Kenneth E. Boulding

Toward a General Theory of Growth

(1953)

 


Note

This paper was presented at the annual meeting of the Canadian Political Science Association in London, Ontario, June 3, 1953.

 


 

The growth phenomenon is found in practically all the sciences and even in most of the arts, because almost all the objects of human study grow - crystals, molecules, cells, plants, animals, children, personalities, knowledge, ideas, cities, cultures, organizations, nations, wealth, and economic systems. It does not follow, of course, from the mere universality of the growth phenomenon that there must be a single unified theory of growth which will cover everything from the growth of a crystal to the growth of an empire. Growth itself is not a simple or a unified phenomenon, and we cannot expect all the many forms of growth to come under the umbrella of a single theory. Nevertheless all growth phenomena have something in common, and what is more important, the classifications of forms of growth and hence of theories of growth seem to cut across most of the conventional boundaries of the sciences. In addition there are a great many problems which are common to many apparently diverse growth phenomena.

It is convenient to start with a threefold classification of growth phenomena. We have first what might be called simple growth, that is, the growth or decline of a single variable or quantity by accretion. or depletion. In all that follows it should be understood that growth may be negative as well as positive, decline being treated merely as negative growth. In the second place we have what might be called populational growth, in which the growing quantity is not regarded as a homogeneous aggregate, but is analysed into an age distribution. Growth is regarded as the excess of "births" (additions to the aggregate) over "deaths" (subtractions from the aggregate), and the analysis of the process is conducted in terms of functions which relate births and deaths to the age distribution. Finally we have what might be called structural growth, in which the aggregate which "grows" consists of a complex structure of interrelated parts and in which the growth process involves change in the relation of the parts. Thus in the growth of a living organism, or of an organization, as the "whole" grows, the form and the parts change: new organs develop, old organs decline, and there is frequently growth in complexity as well as in some over-all magnitudes. Problems of structural growth seem to merge almost imperceptibly into the problems of structural change or development, so that frequently "what grows" is not the over-all size of the structure but the complexity or systematic nature of its parts. Thus the "growth" of a butterfly out of the chrysalis involves an actual decline in over-all magnitudes such as weight or volume, but certainly seems to come under the general heading of phenomena of growth or development.

These three "forms" of growth constitute three different levels of abstraction rather than a classification of actual growth phenomena. Growth phenomena in the real world usually involve all three types. Thus a phenomenon of simple growth such as, for instance, the growth of a capital sum put out to interest, or the growth in the inventory or stocks of a single commodity are in fact part of, and ultimately dependent upon, much more complicated structural processes. Similarly, populational growth as in the case, say, of a human population, never takes place without changes in the organizational structure of the society, that is, in the kinds and the proportions of its "parts" - its organizations, jobs, roles, and so on. Thus all actual growth is structural growth; nevertheless, for some purposes of analysis the structural elements may be neglected and the growing aggregate can be treated as a pure population, and for other purposes even the populational aspects can be neglected and the growth can be treated as simple growth.

II

Turning first then to the analysis of simple growth, the main problem here is that of finding a "law" of growth which will serve to describe the growth curve, that is, which will express the size of the growing variable as a function of time. Perhaps the simplest case of simple growth is growth at a constant rate, for example, the growth of a capital sum at a constant rate of interest. In this case the growth function is the simple exponential Pi = Po (1 + i)2, where Po is the original sum, Pi the amount into which it has grown in t years at a constant rate of growth i, growth being added at the end of every year. If growth is continuous the function becomes Pi = Poeit.

Continuous growth at a constant rate, however, is rare in nature and even in society. Indeed it may be stated that within the realm of common human experience all growth must run into eventually declining rates of growth. As growth proceeds, the growing object must eventually run into conditions which are less and less favourable to growth. If this were not true there would eventually be only one object in the universe and at that point at least, unless the universe itself can grow indefinitely, its growth would have to come to an end. It is not surprising, therefore, that virtually all empirical growth curves exhibit the familiar “ogive” shape, the absolute growth being small at first, rising to a maximum, and then declining eventually to zero as the maximum value of the variable is reached. Many equations for such a curve have been suggested, though none seem to rest on any very secure theoretical foundation. The most familiar is perhaps that of Raymond Pearl, which graphically is a cumulative normal frequency curve. In any such equation the most important constants are (i) one which measures the total amount of growth, that is, the difference between the initial and the maximum value of the variable, and (ii) one which measures the time taken to grow from the initial position to a value reasonably close to the maximum. All that growth equations can do, however, is to describe growth; they are never capable of interpreting or understanding it.

