Carrying Capacities and Density Dependence of Firms in Organizational Populations

In developing the LV model in the previous chapter, we noted that carrying capacities are simple functions of parameters expressing density dependence in rates of birth and death. The existence of a finite carrying capacity depends on the assumption that the birth rate falls with density and that the mortality rate rises with density. Density serves as a surrogate for the difficult-to- observe features of the material and social environment that affect the rates, particularly competition and legitimacy. Now we consider organizational applications of such models.

1. Founding Rates

We think that density affects founding rates through several processes. Knowledge about organizational strategies and structures is often available only to “insiders,” that is, to those already participating in such organizations. This is commonly the case when organizational functioning is shielded from public observation and when essential features of the organizational form have not been codified. In such situations, existing organizations are the only training grounds for knowledgeable organization builders. The number of foundings in such populations depends on the number of jobs in existing organizations that give the requisite training (Brittain and Freeman 1980). When the number of such organizations is small, the founding rate is depressed as a result of the absence of potential organization builders. Marrett (1980) argues that high density increases the founding rate by widening and strengthening the networks that connect individuals with the inclination and skills to succeed in creating a certain kind of organization.

Institutional processes also link density and founding rates. If institu-tionalization means that certain forms attain a taken-for-granted character, then simple prevalence of a form tends to give it legitimacy. When numbers are small, those who attempt to create a form must fight for legitimacy; they must argue both for the special purposes of a proposed organization and for the design of the form. Once many instances of the form exist, the need for elaborate justifications diminishes. Reducing the need for such justifications reduces the cost of organizing. Other things being equal, legitimation of a form increases the founding rate of populations using that form. If, as we argue here, legitimacy increases with prevalence of the form in society, then legitimation processes produce a positive relationship between density and founding rates.

By contrast, competition within and between populations induces a neg- ative relationship between density and founding rates. Given a set of envi- ronmental conditions that set a carrying capacity, the more abundant the competitors, the smaller will be the potential gains from founding an orga- nization at a given level of demand for products and services. Fewer resources are available, and markets are packed tightly in densely populated environments. For these reasons, collectives with the knowledge and skills to build organizations are less likely to make attempts in densely populated environments. Capital markets and other macro structures often reinforce this effect. For example, investors may be reluctant to participate in new ventures in dense markets. Likewise, professional associations often try to restrict entry when density is high.

In general terms, high density implies strong competitive interactions within populations dependent on limited resources for organizing (when levels of such resources have been controlled). As density grows relative to the current level of the carrying capacity, supplies of potential organizers, members, patrons, and resources become exhausted. Moreover, exist-ing organizations respond to increasing competitive pressures by opposing attempts at creating still more organizations. For both of these reasons, the founding rate declines as the number of organizations in the population increases.

Use of a simple model of the process helps convey the main point of our argument (see Hannan 1986b). To keep the exposition simple, assume that legitimacy (L) and competition (C) are the only relevant factors. That is, assume that we have already adjusted for the changing environmental conditions that affect carrying capacities. We propose that the founding rate at time t, λ(t), is proportional to the legitimacy of the population and inversely proportional to the level of competition within the population:

In addition, we think that it is often useful to assume that levels of legitima- tion and competition are functions of density (N):

The crucial modeling questions concern the forms of these relationships. Unfortunately, institutional and ecological theories have not addressed these issues. Our understanding of the process is that both of the relationships in (6.2) are nonlinear.

Consider the dependence of legitimacy of an organizational form on the number of copies of the form in existence. From the perspective of legitimacy as taken-for-grantedness, it seems clear that extreme rarity of a form poses serious problems of legitimacy. If almost no instances of a form exist, how can it be taken as the natural way to achieve some collective end? On the other hand, once a form becomes common, it seems unlikely that increases in numbers will have a great effect on its institutional standing. Therefore, we assume that legitimacy is sensitive to variations in density in the lower range but that there is something like a ceiling effect on the relationship, that is, that density increases legitimacy at a decreasing rate:

In the case of competition, variations in the upper range have more impact on founding rates than variations in the lower range. When num-bers are few, addition of an organization to the population increases the frequency and strength of competitive interactions slightly at most. But when density is high, the addition of an organization strongly increases competition. So we assume that density increases competition at an increasing rate:

Moreover, we propose that the legitimacy process dominates when N is small but that the competition process dominates when N is large. That is, the effect of density on the founding rate is non-monotonic. This is the core idea of the modeling effort.

