spe-

cifically, we blend a generalized Lanchester-type combat model approach (which addresses the

competitive market share dynamics based on the firms' prices, design quality and historical

inertia levels) with a generalized Bass-type diffusion model approach (which incorporates the

cooperative product-life cycle or market penetration/saturation dynamics).

We analyze the partly competitive, partly cooperative, differential game theory model

formulated. Our aim is to examine the pervasive, though not very visible, influence of several

intra- and inter-firm, design quality-related effects on both duopolists' optimal control decisions

and costs and on the structure-conduct-performance linkages within the industry, over the long

haul. Specifically, we will use an open loop approach, that we believe is quite robust, to obtain:

i) each firm's marginal valuations of its intangible assets and liabilities; ii) a deeper understand-

ing of the tax-subsidy adjustments to its marginal cost, the myopia-related adjustments to its

effective discount rates, the adjustments to its effective reference quality level and to its own-

and cross-elasticity coefficients of demand, strategic substitutability/complementarity, and the

onset of saturation; and iii) the first order conditions for its two control variables in an interpret-

able form.

1

1. Introduction

We seek to examine and understand the growing strategic significance of product differentiation

in general, and design quality in particular, as "a powerful competitive weapon…(that) offer(s)

great leverage" (borrowing from Garvin, 1988) to forward looking managers. Yet, what exactly

does investing in design quality do for each firm's ability to compete effectively in the market-

place? How exactly do the firms' decisions and track records vis-à-vis design quality affect, in

the short run and over the long haul, the structure of the industry, the players' conduct and the

outcome of the rivalry, in terms of each firm's prospects for sustained growth or even survival?

Besides market share dynamics, there are other effects that cannot be ignored in this regard, e.g.,

market saturation (how, for instance, does one sell any more saunas—even well-designed ones—

to 5 million Finns who already have access, as per Business Week International, May 19, 1997,

p. 4A4, to 1.6 million saunas?). How do such effects interact to influence the rivals' actions and,

hence, the evolution of the industry as a whole? We seek in this paper to answer such questions.

The following example about two hypothetical firms (A and B, say) might illustrate how

design quality decisions affect the competitive and product life cycle dynamics, and vice versa.

1.1. An Example

Figure 1 depicts the post-launch performance of firms A & B, who started with a 1.5 billion unit

potential market for the "E-Lamp," a durable and efficient lighting product, with the competitive

market share dynamics (the stacked bars denote each firm's share of current industry sales, or the

wedge of the intertemporal pie to be consumed at time t) shown superimposed on the product life

cycle dynamics (the two curves depict the total current industry-wide volumes and the degree of

market saturation). From a planning perspective, if either firm had changed any control variable

(i.e., price, efficiency, etc.) in any year, it would have changed not only in its share of the wedge

for that year (subject to its rival’s response), but also the size of the slices set aside for the future.

2

While firm A’s absolute volumes grew substantially in the pre-saturation phase and then

declined, its share of industry sales had been dropping sharply ab initio (from year 2’s dominant

70% level to less than 15% after year 6). The adverse impact of this market share erosion—

possibly attributable to its inability to retain its initial technological lead, and to the gains made

progressively by firm B—was not very palpable when the market was expanding, but became far

more clearly manifest when the industry-wide sales volumes started shrinking, post-saturation.

Using the traditional, static, “rear-view mirror” approach, firm A could have, say after year 4,

justifiably projected sustained future growth in volume, revenue and perhaps profits, given past

growth in volume, etc., instead of what actually ensued. The dynamic, forward-looking approach

being offered here might have helped firm A: anticipate the drastic setback it would suffer post-

saturation; and plan and act so as avert the "double whammy" from the erosion in market share

and the shrinking residual market on its future prospects. Our approach would have enabled it to:

♦ relate its and its rival’s pricing and design quality choices to its ability to sustain market share

gains (or to even stem the erosion in market share), the rate at which the market is saturated, etc.;

♦ adopt a forward-looking, less myopic perspective on quality by focusing on its impact on

deferred-value, viz., the intangible asset/liability accretion/depletion terms and capital gains/

losses, and not just on the tangible stocks and flows that are conventionally accounted for;

♦ to relate each firm’s perception of the other—from a strategic and dynamic, rather than the

conventional static, perspective—to the cost-benefit trade-offs that are entailed over time; and

♦ to relate the zero-sum market share dynamics and the non-zero-sum product life cycle

dynamics, in general, to the firms’ long term growth or survival prospects, and so on.

The paper is organized as follows: Section 2 provides the background for our model and

explains its contribution to the quality and oligopoly literature. Section 3 formally introduces our

model. Sections 4 and 5 provide the analyses and results, and Section 6 the conclusions.

3

2. Perspectives and Related Literature on Design Quality-based Oligopolistic Competition

Extending Garvin's (1988) approach, we can think of design quality in terms of several product

specific dimensions (performance, features, reliability, durability, serviceability/maintainability,

safety, aesthetics, ease-of-use, recyclability, etc.), specifiable ex ante by the firm. Consumers can

search out (Nelson, 1970), or be given, such product-related information that could then be used

to assess the cost-benefit tradeoffs associated with—and to compare and choose between—the

available products. Conformance quality, in contrast, relates to the customer's experience with

the product and can be thought of as the degree to which the product in usage conforms ex post

to specifications, or as the extent to which it meets or exceeds consumer expectations over time.

Design quality often tends to be multi-dimensional (for instance, performance attributes for

tires are both quantifiable, e.g., load- & speed-rating, longevity, etc., and qualitative, e.g., all-

weather capability, road-handling, etc.) and markets tend to be segmented and not homogeneous.

Rivals, who gravitate away from dimensions that can be aped more easily and towards less easily

replicated ones, can thus gain more and longer-lived relief from head-on competition. To avoid

complications, we will assume that qualitative or categorical attributes can be redefined and

interpreted along a continuum, that a superior product is indeed better than the other along each

dimension, and that multiple non-price product dimensions are reducible, using exogenously

defined weights, to a uni-dimensional composite score, that can serve our purpose more readily.

While our approach can accommodate any product or service, we assume that the rivals

target price- and quality-conscious consumers with their durable products, for which a market

exists, and that consumers reward firms offering more "bang for the buck" with price premia,

market share gains, etc. Our deterministic duopoly model focuses more on each rival's design

quality choices for its durable product at each instant, on its track record in this respect, and on

the dynamic implications for the nature and intensity of the rivalry exhibited in the industry.

4

2.1. Design Quality and Rivalry in the Oligopoly

Traditionally, oligopolists were seen as relying only on pricing to either grab more of the pie, or

retaliate against initiators of such moves. When non-price variables were included (e.g., Shaked

and Sutton, 1982, and Moorthy, 1988), quality and product differentiation were usually given the

same static treatment. Product-differentiation and further market penetration have always had

dynamic, supply- and demand-side implications for the industry structure, the rivals' conduct and

the outcome of this rivalry, but this was largely ignored. Price-competition did get attenuated in

intensity in some models for rival products that were more differentiated from each other, i.e.,

the industry did move structurally towards monopolistic competition and niche-based local

monopolies. The analytical treatment and the explanations, however, were essentially static.

In most cases, the initiator of a more substantial quality related or product differentiation

move knows its rivals may not have a ready response—it can either extract a premium price for a

premium product, or parlay its competitive edge into gains in market share, volume and revenue,

accordingly, before they respond with similarly differentiated products. There is more, however,

to investing in quality and product differentiation than the usually-emphasized static gains or the

mere ability of the initiator of such a move to extract price premia for a while, or the temporary

respite from a price war afforded by it. We focus on the dynamic and not the static aspects of the

problem, and especially the impact of each firm's track record on design quality on the industry's

structure-conduct-outcome linkages. Working within a differential game theoretic framework,

we seek to provide a more comprehensive and, hopefully, coherent rationale for strategic design

quality and product differentiation investments, than has been tried, in our opinion, to date.

2.2. Modeling Contributions And Differences With Existing Literature

The cost-benefit tradeoffs associated with our rivals' design quality investments depend heavily

on the market share and the industry-wide product life cycle dynamics. In the conventionally-

5

used proportional attraction market-share rivalry models (see, e.g., Cooper and Nakanishi, 1988),

firm i's current market share in period t was given by )( jtititit AAAs += , with , or

firm k's attraction factor, as a function of its current advertising expenditure, quality level, etc.

Such an approach is: i) passive (firm j adversely affects its rival only by increasing the industry-

wide attraction in the denominator) and not active; ii) based on current- and not cumulative-

market share dynamics; iii) static and not dynamic (both the cumulative, rather than current,

market share, and its rate of change, play a role in our model); and iv) difficult to manipulate as a

differential game in continuous time; and was hence deemed unsuitable for our purposes. Yet,

even Lanchester's combat model (perhaps, less "flawed" in these respects) had to be modified

and generalized before it could be used to define the competitive zero-sum market share game.

jikAkt ,, =

In what may well be the first such attempt to simultaneously tackle market share and market

penetration dynamics, we blend generalizations of Bass' (1969) diffusion model and Lanchester's

combat model (applied by Kimball, 1957, and Fruchter & Kalish, 1997, to advertising) in order

to analytically differentiate between, and also synthesize as well, the competitive market share

and cooperative product life cycle effects of each firm's actions and states. As in the example,

our firms: jointly decide on how large a wedge of their inter-temporal pie of potential adopters is

to be currently consumed or converted; and simultaneously compete for (based on their current

control actions and states) a larger share of the wedge between themselves. To elaborate:

♦ Unlike in the traditional Lanchester framework, market-share trades depend in our model on

both firms' states, control actions and track records vis-à-vis quality, as well as exogenously

on the potential market size, the product life cycle-specific parameters, the reference quality

levels, etc. Finally, as in Case (1979), ours is a differential game in market shares and not

current volumes, but our accretion and attrition forces are intertwined with both firms' states

and control actions and lead to gains or losses in cumulative and not current market shares.

