Opportunity costs and non-scale free capabilities: profit maximization, corporate scope, and profit...

Strategic Management JournalStrat. Mgmt. J., 31: 780–801 (2010)

Published online EarlyView in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/smj.845

Received 20 February 2008; Final revision received 11 January 2010

OPPORTUNITY COSTS AND NON-SCALE FREECAPABILITIES: PROFIT MAXIMIZATION, CORPORATESCOPE, AND PROFIT MARGINS

DANIEL A. LEVINTHAL1* and BRIAN WU2

1 Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.2 Stephen M. Ross School of Business, University of Michigan, Ann Arbor, Michigan,U.S.A.

The resource-based view on firm diversification, subsequent to Penrose (1959), has focusedprimarily on the fungibility of resources across domains. We make a clear analytical distinctionbetween scale free capabilities and those that are subject to opportunity costs and must beallocated to one use or another, thereby shifting the discourse back to Penrose’s (1959)original argument regarding the stock of organizational capabilities. The existence of resourcesand capabilities that must be allocated across alternative uses implies that profit-maximizingdiversification decisions should be based upon the opportunity cost of their use in one domain oranother. This opportunity cost logic provides a rational explanation for the divergence betweentotal profits and profit margins. Firms make profit-maximizing decisions to increase total profitvia diversification when the industries in which they are currently competing become relativelymature. Due to the spreading of these capabilities across more segments, we may observe thatfirms’ profit-maximizing diversification actions lead to total profit growth but lower averagereturns. The model provides an alternative explanation for empirical observations regardingthe diversification discount. The self-selection effect noted in recent work in corporate financemay not be indicative of inferior capabilities of diversifying firms but of the limited opportunitycontexts in which these firms are operating. Copyright 2010 John Wiley & Sons, Ltd.

INTRODUCTION

The resource-based view of the firm has long rec-ognized that firms diversify in order to exploitfirm-specific resources1 for which factor marketsare imperfect (Penrose, 1959; Teece, 1982). AsMahoney and Pandian (1992) note, this argumentis based both upon the availability of resources, inparticular the degree to which there may be slack

Keywords: corporate diversification; firm capabilities;opportunity cost; diversification discount; industrydynamics; resource-based view of the firm∗ Correspondence to: Daniel A. Levinthal, Wharton School,University of Pennsylvania, 3209 Steinberg-Dietrich Hall,Philadelphia, PA 19104-6370, U.S.A.E-mail: [email protected] ‘Resources’ and ‘capabilities’ are used interchangeably in thispaper.

resources in the firm’s current market context, aswell as the implications of the nature of the firm’sresources for the direction of possible diversifi-cation efforts. The diversification literature alongthe lines of the resource-based view has largelyfocused on this latter point highlighting the fungi-bility of resources, or the degree to which the valueof resources may be diminished as resources areleveraged in settings more distant from the origi-nal context in which the resource (e.g., brand-nameor technical capability) was developed (cf. Mont-gomery and Wernerfelt, 1988). The fungibility ofresources is the basis for the explanation as to whyrelated diversification tends to outperform unre-lated diversifications and, in turn, why firms tend topursue more related diversifications (Bettis, 1981;Markides and Williamson, 1994; Montgomery and

Copyright 2010 John Wiley & Sons, Ltd.

Opportunity Costs and Non-Scale Free Capabilities 781

Wernerfelt, 1988; Robins and Wiersema, 1995;Rumelt, 1974).2

Implicitly, and at times explicitly, resources areoften treated as having a scale free property in thesense that the value of resources is assumed to notbe reduced as a result of the sheer magnitude offirm operations over which they are applied. AsChang (1995: 387) notes, ‘The dominant view indiversification research is that intangible resources,such as technology and marketing skills, encouragefirms to diversify into new businesses in order toexploit the ‘public goods’ nature of information-intensive assets.’ However, as argued in Penrose(1959), the stock of a firm’s resources and thedegree to which they are fungible across productmarkets are both critical in determining diversifi-cation decisions. Many of the resources that mayunderpin a firm’s diversification efforts, such asan effective management team or product devel-opment expertise in a particular domain, have thefeature that they are subject to opportunity costs.At any point in time, these resources must be allo-cated among alternative activities and the use ofthese resources in one activity precludes their usein other settings. While some resources, such as abrand name or patent, may have a public good-likequality, most firm resources or capabilities do not.3

The most familiar example in the business settingis a firm-specific management team (Slater, 1980).While a superior management team can improvethe productivities across all segments, the teamalso has to allocate its limited time and attention(Rosen, 1982).

This issue of the need to allocate capabilitiesacross markets and, as a consequence, the link-ing of diversification efforts to demand conditionsin alternative markets is highlighted by Chandler(1969), who maintains that it was the decline inproduct market activity in the late 1920s and the1930s that precipitated the enormous growth indiversification of industrial firms in the UnitedStates. Chandler (1969: 275) submits that firmssuch as DuPont and General Electric that ‘had

2 It is also important to note that owning a more fungible resourcedoes not necessarily lead to competitive advantage, since it mightbe in more abundant supply (Montgomery and Wernerfelt, 1988)or attract more competition (Adner and Zemsky, 2008).3 Even a resource such as a brand name may not be a pure publicgood as the application of a brand name for one product mightimpact its value in another. Thus, when Gucci wildly appliedits brand name to a number of lower-end products in the 1980s,the value of the brand was argued to have been reduced (Aaker,2004).

accumulated vast resources in skilled manpower,facilities, and equipment’ were under great pres-sure to find new markets as their existing onesceased to grow.

More generally, as Mahoney and Pandian (1992)point out, there is an important line of inquiry run-ning from Uzawa (1969), Chandler (1969; 1977),Rubin (1973), Slater (1980), and Teece (1982) thattakes onboard Penrose’s (1959) concern for thedynamics of resource accumulation by the firm andthe implications of these dynamics for diversifica-tion efforts. In a similar vein, Helfat and Eisenhardt(2004) suggest that firms may reallocate resourcesacross domains over time in order to achieve, whatthey term, ‘inter-temporal’ scope economies.

However, the research literature has not beenclear about the distinction between scale free capa-bilities and those capabilities that must be allocatedamong alternative uses. This distinction is criti-cal because we believe that it is this latter classof capabilities that capture the essence of Pen-rose’s (1959) arguments regarding diversification.If capabilities are all scale free, as is often implic-itly assumed in the literature, issues of opportunitycosts and resource allocation are inconsequential,since scale free capabilities can always be lever-aged in other areas and hence will always have‘excess’ capacity. Thus, it is only resources subjectto an opportunity cost that affect how resourcesshould optimally be allocated.

In this sense, we are making an analyticalreturn to the original sensibility of Penrose’s(1959) capability-based perspective on diversifica-tion. However, it is important to note that Pen-rose (1959) focused on the limit case in whichfirms had ‘excess’ capabilities. This excess derivesfrom two possible sources. One stems from thefact that some resources constitute discrete invest-ments, such as a physical plant. If a manufac-turing facility is not being fully utilized in theproduction of the firm’s current products, thensuch a facility can be a free resource that can beapplied, in part, to other means. The other sourceresults from managerial learning, which enables agiven managerial team to handle a greater rangeof responsibilities over time. Yet, resources, par-ticularly human capital-based resources, tend to befully allocated to particular tasks and initiatives atany point in time. In that sense, such resourcesare not in ‘excess.’ However, there still remainsthe question of what is the best allocation of thetime of a sales force, product development team, or

Copyright 2010 John Wiley & Sons, Ltd. Strat. Mgmt. J., 31: 780–801 (2010)DOI: 10.1002/smj

782 D. A. Levinthal and B. Wu

top management group based on their opportunitycosts.

In addition to extending Penrose’s (1959) notionof excess resources to the more general notion ofopportunity cost, we incorporate a greater consid-eration of the role of the demand environment ininfluencing the opportunity costs associated with afirm’s resources. The emphasis in Penrose’s (1959)work was on the internal growth of resources asthe source of excess capacity in the context ofdemand environment that is implicitly assumed tobe static. We examine how the dynamics of thedemand environment influence the allocation ofa firm’s resources, its diversification efforts, andmeasures of performance. Specifically, since thecriteria of carrying out an activity are based uponthe opportunity cost of applying capabilities in onedomain or another (Rubin, 1973; Slater, 1980), acomplete account of excess capacity of capabil-ities should take into account not only internalgrowth in firm-specific capabilities but also thechange in external opportunities across differentmarkets. Underutilized capacity becomes availablewhen the growth opportunities in the current mar-ket cannot keep pace with the internal growth ofcapabilities. The maturity of the current market rel-ative to other potential markets could either reducethe value of applying non-scale free capabilities inthe current market or raise the opportunity cost ofnot applying some of these capabilities in relatedproduct markets.4 It is in this sense that resourcesbecome ‘underutilized’ or ‘excess.’ Alternatively,if the current market continues to offer sufficientlyfavorable opportunities, it will not be economi-cally rational to divert non-scale free resources intoother industries as long as there is any imperfectfungibility in the value of capabilities when appliedto other domains.

