Structure, Scale, and Scope Structure, Scale, and Scope in the Global Computer Industry

نویسندگان

  • Matthew S. Bothner
  • Peter Bearman
  • Frank Dobbin
  • Stanislav Dobrev
  • Damon Phillips
  • Paul Ingram
  • Toby Stuart
  • Louis R. Pondy
  • Matthew Bothner
چکیده

Scholars in many fields have often noted that size and diversification raise the performance of firms in the most significant sectors of our economy. Using longitudinal network data on the personal computer industry, this paper identifies the main effects and complex interplay of two attributes of a firm’s role in a system of competitive relations: (a) its size relative to that of its structurally equivalent rivals and (b) its level of diversification. The results show first that, net of absolute size, relative size raises sales growth; second, that the main effect of scope is positive; and third, that the effect of scope follows an inverted U-shaped pattern over the distribution of relative size. Diversification thus lowers growth when firms are relatively small, raises growth after a threshold before a maximum, but has a negative effect again for extreme levels of relative size. Theories of diversification, which have been disparate historically, are integrated and jointly find support in the analysis. Structure, Scale, and Scope Structure, Scale, and Scope in the Global Computer Industry Scholars in many fields have often claimed that scale and diversification further the growth of firms in the major sectors of our economy. Fligstein (1990a, 1990b) described the role of scale in the formation of the U.S. steel industry and underscored the impact of diversification on firm growth during the Great Depression. Chandler (1990) traced the early expansion of capitalist firms almost entirely to the advantages of scale and scope. Sugar refiners, aluminum producers, and automakers all benefited from the economies of size. Makers of dyes and pharmaceuticals grew by using the same raw materials and machinery to turn out a portfolio of related products. By Chandler’s account, our economy is the byproduct of firms—General Motors, Ford, and DuPont—that harnessed the benefits of scale and diversification. What is thought to have caused the growth of the early titans has also been seen as crucial for the drivers of today’s economy, such as makers of semiconductors and personal computers. Apple, IBM, and NEC have enjoyed huge scale economies for years (Brock 1975; Scherer 1996). Compaq’s creation of the portable market and Dell’s entry into the PC server market exemplify the importance of widening scope for performance. In light of the importance of scale and scope for the growth of firms—and thus of industries and even whole economies—it is surprising that researchers have not yet considered the effects of scale advantages and diversification jointly in models of firm growth rates. Whether these factors have unique effects—net of other covariates, such as market size or managerial skill—remains unresolved. Although scale and scope advantages are often thought to amplify each other, scholars have not theorized the ways in which they may interact, nor have they explored their joint effects empirically. Doing so may further illuminate how the effects of Structure, Scale, and Scope strategy depend on context, and thus start to resolve tensions in our theories of how firms behave and perform. Using a panel of over 400 computer vendors, in this paper I extend the literature on social structure and economic performance (White 1981; Burt 1992; Podolny, Stuart, and Hannan 1996; Ingram and Roberts 2000) by identifying the main effects and complex interplay of two features of a firm’s position in a network of competitive relations: its size relative to that of its strategically proximate rivals and its scope or diversification across market segments. Consistent with the implications of prior research, the analyses reveal three important findings: first that, adjusting for absolute size, relative size raises sales growth; second, that the main effect of diversification on sales growth is positive; and third, that the effect of scope follows an inverted U-shaped pattern over the distribution of relative size. In the analyses that follow, I take a structural approach by focusing on the locations of firms in a network of competitive relations. This marks a departure from studies in economics, which have modeled firm growth rates without considering the structure of inter-firm rivalry. Economists have focused on “internal growth,” that is, on how the features of individual firms raise their performance. Except for considering the impact of acquisitions (Hannah and Kay 1981), the effects of the conduct and performance of a firm’s competitors have been overlooked. In a classic paper, Edith Penrose (1952:808) noted: “We have every reason to think that the growth of a firm is willed by those who make the decisions of the firm and are themselves part of the firm, and ... no one can describe the development of a given firm or explain how it came to be the size it is except in terms of decisions taken by individual men.” Consistent with this approach, common predictors of growth in the economics literature are size, age, and the number of manufacturing plants (e.g., Evans 1987). Structure, Scale, and Scope Conversely, a main tenet of the sociological approach is that relations among economic actors—managers, firms, industries, and even nations—shape economic outcomes (Granovetter 1985; White 