6

♦ We adjust Bass' (1969) diffusion model to allow for: both Bass-type linear and non-linear

saturation dynamics, and the firms' controls and states to multiplicatively affect the market

penetration rate. In Bass' linear model, the onset of saturation affects market penetration spe-

cifically through the linear saturation factor, and the declining residual market potential. We

generalize the Bass model by using an unspecified, general term )(ψΨ , with ψ as the cur-

rent penetration level, instead of the linear factor, and allow for the coefficients of elasticity

of the market penetration rate w.r.t. the firms' controls and states to endogenously depend on

time, instead of the rather awkward, conventionally-used exponential multipliers (see, e.g.,

Bass and Bultez, 1982), with constant elasticity coefficients as the exponents.

Further, we aim to analyze design quality's dynamic role more thoroughly and as broadly as

possible. Some industries at the fast paced technology end tend to be volatile with frequent

technology- and market-leadership changes (e.g., photolithography; see Henderson, 1993), while

those at the stable and mature end (e.g., steel castings) seem almost impervious to the forces that

foster change. Hence, we have included two quality-related state variables for each firm, viz., its

design quality level, , and its historical inertia level, , a function of its installed base

that reflects its track record w.r.t. design quality, through state equations defined in §3.1 below.

)(tQi )(thi

If the quality ('Q ') effect dominates the inertia (' h ') effect at the more volatile end, a superior

product can get more readily accepted, altering the industry structure and the rivals' fortunes as a

result. Conversely, if the ' ' effect dominates the 'Q ' effect, the installed base could help a more

established but inferior, incumbent, QWERTY-type technology (see Dixit and Nalebuff, 1991) to

hold sway in a mature industry, despite newer, superior DSK-type alternatives being available.

h

2.3. The Strategic Substitutability/Complementarity Effects.

We now examine the role of strategic substitutability/complementarity—a relatively new but not

uncommon and unfortunately not very-well-understood phenomenon. Fudenberg and Tirole

7

(1983) and Bulow, Geanakoplos and Klemperer (1985), henceforth F&T and BGK, respectively,

were among the pioneering analysts of its counterintuitive nature. F&T analyzed firms that chose

to strategically underinvest in advertising goodwill and appear "lean and hungry" rather than as

"fat cats," if investing more reduced the incentives to respond aggressively to rivalry and entry.

BGK related strategic effects to a firm's impact on its rival's marginal (not total) profitability and

saw products as strategic substitutes (complements), depending on whether the more aggressive

a firm was in a market, the lower (higher) were its rivals' marginal profits, respectively.

Our strategic effects are akin in flavor to, and yet essentially different from, those analyzed

by BGK and F&T. Firstly, our effects are dynamic and not static, and have an impact both at an

instant and across time—changes that occur over time or over the product life cycle can thus lead

to our strategic effects having diametrically opposite implications at different times, unlike their

effects which remain constant in sign and direction. Secondly, both exogenous (saturation) and

endogenous (changes in perceived quality-related gains) factors affect the nature and intensity of

the rivalry in our case. Lastly, we distinguish between price-related (as in BGK) and quality- and

inertia-related strategic effects. Price's role in our partly competitive, partly cooperative model is

slightly subsidiary, in that each firm uses it, at each t, more to buttress the gains from the inertia

increment over the entire horizon, i.e., the effect vis-à-vis inertia subsumes that vis-à-vis price.

Unlike in BGK, where strategic substitutes or complements are exogenously defined, or in

F&T, where reaction curves' slopes are exogenously given, our strategic effects have endogenous

as well as exogenous roots (being driven by the firms' actions and also by the inevitable onset of

saturation). Unlike F&T, whose firms sought to deter or accommodate entry, our firms behave

strategically even without entry or exit. Their firms could strategically underinvest in capital,

either to reduce the ferocity of future competition as in BGK's strategic complementarity case, or

to credibly threaten lower post-entry prices as in F&T's goodwill case. In contrast, each of our

8

firms can opportunistically enjoy a "free ride" under its rival's market expansion "umbrella."

Further, the strategic effect could have a demand (or supply)-side impact in F&T and BGK,

shifting the marginal revenue (cost) curve up (down), respectively, as a result. Investments in

quality help our firms gain and retain market share, but also leave less of the residual market

potential for future consumption—thus our marginal revenue curves get shifted upwards over

time as well as depressed in the future. While our firms can thus simultaneously experience both

joint economies and diseconomies in sequential markets, unlike in BGK's learning curve and

"natural resource" examples, it is the net effect which is important.

3. Model Description

In our model, , firm i's current sales rate, depends on: both firms' control and state

variables; the market-share attrition and accretion forces that affect each firm's market share's

susceptibility and retentivity; and on the industry-specific market penetration dynamics. With

, as the potential market size, our model's six state variables are given by: i) both

firms' design quality levels, Q ; ii) both firms' historical inertia levels, ;

iii) firm i's cumulative market share, (since , we need only one market

share-related state variable to define the market share equilibrium for the duopoly as a whole);

and iv) the industry-wide market penetration level, , with as

firm i's cumulative sales volume over [ , and with

)()( tXtx ii&=

],0[),( TttM ∈

2,1),( =iti

(tSi

2,1),( =ithi

)()( tPtSi ∗

)

0

))(1()( tStS ij −=

)()( tXtP i∑= )(tX i =

]t, ))(t) Mt(()( Pt =ψ representing the

current market penetration level. Similarly, the two control variables for firm i , are given

by: i) its product price, ; and ii) , the change in , its design quality level, at .

2,1

t

, =i

)(tpi )(tqi )(tQi

3.1. Building Blocks of the Model

To illustrate how the tradeoffs involved for each firm, both at a point in time and over time, and

hence the firms' prospects for the future, are affected by its, and its rival's, control decisions and

states, we will first elaborate on the model's intra-firm and inter-firm building blocks as follows:

9

Current and Cumulative Sales Volumes: Firm i's current installed base, or its cumulative volume

over , is given by , where is its current

sales rate at s. While we ignore scale economies, learning curve and experience effects, the firms'

future prospects are affected by their current and cumulative sales volumes (cf., a curve and the

area under it, respectively) in several ways in our model, e.g., through their effect on the firms'

inertia levels, and inertia's impact on the market-share and -penetration dynamics, and so on.

[ t,0 ] ∫=t

ii dssxtX0

)()( )()()()()( sPsSsPsSsx iii&& ∗+∗=

Design Quality & Design Quality-related Fixed and Variable Costs: Firm i's design quality-

related state and control variables are linked through q , or alternately, through: )()( tQt ii&=

(1) ,)()0()(0

ττ dqQtQt

iii ∫+=

with as firm i's initial design quality level. We assume that all variable and fixed costs

depend either on the design quality level or the change in it. Each firm bears four types of such

costs: two that depend on control variable (viz., a unit variable cost, c , and a fixed

cost, c ); and two that depend on state variable Q (viz., a unit variable cost, ,

and a fixed cost, c ). Design quality affects the direct material, conversion and variable

overhead costs through and . The two fixed costs capture the continuously

borne design quality dependent costs of obtaining and managing the resources (manpower,

facilities, capital, etc.) normally required in the pre-manufacturing, manufacturing and post-

manufacturing phases. We assume that one-off costs, if any, are amortized over time and that the

firms bear no other costs that do not depend on design quality. If not, let the firms be symmetric

w.r.t. such a cost, and the cost be normalized to zero, accordingly. Lastly, the four costs are

positive, convex and strictly increasing on the margin, or c

0)0( >iQ

))((3 tqii

)(tqi

))(( tQi

))((1 tqii

,2,1,0 =v

)(ti

′vi

))((2 tQc ii

.4

))((4 tQii

(1c i ))(tqi 2c i

,3(.),0(.) >′′> cvi

Path Dependence, Increasing Returns and Historical Inertia: Path dependence refers to the vital

role that the product's "history," the sequence of the rivals' design quality decisions, in this case,

plays in the future evolution of the competitive rivalry. Because "history matters," as Dixit and

10

Nalebuff (1991) put it, and to analyze the impact of such issues on each firm's ability to parlay its

intangible assets into greater gains down the road, we emphasize the role(s) of:

i) firm i's cumulative volume, , and cumulative market share, )(tX i )()()( tPtXtS ii =

)()( tSts ii&≠

, over

those played by its current volume, , and current market share ; )()( tXtx ii&=

ii) each firm's historical inertia level (a weighted installed base, with the design quality surplus

or deficit as the weight), which is affected by the current sales volume and design quality

level, and affects the market share and market penetration/saturation dynamics, in turn;

iii) the current market penetration level (the cumulative sales volume aggregated over all firms),

which influences the industry-wide market penetration/saturation dynamics; and

iv) the durational and terminal effects of changes in the level (or the intangible asset/liability

accretion/decretion terms), and in firm i's marginal valuation (or the capital gains/losses), of

its design quality, inertia, market share and market penetration-related state variables.