Building on these issues concerning firms’ inter-nal resource base and their external product marketenvironment, we develop a basic economic modelthat provides a rational explanation of firms’ diver-sification behavior in trading off profit margins forcorporate growth. Largely ignored by the researchliterature is the fact that profit-maximizing diversi-fication decisions imply that firms seek to increase

4 The relative maturity of the current market could arise eitherfrom the decline of the current market or from the fast growthof other markets. An example of the former case is the defenseindustry after the mid-1980s (Anand and Singh, 1997), while anexample of the latter case is the mature desktop PC market incomparison with the rapidly growing handheld device market.

total profit but not necessarily their profit mar-gin or market-to-book value, with the latter twomeasures being among the more common perfor-mance measures used in the diversification liter-ature (Palich, Cardinal, and Miller, 2000). Firmsmake rational decisions to increase total profit viadiversification when the industries in which theyare currently competing become relatively mature.In this process, however, firms need to allocatetheir non-scale free resources away from the cur-rent business to the new one. Due to the spreadingof these capabilities across more segments, we mayobserve that firms’ profit-maximizing diversifica-tion actions lead to total profit growth but loweraverage returns. In a similar vein, Montgomery andWernerfelt (1988) show that a wider level of diver-sification can lead to lower average rents (Tobin’sq) due to the imperfect fungibility of firm-specificfactors. We find that the decline in average returnsmay arise from the reallocation of capabilities tonew product markets, even in the absence of anyimperfect fungibility of firm-specific capabilities.

We first examine this model in a simple Bertrandsetup in which we demonstrate the basic resultregarding the implication of profit maximizingdiversification on profit margins. We then expandthis to a Cournot model that allows for a moreexplicit treatment of competition. In the Cournotsetting, we can generate cross-sectional results inwhich a firm with inferior capabilities remainsfocused on a single product market and earnshigher profit margins, but less profit, than its morecapable competitor. We also find that as firmsbecome more asymmetric in their capabilities, thenecessary size of the new product market to elicitdiversification on the part of the more capablefirm rises. With greater asymmetry in capabilities,the original market in which both firms competebecomes more attractive, thereby diminishing theincentive to diversify. Thus, the Cournot resultsallow us to expand the considerations of competi-tive effects and also demonstrate the robustness ofour original results to those effects.

The remainder of the paper is organized as fol-lows. The next section further develops the notionof non-scale free capabilities that must be allo-cated among alternative uses and its contrast withscale free capabilities. We then set up and analyzethe formal model by linking capability-based argu-ments regarding diversification and the demandconditions in the markets in which the firm does



and may participate. After developing some gen-eral results under Bertrand and Cournot competi-tion for the relationship among diversification deci-sions, firms’ capabilities, and profit margins, weengage in a numeric analysis of the Cournot modelto further develop the implications of competitiveforces on both diversification decisions and profitmargins. Finally, we discuss the broader implica-tions of these results.

OPPORTUNITY COSTAND DIVERSIFICATION

Table 1 illustrates the contrast between what weterm scale free capabilities and those resources thatare congestible and require allocation to distinctpurposes. In addition to this dimension by whichcapabilities may differ, there is the more traditionalissue of fungibility, or the range of activities overwhich a resource or capability may be applied.The most restricted sort of resources resides in thelower left cell. Highly specific human or physicalcapital not only has the property that its specificitynarrows the domain of activities over which theresource can be applied, but also that its use in oneactivity constrains its possible use in other activi-ties. The cell in the upper left again offers humanand physical capital examples, but in this case thecapital is applicable over a wider domain of activ-ities. While an auditor may not be usefully appliedin a marketing function, he or she can apply theirauditing expertise to a wide variety of enterprises.Similarly, power generation equipment is a generalpurpose capital infrastructure that could be appliedto an enormous range of uses. However, per theissue of opportunity cost, it is important to notethat the use of auditors in one engagement restrictstheir possible use in another; and the application

Table 1. Dimensions of capabilities

Non-scale free (positiveopportunity cost)

Scale free (zeroopportunity cost)

Lowfungibility

Highfungibility E.g., brand-name;

computeroperating system

E.g., personnelwith specifictechnical expertise;steel plant

E.g., team ofauditors; powergenerationequipment

E.g., patent;customerrelationship

of a certain magnitude of kilowatts to one purposereduces the kilowatts available for another. In theright hand side, the resources are not subject toopportunity cost. There are no inherent constraintson how many goods or services can bear a commonbrand name. Nor does the use of a computer pro-gram on one machine preclude its use on another.5

While the fungibility of a brand name or com-puter operating system is not limitless, the rangeof application is generally greater than either a spe-cific piece of intellectual property or the range ofuses of a particular customer relationship.

The extant research literature on diversificationhas tended to focus on scale free capabilities, suchas technical know-how and reputation, which leadto economies of scope or synergies in the diversi-fication process because they ‘display some of thecharacteristics of a public good in that it can some-times be used in many different non-competingapplications without its value in any one appli-cation being substantially impaired’ (Teece, 1980:226). The recognition of scale free capabilitieshas had a profound influence on both academicresearch and industry practice, since it highlightsthe role of knowledge and competence as strate-gic assets (Winter, 1987). Winter and Szulanski’s(2001) study of replication processes provides aparadigmatic example of a scale free capabil-ity, defining the Arrow core as the informationalendowment a firm extracts from an original settingthat can be replicated to other settings. The distinc-tive property of such information-like resource isthat ‘unlike any resource that is rivalrous in use, aninformation-like resource is infinitely leverageable. . . it does not have to be withdrawn from one useto be applied to another’ (Winter and Szulanski,2001: 741).

The critical constraint in the application of scalefree resources is not by definition the scale of theoperations over which they are applied, but ratherthe scope or range of their applicability. Indeed,the issue of resource relatedness and fungibilityis arguably the most studied question in corporatestrategy. Rumelt (1974), in a pioneering exami-nation of this issue, showed that firms pursuingrelated diversifications outperform those pursuinga strategy of unrelated diversification. This basic

5 The producer may restrict the replication of the programthrough site licenses or copy restrictions, but clearly such con-straints are to address issues of value appropriation and are notrelated to technological constraints to the application of a givenprogram to multiple applications.



finding has been reconsidered with a variety ofdifferent measures of relatedness, but the generalresult has stood up (Bettis, 1981; Markides andWilliamson, 1994; Montgomery and Wernerfelt,1988; Robins and Wiersema, 1995).

While Penrose’s (1959) emphasis on a firm’sstock of capabilities as a basis for diversificationhas played a secondary role to the consideration ofthe fungibility of resources, there has been someawareness in the literature that there may be oppor-tunity costs associated with the use of resources. Inhis brief discussion on the limits to diversificationeconomies, Teece (1982: 53) suggests that, ‘. . .know-how is generally not embodied in blueprintsalone; the human factor is critically important intechnology transfer. Accordingly, as the demandsfor sharing know-how increase, bottlenecks in theform of overextended scientists, engineers, andmanagers can be anticipated.’ Recent empiricalwork in both finance and management has pro-vided suggestive evidence of the opportunity costof allocating resources from one product domainto another. Schoar (2002) finds that after a firmdiversifies into a new industry by acquiring a plant,the incumbent plants will incur a decrease in pro-ductivity, while the acquired plant increases pro-ductivity. Similarly, Roberts and McEvily (2005)find that entering a new pharmaceutical prod-uct market reduces a firm’s performance in itscurrent markets. Finally, Hitt et al. (1991: 695)observe that acquisitions tend to reduce both theextent and productivity of firms’ research anddevelopment (R&D) investments, suggesting thatthe ‘resources remaining for managerial alloca-tion may become constrained, causing managersto forgo other investment opportunities.’

Consider the following example of these argu-ments. As the strongest player in the microproces-sor industry, Intel has been experiencing sluggishgrowth in the PC microprocessor market due tosaturated demand and increasing competition fromAdvanced Micro Devices. In order to spur growth,Intel has sought to extend its reach beyond thePC microprocessor industry into mobile phonesand consumer electronics. The maturity of the PCmicroprocessor market has made the opportunitiesof using its capabilities in other industries moreattractive, and, correspondingly, the opportunitycost of staying focused has risen. At the sametime, diversification requires Intel to allocate itsscarce resources into these new segments. Con-sequently, our theory would predict that, on the

whole, Intel’s diversification efforts will increasesales and total profit. However, its average returnwill decline, reflecting both the shifting away offirm-specific resources from the development andmanufacturing of microprocessors for PCs and thepossible reduced efficacy of these same resourcesin the related product markets into which the firmis diversifying.6

Based upon the above reasoning, we developthe following arguments regarding diversificationefforts. First, it is important to distinguish betweendiversification efforts based on scale free and non-scale free capabilities. A scale free resource, suchas brand name, faces limits on the breadth of itsfungibility (i.e., how broadly fungible is a givenbrand name) but not on its extent of application(i.e., the number of markets in which a given brandcan be applied for a given level of fungibility).In contrast, the application of those non-scale freecapabilities is driven by the logic of opportunitycosts. On the margin, is the greatest value of thesefirm-specific capabilities realized within the cur-rent product market context or in diversifying toa new context? This opportunity cost is, in turn,importantly affected by the size, growth, and com-petitive conditions in alternative product markets.Thus, when there are multiple segments, the rangeof diversification activity is constrained by the totalstock of capabilities. This analysis supplements theinsight in the strategy literature that the imperfectfungibility of scale free capabilities restricts cor-porate scope (e.g., Montgomery and Wernerfelt,1988).