[…]

The analysis of structural growth is much more complicated than the analysis of populational growth, and is much more difficult to reduce to a neat set of propositions. Structural growth includes such complex phenomena as the growth of crystal structures, the growth, division, and differentiation of cells, the growth of organisms, of organizations, of language and other mental structures, of buildings, and of societies. We would hardly expect such diverse phenomena to reduce themselves to a uniform simple scheme. Nevertheless it is striking how many of the problems which rise to special prominence in one field of study also carry over into others, and it is not impossible to formulate some principles of great generality.

The first of these we may call the principle of nucleation, following a term which comes originally from physics. Any structure has a minimum size, which is its "nucleus." Once a nucleus has been formed, it is not too difficult to understand how additions to the structure are made. The formation of the nucleus itself, however, presents many problems which are quite different from those involved in the growth of an already established structure. Thus there is a minimum size of crystal, depending on its complexity: in smaller aggregations of atoms than this minimum there are not enough components to make up the minimum structure. In the case of the cell the problem of nucleation is almost completely unsolved: as far as we now know, all living matter grows from living matter. We know something about how the complex societies of subordinate structures which comprise the living nuclei can grow, divide, and differentiate as "going concerns." We know practically nothing about how such an immensely complex organization ever came to be established in the first place, and up to the present we have not been able to reproduce that mysterious initial act of nucleation. In the social sciences the nucleation problem also exists; here it manifests itself as the problem of innovation, in the Schumpeterian sense. Once a form of organization is established, whether a type of production or a kind of enterprise, a cult, a school, a party, or any distinct type of institution, it is not too difficult to imitate. The initial innovation, however, is something of a mystery, which does not fall easily within the smooth rubrics of historical necessity. We do not understand very well what it is that makes the genuine innovators - those mysterious individuals who establish religions, cultures, nations, techniques, and ideas. In society as in physics, however, we find that very small amounts of "impurities" (say one Edison per hundred million) produce effects fantastically disproportionate to their quantity, because of the nucleation principle. It is also not perhaps too farfetched to speak of "super-cooled societies," where the external conditions call for change to a new state of society, but where the rate of nucleation is so small that the change does not take place.

Another principle which emerges from the study of nucleation is that the nucleus does not have to be homogeneous with the structure that grows around it; thus the speck of dust at the heart of the raindrop! Consequently impurity in a universe is a very important factor in explaining change. Examples of this principle are numerous in society. Colleges nucleate around sects, farm organizations around county agents, trade unions around socially minded priests. One cannot be a student of society for long without observing what I have sometimes called the "Pinocchio principle." Some agent or organization sets up a "puppet" in the form of some other kind of organization. Before long however the "puppet" begins to take on a life of its own, and frequently walks right away from its maker. Here we see the principle of "heterogeneous nucleation" actively at work in society. The principle of nucleation applies even to the learning process, which can be thought of as the growth of a "mental structure." Thus in learning a language it is necessary for the language to "nucleate" in the mind of the learner before it can become anything more to him than an unusable bunch of unrelated words. Even in learning economics the student frequently finds that for the first few months the subject makes practically no sense to him, and then quite suddenly he experiences a "conversion"; what had previously been disconnected parts somehow fall into place in his mind, and for the first time he "sees" the subject as an organic whole. The application of this idea to the phenomenon of religious or political conversion would make a most interesting study.

The second general principle of structural development might be called the principle of non-proportional change. As any structure grows, the proportions of its parts and of its significant variables cannot remain constant. It is impossible that is to say, to reproduce all the characteristics of a structure in a scale model of different size. This is because a uniform increase in the linear dimensions of a structure will increase all its areas as the square, and its volume as the cube, of the increase in the linear dimension. Thus a twofold increase in all the lengths of a structure increases its areas by four times and its volumes by eight times. As some of the essential functions and variables of structure depend on its linear dimensions, some on its areal dimensions, and some on its volumetric dimensions, it is impossible to keep the same proportions between all the significant variables and functions as the structure grows.