We allow for changes in inertia to depend on how each firm's design quality level stacks up

against customer expectations, embodied in an industry-specific reference quality level or datum,

and also adjust for noblesse oblige and decreasing marginal returns to inertia, as we show below.

Specifically, with 0>iδ as a firm-specific parameter, the rate of change, at time t, of firm i's

inertia level depends on , its current sales rate, the normalized difference between its design

quality level, , and the datum, , and its current inertia level, h , viz.,

)(tix

)(tQi )(tQr )(ti

+

−=

)(1)(

)()(1)()(

thth

tQtQtxth

ii

i

i

rii δ

& . (2)

The rivals' design quality decisions affect the industry-wide market-share and penetration

dynamics through four state variables, and in four different ways: )(tQi ,2,1),( =ithi

i) Since h , if , respectively, inertia: (a) has a positive or "bouquet" effect, if

the firm chooses to exceed customers' expectations; (b) has a negative or "brickbat" effect, if

0)(<>ti

& )()( tQtQ ri <>

11

the firm's design quality falls short of expectations; or (c) maintains the status quo ante, if

expectations are just met. We further assume that 0)()( >∂ thth ii&∂ and 0)()( 22 <∂∂ thth ii

&

0)( ≥tr&

,

i.e., that inertia-related benefits are positive but declining on the margin, and that firms with

more inertia, or that are more visible, or which have greater revenues and profits at stake,

perforce have to aim to meet or exceed expectations vis-à-vis design quality, because of no-

blesse oblige-type compulsions—else, their prospects for sustained growth would be more

drastically affected by penalties relating to the ensuing loss of inertia or momentum.

ii) Each firm's design quality decisions directly affect its variable and fixed costs, and hence

effective marginal cost (which, along with its effective own- and cross-elasticity coefficients

of demand, determines its product price, and in turn demand, through the effect of price on

market share accretion and attrition rates as well as market penetration/saturation dynamics).

iii) Each firm's design quality and inertia levels affect both firms' market-share accretion and

attrition rates directly; the state equation for market shares, defined below, will capture this.

iv) Lastly, the moving datum, Q (henceforth ), depends on time-dependent

exogenous factors that raise consumers' expectations of further improvements, and on the

industry quality leader's current quality level, Q . Thus, Q , and

))(,( tQt mr )(tQr

max ,ik= ))(()( tQt kjm =

0≥))((/ ∂∂= mrrmQQ QQQQν mr imply that the follower is driven, partly exogenously and

partly by the leader's strategic investments in quality, to meet continually higher standards.

To see how the movement in the datum provides more incentives for each firm to further

invest in quality, let firm i have a current surplus and firm j a deficit, i.e., .

Noblesse oblige would induce firm i to try to improve its design quality and augment its inertia

or momentum. Since

)()()( tQtQtQ jri >>

0)( >∂∂ ir QQ , the added impetus to the datum from firm i's investments

in quality could strategically, possibly permanently, handicap firm j. If firm i adds to its strategic

lead, firm j—whose future is even more adversely affected—may find the cumulative deficit to

12

be even more of a yoke and a restriction on its price or current sales, than if it were a monopolist.

Firm j, burdened with an initial quality deficit, Q , may find it even more optimal

now to suppress initial sales to avoid getting slotted as a low quality producer, since the inertial

effect cannot be shaken off easily and would adversely affect future sales.

)0()0( rj Q<

t)

)(tP&

()(())( MtPt Ψ

Market Penetration & Saturation Dynamics: To tackle product life cycle issues, we have:

as the potential market size and as the cumulative industry-wide sales, at t.

Generalizing Bass' (1969) diffusion model, we have , the market penetration rate, given by:

)(tM

∑=i iXtP ()(

))()(())),(),(()(2,1

tPtMthtQtpdtP kkkk k −= ∑ =& , (3)

where or the fractional rate at which firm k's price, design quality

and historical inertia levels contribute to expanding the market, is defined such that:

,2,1)),(),(),(( =kthtQtpd kkkk

2,1,0,0,0 =>∂∂

>∂∂

<∂∂

khd

Qd

pd

k

k

k

k

k

k . (4)

As saturation effects begin to dominate, reaching out to the more price-sensitive laggards in

the residual market potential, ( , becomes an increasingly unattractive proposition to

the firm, as in any product life cycle model. Also, the multiplicative saturation factor,

))()( tPtM −

)(ψΨ ,

where ))()(()( tMtPt =ψ , is defined so that before (after) the onset of saturation, we have:

0 , if , (5) )(<>Ψ′ ψ )()( ψτψψ

><t

respectively. The critical penetration level, )( ψτψ , is attained at some T<ψτ at which saturation

effects start to dominate. The dampening effect of on further market penetration is

thus offset (reinforced) by the multiplicative saturation factor,

))()(( tPtM −

)(ψΨ , before (after) ψτ , respec-

tively. can, therefore, be interpreted both as an intangible asset in the pre-saturation phase

because of its catalytic effect on potential adopters, and as an intangible liability, post-saturation.

)(tP

Market Share Dynamics: Essentially, the game vis-à-vis cumulative market shares is assumed to

be an adversarial zero-sum one, so that a firm can gain in market share only at the expense of the

13

other. In generalizing Lanchester's approach to modeling market share dynamics, our focus was

on how design quality, inertia and pricing influence the firms' market shares' susceptibility and

retentivity characteristics. We assume both firms' current and previous control actions and states

help accentuate or attenuate (adversely or favorably affect) the vulnerability of each firm's

market share to its rival's moves (thus addressing path dependence, etc.). Specifically, let the

firms trade market shares (or firm i's cumulative market share, )()()( tPtXtS ii =

(),(),(),( hthtQt jiji

, change) at a

rate that depends on both firms' current prices, and design quality and inertia levels, through: (i)

and (ii) ))(),(),(),(),(),(( ththtQtQtptpb jijijii )),),(),(( tQtptpa jii ;2,1, =ji

ji ≠

i

, which are, respectively, the fractional rates at which firm i's cumulative market share

erodes, and at which firm i takes market share away from firm j, because of both firms' quality

and pricing decisions. Let the b 's and 's (without arguments henceforth) be defined such that: ia

,0,0,0,0,0,0,0,0 >∂∂

<∂∂

<∂∂

>∂∂

<∂∂

>∂∂

>∂∂

<∂∂

j

i

i

i

j

i

i

i

j

i

i

i

j

i

i

i

Qb

Qb

Qa

Qa

pb

pb

pa

pa

jijihb

hb

ha

ha

j

i

i

i

j

i

i

i ≠=>∂∂

<∂∂

<∂∂

>∂∂

;2,1,,0,0,0,0 (6)

The rate of change of firm i's cumulative market share, , is therefore given by: )(tSi

))(1()1)(()( tStatbtSttS iiiii −∂+∂−=∂+

which, based on the fact that the firms compete for cumulative market share gains on a zero-sum

basis, viz., reduces (in the limit as ) to: ,0)(,1)( ∑∑ == tStS ii& 0→∂ t

jijitSabtSabtS iijijii ≠=−+++−= ;2,1,)),(1)(()()()( 21

21& . (7)

Our formulation needs only one of the two state equations (7), since the other follows from it.

Firm i's instantaneous sales rate at t, is given by: ),()()()()()( tPtStPtStXtx iiii&&& +==

)).()()()()(())(()()()()()( 21

,;,21 tPtMtMtPddtStPabtXgtx jiiijijikbag ki −Ψ++++−= ∑ ==

(8)

Firm i gets a )( jii ddd + share of current industry sales in the Hall-type proportional attrac-

tion models, e.g., Karnani (1983), based on its relative ability to "attract" new customers. Equa-

14

tion (8) reveals that market shares in our model have an inertial effect, since firm i's share of the

new adopters remains at the level of its cumulative market share, , even as both the firms'

control decisions and states affect the simultaneous trading in market shares. Thus:

)(tSi

,0 ∂S&

♦ the inertia accumulated over [ would help dominant players defend their market- and

technological- leadership against smaller upstarts (as is often true in the real world, except in

technologically volatile high tech. industries, e.g., photolithography);

],0 t

♦ if inertia and not design quality was driving market share dynamics and if Q , being in-

sensitive to Q and to exogenous factors, were to remain quite stable at a

level low enough for both firms to avoid design quality deficits, then—despite lacking design

quality leadership—a QWERTY-type incumbent could use its initial inertia gains to sustain

and defend its share against a later entrant with superior DSK-type design quality; and

(.,.)r

))((max)( , tQt kjikm ==

♦ while the design quality-based rivalry is competitive, there is also a cooperative flavor to the

industry's dynamics, since firm i, regardless of its size and its own expansion related efforts,

does benefit from firm j's contribution to market expansion. Examples of free-ridership are

not uncommon in the real world. For instance, potential market leaders may wait for pioneers

to clarify industry standard-related issues before raising their own stakes in the game.