This analysis also provides new insights intothe diversification discount, the observation thatdiversified firms tend to have a lower valuation,typically measured as the relationship betweenmarket value to book value or Tobin’s q, thanan amalgamation of an equivalent set of focusedfirms (Berger and Ofek, 1995; Lang and Stulz,1994). Agency theorists suggest that diversifica-tion destroys value for reasons such as managers’empire building behavior that aims to increasetheir own status, power, and pecuniary compensa-tion (e.g., Jensen, 1986). Recently, however, therehas been a growing literature in the corporate

6 Of course, another basis for the decline in average return isthe shift from a market in which Intel has a dominant positionto markets that may be more competitive; however, the factthat Intel is entering these more competitive, but more rapidlygrowing, markets is further testimony to the need to reallocatenon-scale free capabilities.



finance field suggesting that a diversification dis-count arises even when firms are value maximiz-ers. Econometrically sophisticated analyses of theprofitability of diversified firms (e.g., Campa andKedia, 2002; Villalonga, 2004) indicate that thereis something systematically different about firmsthat diversify. It is this endogenous selection intothe act of diversification, rather than diversificationper se, that leads to diversification discount:

. . . the failure to control for firm character-istics that lead firms to diversify and be dis-counted may wrongly attribute the discountto diversification instead of the underlyingcharacteristics. For example, consider a firmfacing technological change, which adverselyaffects its competitive advantage in its indus-try. This poorly performing firm will trade ata discount relative to other firms in the indus-try. Such a firm will also have lower oppor-tunity costs of assigning its scarce resourcesin other industries, and this might lead it todiversify. If poorly performing firms tend todiversify, then not taking into account pastperformance and its effect on the decisionto diversify will result in attributing the dis-count to diversification activity, rather thanto the poor performance of the firm (Campaand Kedia, 2002: 1732).

In existing analytical explanations of this empir-ical finding that, controlling for endogeneity indiversification behavior, there is no diversificationdiscount, the act of diversification is interpreted asa ‘signal’ that the firm has relatively few ex antecapabilities and is diversifying due to the corre-spondingly low rates of return in its initial markets(e.g., Gomes and Livdan, 2004). In the presenceof diminishing returns to production, firms withlower productivity will reach their optimal sizein the incumbent segment at a lower size levelthan those firms with higher productivity and, asa result, firms with lower productivity are morelikely to diversify.

We agree with these recent empirical findingsthat there is something systematic about thosefirms that ‘sort’ themselves into a positive diver-sification decision. However, the above analyticalexplanation is not fully consistent with the well-evidenced proposition in the strategy field thatfirms with more relevant capabilities (R&D or mar-keting capabilities) tend to enter a new field earlier

and perform better (e.g., Helfat, 2003; Klepper andSimons, 2000; Mitchell, 1989). In contrast, in thespirit of the long-standing treatment of diversifica-tion in the strategy literature, we suggest that the‘something different’ is not that these firms are a‘bad type’ and are lacking in capabilities. Rather,these are firms with relatively superior capabilities,and the bad ‘signal’ may be a statement about themarket contexts in which these firms are operating,such as demand maturity, rather than a statementabout the firm’s relative lack of capabilities.7 In thepursuit of the best use of the firm’s non-scale freeresources, there is some allocation of resourcesaway from established markets and, at the sacri-fice of profit margins but not total profits, a shiftof these resources to new markets. Thus, both ourmodel and those developed in the corporate financeliterature are consistent with the empirical findingregarding the diversification ‘discount’; however,the two explanations differ in their predictions asto which firms (more or less capable) are likely tobe more or less diversified.

MODEL STRUCTURE

We model a firm’s diversification decision withregard to two market segments indexed by m

(the initial segment m = I and the new segmentm = N). Production in each segment is describedas Qm = γmtmT km, where γm is the firm’s scalefree capabilities, tm is the share of the firm’stotal non-scale free capabilities, T , that must bedivided among activities, and capital km whosereplacement cost per unit is r , which reflects thecurrent market value of capital. For the sake ofsimplicity, we assume that firms are endowedwith a particular capability stock T and do notconsider the cost of developing this, but onlythe opportunity cost of how it is applied acrossdifferent market segments within the firm. Theamount of km needed to produce Qm, given tm,

7 The recent treatments of the diversification discount such asCampa and Kedia (2002) and Villalonga (2004) do control forindustry demand conditions when examining the self-selectioneffect of the act of diversification on the so-called diversificationdiscount. However, due to data limitations, such work must relyon measures of industry at the four-digit level of the StandardIndustrial Classification (SIC) code. This level of aggregationmasks considerable diversity of demand environments at theproduct level. We elaborate on this issue in the discussionsection.



is therefore Qm

γmtmT, with total cost rQm

γmtmT= cm ×

Qm, where

cm ≡ r

γmtmT(1)

Note that (i) scale free capabilities, γm, and non-scale free capabilities, T , are firm specific andsubject to imperfect input markets (Teece, 1982);(ii) the provision of more capabilities reduces bothtotal and marginal cost since it substitutes for thepurchased capital; (iii) cm is decreasing in tm andincreases to infinity as tm approaches zero; and(iv) we specify the following relationship betweenscale free capabilities in the two markets: γN =(1 − δ)γI . This relationship implies that scale freecapabilities, γI , are not perfectly fungible, andthat the effectiveness of scale free capabilities, γI ,diminishes by a factor δ(0 ≤ δ < 1) when γI isapplied to the new segment N .8

It is important to relate this production functionand associated cost function to our underlyingargument regarding scale free and non-scale freecapabilities. Our focus is on the allocation of non-scale free capabilities across market contexts. Thiscapability could be a product development teamthat might be spread across multiple initiatives,as highlighted in Christensen and Bower’s (1996)work on the disk drive industry, the allocationof scarce production capacity as in Burgleman’s(1994) work on Intel, or the attention of the topmanagement team (Rosen, 1982). For the sake ofsimplicity, we are not addressing this allocationwith respect to the scale of activity within a givenbusiness unit. Thus, a business unit is assumedto be able to scale up its activity at a constantmarginal cost. This simplification follows on theworks of Klepper (1996) and Lippman and Rumelt(1982) on industry evolution, which, in order tohighlight the effect of across-firm heterogeneity,postulate a production technology with a constantmarginal cost of production. This assumption ofconstant return to production scale also allows usto demonstrate that diversification can arise from

8 As we point out in Table 1, non-scale free capabilities are alsosubject to the issue of imperfect fungibility, but the distinctivefeature of non-scale free capabilities is that they must be allo-cated across alternative uses based on opportunity costs. In orderto highlight the issue of allocation, we assume that non-scale freecapabilities are perfectly fungible; however, so that the modelmore closely corresponds to Montgomery and Wernerfelt (1988),we allow for the scale free capability to have imperfect fungi-bility across alternative uses.

the change in demand conditions in the absence ofdiminishing return to production scale, in contrastto recent work in corporate finance (cf. Gomes andLivdan, 2004; Maksimovic and Phillips, 2002) inwhich diminishing returns to production scale actsas an underlying driver for diversification.

We denote the demand and price for a givenfirm’s product in segment m as qm and pm respec-tively. Should the firm engage in both activities,its profit is qI (pI − cI ) + qN(pN − cN) or qI (pI −

rγI tI T

) + qN(pN − rγNtNT

). Assuming the opti-mal tI and tN are both strictly positive, the firm’sproblem is denoted as the following equation:

max{(pIqI + pNqN) − r(qI

γI tIT+ qN

γNtNT)|tI

+ tN = 1,r

γNtNT≤ pN, and

r

γI tIT≤ pI } (2)

Note that it may not be optimal for the firm toengage in both activities simultaneously. Further-more, if it is optimal for the firm to be diversifiedinto both activities, then the optimal allocations ofnon-scale free capabilities t∗

I and t∗N are bounded

away from zero. That is, to get into the new activ-ity requires a discrete reduction in the capabilitiesemployed in the initial activity. Indeed, depend-ing on the magnitude of the firm’s capabilities andthe market price, a firm may not be competitivelyviable in a given market.

This above setting offers a general frameworkfor analyzing firms’ diversification decisions andthe associated performance effects based on (non-scale free) capability allocation. In the followingsections, we examine the diversification problemcharacterized in this general framework using thetwo most widely used market models: the Bertrandmodel and the Cournot model, which usefullycomplement each other. The Bertrand analysisallows us to develop the basic insights regardingthe allocation of scarce capabilities across lines ofbusiness and its implications for firm profitabil-ity. Introducing Cournot competition allows us toexamine more directly the impact of competitiveinteraction among firms. However, the Cournotanalysis in this context does not lend itself to afull closed form analytical solution as the reactionfunctions are neither continuous nor monotonic.Thus, we analytically derive the general resultsregarding firm diversification decisions, and thenperform a numerical analysis to characterize morespecific properties regarding profit margins and the



pattern of diversification. Further, we analyze awide array of parameter values to both explorefurther insights of the model and the robustnessof our results.

ANALYSIS: BERTRAND MODEL

Each market is specified to be a Bertrand duopoly,where whenever the firm allocates non-scale freecapabilities such that its marginal cost is lower thanthat of the competitor, the firm serves the wholemarket with demand. The demand side of segmentm consists of sm consumers. Thus, the firm thatcaptures the market produces quantity qm = sm,charges a price of pm equal to the competitor’smarginal cost, and has profits sm(pm − cm), wherecm = r

γmtmTis defined as in Equation 1.

In this Bertrand setting, a given firm diver-sifies into the new segment when demand con-ditions are such that the maximization problem(Equation 2) has an interior solution (t∗

I , t∗N ). The

total profit associated with the diversification strat-egy is sI (pI − r

γI t∗I T

) + sN(pN − rγNt∗

NT), and the

total sales associated with the diversification strat-egy is pIsI + pNsN . Therefore, the profit marginassociated with the diversification strategy, whichwe denote as π ∗, is

π ∗ = sI

pI sI + pNsN

(pI − r

γIT t∗I

)

+ sN

pI sI + pNsN

(pN − r

(1 − δ)γIT t∗N

) (3)

In contrast, when firms focus on the initial

segment, the profit margin issI (pI − r

γIT)

pI sI, while

when firms focus on the new segment the profit

margin issN(pN − r

(1 − δ)γIT)

pNsN. Therefore, the

weighted average of the profit margin of the twofocus strategies, which we denote as πw, withrelative sales pIsI

pI sI + pNsNand pNsN

pI sI + pNsNas

the respective weights, is

πw = sI

pI sI + pNsN

(pI − r

γIT)

+ sN

pI sI + pNsN

(pN − r

(1 − δ)γIT) (4)

πw is a standard benchmark used to comparethe performance of diversified and focused firms

(Berger and Ofek, 1995; Lang and Stulz, 1994).The relation between the profit margin of thediversification strategy (Equation 3) and that offocused strategies (Equation 4) is characterized byEquation 5.