This principle has two important corollaries. The first is that growth of a structure always involves a compensatory change in the relative sizes of its various parts to compensate for the fact that those functions and properties which depend on volume tend increasingly to dominate those dependent on area, and those dependent on area tend increasingly to dominate those depending on length. Large structures therefore tend to be "long" and more convoluted than small structures, in the attempt to increase the proportion of linear and areal dimensions to volumes. It follows therefore that as a structure grows it will also tend to become longer and more convoluted. Architecture and biology - two sciences which are much more closely related than might appear at first sight - provide admirable examples. A one-room schoolhouse, like the bacterium, can afford to be roughly globular and can still maintain effective contact with its environment - getting enough light and nutrition (children) into its interior through its walls. Larger schools, like worms, become long in relation to their volume in order to give every room at least one outside wall. Still larger schools develop wings and courtyards, following the general principle that a structure cannot be more than two rooms thick if it is to have adequate breathing facilities. This is the insect level of architecture (skin-breathing). The invention of artificial ventilation (lungs) and illumination (optic nerves) makes theoretically possible at any rate much larger structures of a "globular" or cubic type, with inside rooms artificially ventilated and lit, just as the development of lungs, bowels, nerves, and brains (all involving extensive convolution to get more area per unit of volume) enabled living matter to transcend the approximately three-inch limit set by the insect (skin-breathing) pattern. In the absence of such devices further growth of the structure involves splitting up into separate buildings (the campus) of which the biological analogy is the termite or bee colony.

The second corollary follows immediately from the first: if the process of compensation for structural disproportion has limits, as in fact seems to be the case, the size of the structure itself is limited by its ultimate inability to compensate for the non-proportional changes. This is the basic principle which underlies the "law of eventually diminishing returns to scale" familiar to economists. Thus as institutions grow they have to maintain larger and larger specialized administrative structures in order to overcome the increasing difficulties of communication between the "edges" or outside surfaces of the organization (the classroom, the parish, the retail outlet) and the central executive. Eventually the cost of these administrative structures begins to outweigh any of the other possible benefits of large scale, such as increasing specialization of the more directly productive parts of the organization, and these structural limitations bring the growth of the organization to an end. One can visualize, for instance, a university of a hundred thousand students in which the entire organization is made up of administrators, leaving no room at all for faculty.

It is interesting to note that the principle of compensation may operate in two very distinct ways - in the direction of attempt to solve the problems posed by large scale or in the direction of an attempt to avoid these problems. Thus the critical problem of large-scale organization is that of the communications system (nerves and blood!). This being a "linear" function tends to become inadequate relative to the "surface" functions of interaction and production as the organization grows. One method of compensation is to increase the proportion of the organization which is devoted to the communications system as the organization grows larger (the modern army, David Reisman has remarked, marches not on its stomach but on the punched card). Another method however is to diminish the need for communication by developing autonomy of the parts or rigid and ritualistic patterns of behaviour. Thus a very large church organization, such as the Roman Catholic Church, not only has to permit a good deal of autonomy to its various parts (in this case, national churches), but can only maintain its structure at the cost of extreme rigidity in its basic operations. The Pope doesn't have to communicate with the priest in the wilds of Bolivia because the mass is removed from the agenda of discussion! In this connection it might be noted also that great size in itself leads to relative invulnerability, and hence very large organizations do not face the same problems of uncertainty and adjustment which face smaller organizations or organisms. The whales and elephants of the universe can afford to have fairly placid dispositions and insensitive exteriors.

A third principle of structural growth follows somewhat from the second, but is sufficiently distinct to warrant a separate status. This might be called the D'Arcy Thompson principle, after its most famous exponent (D'Arcy W. Thompson, On Growth and Form, 2nd ed., Cambridge, 1952). It is the principle that at any moment the form of any object, organism, or organization is a result of its laws of growth up to that moment. Something which grows uniformly in all directions will be a sphere. Something which grows faster in one direction than in others will be "long." Something which grows faster on one side than on the other will twist into some sort of spiral. The shape of an egg is related to its flow down the oviduct. Examples can be multiplied almost indefinitely, and the contemplation of these beautiful and subtle relationships is perhaps one of the most refined delights of the human mind. It is clear that the principle applies in general not only to organisms, but to organizations, though its applications here are more complex and perhaps less secure. An aggressive chairman or department head will cause his department to grow relative to the total structure. Scientific inquiries follow growth patterns which are laid down by previous studies and by the interests of scientists. Economic and technological development follows patterns which in turn determine the structure of an economy. Law grows like a great coral reef on the skeletons of dead cases.

