CHAPTER 5 Modern Capital Theory

Wicksell and others attempted to defend and extend Böhm-Bawerk’s approach (Lutz 1967; Ebeling 1997). With the advent of the Keynesian revolution, however, interest in capital theory waned. It revived slowly in the post-war period. In retrospect we may identify two main lines of development. One is the familiar neoclassical approach, in which capital simply came to be understood as an amorphous stock of production potential equal to K as an argument in a production function. The other is the resurgence, emanating from the contributions of Joan Robinson (1956) and Pierro Sraffa (1960), of the Ricardian classical approach (the neo-Ricardian School), in which capital and labor are not continuously substitutable for each other, and the earnings of labor relative to capital (the prime focus of this literature) are seen to be determined by “social” rather than economic conditions. Modern capital theory controversy consists largely in the clash of these two perspectives. Ironically, as explained above, both of these approaches can be traced to the Ricardian aspects of Böhm-Bawerk’s work.

The Production Function Approach

The Production Function as Metaphor

The production function is a metaphorical device (Lewin 1995:288–290). It is a mathematical shorthand expression for an input-output process.¹ Its use was motivated primarily by an attempt to account for the way in which economies grow. It is the basis of modern growth theory and of growth accounting; of the attempt to answer the question: What factors account for the observed growth in the economy, and to what extent? As such it also answers the question: What explains the earnings of the various inputs and their owners? Aggregate output Q is seen to result invariably and inexorably from the application of aggregate inputs K and N. All three have been identified with various statistical aggregates. The classic treatment is Solow’s seminal article.

Q = A(t) · f(K,N)

(5.1)

where the “multiplicative factor A(t) measures the cumulated effect of shifts over time” (Solow 1956:402). The shifts in the production function to which he refers imply “technical change.”

As with the formulations associated with Böhm-Bawerk’s theory, it is possible to get carried away with the technical aspects of the production function and to spend time in detailed examination of its various possible forms (Cobb Douglas, CES, etc.) and their implications. A more charitable, and perhaps more enlightening, way to interpret the growth theory literature is as “an invitation to a conversation.” The conversation is about the best way to describe “economic progress.” This is clear from the very start. In the above formulation, Solow is unable to account for the growth observed in (measured) output by considering inputs of (measured) K and N alone. As a result he must look to something else to explain the “residual.” In this case it is A(t), the “technical change” parameter. So growth is a result of inputs of capital, labor, and technical progress. The subsequent conversation is basically about what this means and what these things (capital, labor, and technical progress) really are. The conversation has, indeed, been considerably broadened in recent years with a revival of growth theory which has concentrated on these questions. Because it has turned to an explanation of technical progress in terms of economically motivated decisions, rather than as an “exogenous” (unexplained) shift parameter, it has been called “endogenous growth theory” (for surveys see Grossman and Helpman 1990, 1994; Lucas 1990; Romer 1990, 1994; Solow 1994). In the process, the meaning and nature of the production function and its arguments (capital, labor, and technical change) have come under closer scrutiny. We can examine this further by taking a closer look at the implications of the production function approach.

Constant Returns to Scale and Endogenous Growth Theory

Essentially the production function depicts a process of physical transformation of inputs into outputs. To be of any practical use, the form of this transformation must be indicated. That is, it must be able to specify how the output varies in response to changes in the inputs. The notion of constant returns to scale (CRS) comes to mind. It seems logical that if all of the relevant inputs were doubled, the output should, as a result, double. And if CRS does prevail, it then follows that returns to any one input factor that can be continuously varied while the others are held constant will diminish. Thus the earnings of the factor inputs (and, by implication, of their owners) can be explained by assuming that they are paid in terms of the value of their declining marginal products.

CRS rests, as Romer (1990:98, 1994:12) has put it, on the notion of replication. If all of the relevant inputs are correctly identified, then it is possible, in principle, to replicate (therefore duplicate) the process.

The most basic premise in our scientific reasoning about the physical world is that it is possible to replicate any sequence of events by replicating the relevant initial conditions. (This is both a statement of faith and a definition of the relevant conditions.) For production theory, this means that it is possible to double the output of any production process by doubling all of the rival inputs.