Lastly, ,0>∂ ii QS&∂ ,0>∂∂ iQP& ,0>∂∂ ii hS& >∂∂ ihP& ,0<∂ ii p 0<∂∂ ipP&

and ,0<∂∂ ii SS& follow from equations (4), (6), and (7), and these imply that:

♦ investing in design quality and inertia could help a firm stem the erosion of market share or

even reverse the long term trend through more sustainable or defensible market share gains;

♦ such an approach also accelerates the expansion or further penetration of the market;

♦ a price-cut accelerates the expansion of the market, and absent an adequate response from the

initiator's rival(s), could translate into short-term market share gains that (unless it is a part of

a quality-based strategy) may not be readily sustainable or defensible in the long run; and

15

♦ lastly, the higher its cumulative market share, the less easy is it for firm i to parlay price

reductions and quality improvement into further market share gains. Forces that lead to the

accretion effect being overcome by the attrition effect, on the margin, thus also induce domi-

nant firms to look elsewhere (further market penetration) for sustained growth in volume, etc.

3.2. Firm i's Optimization Problem.

Based on the above, we can now formulate firm i's optimization problem as follows: with its

current sales/production volume at t given by firm i seeks

to maximize its total discounted profits over the [ interval, viz.,

),()()()()()( tPtStPtStXtx iiii&&& +==

],0 T

(9) [ ] ,2,1,))(())(()())(())(()(max0 4321)(),(

=−−−−∫ − idttQctqctxtQctqctpeT

iiiiiiiiiirt

tqtp ii

subject to equations 1, 2, 3 and 7, the six state equations for: (i) both firms' design quality levels,

; (ii) both firms' historical inertia levels, ; (iii) the industry wide

market-penetration/saturation level, ; and (iv) firm i's market share at t, ; respectively.

4. Analysis And Results

2,1),( =itQi 2,1),( =ithi

)(tP )(tSi

For let firm i's marginal valuation, in current value terms, of: i) its own (and its

rival's) design quality level be given by

,;2,1, jiji ≠=

)(tiiω ( )(tijω , respectively); ii) its own (and its rival's)

historical inertia level be given by )(tiiζ ( )(tijζ , respectively); and iii) its market share and the

market penetration level be given by )(tiλ and )(tiµ , respectively. These reflect how marginal

changes in each firm's valuation of each of the six state variables (or, its intangible assets and

liabilities) affect the firms' behavior and the outcome of the rivalry, e.g., 0)( ≤tijω and 0)( ≤tijζ

capture the adverse impact of firm j's design quality and inertia gains, respectively, on firm i's

future prospects. With the market share accretion and attrition factors, and , and the market

expansion related factors, , 2 , defined in (4) and (6) as functions of prices, design quality

and inertia levels, firm i's current value Hamiltonian for its optimization problem (9) is given by:

i iba

id ,1=i

16

)()()()())(())(()())(())(()( 4321 tQttQttQctqctXtQctqctp jijiiiiiiiiiiiiii &&& ωω ++−−−−=H

+

−+

)(1)(

)()(

1)()(th

thtQtQ

tXtii

i

i

riii δ

ζ &

+

−+

)(1)(

)()(

1)()(th

thtQtQ

tXtjj

j

j

rjij δ

ζ &

))(1)(()()()( 21

21 tSabtSabt iijijii −+++−+ λ

jijitPtMtMtPddt jii ≠=−Ψ++ ;2,1,,))()(())()(()()(µ . (10)

Here, relates firm i's instantaneous sales rate at t to its cumulative

market share, the market penetration level and the rates at which these change. We now analyze

how firm i's marginal valuation of each state variable evolves over time as a function of i) the

marginal valuation of its other state variables; ii) its fixed and variable cost structure; iii) its

elasticities of demand w.r.t. its own (and its rival's) price, design quality and inertia levels;

iv) the market-share and market-penetration dynamics; and on other aspects of the problem.

)()()()()( tPtStPtStX iii&&& +=

Further, let firm ii be the quality leader at time t, so that , the datum, is

affected only by , and not by , the follower's

quality level. Since we seek only to characterize the solution to the problem, we will assume in

the interest of brevity that firm i retains design quality leadership at each instant over [ .

,2,1, =

()( Qt mi =

))(,( tQtQ mr

ij ≠= ;2,1)(max) tQtQ kk= jtQ j ),(

],0 T

4.1. Marginal Valuation Functions and Strategic Interaction Effects.

Result 1: Law of motion for firm i's marginal valuation of its own design quality )(tiiω .

Firm i's marginal valuation, at t, of the design quality of its product, is given by:

(11) ,2,1),()()()( )( =++= −− ittTet sdii

cdiiii

tTrii

i ωωωω

where the three terms of the r.h.s. of (11) are, respectively:

i) the terminal component, with as firm i's terminal quality level's salvage value; 0)( ≥Γ= iQii Tω

ii) the cross-durational strategic substitutability/complementarity-related component given by:

dssXsh

shsQsQs

sQs

et jjj

j

j

rij

i

adjjiT

t

tsrcdii

i )())(1(

)()()(1)(

)()(

)( )( &

+

−= ∫ −−

δζ

νω ;

17

iii) and the self-durational component which reflects the impact of the design quality increment

on the effective contribution margin and effective incremental fixed costs, and is given by:

))(())(()()()()( 21

)( sQcsqcspsQs

et iiiiii

adjiiT

ttsrsd

iii

+−= ∫ −− νω

,))(()())(( 42 dssQcsXsQc iiiii

′−

′− &)()(1

)()(

)(11

)()(1)( / sCsh

shs

ssQsQs Q

tsii

iadjii

adjQQ

i

rii

ii

ir

−

+

−−−−

δνν

ζ

with:

- , firm i's adjusted own-elasticity of demand w.r.t. its quality, given as a weighted sum of

firm i's elasticities of market-share and -expansion w.r.t. its quality,

adjiiν

ii QS /&ν and iQP /&ν , by (A.2);

- , firm j's adjusted cross-elasticity of demand w.r.t. firm i's quality, given as a weighted

sum of firm j's market-share's cross elasticity w.r.t. firm i's quality,

adjjiν

ij QS /&ν , and iQP /&ν by (A.5);

- , the adjusted coefficient of the reference quality level's sensitivity to increases in the

leader's, i.e., firm i's, quality level, which (with

adjQQ ir /ν

]1,0[/ ∈∂∂= irriQQ QQQQir

ν as the datum's

unadjusted sensitivity to increases in the leader's quality level, and after adjusting for the added

edge the leader enjoys because of the effect of the incremental movement in the datum on the

follower, now held to a higher standard and forced to run faster to even stay in place) is given by:

iririr QQjjjiiii

iiijjijQQ

adjQQ tQththtXt

tQththtXt/// )())(1)(()()(

)())(1)(()()(1 ν

δζδζ

νν <

++

+=&

&;

- and with C , the subsidy/tax accruing to firm i ascribable to its design quality related gains,

obtained by substituting

i

i

Qts /

ikQi ==∆=Φ &, ν in equation (12) which generalizes the subsidy/tax

as a weighted sum of its marginal valuations of its market share and the saturation level at t, viz.,

2,1,,0)()(

/// =

∆+∆

+∆

+∆=

<>ΦΦΦ ki

PSSPPSS

PSSPSP

PC adj

ik

iiiP

i

iadjik

iiiSits

i

i

&&&&&&&& µλ . (12)

Discussion:

Result 1A: The Strategic Substitutability/Complementarity (SS/C) effect and )(tiiω .

The sign of , the cross-durational component of firm i's marginal valuation of its own )(tcdiiω

18

design quality, depends on whether its product is a strategic substitute for, or complement to, its

rival's, i.e., on whether the product of the three terms in the numerator within the integral, viz.,

♦ (which is for the strategic substitutability/complementarity case, respectively), )(tadjjiν ,0

><

♦ )(sijζ (which is negative), and

♦ (which is depending on whether firm j faces a quality surplus or deficit), )(sh j& ,0

<>

is positive or negative, respectively (see Table 1, below, for more details on the strategic effect).

Result 1B:

The self-durational component reflects design quality's marginal impact on the net rate at which

profit will accrue, over ( , to firm i, which is measured in terms of the difference between: ],Tt

♦ the first demand-side component which equals the incremental volume times the contribution

margin, with the unit marginal cost, , adjusted down- or up-wards through: (.))((.))( 21 iiii Qcqc +

- an inertia-related subsidy (tax), for the quality surplus (deficit) case, i.e., if ,

respectively, where

(.)(.) effri QQ

<>

,(.)(.))(.)11(.)((.) /adjii

adjQQ

adjiir

effr ir

Q ννν +−=Q the effective reference

quality level, is adjusted twice: first, multiplicatively downwards, through the )(.)11( adjiiν−

term, because a design quality improvement directly affects the firm's current quality surplus

or deficit; and second, upwards, by increasing the multiplicative factor by ,(.)(.) adjiii

ν/adj

QQrν to

compensate for the quality leader, in turn, being itself measured in the future against a more

stringent yardstick because of the impetus from its own quality gains to the datum; and

- a quality related subsidy (tax), C , in the pre- (post-) saturation phase, respectively; (.)/i

i

Qts

♦ and design quality's supply-side impact on firm i's profits, comprising of the existing volume

times , the incremental unit cost, and , the fixed cost increment borne over . )(2 ii Qc′ )(4 ii Qc′ ],( Tt

Result 2: Law of motion for firm i's marginal valuation of its rival's design quality, )(tijω .