Growth creates form, but form limits growth. This mutuality of relationship between growth and form is perhaps the most essential key to the understanding of structural growth. We have seen how growth compels adjustments to changes in relative proportions. It is also true that occasionally growth stops because of “closure” - because the growth itself seals off all the growing edges. We see this frequently in the world of ideas where self-contained ideologies (such as Marxism) exhibit closure in the sense that any development outside the narrow circle of the self-contained system is inhibited. It is often the “loose ends” of systems that are their effective growing points - too tight or too tidy an organization may make for stability, but it does not make for growth. This is perhaps one of the most cogent arguments for toleration in the political sphere; the morbid passion for tidiness is one of the greatest enemies of human development.

There is, I believe, a fourth principle of structural growth, detailed evidence for which it is difficult to find in the biological sciences, but which is clearly apparent in the growth of man-made structures, especially, oddly enough, in building. It may perhaps be called the "carpenter principle." In building any large structure out of small parts one of two things must be true if the structure is not to be hopelessly misshapen. Either the dimensions of the parts must be extremely accurate, or there must be something like a carpenter or a bricklayer following a "blueprint" who can adjust the dimensions of the structure as it goes along. If, for instance, we are building a structure out of a thousand identical parts, the tolerable variability of each part would have to be about one thousandth of the tolerable variability of the whole structure, if we were simply adding part to part without making any adjustments on the way. It is only possible, however, to build a brick wall or a wooden house, with bricks and with boards that are in themselves highly variable, if there is exercised during the process of growth a "skill" of adjustment, that is, of adjusting the structure as it grows in conformity with some plan or requirement. It may perhaps be hazarded that the genes perform some such function in the growth of living organisms, though what is the machinery by which these little carpenters of the body operate is still unknown. In social and intellectual structures, however, the principle is of great importance, for the whole development of these structures may be affected profoundly by the existence of a “plan” and of an apparatus by which the growing structure is constantly conformed to the plan. The construction of a building, a machine, or a bridge is an obvious example. The process of academic learning is another example. In the learning of a language, for instance, the existence of a grammar exercises a profound effect on the capacity of the student to develop originality. In any subject, the presence of a textbook exercises somewhat the same influence as a grammar on the process of learning, and the instructor is the intellectual carpenter, trimming the student's mind off at frequent intervals by means of quizzes and examinations. What we have here is essentially a homeostatic process, the divergence which excites action being the divergence at any time between the actual condition of the structure and the "planned" condition.

Even where a detailed “plan” of development is not available the existence of some kind of ideal structure with which present reality may be compared exercises a profound influence on growth, especially in the sciences. One need only cite, for instance, the influence exerted on the progress of chemistry by the existence of the periodic table of elements, an ideal structure which in the beginning had many “holes” (undiscovered elements), but which was gradually filled up as the process of chemical research was directed into filling the obvious gaps in the structure. It may, indeed, be regarded as one of the prime functions of "theory" in any field to set up these ideal structures which are in fact incomplete. In this connection it may be observed that one of the principal advantages which may be derived from interdisciplinary research is the development of a “general theory” which will fulfil this function of the "ideal structure" for knowledge as a whole.

A fifth principle of structural growth emerges fairly clearly from economics, though here also its application to the biological or physical sciences is quite unproved. This may be called the "principle of equal advantage." It governs the distribution of the "substance" of a structure among the various parts of the structure. We assume first that the "atoms" of a structure can be ordered according to a parameter which we label, for want of a better term "advantage," this being defined operationally as a "potential" such that units will tend to flow towards locations of higher, and away from locations of lower, advantage. We then postulate that the advantage of a unit in any location is an inverse function of the relative quantity of units in that location - that is, the fewer the merrier! This implies a concept something like the "demand" for any given location - demand in this case fulfils something of the function of the "plan" under the fourth proposition. We then postulate a principle that "demand" tends to be satisfied, that is, if there are differences in advantage to units ("industries") in different locations, units will tend to move from the low-advantage locations where there are "too many" to the high-advantage locations where there are "too few."