The next step in moving toward empirical research is to build a model using particular functional forms. We use simple functional forms that are consistent with these assumptions and with the restriction that a transition rate must be non-negative. In the case of legitimacy, we propose that legitimacy increases with density according to a power law:

As long as the inequalities on parameters are met, the level of legitimacy increases with density but at a decreasing rate. In the case of the level of competition, we assume an exponential relation between competition and the square of density:

As long as both parameters are positive, the level of competition increases at an increasing rate as required.

Inserting equations (6.3) and (6.4) into equation (6.1), the simplified model of the founding rate, yields the basic parametric model:

where Φ(t) = a(t)a/y. As we noted earlier, we chose this model because its qualitative behavior agrees with the theory. In particular, it implies that there is a non-monotonic relationship between density and the founding rate. To see this, note that

There is a point of inflection (a maximum) at

The founding rate rises as density increases until N reaches the level indicated in (6.6); from that point on, the founding rate falls with increasing density.

In empirical testing the key issue is whether this non-monotonic model represents an improvement over a simpler model with only monotone dependence on density. A secondary but still interesting issue is whether the point of inflection falls within the observed range of density. So in estimating models with this form, we check first to see that the estimated parameters have the predicted signs and second whether the implied behavior of the process over the range of density is non-monotonic. If the founding rate rises initially and then falls with increasing density, the process implies the existence of a carrying capacity for the population.22 So it is important in evaluating this theoretical approach to learn whether founding rates vary with density and, if they do, whether the dependence is nonmonotonic.

This model of density dependence in founding rates is more complicated than the corresponding assumption in the LV model. As we explained in the previous chapter, the LV model is built on the assumption of linear density dependence in vital rates, whereas our model implies non-monotonic dependence on density. This added complexity reflects the greater complexity of organizational ecology.

Next we consider how density affects organizational mortality. Because we suspect that density affects the two main components of the mortality rate differently, so we consider the rate of disbanding and the rate of merger separately.

2. Disbanding Rates

We assume that the processes by which density affects founding rates also apply to disbanding rates, with some modifications. The modifications follow from the observation that founding rates pertain to attempts and the absence of attempts at creating organizations while disbanding rates per-tain to existing organizations. Disbanding rates presumably depend on properties of individual organizations such as age and size as well as on population characteristics such as density. But founding rates depend only on population characteristics, since a founding that does not occur cannot be associated with an organization’s characteristics.

Some of the processes by which density affects disbanding rates operate mainly at low densities and others at higher densities. At low densities, the growth of populations of organizations is constrained by the novelty and rarity of the form. The fact that there are few organizations in the population presumably makes it difficult to convince key actors such as banks and government agencies to transfer material and symbolic resources to orga- nizations in the population. It may likewise be difficult to convince talented people to join such organizations and to remain in them.

Earlier we suggested that rarity of an organizational form undermines its legitimacy. Most organization theorists assume that legitimacy decreases disbanding rates (Meyer and Rowan 1977). Thus, increasing density will lower disbanding rates by increasing the legitimacy of the form and of populations using the form. Low density also hampers attempts at coordinated political action to protect and defend claims of the population or of some of its members. Increases in numbers alleviate these problems. Growth in numbers of organizations gives force to claims of institutional standing and also provides economies of scale in political and legal action. That is, increases in numbers lower the disbanding rate.

At high density, competitive interactions intensify. Growth in numbers increases the likelihood and intensity of both direct competition between pairs of organizations and diffuse competition among all or many of them. Individual organizations can easily avoid direct competition for members and other scarce resources when there are few organizations in the system. As the number of potential competitors grows, however, avoidance becomes more difficult.