1981; Burt 1992; Smith and White 1992). White’s (1981; 2001) model of markets, for example, rests on the claim that the profitability and growth of any firm is driven by the procurement and production behavior of its competitors. Correspondingly, Smelser and Swedberg (1994:6) noted that economic sociology is marked by the notion that “other actors either facilitate, deflect, or constrain individuals’ actions in the market.” This approach has recently been developed by Uzzi (1996; 1999) in studies pinpointing optimal network structures and by Ingram and Roberts (2000) in a study clarifying how friendships among rival managers increase firm performance. Efforts to clarify how inter-firm connections affect performance have also informed most if not all analyses of firm growth and decline in sociology. The kinds of external factors thought to shape growth have been many. They range from global metrics of competition, such as industry density (Barron, West, and Hannan 1994), to localized metrics of status and technological crowding (Podolny et al. 1996), to measures as diverse as the depth of competition faced by a firm over its history (Barnett and Sorenson 1998) to the size of its alliance partners (Stuart 2000). The theme that unites these studies is an effort to show how social interactions— competitive, deferential, or collaborative—between organizations shape their future prospects. Extending this stream of research, I begin by estimating the effect of relative size to see if scale advantages raise growth. But that aim begs the question, relative to whom? Clarifying who competes with whom and to what degree is crucial for any field that studies competition. Economics, corporate strategy, organizational ecology, and structural sociology all have ways of demarcating the limits of rivalry. Structure, Scale, and Scope In economics, markets have been circumscribed on the basis of gaps in chains of substitute goods (Robinson 1933), price correlations (Slade 1986), and spatial proximity (Eaton and Lipsey 1989). Students of corporate strategy have used the notion of “strategic groups,” which have been identified by similarities in performance (Porter 1979), conduct (Oster 1982), and the beliefs managers have about who their peers are (Porac and Rosa 1996). Ecologists have argued that the rivalry between two firms rises with their proximity on a given dimension, such as size (Hannan and Freeman 1977), technology (Barnett 1990), or geographical space (Baum and Mezias 1992). Network analysts have also used patterns of resource dependence to define competition. Burt’s analyses of aggregate markets use measures of structural equivalence to capture competitive boundaries (Burt and Carlton 1989). Market sectors—such as those of electronic components and scientific instruments—are defined as structurally equivalent because of their similar profiles of purchasing from, and selling to, other sectors. Sectors in “compete in the sense of depending on similar levels of purchases from the same supplier markets and similar levels of sales to the same customer markets” (Burt 1992:88). Using Burt and Carlton’s (1989) approach, I identify a firm’s closest rivals by comparing patterns of shipping computers across markets segments, which are defined by technology, geography, and distribution channel. Gateway and Dell are thus structurally, or strategically, similar since they both mostly sell desktops in the U.S. by direct methods. This tack is well suited to the realities of the computer industry—in which PC vendors compete with those of similar strategies, who are in their “market space”—and it yields a metric of conduct that is “clean” performance itself. In measuring relative size, I place ego’s sales over the weighted average of the sales of its peers, where the weights are the varying levels of structural equivalence between a firm and its rivals. This approach reflects the stratified nature of most networks (Stuart Structure, Scale, and Scope 1998) and is suited to the coexistence of competitive overlap and segmentation characteristic of the computer industry. As I later discuss, in identifying the effect of relative size on growth, this paper adds to literatures in economics (Sutton 1997) and sociology (Carroll and Hannan 2000), which have long sought to clarify the nature of the size-growth link, but have yet to reach consensus (in either discipline) about the effect of size. But I also use relative size to clarify the complex effects of diversification. The “specialize or diversify” question is in no way unique to the executive suites of large organizations; rather, it is ubiquitous to social life, for it confronts individuals and organizations of almost every kind. Entry-level workers have to choose between “borrowing” the network of a sponsor and spreading their time and energy over a range of contacts (Burt 1998). Social and political organizations must weigh the benefits of sticking with their stated mission against those of diversifying into new activities, which may be vital for their longevity (Zald and Denton 1963). Even as cities court multinational corporations, they have to balance focus on a core competence with the task of building the cultural institutions that keep firms and workers in the long run (Kanter 1995). Since Lawrence and Lorsch’s (1967) work, sociologists often have claimed that such questions of design and strategy have a contingent answer—that what a firm should do hinges on the nature of its task environment. Freeman and Hannan (1983), for example, developed a contingent theory of niche width, which they defined as the diversity of activities in which a firm involves itself. They showed that broad scope raised life chances only in contexts that were highly variable and marked by seasonal change. Davis, Diekmann, and Tinsley (1994) linked the impact of diversification to the institutional environment, demonstrating that the once sacrosanct strategy of growth by merger was risky in the takeover frenzy of the 1980s. Other studies have Structure, Scale, and Scope also started from assumptions about the nature of the environment and then hypothesized the effect of diversification (e.g., Haveman 1992). I also take a contingent approach. But rather than assume that the effect of scope varies by industry or time, I expect it to depend on a key attribute of a firm’s role, namely its relative size. This approach yields brings together otherwise disparate, and seemingly conflicting, views of diversification and its consequences. Some theories suggest that organizations should specialize to elude competition (White 1981, 2001; Carroll 1985); others imply that firms should specialize to reduce barriers to learning (e.g., Teece 1980); still others, that they should diversify in order to grow (e.g., Chandler 1990). An aim of this paper is to clarify when each of these different viewpoints apply. In the next section, I set forth hypotheses whose aim is to clarify the effects of relative size, scope, and their interaction. In section two, I describe the data and the measures of substantive interest. I move to the modeling strategy and control variables in section three. The results of nine within-firm models of quarterly sales growth appear in section four. In section five, I conclude by discussing the results and their implications for future research. 1. Theory and Hypotheses 1.1. Relative size The effect of size on growth has been the object of study for decades. Scholars have repeatedly tested Gibrat’s (1931) “law of proportionate effect,” which claims that a simple stochastic process accounts for the skewed size distributions seen in many industries (see Sutton [1997] and Carroll and Hannan [2000] for reviews). Gibrat asserted that growth in absolute terms (of revenue, assets, or employees) was a function of prior size multiplied by random error. Even if all firms were of equal size at an industry’s birth, it is easy to imagine that small random Structure, Scale, and Scope differences in growth could yield a severely skewed distribution (i.e., market concentration) in due time. In that spirit, one of the main causal mechanisms thought to underlie Gibrat’s law is luck (Scherer 1970); another is information (Jovanovich and Rob 1987), in that larger firms may have better knowledge of buyer tastes, make better products, and thus grow more than their smaller counterparts. But many, if not most, studies in economics and sociology point to the failure of Gibrat’s law (e.g., Kumar 1985; Barron et al. 1994), showing that smaller firms grow at a faster rate than larger firms. Explanations of this pattern range in emphasis from the problems of panel attrition (Mansfield 1962), to the claim that size may be an asset only in certain industries (Jovanovich and Rob 1987), to attention to how growth is defined (Carroll and Hannan 2000). Carroll and Hannan (2000:318) noted that even if Gibrat’s law fails, large firms can often grow more than smaller firms in absolute terms. They also suggested that relative size, net of absolute size, may yield new insights about firm growth. Considering relative size as a predictor of growth may add to the literatures on size and growth, whose lack of consensus in economics Sutton (1997:42) expressed in his review, stating that, “there is no obvious rationale for positing any general relationship between a firm’s size and its expected growth rate.” Early insights on the relative size-growth link are evident in the work of Hannan and Ranger-Moore (1990), who used simulations to explore the role of size-localized competition in the evolution of firm size distributions. Measuring the absence of competition by ego’s distance on size from all others in the sample, simulations showed that monopolization occurred quickly, particularly once the large firm broke away from its rivals. While they did not frame their study around the effects of relative size, their results imply that growth may rise with relative size in actual industries. Structure, Scale, and Scope Considering relative size directly, Hannan et al. (1998) estimated the effects of relative size on survival in the British, French, German, and American automobile industries. Adjusting for absolute size, they showed that size relative to the largest firm in the industry lowered the likelihood of death, a result held across all four national markets. Carroll and Swaminathan (2000) showed that firms in the U.S. beer industry with many and large competitors were more likely to perish. They argued as well that a relative measure of size is fitting whenever firms compete on scale. The higher cost structures of smaller firms endanger their survival precisely because of their larger, more efficient rivals, in whose absence these smaller firms would be better off. It is because of their low position in a hierarchy defined by size that smaller organizations face a greater risk of extinction (Carroll and Swaminathan 2000). Carroll and Hannan (2000) also noted three closely related domains in which relative