Firm i's marginal valuation, at t, of its rival's design quality level, is given by:

19

, (13) jijittTet sdij

cdijij

tTrij

i ≠=++= −− ;2,1,),()()()( )( ωωωω

where the three terms on the r.h.s. of (13) are, respectively:

i) the terminal component, with 0)( =Tijω ;

ii) the cross-durational strategic substitutability/complementarity component, given by

dssXsCsCspsQs

et iQ

tsadjii

j

adjijT

t

tsrcdij

j

i

i )()()()()()(

)( /)( &

−−= ∫ −− ν

ω ;

iii) and the self-durational component, given by:

dssXsh

shsQ

ssQs

sQs

et jjj

j

j

adjjjr

ijj

adjjjT

t

tsrsdij

i )()(1

)()(

))(11)((1)(

)()(

)( )( &

+

−−= ∫ −−

δν

ζν

ω

which is —making firm j more or less of a quality-based threat to firm i—if firm j effectively

enjoys a quality surplus or deficit, i.e., if

0><

))(11)(()( ssQs adjjjrj ν−

<>Q , respectively, with:

- , firm i's adjusted cross-elasticity of demand w.r.t. firm j's design quality, given as a

weighted sum of

adjijν

ji QS /&ν and jQP /&ν , by equation (A.5), with i and j interchanged;

- , firm j's adjusted own-elasticity of demand w.r.t. its quality, with j replacing i in (A.2); adjjjν

- firm i's adjusted marginal cost,

2,1,)(1

)()()(1)())(())(()( 21 =

+

−−+= i

thth

tQtQttQctqctC

ii

i

i

riiiiii

adji δ

ζ , (14)

that is obtained by adjusting c , the unit marginal cost in the usual accounting

sense, downwards or upwards (if Q , respectively) to allow for the intra-firm demand-

side effects of the marginal unit's +ve or -ve contribution to firm i's inertia over ( ; and with

(.))((.))( 21 iiii Qcq +

)()( tQt ri <>

],Tt

- , the cross-subsidy/tax adjustment to firm i's effective margin because of firm j's design

quality-related gains, obtained by substituting

j

i

QtsC /

ν=∆=Φ ,jQ and in equation (12). ijjk ≠= ,

Discussion: (The working paper, available from the second author, has more details.)

Result 3: Law of motion for firm i's marginal valuation of its own inertia level, ζii(t).

Firm i's marginal valuation, at t, of its own inertia level is given by:

20

(15) ,2,1),()()()( )( =++= −− ittTet sdii

cdiiii

tTrii

ii ζζζζζ

where the three terms on the r.h.s. of (15) are, respectively:

i) the terminal component, with firm i salvaging its terminal inertia level at ; 0)( ≥Γ= ihii Tζ

ii) the cross-durational strategic substitutability/complementarity-related component, given by:

dssXsh

shsQsQs

shs

et jjj

j

j

rij

i

adjjiT

t

tsrcdii

ii )())(1(

)()()(1)(

)()(

)( )( &

+

−= ∫ −−

δζ

ηζ

ζ

,

iii) and the self-durational component which reflects the impact of the design quality increment

on the effective contribution margin and effective incremental fixed costs and is given by:

( ) dssXsCsQcsqcspshs

et ih

tsiiiiii

adjiiT

ttsrsd

iii

iii

−+−= ∫ −− )()())(())(()()()(

)( /21)( &ηζ ζ

;

with:

- , firm i's adjusted own-elasticity of demand w.r.t. its inertia level, given by (A.3) in the

Appendix as a weighted sum of

adjiiη

ii hS /&η and ihP /&η , firm i's own-elasticities of market share and

market penetration w.r.t. its inertia level, respectively;

- , firm j's adjusted cross-elasticity of demand w.r.t. firm i's inertia, given as the weighted

sum of

adjjiη

ij hS /&η , the elasticity of firm j's market-share w.r.t. firm i's inertia, and ihP /&η , in (A.6); and

- as the subsidy/tax adjustment to firm i's effective margin, because of its incremental

inertia-related gains, obtained by substituting

i

i

htsC /

η=∆=Φ ,ih and k , in equation (12). i=

Result 3A: The SS/C effect and firm i's marginal valuation of its own inertia level, )(tiiζ .

The sign of the cross-durational component of firm i's marginal valuation, at t, of its own inertia

level depends on whether the product of the three terms within the integral, viz.,

- (which is , for the strategic substitutability/complementarity case, respectively); (.)adjjiη 0

><

- (.)iiζ (which is negative); and

- (which, for the case that firm j faces a quality surplus or deficit, is , respectively), (.)ih& 0<>

is positive or negative, respectively (Table 1 has more details on inertia-related strategic effects).

21

Result 3B: Effective discount rate for firm i's marginal valuation of its own inertia level.

The effective discount rate for inertia related gains, , is given by: ζiir

.))(1)((

)())(1)((1)()( ds

shshsh

shstrtrt

iii

iii

adjiiiii ∫

+

++−−=−τζ

δδηττ

& (16)

Discussion:

i) , the self-durational component that captures the inertia increment's demand-side impact

on firm i's net profitability over , is evaluated as the incremental volume times the existing

contribution margin (with , the unit marginal cost, adjusted only through

an inertia related subsidy or tax, C in the pre- or post-saturation phase, respectively).

sdiiζ

],( Tt

(.))iq +

/ <>i

i

hts

(.))(( 21 iii Qcc

,0

ii) Result 3B implies that the effective rate at which firm i discounts the impact of its inertia gain

/loss on its own future prospects from τ in the future, , can be , its actual discount rate, if

respectively, i.e., if firm i's actions over [

ζiir

],

ir><

(.),(.) ri QQ<> τt serve to augment (deplete) its inertia,

and enable (force) it to become less (more, respectively) myopic than it would normally be.

Further, the multiplicative factor in (16), , serves to amplify the

impact of an inertia gain or loss on firm i's future. While each firm can easily and readily match

its rival's price moves and thus trigger off internecine price-wars that adversely affect both firms'

bottom lines, the firm that opts to invest in design quality and inertia would have a more sustain-

able, longer-lived advantage that enables it to compete more effectively in the future and also to

be more patient about recovering its investment. Each firm's ability or inability to find and retain

a niche not only has numerous elasticity and tax/subsidy implications, but also affects the extent

to which it (feeling more, or less, secure about the future) becomes less or more myopic about

future revenue and cost streams, thus influencing its, and its rival's, prospects for the future.

1(.))1(.)(1 >++ iiadjii hδη

Result 4: Law of motion for firm i's marginal valuation of its rival's inertia level, )(tijζ .

Firm i's marginal valuation of its rival's inertia level at t, evolves according to:

(17) ,;2,1,),()()()( )( jijittTet sdij

cdijij

tTrij

ij ≠=++= −− ζζζζζ

22

where the three terms on the r.h.s. of (17) are, respectively:

i) the terminal component, with 0)( =Tijζ ;

ii) the cross-durational component, which reflects the impact of firm j's inertia gains on firm i's

effective contribution margin, given by:

( ) dssXsCsQcsqcspshs

et ih

tsiiiiij

adjijT

ttsrcd

ijj

iij

−+−= ∫ −− )()())(())(()()()(

)( /21)( &η

ζ ζ;

iii) and the self-durational component, which is given by:

dssXsh

shsQsQs

shs

et iii

i

i

rii

j

adjijT

t

tsrsdij

ij )())(1(

)()()(1)(

)()(

)( )( &

+

−= ∫ −−

δζ

ηζ

ζ

(with self- and cross- as seen from firm j's point of view, and not from firm i's), where:

- , the adjusted cross-elasticity of firm i's demand w.r.t. firm j's inertia level, is given by

equation (A.6), with i and j interchanged;

adjijη

- , firm j's adjusted own-elasticity of demand w.r.t. its quality, with j replacing i in (A.3), adjjjη

- and where C , the subsidy/tax adjustment to firm i's effective margin, on account of firm j's

inertia-increment, is obtained by substituting

j

i

hts /

η=∆= ,jhΦ and in equation (12). ijjk ≠= ,

Result 4A: Effective discount rate for firm i's marginal valuation of its rival's inertia level.

The effective rate, , at which firm i discounts the durational impact of its rival's inertia on its

own future growth or survival prospects, from some instant τ in the future, is given by:

ζijr

.))(1)((

)())(1)((1)()( ds

shshsh

shstrtrt

jjj

jjj

adjjjiij ∫

+++−−=−

τζ

δδηττ

& (18)

Discussion: (The working paper, available from the second author, has more details.)

Result 5: Law of motion for firm i's marginal valuation of its market share, )(tiλ .