π ∗ = πw − 2

√sI sN

pI sI + pNsN

r√(1 − δ)γIT

(5)

As a result, we have the following proposition.

Proposition 1: The profit margin of the profit-maximizing diversification strategy for a firmwith capability stock T is lower than theweighted average of the profit margin of focus-ing this capability stock T in each of the twomarkets. The difference is characterized byEquation 5.

(See Appendix 1 for a proof.)

Proposition 1 allows us to identify the sources ofthe declining profit margin associated with diver-sification by decomposing the profit margin of thediversification strategy into two parts: the weightedaverage of the two focus strategies and the discountdue to the spreading of non-scale free capabilities.The second term of πw in Equation 4 indicateshow the profit margin of the diversification strategydeclines with δ, the degree to which the effective-ness of scale free capabilities diminishes in the newsegment. This is the case studied in Montgomeryand Wernerfelt (1988), who account for a declinein profit margin or Tobin’s q as firms apply, whatwe term, scale free capabilities in increasingly dis-tant markets.

Adding to this consideration of imperfect fun-gibility, the second term of Equation 5 capturesthe discount due to the spreading of non-scalefree capabilities. Therefore, Equation 5 provides amore complete picture of the diversification dis-count from the resource-based view by incorpo-rating the effect of non-scale free resources thatneed to be allocated across applications. Proposi-tion 1 suggests that the existence of a diversifi-cation discount does not necessarily result fromagency behavior that deviates from profit maxi-mization. The spreading of non-scale free capabil-ities across more applications based on opportunitycosts implies that the profit margin will be ‘sacri-ficed’ to some extent in the pursuit of total profitmaximization.



In parallel to this discussion of the impact ofdiversification on the firm’s profit margins, we canalso show that rational diversification efforts leadto lower Tobin’s q, a widely used measure in theempirical analysis of diversification performance.Given that there is no short-run variable productioncost (e.g., depreciation and labor) in the model, themarket value of the firm in this stylized one-periodmodel can be represented by operating profit pmsm,which is the earnings stream that is generatedfrom the firm’s capital stock of r

sm

γmT. Therefore,

Tobin’s q, defined as the market value of thefirm divided by the replacement cost of capital(Lindenberg and Ross, 1981; Winter, 1995), canbe represented as q = pmsm

rsm

γmT

.

We denote the Tobin’s q associated with thediversification strategy as qDIV and the Tobin’sq associated with the weighted average of twofocused strategies as qFOC . Analogous to the rela-tionship in Equation 5, the ratio of qFOC andqDIV is

qFOC

qDIV

= 1 +2√

sI sN

1 − δ

sI + sN

1 − δ

(6)

As a result, we have the following corollary.

Corollary 1: The Tobin’s q of the profit-maximi-zing diversification strategy for a firm with capa-bility stock T is lower than the weighted averageof the Tobin’s q of focusing this capability stockT in each of the two markets. The difference, inratio terms, is characterized by Equation 6.

(See Appendix 2 for a proof.)

Therefore, whether the average return to capitalis measured as economic profit margin(Equation 5) or as Tobin’s q (Equation 6), we seethat profit-maximizing diversification leads to areduction in these common measures of firm per-formance due to the spreading of firm resourcesover multiple product markets.

ANALYSIS: COURNOT MODEL

The prior analysis examined the profit maximizingtrade-off between total profits and profit marginsbased on a Bertrand model. This analysis models

the diversification of a focal firm that faces com-petitive constraints, but does not explicitly modelan endogenous competitive process. In this section,we address this gap by introducing a model ofCournot competition. We develop this analysisusing the same analytical structure for firms’ capa-bilities and their associated cost functions, but theCournot structure requires us to more fully specifya demand function.

Demand in the initial market is specified as:pI = aI − bI (q1I + q2I ), where q1I and q2I are theoutput of firms 1 and 2 in the initial market.Similarly, in the new market, demand is: pN =aN − bN(q1N + q2N), where q1N and q2N are theoutput of firms 1 and 2 in the new market.

As before (Equation 1), firms are characterizedby their non-scale free capability T1 and T2, respec-tively. However, we set the value of scale freecapabilities γ for the two competitors to be thesame as we are focusing on the role of non-scalefree capabilities. The marginal production costs forfirms 1 and 2 in market I and N are given by c1I =

1γ t1I T1

, c2I = 1γ t2I T2

, c1N = 1(1 − δ)γ (1 − t1I )T1

,

and c2N = 1(1 − δ)γ (1 − t2I )T2

, where t1I and t2I

are the fraction of non-scale free capability of firms1 and 2 invested on the initial market, respectively,and (1 − δ) is the effectiveness of scale free capa-bilities, γ , when γ is applied to the new segmentN(0 ≤ δ < 1). Thus, firms’ profits in each mar-ket are given by πim = (pm − cim)qim, for i = 1, 2;m = I, N .

The sequencing of the firms’ choice problem isthat firms first choose their non-scale free capabil-ity allocations, characterized by t1I and t2I . Second,given the chosen t1I and t2I , firms choose their pro-duction quantities q1I and q2I in the initial marketand q1N and q2N in the new market. The associ-ated optimization problem can be well specified viabackward induction. However, in this context, theproblem does not lend itself to a general analyticalsolution. Indeed, we will show in the subsequentanalysis that the best response functions that under-pin the equilibrium analysis may, in some cases,be noncontinuous and nonmonotonic.9

9 In this respect, it is worth noting that our model is related tobut distinct from a classic industrial organization model devel-oped by Kreps and Scheinkman (1983), which showed that atwo-stage model of production capacity investment followed byBertrand price competition can be treated in a relatively straight-forward manner as being equivalent to a one-stage model of



The best response curves are not continuous fortwo reasons. First, the optimal allocations of non-scale free capabilities are bounded away from zero,that is, to get into the new activity requires a dis-crete reduction in the capabilities employed in theinitial activity. It would never be optimal for a firmto allocate less than some epsilon to the new mar-ket. The firm must allocate sufficient capabilitiesinto the new market such that it can establish com-petitive viability in this market. Furthermore, thefirm must make sure the gains in the new market towhich these capabilities are being applied is largerthan the losses incurred by allocating these capabil-ities away from the old market. As a result, the bestresponse curve always has a jump from zero at thepoint of diversification to some discrete magnitude.

The second reason that the best response curvesmay not be continuous is that the best responsecurves have small jumps at the point where thecompetitor either makes a discrete move into or outof a product market in response to its competitor’schange in allocation of capabilities across markets.Note that this second reason for the best responsecurves not being continuous is also the reasonwhy the best response curves are not monotonic(see Appendix 3 for an illustration of the aboveargument and a more detailed explanation).

Even in the presence of the analytical challengesidentified above, we are still able to derive thefollowing propositions. To capture the change inrelative demand, without loss of generality, wehold constant the size of the initial market aI

Cournot competition. The first-stage decision variable is pro-duction capacity in Kreps and Scheinkman’s (1983) model, butin the context of our model it is the allocation of capabili-ties. The allocated capabilities determine the level of averagecost in a given market (see Equation 2). Also, in contrast toKreps and Scheinkman (1983), we assume there is no con-straint to production capacity and thus production capacity canscale up simultaneously with the output decision. Under thisspecification, the sequencing of the firms’ choice problem isthat firms first choose their non-scale free capability alloca-tions, which in turn determine firms’ cost level in each market.Second, given the chosen capability allocations and the associ-ated cost levels, firms either choose their prices, if under theBertrand competition, or choose production quantities, if underCournot competition. The firm’s optimization problem can thenbe analyzed via backward induction. In sum, our model differsfrom Kreps and Scheinkman’s (1983) model in two importantways. First, Kreps and Scheinkman’s (1983) model examinesonly one market, while we model the allocation of capabilitiesacross multiple markets along with diversification decisions. Sec-ond, Kreps and Scheinkman’s (1983) model examines Cournotcompetition assuming homogeneous firm capabilities, while ourmodel examines how capabilities in each market are endoge-nously determined prior to Cournot competition.

and only vary the size of the new market aN .Let P1 ≡ P(aN, T1, T2, t1I , t2I ) be firm 1’s profitfunction given the new market size aN , firm 1’scapability T1, the competitor firm 2’s capability T2,firm 1’s allocation t1I , and the competitor firm 2’sallocation t2I . Similarly, P2 ≡ P(aN, T2, T1, t2I , t1I )

represents firm 2’s profit function given the newmarket size aN , firm 2’s capability T2, the com-petitor firm 1’s capability T1, firm 2’s allocationt2I , and the competitor firm 1’s allocation t1I . Weuse t∗

1I and t∗2I to denote firms’ allocation in equi-

librium. Given an initial setting (aN = 0) in whichboth firms compete in the initial market, we areable to state the following propositions:

Proposition 2a: For i = 1, 2, there exists anaN > 0, η > 0, and ε > 0, such that, whenthe size of the new market aN gets sufficientlylarge but not too large ( aN < aN < aN + η)and the two firms’ capability asymmetry issmall enough (

∣∣Ti − T(3−i)

∣∣ < ε), there existsa pure strategy Nash equilibrium in whicheither firm may first diversify ( t∗

(3−i)I < 1)while the other firm stays focused on the ini-tial market ( t∗

iI = 1).