size should matter: Beyond superior efficiency in production, they suggested that relative size could also yield greater influence over suppliers and distributors. While these insights apply widely, they particularly concern the performance of computer firms, which compete in an environment marked by scale advantages in purchasing, production, and distribution. Upstream, the larger vendors enjoy significant discounts, especially when buying software, processors, disk drives, and keyboards. Given their bargaining power over suppliers, they are also less likely to have to assume inventory risk. In manufacturing, the advantages of scale are especially important (Brock 1975; Scherer 1996, pp. 244-6), because of the amortization of development costs. While only an ex post account, Michael Dell’s description of Dell’s trajectory emphasized the crucial role of scale, explicitly noting the importance of catching up with its rivals: “I realized we had to decide whether to stay the size we were—and face the consequences— Structure, Scale, and Scope or go for big time growth. Though we were at $1 billion in sales at the time, it didn’t really matter. We were not growing in increments that would allow us to compete on a global level when the market started to consolidate, and it was clearly going to—soon. If we stayed the size we were, we wouldn’t be able to amortize our development costs over a large enough volume, and our cost structure would be too high. We’d run the risk of being uncompetitive, and we could easily get left in the dust” (1999:43). Lastly, downstream, larger firms get the best space on the shelves of retailers and are the first priority of major distributors. Given the benefits of size, I expect that relative size is associated with future growth. H1: Sales growth rises with relative size. 1.2. Market scope The next hypothesis marks a shift away from a firm’s relative standing by considering the structure of its ties to its buyers. Central to any firm’s strategy, regardless of industrial setting, are decisions about how to allocate goods or services over a set of possible market segments. Some firms, such as IBM, ship outputs of all form-factors, through many channels, and nearly evenly across the geographical markets in which they have a presence. Sanyo, on the other hand, has focused primarily on Japan and the U.K., with modest forays into the French market starting in 1996. Many studies have shown that diversifying—shifting away from Sanyo’s strategy, toward IBM’s—raises firm performance. Various researchers have focused on economies of scope, which arise when making two goods jointly is cheaper than making them separately (e.g., Willig 1979). Scholars of corporate strategy have taken this notion beyond the case of producing similar Structure, Scale, and Scope goods to embrace all instances in which a firm parlays the skills or resources it acquired in one domain into another. Chandler (1990), for example, stressed the importance of economies of scope for the growth of large industrial enterprises, noting that capabilities honed in one market enabled firms to expand successfully into new ones. Haveman (1992) offered a complementary perspective on the link between scope and competencies, noting that related diversification yields added resources, which then sharpen a firm’s set of capabilities in its original area of competence. Haveman’s analysis of the effects of diversification in the savings and loan industry showed that broadening scope increased a firm’s life chances. In many industries, firms often grow by widening their reach into new markets, selling through new channels, and by expanding technologically. As they widen in scope, they can purchase components more cheaply—suppliers often want their products to reach a broad range of buyers—and they can exploit skills that they built up in earlier areas of focus in new ones. Consequently, I expect that: H2: Sales growth rises with market scope. 1.3. The interaction of market scope and relative size In spite of the support for hypothesis two, prior research also suggests that the impact of diversifying may be subtler. Several lines of work imply that scope is beneficial only under certain conditions. Sociologists often claim that specialization raises the life chances of organizations, but rely on very different accounts of why this might be the case. Studies that look outside the firm imply that diversification is a means of eluding competition. Conversely, other lines of work, which look within the firm, suggest that organizations should steer clear of diversification to avoid diseconomies of scope. Are these different mechanisms reconcilable? If so, it is then possible to square them with the claim that firms should diversify to exploit the Structure, Scale, and Scope advantages of scope? To anticipate my response, I first describe research whose premise is that firms specialize to differentiate; next, I consider studies that imply that they should not diversify because of the complexities of scope; I then bring these views together by suggesting reasons to expect the effect of scope to vary curvilinearly over the distribution of relative size. The idea that specialization reduces competitive pressure has appeared in several literatures. In an early institutionalist work, Selznick (1957:42-44) noted the importance of a “distinctive competence” when dealing with the inimical interests of other organizations. His classic (1949) study of the Tennessee Valley Authority