Firm i's marginal valuation of its cumulative market share, at time t, is given by:

(19) ,2,1),()()()( )( =++= −− ittTet sdi

cdii

tTri

i λλλλλ

where the three terms on the r.h.s. of (19) are, respectively:

23

i) the terminal component, with as firm i's terminal market share's salvage value; 0)( ≥Γ= iSi Tλ

ii) the cross-durational component, that captures strategic substitutability/complementarity

related implications of firm i's market share gains for its rival, firm j, and is given by:

dssPbabasPsh

shsQsQset jjii

jj

j

j

rij

T

t

tsrcdi

i

+++−

+

−−= ∫ −− )()()(

))(1()(

)()(1)()( 2

1)( &δ

ζλλ

,

iii) and (with the adjusted marginal cost, , given by (14)) the self-durational component,

which reflects the impact of firm i's market share gains on its contribution, and is given by:

)(tC adji

[ ]dssPbabasPsCspet jjiiadjii

T

t

tsrsdi

i )()()()()()( 21)( +++−−= ∫ −− &λ

λ .

Result 5A: The SS/C effect and firm i's marginal valuation of its market share, )(tiλ .

The sign of )(tiλ 's cross-durational component, and the strategic effect of firm i's market share

gains on firm j, depends on whether the product of the three numerator terms within the integral:

♦ )()()( 21 sPbabasP jjii +++−& (which is for , or for the pre- or post-saturation

case, respectively, with the onset of saturation occurring here at

0<>

λττ<>

T≤λτ , given implicitly by:

;;2,1,),()())(1))((()( 21 jijibabadd jjiiji ≠=+++=−Ψ+ λλλ τψτψτψ (20)

♦ (.)ijζ (which is negative), and

♦ (which is , if firm j faces a quality surplus or deficit, respectively), (.)jh& 0<>

is negative or positive, respectively (see Table 2, below, for more details on the strategic effect).

Result 5B: Effective discount rate for firm i's marginal valuation of its market share.

With as the actual discount rate, the effective rate at which firm i discounts its terminal and

durational benefits from its market share gain from some instant τ in the future is given by:

ir

[ dsbabatrtrt jjiiii ∫ ++++−=−τλ ττ (.)(.)(.)(.))()( 2

1 ] (21)

(with each of the , etc., in equation (21), as a function of and ). (.)ia )(),( sQsp kk jikshk ,),( =

Discussion:

First, (21) implies that , i.e., the rivals are myopic in assessing the future benefits of gains ii rr >λ

24

in market share. Intuitively, the adversarial, zero-sum nature of the game in market shares would

seem to make the market share increment (indubitably a valuable intangible asset, over time):

♦ costly to obtain: an increase in volume through a small gain in its market share, costs firm i

more than the same increase through the more collaborative market penetration route;

♦ costly to maintain: equation (6) implies that conserving past market share gains can entail

further competitive strife, through price-cuts or improving quality further—since this tends

to dissipate current profits, a dominant firm may be more inclined to "harvest" this asset than

its other intangible assets, e.g., design quality, inertia, market penetration, etc.; and

♦ more vulnerable, at higher market share levels, if firm j—absent any network externality or

band-wagon effects—chooses to match firm i's pricing- or quality-related moves, at each t or

with a lag: this could help consumers, but jeopardize both players' future prospects because

the industry-wide profit-pie shrinks as a result, while their shares remain roughly unchanged.

Second, the r.h.s. of (20), an implicit equation in )( λτψ , the degree of firm-specific satura-

tion, could imply that at, or even before, T≤λτ , one or both firms, and especially the dominant

one, might choose to either exit or else "harvest" existing market share(s). Alternately, both

firms might optimally "collaborate" in tacitly letting post-saturation prices stiffen, rather than

lower prices to further penetrate an already saturated market, and thus volitionally "ration" the

fixed intertemporal pie. This would also seem to be in step with: i) the other post-saturation

changes in each firm's perceptions of the residual market's attractiveness and in its willingness to

either lessen or increase the intensity of the rivalry (see equations 23 & 24); ii) the rationale,

given below, for letting 0)( <Tiµ , i.e., for assigning a negative salvage value to the terminal

penetration level, , (or a positive salvage value to , the untapped market

potential); and iii) the existing models (e.g., Ghemawat, 1985) of declining or mature markets.

)(TP )()( TPTM −

The impact of saturation on the cross- and self-durational components thus underscores the

25

need to reexamine market share's usefulness as an asset over the entire planning interval.

Result 6: Law of motion for firm i's marginal valuation of the market penetration level, )(tiµ .

Firm i's marginal valuation of the industry-wide market penetration level is given by:

(22) ,2,1),()()()( )( =++= −− ittTet sdi

cdii

tTri

i µµµµµ

where the three terms on the r.h.s. of (22) are, respectively:

i) the terminal component, with as the terminal penetration level's salvage value; 0)( ≤Γ= iPi Tµ

ii) the cross-durational strategic interaction effect related component, given by:

( )

Ψ−−Ψ′+−+−= ∫ −− dssssddsSsS

sXsh

set jiiij

jij

T

t

tsrcdi

i ))(())(1))((()))((1()()()(

)()( )( ψψψζµµ &

&

&

iii) and the self-durational component, which reflects the impact of the incremental industry-

wide penetration on firm i's effective contribution margin, given by

( ) [ ]dssssddsSsSsCspet jiiiadjii

T

t

tsrsdi

i ))(())(1))((())(()()()()( )( ψψψµµ

Ψ−−Ψ′++−= ∫ −− &

Result 6A: Effective discount rate for firm i's marginal valuation of market penetration.

The effective rate, , at which firm i discounts its gains from further market penetration, from

some instant τ in the future to the current time instant, is defined by:

µir

[ ] dssssddtrtrt jiii ∫ Ψ−−Ψ′++−=−τµ ψψψττ ))(())(1))((()()()( 2

1 (23)

where ))()(()( sMsPs =ψ is the penetration level at , and is the firm's actual discount rate. s ir

Discussion:

i) is more likely to hold for a long enough horizon, so that , the residual

market potential, would be salvaged at $ per unit, and not the industry-wide cumulative

sales. The current penetration level thus serves, pre-saturation, as an intangible asset because of

0<Γ iP ))()(( TPTM −

)( iPΓ−

its catalytic effect on potential adopters, and, post-saturation, as an intangible liability.

ii) Notice that unlike which exceeds , the effective rate, r , at which firm i discounts its

gains from further market penetration, can be less or greater than the actual discount rate, i.e.,

λir ir

µi

26

ii rr><µ , corresponding to t , where the industry-specific saturation point µτ

><

µτ is given by:

))(())(1))((( µµµ τψτψτψ Ψ=−Ψ′ , (24)

whose solution is basically akin to the Bass-type critical saturation level.2 Firm i is thus more, or

less, myopic than usual vis-à-vis its gains from further market penetration, depending on whether

the effects of saturation have already become manifest, or not. The self- and cross-durational

terms reflect changes in the firms' perceptions about the residual market's attractiveness. From

the -bracketed terms in and , we see the effective market share attrition rates

falling, or rising, for t , respectively. That is, firm i's marginal valuation of penetration

increases as its own effective attrition rate falls, and decreases (since

)(tcdiµ

µ

)(tsdiµ

τ><

0<ijζ ) as firm j's effec-

tive attrition rate simultaneously drops over [ ],0 µτ , and vice versa over [ ]T,µτ , providing the

cooperation expected between the rivals (vis-à-vis expanding the pie) with a competitive tinge.

5. The Firms' Optimal Pricing and Design Quality Decisions

We now obtain and interpret the firms' optimal control decisions, assuming the firms seek Nash

(1951) equilibria to their open loop game, as in Holt and Mukundan (1972) and Fershtman and

Muller (1984). Following Mangasarian (1966), Seierstad and Sydsæter (1977) and Kamien and

Schwartz (1981), the first order conditions are necessary and sufficient, since the Hamiltonian is

jointly concave in the state and control variables. Using transversality conditions and the mar-

ginal valuation trajectories given above, we solve the first order differential equations in each

firm's control variables simultaneously, and characterize the optimal solutions in what follows.

Obtaining the control trajectories and analysis of the closed loop game are left to future research.

5.1. Nash Equilibrium in Prices

2 With ψ as the saturation level, the solution to: ))(1( ψψψ ba +−=& , which defines the Bass model dynamics, is

given by ])(1[]1[)( )()( tbatba eabet +−+− +−=ψ , yielding ])(,0max[~21 b= ba−ψ as the critical saturation level. We

note that this can also be extracted from equation (24) with Bass's linear saturation function, ψψ ba +=)(Ψ .