Proposition 2b: For i = 1, 2, fixing Ti asthe less capable firm’s capability level, thereexists a T > 0,

�aN > 0, and η > 0 such

that, when the size of the new market aN

gets sufficiently large (�aN < aN <

�aN + η)

and when the two firms’ capability asym-metry is sufficiently large ( T(3−i) > T ), thereexists a unique pure strategy Nash equilib-rium in which only the more capable firmwill first diversify ( 0 < t∗

(3−i)I < 1) while theless capable firm stays focused on the initialmarket ( t∗

iI = 1).

Proposition 2c: For i = 1, 2, given aI as thesize of the initial market and T3−i as the morecapable firm’s capability level, there exists aT > 0 such that the firm without sufficientcapabilities ( 0 < Ti < T ) will either focuson the initial market ( t∗

iI = 1) or switch allresources from one market to the other ( t∗

iI =0) in any pure strategy Nash equilibrium.

(See Appendix 4 for proofs.)

These three propositions highlight how firms’diversification decisions differ when the size of the



new market increases, depending on their capabil-ity asymmetry. Proposition 2a highlights the roleof competitive conditions being endogenous onthe reallocation of resources across product mar-kets. In this case, when one firm diversifies totake advantage of the new market opportunity,the current market becomes more attractive as thediversifying firm withdraws capabilities from thismarket. Therefore, there exists an equilibrium inwhich either the more capable firm or the less capa-ble firm diversifies first while the other firm staysfocused on the initial market.10 In addition, it isnot an equilibrium for both firms to start diversify-ing simultaneously as the relative demand changes,even when these two firms have equal capabilities.

However, while Proposition 2a provides a base-line result when firms are relatively symmetric,the core issue that is of primary interest to thestrategy field is when firms are, to a significantdegree, asymmetric in their capabilities. Proposi-tion 2b demonstrates that with sufficient asym-metry in capabilities, there exists a unique purestrategy equilibrium in which the more capablefirm diversifies first. In this case, for the morecapable firm, the need to reallocate capabilities tothe new market due to opportunity costs dominatesthe strategic consideration of waiting for the lesscapable firm to diversify first and thus alleviatecompetition in the initial segment.

Proposition 2(c) indicates that a minimumamount of capabilities are necessary for the diver-sification strategy ever to be the optimal strat-egy, irrespective of relative demand conditionsacross alternative markets. When relative demandchanges, the more capable firm goes through adiversification process; however, without suffi-cient capabilities, the less capable firm may neverbecome diversified. It will focus on the initial mar-ket when the relative market size of the new marketis small. When the relative market size of the newmarket gets larger, the less capable firm may com-pletely switch to the new market but will neversimultaneously engage both markets. This propo-sition further highlights the critical role of non-scale free resources in determining the scope ofthe firm.

10 There could be two pure strategy Nash equilibria in Proposi-tion 2a ((t∗

iI = 1, 0 < t∗(3−i)I < 1)or (t∗

iI = 1, t∗(3−i)I = 0)). How-

ever, in either case, Proposition 2a holds. Either firm can diver-sify first (diversify partly into the new market (0 < t∗

(3−i)I < 1)or completely switch to the new market (t∗

(3−i)I = 0)), while theother firm stays focused in the initial market (t∗

iI = 1).

In the following numerical analysis, we examinemore fully the impact of asymmetry of capabilitieson the diversification decisions and, in particular,the threshold of relative size of the new and theinitial market that triggers diversification. In addi-tion, we examine the impact on profit margins ofdiversification. This analysis not only generalizesProposition 1 to the Cournot case, but it allowsus to examine the cross-sectional implications ofdiversification behavior on the profit margins forfirms with varying capability levels.

NUMERICAL ANALYSISOF COMPETITIVE INTERACTION

For our numerical analysis, we set the baselineparameter values as follows: aI = 10, bI = bN =1, scale free capabilities γ = 1, and the fungibilityof scale free capabilities (1 − δ) = 0.8. In thesubsequent analysis, we highlight the impact ofvarying levels of firm capabilities and market size(aN , the size of the new market). This later variableallows us to capture relative demand maturitybased on the increase in the size of the newmarket aN . 11 Holding constant the new marketand varying the initial market would lead to thesame results.

Implications of numerical investigationfor diversification decisions

In the Cournot analysis, we are able to analyti-cally show that when two firms’ capability asym-metry is sufficiently large and the new marketsize is of sufficient size, the more capable firmwill diversify first while the less capable firmstays focused on the initial market. Building onthis property, we conduct the following numeri-cal analysis to examine how the demand thresholdto diversify changes with the degree of capabilityasymmetry. In so doing, we identify some inter-esting competitive effects of asymmetry in firms’capabilities.

In Figure 1, aN is the demand threshold wherethe more capable firm (with capabilities T1) first

11 Note that while we keep scale free capabilities γ and theirfungibility (1 − δ) in the model to stay connected with theexisting literature, we do not explore these two parameters inthe analysis since the focus of this paper is non-scale freecapabilities. See MacDonald and Ryall (2004: 1330–1331) andAdner and Zemsky (2008) for analytical examinations of scalefree capabilities and the effect of their fungibility on profitability.



Figure 1. Capability asymmetry and demand threshold. This figure is available in color online atwww.interscience.wiley.com/journal/smj

chooses to diversify, while the less capable firm(with capabilities T2) is still focused on the initialmarket. The downward movement along a givencurve captures the ‘capacity effect,’ meaning thatas the focal firm’s own capabilities (T1) increase,its threshold to diversify becomes smaller, or firm1 reaches the threshold to diversify earlier. Theshift in curves from square (T2 = 1.5) to dia-mond (T2 = 0.5) captures the ‘competition effect,’meaning that as the competitor’s capabilities (T2)decrease, the threshold for the more capable firm(with capabilities T1) to start diversifying becomesgreater. This means that the more capable firmdiversifies later (for a larger size of the newmarket), since the current market becomes moreattractive.

Implications of numerical investigationfor profit margin and diversification

Having characterized diversification behaviorunder Cournot competition, we now address thecentral results regarding the impact of diversifica-tion on profit margins. In particular, the numericalanalysis examines whether Proposition 1 regard-ing the trade-off between total profits and profitmargins holds under Cournot competition.

In Proposition 1, under the Bertrand setting, wecompare the profit margin of a diversified firmwith a hypothetical benchmark of the weightedaverage of the firm focusing separately on thetwo markets. However, in the Bertrand setting,since there is essentially a single firm, we can-not make cross-sectional comparisons of firms

with heterogeneous capabilities. The followinganalysis in the Cournot setting addresses thisissue. With the same baseline parameter val-ues of aI = 10, bI = bN = 1, 1 − δ = 0.8, welet the more capable firm have capabilities T1 =2 and the less capable one have capabilitiesT2 = 1.5.

Examining profit margins as the size of the newmarket varies, we see a number of interestingproperties (Figure 2). First, consistent with Propo-sition 1, for both the more and less capable firm,there is a substantial drop in profit margin whenthe relative demand level in the new market passesthe threshold sufficient to elicit diversification andmarket entry. Second, consistent with Proposition2, given the significant heterogeneity in capabili-ties, the first firm to diversify is the more capablefirm. In the interim range of market demand forwhich it is optimal for the high-capability firmto diversify but for the low capability firm toremain focused on the initial market, we see thatthe more capable firm may end up with a lowerprofit margin than the less capable firm.12 A firm’srational diversification decision is driven by itspursuit of profit maximization rather than profitmargin maximization. Thus, profit margin may notbe a good proxy indicator for differences in capa-bilities among firms that vary in their degree ofdiversification.

12 Two working papers show a similar result by assuming mul-tiple equal-sized markets that are perfectly competitive (San-talo, 2002) or monopolistically competitive (Nocke and Yeaple,2008).



Figure 2. Change in profit margin along with relative demand (aI = 10, bI = bN = 1, T1 = 2, T2 = 1.5, and(1 − δ) = 0.8).

This figure is available in color online at www.interscience.wiley.com/journal/smj

Figure 3. Change in profit margin along with relative demand (aI = 10, bI = bN = 1, T1 = 2, T2 = 0.5, and(1 − δ) = 0.8).

This figure is available in color online at www.interscience.wiley.com/journal/smj

Another finding is that while both firms’ profitmargins drop substantially when the relativedemand level in the new market passes the thresh-old sufficient to elicit diversification, their profitmargins will gradually increase as the relativedemand grows further. Moreover, when the de-mand level in the new market reaches a levelsufficient for the low-capability firm to diversifyas well, we see the cross-sectional property thatone generally intuits that the more capable firmearns a higher profit margin.

Finally, note that if the capability differencesamong the two firms is rather extreme, it may bethe case that the more capable firm maintains ahigher profit margin for all demand environments,even in that intermediate level of new marketdemand that generates diversification behavior onthe part of the more capable firm but a focus on theoriginal market on the part of the less capable firm.Figure 3 provides an illustration of this with: aI =10, bI = bN = 1, 1 − δ = 0.8, T1 = 2, T2 = 0.5.



DISCUSSION: IMPLICATIONSOF OPPORTUNITY COSTSAND DIVERSIFICATION

While the contemporary literature on diversifica-tion from a resource-based view builds upon theidea of excess firm capabilities developed by Pen-rose (1959), the emphasis has been on the fungi-bility of resources across domains. Making a clearanalytical distinction between scale free capabili-ties and those that are subject to opportunity costsand must be allocated to one use or another helps toshift the discourse back to Penrose’s (1959) focuson the stock of organizational capabilities. Theexistence of non-scale free capabilities implies thatprofit-maximizing diversification decisions shouldbe based on the opportunity cost of their usein one domain or another, which is, in turn,determined by the relative size of different marketsegments and the degree to which the effective-ness of capabilities diminishes across markets. Wefurther identify the demand thresholds for firms todiversify as a function of their capabilities, whichallows us to infer the effect of heterogeneous capa-bilities on the order of diversification in the face ofcompetitive interactions. The recognition of capa-bilities that must be allocated across multiple seg-ments based on opportunity costs also provides aprofit-maximizing explanation for the divergencebetween total profits and profit margins and, inturn, an alternative explanation of the diversifica-tion discount.