described how the agency narrowed its focus, giving up many of its original goals to the control of local interests to preserve its most important programs, such as the development of electric power facilities. More recently, ecologists have noted that specialized firms enjoy advantages when competing with their diversified counterparts. These include the absence of organizational slack and the many benefits of focusing resources and time on a particular segment of the market (Freeman and Hannan 1983; Carroll 1985). Carroll’s model hinges on the idea that competition makes specialization necessary. A firm’s focus on a single customer segment means it has differentiated itself from its competitors. In related work, Baum and Haveman (1997) developed a model of organizational foundings, which focused on the choices of new firms to pursue distinctive locations congenial to their longevity. Taking a formal approach, White’s (1981; 2001) theory of markets rests on the notion that firms establish inimitable identities to turn a profit and survive. Each firm specializes by procuring a unique bundle of inputs so that its cost structure and price points attract a set of consumers whose tastes differ from those buying from its rivals. White’s model, which draws on Chamberlin’s (1962) theory of product differentiation, carries strategic implications based on the benefits of a distinctive reputation and set of capabilities. Central to the model is the idea that Structure, Scale, and Scope specialization raises a firm’s life chances by conferring first-mover advantages in its current role, reducing the competition it would otherwise face if it migrated onto the roles of other firms. Conversely, other studies offer a very different reason for not diversifying. They focus not on the shelter found by specializing, but on the negative effect of diversification on a firm’s ability to perform its tasks and to learn. This could be called an intra-firm perspective on the effects of scope. Teece (1980:232-233) argued that the advantages of scope have limits, noting that as a firm transfers the knowledge it gained in one market to its other units, evaluating and acting on that knowledge may grow increasingly costly. Such costs surface because of “a congestion factor that may attend the transfer process... know-how is generally not embodied in blueprints alone: the human factor is critically important. Accordingly, as the demands for sharing know-how increases, bottlenecks in the form of over-extended scientists, engineers, and manufacturing/marketing personnel can be anticipated.” In related work, Chandler (1990) noted that a firm could not achieve scope economies just by leveraging its technologies and resources; managers who are highly skilled and arranged in a hierarchy of roles figure centrally in his account. Consequently, because of the human propensity for error, diseconomies of scope are always a looming possibility. Similarly, Barnett, Greve, and Park (1994) discussed the costs of diversification in terms of firm-level learning. Their insights imply that as a firm diversifies, it will face weaker incentives to better its practices due stronger protection from competition; suffer losses in local adaptability because of its growing lack of focus; and have greater difficulty using the skills it acquired in one market in others. The literature thus contains three views of diversification: (1) it furthers performance because of scope economies; (2) it weakens performance by removing protection from rivalry; and (3) worsens performance by creating inefficiencies within the firm. Can relative size clarify the matter? Structure, Scale, and Scope Earlier, I stressed the benefits of size in terms of scale economies. But since Weber (1924), sociologists have also studied the link between size and bureaucracy and in turn the negative effect of size on performance (Whetten 1987). Michel’s (1966) classic study showed that large size causes even the most unlikely organizations to ossify. More recently, Haveman (1993) found support for the claim that changes in strategy are difficult for large firms due to creeping bureaucracy and attendant inflexibilities. Barron (1999) found evidence of a “liability of bigness” in models of firm growth, showing that the negative effect of absolute size on growth worsened with the number of firms in the industry. This research suggests that size may elicit and intensify the inefficiencies that result from diversification. In her influential book, Penrose (1959:206) noted that, “The large and diversified firms, although undoubtedly wielding much power and occupying strong monopolistic positions in some areas, do not, as far as we can see, hold their position without extensive managerial effort.” Especially in the computer industry, firms who are exceedingly large relative to their rivals, such as Compaq, Apple, and IBM, may become vulnerable to those rivals the more they themselves widen their reach. By diversifying, they may face problems with coordinating tasks, obstacles to learning, and inertia, which in turn weaken their position relative to their smaller counterparts and thus lower their rates of growth. But I expect that the “interference” of relative size with scope will occur only for the largest values of relative size. Consistent with hypothesis two, increments in scope should raise performance of at least certain kinds of firms. Comparably large firms may be able to expand by moving into new markets. When vendors