27

Result 7: Firm i's optimal pricing decision, pi(t), for the open loop case:

Firm i's optimal price, , is given (as a mark-up on the effective marginal cost) by: 2,1),( =itpi

2,1,)()()()1(

)( // =−−−

= itCtCtCtp ij

i

i

i

SSCts

pts

adjiadj

ii

adjii

i εε , (25)

with:

- , firm i's adjusted own-elasticity of demand w.r.t. price as a weighted sum of the market

share and market penetration elasticities w.r.t. its price (see equation (A.1) in the Appendix); and

adjiiε

- , the adjusted cross-elasticity of firm j's demand w.r.t. firm i's price, obtained as the

weighted sum of the cross-elasticity of firm j's market-share and market penetration w.r.t. firm i's

price, in equation (A.4) in the Appendix.

adjjiε

Firm i's effective marginal cost, , consists of three components

(the first incorporates intra-firm effects; the others adjust for inter-firm effects), viz.,

)()()( // tCtCtC ij

i

i

i

SSCts

pts

adji −−

♦ the marginal unit's design quality related favorable or adverse contribution over [t,T] to the

firm's inertia level is captured on an intra-firm basis by C , firm i's adjusted marginal

cost (equation 14 reveals that , firm i's "actual" unit marginal cost, is

adjusted downwards or upwards through an inertia-related subsidy or tax, for the case that

firm has a current design quality surplus or deficit, i.e., if , respectively);

)(tadji

)( QtQi <>

))(())(( 21 tQctqc iiii +

)(tr

♦ , the subsidy derived (tax borne) by firm i, attributable to the marginal unit's pricing-

related contribution, obtained by substituting

i

i

ptsC /

ε=∆=Φ ,ip and in equation (12); ik =

♦ and lastly, firm i factors strategic substitutability/complementarity into its pricing through:

jijitX

thttC

i

jijadjii

adjjiSSC

tsij

i≠==

<> ;2,1,,0

)()()(

)(/ &

&ζεε

. (26)

Result 7A: The SS/C effect and firm i's optimal pricing decision : )(tpi

From equation (26), it follows that strategic substitutability/complementarity has a subsidy or tax

effect, depending on whether the product of the three numerator terms, viz.,

28

♦ (if firm i's product is a substitute for, or complement to, firm j's, vis-à-vis price, ); adjjiε adj

jiε 0><

♦ ijζ (which is negative); and

♦ (which, for the case that firm j faces a quality surplus or deficit, is respectively); jh& ,0<>

is positive or negative, respectively. (See Table 3, below for details.)

Though pricing is only part of the larger paradigm here, it remains a crucial determinant of

demand for both firms. Instead of using it to posture or signal aggressive intent, our rivals use

pricing to further hone the competitive edge provided by their investments in design quality and

inertia. Foreign rivals, who have banked on market share gains from quality improvements being

more defensible and sustainable, than on price-cutting alone, have succeeded in wresting and

retaining market share in several key U.S. industries. Some U.S. firms, who played the quality

game according to the complete set of rules, instead of petitioning under anti-dumping laws or

banking on a stronger yen, etc., to stay viable, have indeed held their ground in the marketplace.

Specifically, firm i's use of price as a strategic weapon in its efforts to attain and retain

market share gains and as an instrument for market expansion, has a four-fold effect:

♦ Firstly, to reflect the marginal unit's contribution to the firm's inertia level from an intra-firm

perspective, firm i passes on a subsidy or tax to the consumer (if , respectively). )()( tQtQ ri <>

♦ Secondly, firm i, aiming for a larger share of a larger pie in the future, can afford to pass on

a subsidy, C , to the consumer, as long as the marginal unit contributes vis-à-vis market-

share and market-expansion. Post-saturation, as the residual market shrinks in size, this term

could switch in sign, and as a self-imposed tax serve to limit the rate at which firm i (and

firm j, similarly, because of C ) consumes its share of the depleting inter-temporal pie.

)(/ ti

i

pts

)(/ tj

j

pts

♦ Thirdly, as each firm seeks to align its pricing strategy with the changes in its perception of

its relationship with its rival, strategic substitutability/complementarity becomes even more

29

central to the rivalry. C (see Table 3, below) is probably the most crucial determinant

of firm i's optimal price, vis-à-vis its impact on both firms' long term prospects.

)(/ tij

i

SSCts

♦ Lastly, design quality and inertia investments make demand for firm i's product effectively

more elastic and hence more responsive to price cuts. Since the price-elasticity of its market

share (now less vulnerable to attrition, even as accretion effects become more pronounced)

and of market expansion are both favorably affected, both terms in the weighted sum, ,

tend to increase with firm i's design quality and inertia gains. Also, the price-elasticity effect

and the other elasticity and tax/subsidy effects mutually reinforce each other. Hence, by stay-

ing the quality course, the firm can parlay lower prices into even larger volume gains and

thus aim to get a larger share of a larger wedge of the intertemporal pie in the future. Con-

versely, if only firm i chose to neglect design quality and inertia, similar effects—but oppo-

site in direction—could exacerbate its problems with the consumer and hasten its exit.

adjiiε

Finally, if one or more of the terms switch in sign making it imperative for the dominant

firm, say, to focus on harvesting its market share, it can use the same relationship between the

elasticity effect and the marginal cost effects to translate higher prices into revenue gains by

ceding market share and hence sales volume to its rival. In either case, price continues to play a

strategic role, but only as part of the larger design quality/inertia paradigm, and not on its own.

5.2. Nash Equilibrium in the Changes in Design Quality Levels

Result 8: Optimal change in firm i's design quality level, . )(tqi

From the f.o.c. w.r.t. , changes in firm i's design quality level are implicitly given by: )()( tQtq ii&=

. (27) 2,1),())(()())(( 31 ==′+′ ittqctXtqc iiiiiii ω&

Here, discounted future gains from improving quality, captured by the terminal, cross- and self-

durational components of )(tiiω , firm i's marginal valuation of its quality level (cf. equation 11),

serve to exactly offset the quality improvement's total cost impact (c , or ))(()())(( 31 tqctXtq iiiii ′+′ &

30

the sum of the variable and fixed cost components, respectively). If these discounted future gains

were to decline monotonically, i.e., if ],0[,0)( Tttii ∈∀<ω&

q

],0 T

, then one might expect the change in

the firm's quality level to similarly decline from a high of to a low of , over [ . )0(i )(Tqi

0

],0 T

]

ii t →)(ω

i

0>

Tt →

=Γ= iQ

0)( >tqi

Ψ′ ψτ><

)( λτψ

Yet, firm i's demand, , does depend on both firms' prices, design quality and inertia

levels, as well as on the market share and market penetration dynamics. If firm i, burdened with

a quality deficit, were to suppress volume (to avoid dissipating inertia by selling substandard

units), could hold over part of [ and the firm could invest progressively more in

improving design quality. Further, the firm, induced by the demand-side benefits accruing in the

future, could progressively invest more in design quality gains if

)(tX i&

0)( >tqi&

)( >tiiω&

,0 T

holds over part of

. Design quality, an intangible or soft asset, could appreciate in value, unlike hard assets,

e.g., plant and equipment, that depreciate always because of wear and tear, etc., and the capital

gain, rather than loss, could induce the firm to invest more over part of [ , accordingly.

],0[ T

As the residual market becomes increasingly saturated and unattractive, the drop in firm i's

marginal value of its quality level becomes increasingly pivotal (since as ,

however. Hence must hold as the industry matures, irrespective of the q -trajectory's

pre-saturation slope, implying that the firm will gradually invest less in improving quality, as

. For the salvageable terminal design quality case, i.e., if holds, it follows

that must hold ∀ . Even for the non-salvageable case, i.e., if ,

positive durational benefits would ensure that holds for t . In short, firms would

invest in design quality gains over the entire product-life cycle (except at

iQΓ

)(t

Γ iQ

(iiω

)

0

0)( <tqi&

Tt →

qi

)( =ii tω

T<

0)( >t ],0[ Tt ∈ )t

T in the latter case).

5.3. Saturation Related Effects

The time instant associated with the onset of saturation can be characterized in many ways, e.g.,

♦ ψτ is the Bass-type saturation time instant from (5), with for t , respectively; 0)(<>ψ

♦ T≤λτ is the firm-specific saturation time instant from (20), an implicit equation in ;

31

♦ T≤µτ is an industry-specific saturation instant from (24), after which: i) firm i becomes

more myopic ( for ), and more reluctant to saturate the market further; and ii) the

strategic relationship and competitive dynamics change significantly (see Table 3); and lastly

ii rr><µ

µτ><t

♦ T≤Φτ , obtained implicitly by substituting or , etc., in (12) in each case,

could be the instants at which some or all of the C switch from pre-saturation subsidies

to post-saturation taxes, so that holds for (this holds only if terminal market

share and penetration are salvageable, since

jiji hhQQ ,,,=Φ

)(/ tits

Φ

Φ><τt

0)( >Ti

ip

0)(/ <>Φ tC

its

λ and 0(Ti ) <µ , from transversality).

Firm i finds it optimal to thus initially expand the market or limit the rate of consumption of

the depleting pie, i.e., to subsidize or tax the marginal unit, depending on whether or not its

favorable impact on market share outweighs its adverse impact on the shrinking residual market.

Yet, the onset of saturation seems to be a gradually unfolding process, with several critical

saturation-related mile-stones, and not a discrete event (in the traditional Bass sense) at some

, at which the market switches from growing-and-attractive to mature-and-unattractive. T≤τ

6. CONCLUSIONS

This paper sought to analyze how design quality and inertia affect the market share and product

life cycle dynamics in a duopoly. Using 2 control- and 6 state-variables per firm, our analytical

differential game theoretic framework, a blend of Lanchester’s combat- and Bass' diffusion-

dynamics, revealed how the firms': effective marginal costs; effective discount rates; own- and

cross-elasticity coefficients and hence mark-ups; and so on, were adjusted over time and how

each firm’s marginal valuations of its intangible assets/liabilities were affected as a result. These

results and the first order conditions for the pricing and design quality-related control variables,

helped us analyze the broad implications, from the structure-conduct-outcome perspective, of

each firm’s quality and pricing related decisions for its future prospects, in terms of its ability to

effectively anticipate, respond to, and to acquire and retain a competitive edge over, its rival.