Our model suggests an alternative self-selectionmechanism that can account for the observation ofa cross-sectional diversification discount (Bergerand Ofek, 1995; Lang and Stulz, 1994). Firms withsuperior capabilities in a low-value existing mar-ket context enhance their profits by diversifying,but at the same time incur lower average returndue to the spread of non-scale free capabilitiesacross applications. Therefore, it may not be, assuggested by Gomes and Livdan (2004), that thosefirms with fewer capabilities (lower productivity)diversify first and this sorting of ‘bad types’ intodiversification events explains the observed cross-sectional diversification discount; rather, it couldbe that those firms with more capabilities diversifyfirst and that this diversification activity decreasesaverage returns.

In this sense, our model can help reconcile theconflict between the existing self-selection expla-nations that rely on the assumptions of comparative

productivity differences and diminishing returns toproduction scale (low productivity firms diversify-ing first) and the proposition well established in thestrategy field that firms with more relevant capabil-ities tend to enter a new field earlier (e.g., Klepperand Simons, 2000; Mitchell, 1989). Critical to ourargument is the opportunity cost of applying non-scale free capabilities in less favorable opportuni-ties. Our argument suggests that diversifying firmsare ‘good types’ (i.e., high capabilities) operatingin ‘bad’ market contexts.

This argument suggests that a cross-sectionaldiversification discount may arise when firms par-ticipate in distinct niches in the same broadlydefined industry. Different firms may experiencedifferent degrees of market maturity, and thoseoperating in more mature submarkets are morelikely to diversify and do so earlier. Alternatively,a ‘generalist’ firm (Hannan and Freeman, 1989)may respond to the demand maturity earlier bydiversifying because it has greater exposure tothe overall market conditions, while a ‘special-ist’ may not do so if its demand conditions areless affected by the overall market maturity. Ineither case, such profit-maximizing diversifyingfirms suffer the triple blow of facing a less attrac-tive demand environment with the decline in sizeof their original market, the diminished effective-ness of their capabilities as these capabilities areapplied to related, but distinct, product markets,and the spreading of non-scale free capabilitiesacross more segments.

The current theoretical model provides a con-ceptual basis for subsequent empirical analysisto sort out the different arguments regarding theself-selection mechanism in the diversification pro-cess. We make distinct empirical predictions fromthe existing corporate finance literature regardingwhich firms (more or less capable) are more or lessdiversified. Existing industry-level studies, such asKlepper and Simons’s (2000) work on the TVreceiver industry, are broadly consistent with thearguments developed here. As commercial broad-casting began after World War II, the demand forTV receivers took off rapidly and attracted a floodof entrants (through 1989 a total of 177 U.S. firms),many of which came from the radio industry. Klep-per and Simons (2000) find that a greater degreeof radio experience, measured by firm size, typesof radios, and years of production, significantlyincreased the likelihood and speed of entry. Thus,radio producers appear to diversify into the TV



receiver industry as a response to the growth of theTV market, and the relative maturity of the radiomarket; furthermore, if we interpret more expe-rience as evidence of more capabilities, then theresults suggest that firms with more capabilitiestend to diversify earlier.

As a more general methodological note, it isworth observing that research on the relationshipbetween prior experience/capabilities and entrybased on a fine-grained industry classification isable to offer more refined measures of firms’skills and capabilities than cross-industry analysisthat inevitably must rely on more coarse-graineddata. Thus, the analysis that contrasts de novoentrants versus de alia entrants, such as Klep-per and Simons (2000), Carroll et al. (1996), andHelfat and Lieberman (2002), offers an importantwindow to a capability-based logic of diversifi-cation. Along these lines, more refined empiricalanalyses allow for measures of market demandthat more closely correspond to the actual productmarket conditions that firms face. Even industryclassification at the four-digit SIC level may incor-porate many rather distinct submarkets with quitedifferent demand patterns.13 This more refined sortof empirical analysis appears necessary to furtherunpack the critical elements of firm heterogeneitythat results in firms being ‘sorted’ into diversifi-cation activity. Is the sorting into diversificationactivity based on exogenous market maturity anda high level of non-scale free capabilities thathave lost their value in their current applicationas suggested here; or is the differential sortinginto diversification driven by low levels of firmcapabilities and correspondingly relatively ex anteweak performance as suggested by recent writingsin the corporate finance literature (e.g., Gomes andLivdan, 2004)?

Identifying the crucial role of opportunity costof resource allocation provides an alternative,economic-based explanation for the reluctance ofestablished firms to aggressively enter new do-mains as identified by Christensen (1997) in the

13 As an illustration of the heterogeneity within a four-digitSIC category, consider the cardiovascular medical device indus-try. Although this industry is mainly underneath primary SIC3841 (surgical and medical instruments and apparatus) and3845 (electromedical and electrotherapeutic apparatus), the rel-evant demand conditions for a given manufacturer are far morenuanced than a four-digit measure would provide because thereexist eight independent product submarkets, such as stents, pace-makers, and heart valves, that have experienced very differentindustry life cycles (Wu, 2009).

context of his work on the disk drive industry. Inthe face of brutal competition with rapid modelintroductions, the established firms in the diskdrive industry faced a critical choice of whether toallocate their valuable product development teamsto their existing technological platforms, or tointroduce drives with an alternative format. Basedon the relative size of the market for their currentfamily of drives, as compared to the emerging mar-ket for smaller drives and ex ante assessments ofthe growth rates of these alternative technologies,straightforward opportunity cost logic could arguefor the apparent inertia of the established firms.

The notion of opportunity cost is a powerful con-cept. Perhaps its basic and pervasive nature maycause us to ignore, or at least underplay, its role.Diversification is not merely driven by supply-sideconsiderations of rare and distinctive resources, butis equally impacted by the market opportunities towhich these resources may be applied. Ultimately,we need to develop explicit characterizations ofthese ‘supply-side’ dynamics of firms’ capabili-ties in conjunction with analyses of the competi-tive dynamics of product market competition. Boththe investment in capabilities, as well as theirallocation across contexts, is a function of firms’perception of their demand environments and thecompetitive conditions they face. Recent work hasrightly highlighted the endogeneity of diversifica-tion decisions. However, in capturing the basisof this endogeneity, it is important to leverageour existing insights regarding corporate diversi-fication. The economic logic of the allocation ofnon-scale free capabilities over alternative domainsbuilds on traditional capability-based views of thefirm and provides a basis for endogenous diver-sification decisions consistent with existing andemerging empirical findings.

ACKNOWLEDGEMENTS

We thank the Mack Center for Emerging Tech-nologies for generously supporting this research.We have benefited from comments on prior draftsby Ron Adner, David Collis, Glenn MacDon-ald, Costas Markides, Nicolaj Siggelkow, HarbirSingh, Sid Winter, SMJ Associate Editor JosephMahoney, and two anonymous referees, as well asseminar audiences at the Tuck School, DartmouthCollege, the Atlanta Competitive Advantage Con-



ference, the Academy of Management, and theHarvard Strategy Conference.

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APPENDIX 1

In proving Proposition 1, we first solve the inte-rior solution (t∗

I , t∗N ) to the maximization problem

(Equation 2) in the text when demand conditionsare such that it is optimal for the firm to diversify:14

t∗I =

√(1 − δ)sI√

(1 − δ)sI + √sN

and t∗N

=√

sN√(1 − δ)sI + √

sN

Inserting (t∗I , t

∗N ) into the profit margin of the

diversification strategy π ∗ in Equation 3 in thetext, we can transform Equation 3 as

π ∗

pIsI + pNsN

= [sI

pI sI + pNsN

(pI − r

γIT)

+ sN

pI sI + pNsN

(pN − r

(1 − δ)γIT)]

− 2

√sI sN

pI sI + pNsN

r√(1 − δ)γIT

Notice that the first term (in the square brackets)in the above equation is exactly the weighted

qDIV = pIsI + pNsN

r(k∗I + k∗

N)= pIsI + pNsN

r(kI

1

t∗I

+ kN

1

t∗N

)

= pIsI + pNsN

r(kI + kN + kI

√sN√

(1 − δ)sI

+ kN

√(1 − δ)sI√

sN

)

= pIsI + pNsN

r(sI

γIT+ sI

(1 − δ)γIT+ sI

γIT

√sN√

(1 − δ)sI

+ sN

(1 − δ)γIT

√(1 − δ)sI√

sN

)

= pI sI + pNsN

r

γIT(sI + sN

1 − δ+ 2

√sI sN

1 − δ)

14 It should be noted that while competitors’ cost efficiency doesnot affect the continuous allocation of non-scale free capabilitiesto a certain segment in the Bertrand setting, it does influence theboundary conditions that determine whether a firm chooses toenter a segment.

average of two focus strategies in Equation 4 inthe text. Therefore, Equation 5 in the text, andthus Proposition 1, is proved. Q.E.D.

APPENDIX 2

The Tobin’s q associated with the diversificationstrategy is qDIV = pI sI + pNsN

r(k∗I + k∗

N)where capital in

each market is respectively k∗I = sI

γIT t∗I

and k∗N =

sI

(1 − δ)γIT t∗N

.