consider diversifying, one of the first questions managers ask is whether they have the scale advantages necessary to compete with the members of those markets. I also expect that because of scale-based competition, the best course of action for relatively small firms is to target a specific segment. Specialization for them is optimal Structure, Scale, and Scope insofar as it enables them to gain a competitive advantage over larger, strategically proximate firms by learning their markets better. Considered together, these observations suggest the effect of scope on sales growth will follow an inverted U-shaped pattern over the distribution of relative size. I expect that the small firms suffer from increments in scope, moderately large firms benefit from these shifts, and, finally, that the growth of exceedingly large firms also declines when they diversify further: H3: The effect of scope on sales growth follows an IUS pattern over the distribution of relative size. 2. Data and Measures The International Data Corporation (IDC) assembled the data I use in this paper. IDC is the largest data consultancy worldwide to information technology firms and industries. With over 575 analysts and research centers in 43 countries, IDC collected shipment and selling price data for over 400 vendors since the start of its quarterly tracking program. While not complete, its coverage of the global PC industry is nearly exhaustive. The vendors tracked accounted for 83% of the worldwide PC sales over the course of my observation window, which starts with the first quarter of 1995 and ends at the first quarter of 1999. Most consumers of the data are makers of PCs, who use the data to locate their positions relative to their rivals, follow trends in specific segments, and make decisions about market entry. IDC reports quarterly sales as well as shipment breakdowns for each vendor by national market, technological emphasis, and distribution channel. It tracked fifty-seven national markets, ranging from Canada and the U.K. to Japan and Chile. This number includes five aggregated regional markets, such as “rest of Asia Pacific” and “rest of Latin America,” to cover areas in which demand is not individually tracked. Structure, Scale, and Scope Coupled with a firm’s implication in national markets, its choice of “form-factor” technology and channel defines its market position. These “form-factor” types—literally concerning the “form,” or appearance, of the product—are desktops, notebooks, sub-notebooks, and servers. IDC also codes the units shipped by each firm as belonging to one of five channels: (1) direct inbound, (2) direct outbound, (3) reseller, (4), retail, and (5) other. Machines flow through the direct inbound channel if the buyer initiates the transaction by phone, Internet, or a vendor-specific catalog. Conversely, the direct outbound channel is marked by the use of a sophisticated in-house sales force. A well-known example is the IBM representative selling to large corporations. When buyers require unusually specialized solutions, they often turn to the reseller channel. In IDC’s coding scheme, this category includes dealers, system integrators, and value-added resellers. The retail channel consists of well-known chains, such as Circuit City in the U.S. and Dixons in the U.K. Lastly, the fifth “other” channel combines a number of distinct outlets, such as catalog sales and military exchanges. With IDC’s coverage of 57 national markets, 4 form-factor categories, and 5 channels, there are 1,140 possible segments in which vendors ship PCs. A virtue of this dataset is that it includes time-varying information on each firm’s strategy. Such data are sufficient for defining each firm’s unique set of closest competitors, which I do with the techniques of social network analysis (Burt and Carlton 1989). My measure of relative size hinges on structural equivalence between firms having market contact. Market contact occurs between firms i and j at time t if they “meet” by selling jointly in at least one of 1,140 possible technology-by-channel-by-nation market segments. Even in the same country, if firm i only sells PC servers—entirely through the direct outbound channel—whereas j sells only notebooks—solely through retail chains—then i and j have zero contact. Under that condition, I assume that they do not affect each other’s rates of growth. After Structure, Scale, and Scope defining market contact as a binary outcome, the next step in quantifying relative size is to weight by the degree of structural equivalence between firms i and k. I denote i’s alters by k, rather than j, because i is likely to have contact with less than the total number of firms in the panel at t, so that max(k) ≤ max(j). Consider a well-known vendor, such as IBM, for illustration. IBM shares segments with Compaq, which suggests that one’s level of sales at t affects the other’s rate of growth at t+1. But IBM overlaps in segments with scores of other firms k at t—all of which are more or less structurally equivalent to IBM than Compaq. Dell and Everex are also taken to bear on IBM’s performance, for instance. Consequently, in quantifying IBM’s relative size, I follow a known strategy in social network analysis (Burt 1987; Strang and Tuma 1993) by allowing the sales of these firms k to receive weights proportional to their degree of equivalence to IBM. The relative size of the ith firm at time t thus takes the form:

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تاریخ انتشار 2002