For example, labor unions have competed for the services of skilled organizers and dedicated staff, political support and influence, attention from the news media, and so forth. In the late nineteenth and early twentieth centuries, such competition was often virulent. Sometimes it involved direct rivalry, as when two or more unions seeking to organize the same workers competed for support of and membership in a national federation, such as the AFL and CIO. More often the competition was diffuse—it had more the character of congestion than rivalry. As the number of unions grew large, more of the resources used to build and sustain unions were claimed by unions that could defend themselves against raids. Such diffuse competition lowers the life chances of new unions and also affects the life chances of existing unions. In other words, when density is already high, further growth increases disbanding rates, after controlling for the environmental conditions that affect carrying capacities.

We assume that the disbanding rate at age u, which we denote by μ(u), is proportional to the level of competition and inversely proportional to the legitimacy of the organizational form at the time that the organization reaches age u:

As in the case of the founding rate, we assume that competition increases with density at an increasing rate but that legitimacy increases with density at a decreasing rate.

There is no obvious reason for not using the parametric assumptions regarding the dependence of legitimacy and competition on density that we used for the founding process. However, it is unlikely that the parameters of each part of the process are identical for the two rates. So we merely repeat the assumed relationships using a different set of parameters to acknowledge possible differences between the processes:

With these assumptions the model for the disbanding rate is

where . What matters are the functional form and signs of the parameters. Again the crucial qualitative feature of the model is that the effect of density on disbanding rates is non-monotonic. The model implies that the disbanding rate falls with increasing density up to the carrying capacity and then rises with increasing density. (Note again the departure from the LV assumption of linear density dependence.)

We have found it difficult to obtain convergent estimates of the model in (6.7) using some of the data described in the next chapter. Therefore, we shifted to an alternative specification whose parameters have essentially the same qualitative interpretation:

Differentiating the rate with respect to density, we have

Again the model implies a point of inflection (a minimum), which this time is given by

Below this level of density the rate declines with increasing density; above this level the rate increases with increasing density.

It is worth noting that the argument pertaining to low founding rates and high disbanding rates at low densities mixes two kinds of processes. The first kind of process involves what we might call a strong form of low- density dependence. In this case, the fact of low density implies low founding rates and high disbanding rates at any age of the population. Examples of processes of this type are those that emphasize the role of existing organizations as training grounds for potential organization builders and those that call attention to the importance of numbers in collective actions to defend the interests of the organizations in the population. The second kind of process involves a weak form of density dependence in the sense that density stands as a surrogate for processes that unfold over the history of a population. The prime example of this type is the argument that organizational forms acquire a taken-for-granted status as their numbers grow large.

The distinction between the two kinds of density-dependence processes becomes important in those cases where density crashes after an organiza- tional form and population have acquired institutional standing. Consider the examples of the populations of wineries and breweries in the United States after Prohibition (Delacroix and Soit 1988) or the population of newspapers in Argentina after the protracted repression of the Rosas’ dictatorship (Delacroix and Carroll 1983). Does low density in such situations imply low founding rates and high disbanding rates net of the level of demand for the products and services provided by the population? It seems to us that the strong-form processes hold but that the weak-form processes may not. Whether a population of organizations facing these circumstances has low legitimacy may depend on the length of the period of low density. At the repeal of Prohibition many individuals still possessed skills in brewing and wine making, and the organizational forms were still widely known to potential investors and consumers. But what if Prohibition had lasted for several generations? We have not yet seen an analysis of an organizational population that answers this question.

These questions do not arise in our empirical studies because the density of the populations we studied grew monotonically over much of the history of the industries and then declined moderately. So we do not observe the combination of low density far from the initiation of the population.