Note that (i) if all capabilities are focused inthe initial market, then capital is kI = sI

γIT; (ii)

if all capabilities are focused in the new market,then capital is kN = sI

(1 − δ)γIT. Therefore, k∗

I =1t∗I

kI and k∗N = 1

t∗N

kN . Moreover, remember t∗I =√

(1 − δ)sI√(1 − δ)sI + √

sN

and t∗N =

√sN√

(1 − δ)sI + √sN

,

so we have 1t∗I

= 1 +√

sN√(1 − δ)sI

and 1t∗N

= 1 +√(1 − δ)sI√

sN

.

Therefore, the Tobin’s q associated with thediversification strategy can be transformed as:

Next, we specify the weighted average of twofocused strategies, with capital required when all



capabilities are focused in one market, kI = sI

γIT

and kN = sI

(1 − δ)γIT, as weights, as:

qFOC = rkI

rkI + rkN

pI sI

rkI

+ rkN

rkI + rkN

pNsN

rkN

= pIsI + pNsN

r(kI + kN)= pIsI + pNsN

r

γIT(sI + sN

1 − δ)

Finally, we compare the Tobin’s q values asso-ciated with these two strategies by examining theirratio:

qFOC

qDIV

= 1 +2√

sI sN

1 − δ

sI + sN

1 − δ

Therefore, there exists a discount factor

2√

sI sN

1 − δ

sI + sN

1 − δ

.Q.E.D.

APPENDIX 3

The above figure indicates the best responsecurves when relative demand conditions are suchthat both firms diversify in equilibrium. The solidline illustrates the reasons why firm 1’s bestresponse curve is neither continuous nor mono-tonic. Firm 1’s best response curve has smalljumps both when t2I is around 0.2 and 0.8. The‘jump’ around the value of t2I = 0.2 stems from

the fact that when t2I < 0.2, firm 2’s cost value issufficiently high in the initial market that it is notable to produce in the initial market. As a result,firm 1’s best response allocation of capabilities hasa jump up as t2I increases and passes 0.2, becausefirm 2 begins to produce in the initial market. Sim-ilarly, when t2I > 0.82, firm 2’s cost is too high inthe new market to be viable and produce a nonzeroquantity. As a result, firm 1’s best response allo-cation of capabilities jumps up as t2I increases andpasses 0.82, because at that point firm 2 exits par-ticipation in the new market and, as a result, firm1 does not need to put so many capabilities intothe new market.

Similarly, the dotted line illustrates the reasonswhy firm 2’s best response curve is neither contin-uous nor monotonic. When t1I = 0.6, firm 2’s bestresponse curve jumps from 0 to 0.42. This jumpstems from firm 2’s entry into the new market. Inaddition, firm 2’s best response curve has a smalljump when t1I = 0.1. This ‘jump’ stems from thefact that when t1I < 0.1, firm 1’s cost value is suf-ficiently high in the initial market that it is not ableto produce in the initial market. As a result, firm2’s best response allocation has a jump up as t1I

increases and passes 0.1, because firm 1 begins toproduce in the initial market.

APPENDIX 4

Note that to capture the change in relative demandwithout loss of generality for all the analyses, wehold constant the size of the initial market aI

and only vary the size of the new market aN .Let P1 ≡ P(aN, T1, T2, t1I , t2I ) be firm 1’s profitfunction given the new market size aN , firm 1’scapability T1, the competitor firm 2’s capability T2,firm 1’s allocation t1I , and the competitor firm 2’sallocation t2I . Similarly, P2 ≡ P(aN, T2, T1, t2I , t1I )

represents firm 2’s profit function given the newmarket size aN , firm 2’s capability T2, the com-petitor firm 1’s capability T1, firm 2’s allocationt2I , and the competitor firm 1’s allocation t1I .Note that P(aN, T1, T2, t1I , t2I ) = π1I + π1N andP(aN, T2, T1, t2I , t1I ) = π2I + π2N , where,

πim = bm(qim)2 and qim

=

am + c(3−i)m − 2cim

3bm, if q(3−i)m > 0

am − cim

2bm, if q(3−i)m = 0



for i = 1, 2; m = I, N (A1)

Below we restate Proposition 2a by character-izing the properties of the best response curves.Moreover, without loss of generality, we fix thecapability level of firm 1 and vary the capabilitylevel of firm 2 (T2 = T ).

Proposition 2a: Let T2 = T . There existsan aN > 0, η > 0, and ε > 0, such thatP(aN, T1, T , t1I , 1) reaches maximum atsome t1I = t (aN, T1) < 1 and P(aN, T , T1,

t2I , t (aN, T1)) has a unique maximum at t2I =1, whenever aN < aN < aN + η and|T1 − T | < ε.

We start by proving when aN is small enough,there is a unique equilibrium (t1I = 1, t2I = 1), orboth firms focus on the initial market.

Lemma 1: Given any T1 and T2, there existsan aN such that when aN < aN , P1 has a uniquemaximum at t1I = 1 for all t2I and P2 has a uniquemaximum at t2I = 1 for all t1I .

Proof: We provide the proof for P1, and thatfor P2 is analogous. Note that q1N = 0 and thusπ1N = bN(q1N)2 = 0 for all t2I whenever t1I ≥ 1 −

1aN(1 − δ)γ T1

because c1N = 1(1 − δ)γ (1 − t1I )T1≥ aN ≥ pN ; in other words, firm 1’s marginal

cost in the new market is always greater thanthe maximal possible price unless firm 1 allocatesenough capability to the new market (i.e., 1 −t1I ≥ 1

aN(1 − δ)γ T1). This result together with the

fact that q1I is strictly increasing in t1I on [1 −1

aN(1 − δ)γ T1, 1] implies that P1 = π1I + π1N =

π1I + 0 = bI (q1I )2 is strictly increasing in t1I on

[1 − 1aN(1 − δ)γ T1

, 1].Note that when aN = 0 we have q1N = 0 and

hence that P1 = bI (q1I )2 is increasing in t1I on

[0, 1] (since q1I is increasing in t1I ). Thus, whenaN = 0, P1 has a unique maximum at t1I = 1 for allt2I . This result implies that there exists an aN suchthat when aN < aNP1 has a unique maximum att1I = 1 for all t2I , provided that P1 is continuouslyincreasing in aN for t1I ∈ [0, 1 − 1

aN(1 − δ)γ T1]

and is constant with respect to the changes of aN

for t1I ∈ [1 − 1aN(1 − δ)γ T1

, 1] (this later condi-tion implies that P1 is still strictly increasing in t1I

on [1 − 1aN(1 − δ)γ T1

, 1], as proved in the previ-ous paragraph). Q.E.D.

Lemma 1 immediately implies that when aN

is small enough, (t1I , t2I ) = (1, 1), or both firmsfocusing on the initial market, is the unique equi-librium.

We then prove that when the new market getssufficiently large and the two firms’ capabilityasymmetry is small enough, either firm may firstdiversify while the other stays focused at the initialmarket.

In so doing, we first prove a lemma to showthat if it is optimal for firm 1 to choose t1I =1 given some t2I , it must be still optimal forfirm 1 to choose t1I = 1 given some t2I < t2I .That is,

Lemma 2: ∂2P1∂t2I ∂t1I

= 2bI∂q1I

∂t2I

∂q1I

∂t1I+ 2bN

∂q1N

∂t2I∂q1N

∂t1I< 0.

Proof: ∂P1∂t1I

= 2bIq1I∂q1I

∂t1I+ 2bNq1N

∂q1N

∂t1I,

∂2P1

∂t1I ∂t2I

= 2bI

∂q1I

∂t1I

∂q1I

∂t2I

+ 2bIq1I

∂2q1I

∂t1I ∂t2I

+ 2bN

∂q1N

∂t1I

∂q1N

∂t2I

+ 2bNq1N

∂2q1N

∂t1I ∂t2I

Lemma 2 follows because ∂2q1I

∂t2I ∂t1I= 0 and

∂2q1N

∂t2I ∂t1I= 0; ∂q1I

∂t2I

∂q1I

∂t1I< 0 and ∂q1N

∂t2I

∂q1N

∂t1I< 0

(which is easy to check from Equation A1). Q.E.D.Therefore, if it is optimal to focus on the initial

market when the competitor focuses on the initialmarket (t2I = 1), it must be still optimal to focusif the competitor diversifies into the new market.Similarly this lemma applies to firm 2.

Lemma 3: Given T > 0, when the two firmshave the same level of capabilities, that is T1 =T2 = T , there exists an aN > 0 such that P (aN, T ,

T , t1I , 1) has at least two maxima including one att1I = 1. P(aN, T , T , t1I , 1) has a unique maximumat t1I = 1 whenever aN < aN , and P(aN, T , T , t1I ,

1) reaches maximum at some t1I = t1I (aN) < 1whenever aN > aN .

Proof: According to the argument in Lemma1, it is easy to check that P(0, T , T , t1I , 1) isincreasing in t1I over [0,1] and is strictly increasingin t1I when t1I is close to 1. Thus, when aN

is small enough, P(aN, T , T , t1I , 1) has a uniquemaximum at t1I = 1. Also, it is straightforward tocheck that P(aN, T , T , t1I , 1) > P (aN, T , T , 1, 1)

for any t1I < 1 when aN is big enough. Thisresult together with the fact that P(aN, T , T , t1I , 1)



is continuous in aN implies that there exists athreshold aN > 0 such that P (aN, T , T , t1I , 1) hasat least two maxima including one at t1I = 1.