(Romer 1990:98, italics added)

The notion of physical causation (determinism) is at the very basis of production theory. This notion is surely, as a principle, not open to dispute. It is almost tautological. If all the relevant conditions (including the necessary individual actions, following upon conscious decisions) that gave rise to (“caused”) any situation, were to somehow be reinstated, then the very same situation would—almost by definition—arise again. This is held to be true without exception, except for the passage of time. In the physical sciences, when dealing with easily verifiable and classifiable events (like the full moon, the emergence of a homogeneous product from a production line), the number of relevant (initial) conditions is manageably small. Replication, identification, or production of the “same” event is thus quite simple. In the social sciences, however, everything depends on correctly identifying these relevant conditions. Although simple, well-understood, physical processes, like some production processes, are easily replicated, the transition from these to the aggregate economy level is extremely problematic.

At the very simplest level there is the insurmountable problem of aggregation of the diverse outputs and inputs and the correspondence of the aggregate statistical values to the theoretical symbols (supposedly in purely physical terms). Yet, as indicated above, it is perhaps not necessary to take these formulations so literally. Looking at the production function as a metaphorical device inviting conversation and speculation, the above considerations suggest that the conversation is about the “relevant initial conditions.” This can be seen (and has been clear, for example, in application of the Hecksher–Ohlin theory of international trade for many years) by considering the relationship between factors of production and technical progress.

Consider a CRS production function in three arguments—land, labor, and capital (L, N, and K)—so that:

Q = f(K, N, L) and λQ = f(λK, λN, λL)

(5.2)

where λ is a positive scalar. Call this a complete production function. It is complete in the sense that it includes every relevant and necessary input for the production of the product. For a complete production function it is possible to write:

gQ = s1gK + s2gN + s3gL

(5.3)

where g indicates the proportional rate of growth of the symbol and s_i (i = 1, . . ., 3) is the “share” of the factor in (contribution to) the growth of the output gQ.² It must be true that s₁ + s₂ + s₃ = 1, so that the factor shares fully exhaust the product (according to Euler’s law, if the factors are paid according to their shares, that is, according to their marginal products, their combined earnings would equal the total product). Now if one were to mistakenly omit one of the arguments, say land, L, and write the function

Q=θ(K, N)

(5.4)

then this function would have diminishing returns to scale. Call this a partial production function. It is inconceivable that any actually observed (measured) production function should not in some way be a partial production function. When we use a production function to make inferences from observed statistics, we are, no doubt, hoping that the partial function that we have postulated behaves, in some crucial respects, like a complete one. Doubling K and N would less than double Q since L is not doubled. Some “growth” would then be unaccounted for. If we attributed it to a shift parameter A(t), as in the Solovian function,

Q = A(t) . θ(K,N)

(5.5)

then it would appear as though growth were in part due to some “exogenous” cause (technical progress), when in fact it is due to the productivity of L. The same exercise can be applied to a situation in which some input, call it H, acts on output Q by enhancing the productivity of N. Then a complete function,

Q = f(K, N, H)

(5.6)

that omits H, as in equation (5.4) above, will be “shifted” by “external” changes in H.

A related consideration is the question of nonrival inputs (for example Romer 1990:97, 1994:12). Nonrival inputs, of which there are many examples, “are valuable inputs in production that can be used simultaneously in more than one activity” (Romer 1990:97). Chemical processes, computer chip design, a mechanical drawing, a metallurgical (or other) formula, computer software, etc., are examples of nonrival inputs (ibid.) They may be excludable (appropriable) or not. If H is a set of nonrival inputs and R is a set of rival inputs (like K, N), then

Q= f(H, R)

5(.7)

has the properties that

f(λH, λR) > f(H, λR) = λf(H, R)

(5.8)

that is, there are increasing returns to scale because of the “external” benefits to the private accumulation of H. The A(t) in Solow’s basic equation (5.1) above can also be understood as the expression of nonrival inputs. Identifying and talking about them renders them “endogenous.”

Growth Theory, Input Categorization, Knowledge, and Equilibrium

The implications of this discussion should be clear. First, tautologically, all production functions are CRS when specified correctly, that is when all of the relevant arguments are included. Thus technical progress can, in principle, be reduced to the discovery of a productive input, or to the accumulation of knowledge of how to use existing inputs, and so on. In principle, it is possible to always account for any output by a correct identification of all of the relevant inputs. Second, and more important, omission of any relevant input implies that CRS becomes much less likely. Solow has responded that CRS is not necessary.

[T]he model can get along perfectly well without constant returns to scale. The occasional expression of belief to the contrary is just a misconception. The assumption of constant returns to scale is a considerable simplification, both because it saves a dimension by allowing the whole analysis to be conducted in terms of ratios and because it permits the further simplification that the basic market form is competitive. But it is not essential to the working of the model.