3. Merger Rates

We also explore the effects of density on rates of merger among organizations in a population. However, we do not have any a priori model of the process. Merger is a more complicated kind of event than disbanding. Among other things, organizations merge both when they are failing and when they are succeeding. That is, some mergers are last-gasp attempts to avoid collapse of an organization; others are attempts to incorporate successful technologies and organizational forms, as when established giant firms acquire newer, innovative firms.

The semiconductor industry provides many examples of mergers in which firms outside the industry that are seeking access to technology acquire independent semiconductor companies. Examples are General Electric’s acquisition of Intersil and Honeywell’s acquisition of Synertek. It makes little sense to buy a badly run company if this is the motivation. Other mergers reflect the low valuation of merger targets—firms that are faring poorly can often be acquired cheaply. So the merger rate depends on organizational outcomes in a very complicated way.

We suspect that the merger rate also depends on density in a complicated way. In particular, merger rates may be affected by competition and legitimation processes that parallel those discussed for disbanding rates. But there is an additional consideration: as density increases, the number of potential partners for merger increases. Since availability of potential partners seems likely to increase the rate, the merger rate is likely to increase with increasing density in the lower range. But we suggested earlier that the disbanding rate decreases with increasing density in this range. Because of this complexity, we have not developed a model of the merger process.

4. Competition among Populations

The various processes responsible for density-dependence within populations have strong parallels in processes occurring between populations.

Just as the addition of an organization to a population affects the founding and disbanding rates in that population, the addition of an organization to a competing population may also affect those rates. These are both forms of density-dependence. The only distinction is whether the effect occurs within a population or across the boundary between populations.

In conceptualizing interactions among organizational populations, we retain the classic sociological distinction between competition and conflict. As Durkheim and Simmel insisted, conflict is a social relation that requires interaction between the parties. Parties to a conflict take each other into account. Competition, by contrast, is often indirect. The growth of one population of organizations may depress the growth of another even though the members of the two populations never interact directly. In fact, the members of the populations may not even be aware that they stand in a competitive relation if they compete indirectly for resources. In one classic form of competitive situation, the undominated market of many buyers and sellers, no actor needs to know the identity of its competitors.

Our analyses rely on competition between populations for limited material and social resources for building and sustaining organizations. We do not assume anything like a fully competitive market. Our theories hold even when the number of competitors is sufficiently small that the actions of one competitor can change the terms of trade significantly. In these circumstances, competition often causes conflict. There is no reason for us to assume the absence of conflict. In fact, we think that the usual state of affairs is for intense, localized competition to turn into conflict. However, we do not assume the existence of conflict as a precondition for our arguments.

We investigate whether the density of other populations affects the founding and disbanding rates of a target population. If such a link exists and the density of population B decreases the founding rate and/or increases the disbanding rate of population A, we infer that population A competes with B. If, in turn, the density of A depresses the founding rate and/or raises the disbanding rate of B, we have an example of classic reciprocal competition. But if the density of A raises the founding rate or depresses the disbanding rate of B, the interaction has the predator-prey form. Several other cases are possible, including symbiosis, in which the density of each population increases the founding rates and/or decreases the disbanding rates of the other.

Although there may be particular cases in which the legitimation of one population depends on the size of some other population, it seems unlikely that this is the case generally. However, whenever two populations seek to exploit the same limited resources, the density of each affects the strength of competitive interactions, as we noted earlier in discussing the LV model. Therefore, in developing a multi-population model, we specify only competitive effects between populations. It seems likely, as Lotka and Volterra assumed for the biotic case, that the strength of competitive interactions increases monotonically with density. So assume that the strength of competitive pressures on the first population is monotonic.

A simple parametric model consistent with these assumptions (and with the constraint that transition rates must be non-negative) is

This assumption, when combined with the assumption made earlier about legitimacy, implies the following form of density-dependence in the founding rate:

The cross-effect of density (β2) captures the effect of inter-population competition. Whenever the two populations use the same resources, the cross-effect is negative for both populations. We also add such cross- effects into models for mortality in the same way.

Source: Hannan Michael T., Freeman John (1993), Organizational Ecology, Harvard University Press; Reprint edition.

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