Note that: ∂2

∂aN∂t1IP (aN, T , T , t1I , 1) < 0,

which implies that P(aN, T , T , t1I , 1) has a uniquemaximum at t1I = 1 whenever aN < aN and P(aN,

T , T , t1I , 1) reaches maximum at some t1I = t1I

(aN) < 1 whenever aN > aN . Q.E.D.Lemma 2 and Lemma 3 together imply the

following corollary for all t2I < 1.Corollary 1: P (aN, T , T , t1I , t2I ) has a unique

maximum at t1I = 1 for all t2I < 1.Now we can prove Proposition 2(a):Proof: Proposition 2a follows from Lemma 1,

Lemma 3, and Corollary 1 and the fact that P isdifferentiable with respect to all arguments. Q.E.D.

Note that P(aN, T1, T , t1I , 1) reaches maximumat somet1I = t (aN, T1) < 1. This maximum is notnecessarily unique. There can be a unique max-imum at t1I = 0, or a unique maximum at 0 <

t1I < 1, or two maxima at both t1I = 0 and some0 < t1I < 1. However, in any case, Proposition 2ais proved in that, in equilibrium, either firm candiversify first (diversifies partly into the new mar-ket or completely switches to the new market)while the other firm stays focused in the initialmarket.

Below we restate Proposition 2b by character-izing the properties of the best response curves.Moreover, without loss of generality, we let firm2 be the less capable firm.

Proposition 2b: Fixing T2, there exists a T >

0,�aN > 0, aN > 0,

�aN < aN , and η > 0 such

that, when T1 > T ,

i) P(aN, T1, T2, t1I , 1) has a unique maxi-mum at t1I = 1 for all aN <

�aN

ii) P(aN, T1, T2, t1I , 1) has a unique maxi-mum at some 0 < t1I < 1 for all aN ∈(�aN,

�aN + η), and

iii) P(aN, T2, T1, t2I , 1) has a unique maxi-mum at t2I = 1 for all aN < aN

Proof: It is straightforward to see that, for firm1, there exists an

�aN > 0 such that P(

�aN, T1, T2,

t1I , 1) has a maximum at t1I = 1 and a maximumat some t 1I < 1. Similar to the proof of Lemma 3,

∂2

∂aN∂t1IP (aN, T1, T2, t1I , 1) < 0, so P(aN, T1, T2,

t1I , 1) has a unique maximum at t1I = 1 for all

aN <�aN , and P(aN, T1, T2, t1I , 1) reaches maxi-

mum at some t1I < 1 for all aN ∈ (�aN,

�aN + η)

(not necessarily unique, since there can be twomaxima at both t1I = 0 and some0 < t1I < 1).

Moreover,�aN > 0 satisfies P(

�aN, T1, T2,

t1I , 1) = P(�aN, T1, T2, 1, 1). That is,(

aI + 1

γ T2− 2

γ t1I T1

)2

/9bI

+(

�aN − 1

(1 − δ)γ (1 − t 1I )T1

)2

/4bN

=

(aI + 1

γ T2− 2

γ T1

)2

9bI

.

It is easy to check that�aN approaches zero

as T1 increases. In addition, as T1 increases,P(

�aN, T1, T2, t1I , 1) has a unique maximum at

some 0 < t1I < 1, since t1I = 0 can never be amaximum when

�aN approaches zero. This fact

implies that P(aN, T1, T2, t1I , 1) has a unique max-imum at some 0 < t1I < 1 for all aN ∈ (

�aN,

�aN +

η).Similarly, for firm 2, it is straightforward to see

that there exists an aN > 0 such that P(aN , T2, T1,

t2I , 1) has a maximum at t2I = 1 and a maximumat some t2I < 1. Namely, P(aN , T2, T1, t2, 1) =P(aN, T2, T1, 1, 1). That is:(

aI + 1

T1− 2

γ t2I T2

)2

9bI

+

(aN − 1

(1 − δ)γ (1 − t2I )T2

)2

4bN

=

(aI + 1

γ T1− 2

γ T2

)2

9bI

.

Since ∂2

∂aN∂t2IP (aN, T2, T1, t2I , 1) < 0, P(aN,

T2, T1, t2I , 1) has a unique maximum at t2I = 1for all aN < aN . In addition, it is straightfor-ward to check that aN approaches a constant asT1 increases. Since

�aN approaches zero as T1

increases,�aN < aN when T1 is large enough. Since

P1 is continuously increasing in T1 and P2 is con-tinuously decreasing in T1, there exists a T > 0;



for all�aN < aN <

�aN + η < aN , it is a unique

equilibrium that 0 < t1I < 1 and t2I = 1 whenT1 > T . Q.E.D.

Below we restate Proposition 2c by character-izing the properties of the best response curves.Moreover, without loss of generality, we let firm2 be the less capable firm.

Proposition 2c: Given aI and T1, there existsT > 0 such that for any 0 < T2 < T and 0 ≤t1I ≤ 1, P(aN, T1, T2, t1I , t2I ) achieves maxi-mum at t2I = 0 or t2I = 1.

Proof: Recall that the total profit of firm 2 isP(aN, T2, T1, t2I , t1I ) = π2I + π2N , where

π2m = bm(q2m)2 and q2m

=

max(am + c1m − 2c2m

3bm, 0), if q1m > 0

max(am − c2m

2bm, 0), if q1m = 0

for m = I, N

Given that c1I = 1γ t1I T1

, c2I = 1γ t2I T2

,

c1N = 1(1 − δ)γ (1 − t1I )T1

, and c2N =1

(1 − δ)γ (1 − t2I )T2, it is easy to check the fol-

lowing are true:

1) q2I (and hence π2I ) is increasing in t2I andT2. There exists t I

2I ∈ [0, 1] such that q2I = 0(and hence π2I = 0) for all t2I ∈ [0, t I

2I ). Thethreshold t I

2I decreases in T2 and equals 1 whenT2 is small enough. t I

2I does not change with aN .2) q2N (and hence π2N ) is decreasing in t2I , but

increasing in T2. There exists tN2I ∈ [0, 1] such

that q2N = 0 (and hence π2N = 0) for all t2I ∈(tN

2I , 1]. The threshold tN2I increases in aN and

approaches 1 as aN approaches infinity. tN2I is

increasing in T2.

Therefore, there exists an aN > 0 and a thresh-old T (aN) > 0 such that tN

2I ∈ (0, 1) and when-ever 0 < T2 < T (aN) we have t I

2I > tN2I (since t I

2I

is decreasing in and tN2I is increasing in T2 as

explained above) and π2I (t2I = 1) < π2N(t2I = 0)

(since π2I (t2I = 1) is independent of aN whileπ2N(t2I = 0) is increasing in aN as explainedabove). This fact implies that given any aI , T1,and t1I , firm 2’s total profit P(aN, T2, T1, t2I , t1I ) =π2I + π2N achieves maximum at t2I = 0 or t2I = 1for all aN > 0 whenever T2 < T (aN). This resultclearly holds when aN = aN , since, by the abovedefinition of aN and T (aN), P(aN, T2, T1, t2I , t1I ) =π2I + π2N achieves maximum at t2I = 0. Next weprove this result for aN < aN and aN > aN respec-tively.

When aN < aN , we have t I2I > tN

2I , since t I2I

does not change with aN and tN2I increases in

aN . Therefore, P(aN, T2, T1, t2I , t1I ) = π2I + π2N

achieves maximum at t2I = 0 or t2I = 1 becauseP(aN, T2, T1, t2I , t1I ) = π2I + π2N is strictly de-creasing in t2I on [0, tN

2I ), is zero on [tN2I , t

I2I ], and

is strictly increasing in t2I on (t I2I , 1].

When aN > aN , P(aN, T2, T1, 0, t1I ) = π2N(aN,

t2I = 0) is strictly greater than P(aN, T2, T1, t2I ,

t1I ) = π2N(aN, t2I ) + π2I (t2I ) for any t2I ∈ (0, 1],because π2N(aN, t2I = 0) − [π2N(aN, t2I ) + π2I

(t2I )] equals π2N(aN, t2I = 0) − π2N(aN, t2I ) > 0for all t2I ∈ (0, t I

2I ], and is greater than π2N(aN,

t2I = 0) − [π2N(aN, tI2I ) + π2I (t2I = 1)] > 0 for

t2I ∈ (t I2I , 1] (π2N(aN, t2I ) < π2N(aN, tI

2I ) for t2I ∈(t I

2I , 1] since π2N(aN, t2I ) is decreasing in t2I fort2I ∈ (t I

2I , 1]; π2I (t2I ) ≤ π2I (t2I = 1) for t2I ∈(t I

2I , 1] since π2I (t2I ) is increasing in t2I for t2I ∈(t I

2I , 1]).Note that π2N(aN, t2I = 0) − [π2N(aN, tI

2I )+π2I (t2I = 1)] > 0 is because π2N(aN, t2I = 0) −[π2N(aN, tI

2I ) + π2I (t2I = 1)] > π2N(aN, t2I = 0) −[π2N(aN , tI

2I ) + π2I (t2I = 1)] = π2N(aN, t2I = 0) −[0 + π2I (t2I = 1)] > 0, where the first inequal-

ity holds because ∂2

∂aN∂t2IP (aN, T2, T1, t2I , 1) <

0 and aN > aN , the equality holds due to thedefinition of aN (when aN = aN , t I

2I > tN2I and

π2N(aN, t2I ) = 0 for all t2I ∈ (tN2I , 1]), and the last

inequality holds due to the definition of aN andT (aN).

Therefore, T (aN) serves as an upper bound suchthat, given aI and T1, the total profit of firm 2achieves maximum at t2I = 0 or t2I = 1 for any0 < T2 < T (aN) and 0 ≤ t1I ≤ 1. Q.E.D.


Opportunity costs and non-scale free capabilities: profit maximization, corporate scope, and profit...

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Transcript of Opportunity costs and non-scale free capabilities: profit maximization, corporate scope, and profit...