In economic systems the principle of equal advantage, and the fact that the "advantage" parameter can be approximately related to monetary reward, enables us to give reasonably satisfactory explanations of two phenomena which are common in biology, but very little understood. These are the phenomena known as functional substitution and regeneration. If one organ of a living organization is removed, there is an observable tendency for other organs to take over the function of the missing organ. At lower levels of life, also, the organism seems to have a remarkable capacity for the regeneration of lost parts; tadpoles can grow new tails and starfish new limbs. The healing of wounds and of broken bones is an example of the same phenomenon at a somewhat less spectacular level. In the economic system it is possible to observe fairly closely what happens when an "organ" (industry) is cut off, say, by prohibition. In the first place if regeneration of the severed industry is prevented by the cauterizing action of law, other "industries" will begin to take over the functions of the destroyed industry (speakeasies, bathtub gin, etc.). If now the prohibition is removed, the old industry rapidly grows back again into the body economic, under the stimulus of profit (advantage). One may again hazard a guess that biologists will have to look for a variable akin to profit at the level of the cell if they are fully to understand the phenomenon of functional substitution and regeneration.

It is interesting to observe that the "carpenter principle" and the "advantage principle" are to some extent alternative ways of effecting the organization of a society, the first, of course, corresponding to the communist planned economy and the second to the capitalist market economy. In the planned economy, growth is organized by the principle of conformity to the plan: industries which are lagging behind the plan get extra attention to make them catch up, much as a child is taught in an authoritarian educational system. In the market economy, growth is organized by the principle of advantage: it takes place in the directions that "pay off" to the individuals who initiate it or are able to take advantage of its initiation, the “pay off” being derived from "profit," that is, an excess of value of product over cost. There is a certain analogy here perhaps with "progressive" education.

We have not, of course, exhausted a subject so ramified and so universal as growth in five summary propositions, though it may be hoped that there is here presented some indication that general theories of growth are possible. It remains in the compass of a short paper to indicate some of the loose - and therefore growing! - ends of the subject. We have not, for instance, considered the problem of the possibility of equilibrium rates of growth in an organism or system such that higher (or lower) growth rates may seriously disturb the functioning of the system even to the point of its collapse and "death." There is evidence in the plant world that too rapid a rate of growth kills the organism; indeed, some very effective weed killers have been developed on this principle, using growth hormones. The growth theories of Harrod and Domar suggest that in the economic system there are "appropriate" rates of growth of a system as a whole which will yield continuous full employment. These theories also suggest, however, that there is nothing inherent in the nature of an unstabilized market economy which will guarantee these appropriate rates of growth. There is a suggestion, indeed, that under some circumstances a continuous equilibrium rate of growth may be impossible, because certain elements, consumption, for example, do not keep pace with the rise in capacity and so force accelerated growth on other elements, such as investment, if the system is to maintain full employment. These problems may all turn out to be problems in compensation for structural changes through increasing size, such as we have noted earlier, but particularly where equilibrium seems to require acceleration in certain growth rates some quite peculiar problems may be involved.

The problems of the transition from rapidly growing systems to more or less stationary ones are also very general, and need careful study at many levels. The character of a system frequently has to change not merely because it gets big, but because it stops growing. Thus when a religious movement passes from its initial phase of rapid expansion into the phase of slow or even negative growth it has to make profound adaptations: what is appropriate organization in a "movement" is not necessarily appropriate in a "sect." Any system which passes from a rapidly growing phase to a more stationary one, whether it is a religion, a labour organization, a business firm, a nation, an economic system, or a civilization comes face to face with somewhat the same kind of adjustment. David Reisman makes the growth phase of a culture the principal determinant even of the typical character of its individuals, attributing "tradition-directed" behaviour to low-level, slowly growing, or stationary populations with high growth potential, "inner-directed" behaviour to rapidly growing populations, and "other-directed" behaviour to high-level, slow-growing, or stationary populations. These connections may seem a little far-fetched, but there can be no doubt that the type of growth which any system exhibits will affect most if not all of its major characteristics.

In conclusion one may hazard a guess as to the growth patterns of the sciences. The remarkable universality of the principles enunciated here in regard to a general theory of growth indicates that perhaps there is emerging from the welter of the sciences something like a "general theory," something which is a little less general and has a little more empirical content than mathematics but which is more general and therefore, of course, has less content than the content of specific sciences. Mathematics is itself, of course, a "general theory" in that it applies wherever quantitative concepts and relationships are encountered. The sort of general theory which I have in mind, however, is a generalization from aspects of experience which include more than mere abstract quantity and which are common to many or even to all of the "universes of discourse" which constitute the various sciences. Growth is one such aspect: organization is another; interaction is another. When, and if, such a general theory comes to be written it will be surprising if the general theory of growth does not constitute an important chapter.

 


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