32

Financial data often are misleading metrics of strategic health, and our work suggests going

beyond them to measure corporate performance. To return to firm A’s manager's inability to

predict and avert the seemingly inevitable downturn in its E-Lamp business, one could ask: If all

this were already part of conventional wisdom, would she have been as myopic about investing

in design quality and inertia? Would she have ignored the strategic impact of such investments

and of the industry- and market-specific forces, e.g., saturation effects, on both firms’ long term

growth or survival prospects? Would she not have tried: to get a sense of how satisfied firm A's

customers were; to study its competitive cost position; to benchmark its product against firm

B’s; etc., and would these underlying measures not have proved more reliable as prognosticators

of its strategic health than its financial results, particularly in a time of strategic change?

No one, including firm A's manager, can afford see the future only as the past extrapolated in

today’s volatile, globally competitive, high-tech, economic environment. We would like to think

that our insights would have made a difference, and that firm A's manager, thus equipped, might

have: i) been able to "see" how both firms’ optimal control decisions at any instant pervasively

influence firm A’s ability to compete viably and profitably down the road; ii) been less myopic,

than otherwise, about realizing the intangible, deferred value from her investments in quality and

inertia as bottom line dollars; iii) seen the relevance of, and anticipated the impact of, changes in

the marketplace and in firm B’s competitiveness on both firms' intangible assets/liabilities, and

on firm A’s future prospects; and iv) managed firm A, operationally and strategically, as a result,

so as to ensure long term sustainable growth by reappraising the course that it had taken till then,

and by trying harder, for instance, to ensure that it stayed technologically ahead or even abreast

of firm B, instead of letting it get as completely marginalized, as it was otherwise slated to be.

33

Abbreviated Appendix: (Extended version available from second author)

Suppressing time arguments, we first characterize the own- and cross-elasticity effects. As an example,

we derive , firm i's adjusted own-elasticity of demand w.r.t. its price, as a weighted sum of the

elasticities of its market share and the market penetration level w.r.t. its price. Specifically, we have:

adjiiε

,2,1,0)()( // =>

++

+=

∂∂

−= iPSSP

PSPSSP

SPpX

Xp

ii

ipP

ii

ipS

i

i

i

iadjii iii &&

&

&&

&&

& && εεε (A.1)

with ii pS /&ε , the elasticity of firm i's market-share w.r.t. its price, given by:

0)1(21

21

/ >

−

∂∂

+∂∂

+

∂∂

+∂∂

−−=∂∂

−= ii

i

i

ji

i

j

i

i

i

i

i

i

i

ipS S

pa

pb

Spa

pb

Sp

pS

Sp

ii &

&

&&ε (A.1a)

(since ii pb ∂∂ and ij pa ∂∂ are positive, and ij pb ∂∂ and ii pa ∂∂ are negative), and ipP /&ε , the elastic-

ity of the industry-wide market-expansion w.r.t. firm i's price, given by:

0/ >∂∂

+−=

∂∂−=

i

i

ji

i

i

ipP p

ddd

ppP

Pp

i

&

&&ε (since 0<∂ ii pd∂ ). (A.1b)

With the signs of iiiiii QdQbQa ∂∂∂∂∂∂ ,,

ε

ij pS /&

, etc., given in equations (4) and (6), we can substitute

& for in equation (A.1), sans the minus sign, and get (A.2) & (A.3) for & , or firm i's

adjusted elasticity of demand w.r.t. its own design quality level and historical inertia, respectively.

Similarly, we obtain (A.4), (A.5) & (A.6) for the adjusted cross-elasticities

of firm j's demand w.r.t. firm i's price, design quality & inertia level, respectively. Since the elasticities of

firm j's market share (

iQ

ih ip adjiiν adj

iiη

,;2,1,,&, jijiadjji

adjji

adjji ≠=ην

ε ,...), and of the market penetration level (ipP /&ε ,...) w.r.t. firm i's price, etc., are

and > , respectively, their weighted sums can be , implying that product i can be a net substitute

for, or complement to product j, depending on whether or not firm j's market share gains from an incre-

ment in, e.g., firm i's price outweigh the detrimental impact on both firms' market expansion efforts, etc.

0< 0 0><

Section 4: Marginal Valuation Functions.

From the Hamiltonian in equation (10) we can obtain the marginal valuation function trajectories. As an

example, we derive the law of motion for )(tiiω , firm i's marginal valuation of its own quality level.

Law of motion for )(tiiω : Equation (11) follows from ,2,1,)()( =∂∂−= iQtrt iiiiii Hωω& or from:

))(())(()()()()(

)(1)(

)()(1)(

)()(

)()( 21 tQctqctptQttX

thth

tQtQt

tQt

trt iiiiii

adjii

jjj

j

j

rij

i

adjji

iiiii

+−−

+

−−=

νδ

ζν

ωω &&

′−

′−

−

+

−−−− ))(()())(()(

)(1)(

)()(1

1)()(1)( 42/ tQctXtQctCth

tht

ttQtQt iiiii

Qts

ii

iadjii

adjQQ

i

rii

i

i

ir &δν

νζ . (A.7)

34

Laws of motion for )(tijω , )(tiiζ , )(tijζ , )(tiλ & )(tiµ : Similarly, equations (A.8), (A.9), (A.10), (A.11)

& (A.12), which yield equations (13), (16), (19), (22-24) and (25-27)—which, respectively, describe firm

i's marginal valuation of its rival's design quality, its inertia, its rivals' inertia, its market share and the

saturation level—in turn, follow from ,Θ∂Ω H)( ∂−t)( = rt iΩ where & )(),(),(),()( ttttt iijiiij λζζω=Ω &

),(tiµ and & in each case, respectively. )(tSi),(), th j(),( tht iQ j=Θ ,;, jii ≠2,1j =),(tP——————————————————————————————————————

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Review of Economic Studies (1982), 3-13. ——————————————————————————————————————————- Figure 1: Hypothetical Product Life Cycle and Market Share Dynamics for the "E-Lamp."

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8Time after Launch, years

0.%

10%

20%

30%

40%

50%

60%

70%

80%

Cur

rent

Sal

es, m

illio

n un

its p

.a.

Degree of Saturation

Current Industry-wide Sales Degree of Saturation

9

Firm B’sVolume

Firm A’sVolume

36

Table 1: Strategic Substitutability/Complementarity and the Cross-Durational Components of Firm i's Marginal Valuations of Design Quality and Inertia.

)(tadjjiν or

)(tadjjiη

)(tijζ )(th j& )(tcd

iiω or )(tcd

iiζ Net Effect on Firm i's Choice for Investing in Design Quality or Inertia

<0 <0 >0 ↑ (pressure)

Firm i must invest more in quality or inertia to match or exceed firm j's quality surplus related gains (strategic substitutability)

<0 <0 <0 ↓ (relief)

Can afford to invest less in quality or inertia since firm j's quality deficit will hurt it in the long run (strategic substitutability)

>0 <0 >0 ↓ (relief)

Firm i benefits from firm j's market expansion efforts and hence can afford to invest less in quality or inertia (strategic complementarity)

>0 <0 <0 ↑ (pressure)

Must invest more in quality or inertia to offset adverse impact of firm j's quality deficit on market expansion (strategic complementarity)

Table 2: Effect of Strategic Substitutability/Complementarity on the Cross-Durational Component of Firm i's Marginal Valuation of Market Share.

λτ><t )(tijζ )(th j

& )(tcdiiλ Net Effect on Firm i's Market Share Strategy

t<τλ <0 >0 ↑ (pressure)

Firm i must invest more during pre-saturation phase in market share to counter inroads made by firm j through its quality surplus

t<τλ <0 <0 ↓ (relief)

Can afford during pre-saturation phase to strive less harder for market share gains since firm j's quality deficit will hurt it in the long run

t>τλ <0 >0 ↓ (relief)

Given firm j's post-saturation aggressiveness, residual potential unattractive enough for firm i to cede rather than defend market share

t>τλ <0 <0 ↑ (pressure)

Worthwhile to try during post-saturation phase to accelerate firm j's exit or demise, given latter has problems defending its existing share

Table 3: Effect of Strategic Substitutability/Complementarity on Firm i's Marginal Cost and hence Price.

adjjiε )(tijζ )(th j

& )(/ tC j

i

SSCts Net Effect on Firm i's Price

<0 <0 >0 >0 (subsidy)

Reduce price to stave off possibility of losing market share in the long run to strategic substitute, as firm j gains in inertia

<0 <0 <0 <0 (tax)

Can afford to raise price since erosion of firm j's inertia will hurt strategic substitute in the long run

>0 <0 >0 <0 (tax)

Can afford to raise price since strategic complementarity enables firm i to benefit from firm j's market expansion efforts

>0 <0 <0 >0 (subsidy)

Reduce price to offset adverse impact of the loss of inertia for firm j's product (the strategic complement) on market expansion