(Solow 1994:48)

In other words, in the absence of CRS one can still use the notion of the production function to discuss the nature and causes of economic growth, but the content of the conversation will be somewhat different. Indeed this is what has happened.

Third, this discussion suggests that although one may go to great pains to include in one’s measurements all of the relevant inputs, one may not be able to do so adequately because of the multiple dimensions (the unavoidable qualitative aspects) of the identified factors. So, to be more specific, when one includes labor as a factor of production and endeavors to measure it by counting the number of people working and while adjusting for the productivity of different types of labor, one may miss some vital “human capital.” Further, we need not dwell on the difficulties of collapsing the multitude of capital items into a category called K. And with regard to land, the simple fact that “climate” plays a vital and yet elusive role in many productive processes illustrates the problem of the existence of unique, fixed factors. One is drawn back to the problem of aggregation and heterogeneity. The problem is to try to find a satisfying categorization of inputs and outputs such that economic growth (progress) is considered to be explained.

Two broad empirical observations feature in the literature as being important in stimulating economists to re-examine the traditional categorizations (Lucas 1990; Romer 1994). The first is the lack of convergence between rich and poor countries in economic performance. The second is the fact that human capital flows toward the wealthy economies in pursuit of higher returns. Consider a complete production function in two countries, identical in every respect. In the absence of barriers to factor mobility and competitive factor pricing (so that each factor earns a rent equal to its known marginal product), factor input ratios should tend to equality as capital flows to its highest earning location. This should result in capital flowing to the poorer countries where it is scarce, in turn producing a higher rate of growth in the poorer countries. The fact that this does not occur suggests that something is being left out of consideration. The same considerations apply to the flow of human capital. As Romer has put it, “If the same technology were available in all countries, human capital would not move from places where it is scarce to places where it is abundant and the same worker would not earn a higher wage after moving from the Philippines to the U.S.” (Romer 1994:11).

Various explanations have been given. In one way or another they involve the broad notion of “differences in technology,” which means, in terms of this framework, differences in the production functions or the availability of the inputs. But rather than stop there, growth theorists have tried to consider the forces behind the technology differences. Again Romer: “Technological advance comes from things that people do” (ibid.: 12). Technical change, once considered beyond the scope of the discussion, has now emerged as an urgent research topic. It has finally become apparent that economic advance is the result of detailed and difficult, and frequently serendipitous, experimentation leading to dramatic but piecemeal innovations, and ways have been sought to incorporate this into the analysis of growth. The relationship of human capital to R & D expenditures and to public goods has been considered. In particular, it has been noted that innovation (of products and techniques) is inextricably bound up with the manufacturing and distribution process (learning by doing), so that one producer’s experience may benefit another. There are external effects to the production process that manifest in the accumulation of “social” knowledge, which is a nonrival input. This suggests, among other things, that private incentives for “sufficient” investment in R & D may be lacking because of free-rider problems, but these conclusions have been tempered by the extent of our ignorance in this area (ibid.:19–20). (Our examination of the nature of knowledge and of innovation will suggest that the whole production function framework, as helpful as it may be as an organizer of ideas, is inadequate for this type of assessment.) The existence of external effects, as we have seen, may mean the presence of increasing returns to scale and increasing returns to single factors, like capital. In these circumstances, the accumulation of capital may be self-reinforcing, that is, it may not result in a decline in its marginal product. This is one way to “explain” the lack of convergence noted above.³

Limitations of the Production Function Framework

The limitations of the production function framework are related to its existence inside of an equilibrium world. It is in equilibrium in that the production function is presumed to represent knowledge that is available not only to the theorist but also, in some way, to the economic agents of the model. The outputs are assumed to follow in a technically known way from the application of the inputs, and the value of the outputs is likewise known, so that the inputs are paid the value of their marginal products. There is no room for or analysis of differences in individual valuations of inputs and outputs. (It will not do to assume that things are only known “probabilistically,” since this presumes that a finite number of possible outcomes and their distribution is known.) There is no competition “as a discovery process.” As noted, growth theorists have tried to extend this inherently static approach in an attempt to incorporate technological change and innovation. They have done this by considering R&D, for example, as another (in part nonrival) input, like H, with a known measurable marginal product. In so far as R & D leads to the discovery of “new” techniques and products, this is a contradiction in terms. We cannot have future knowledge in the present. We may have a general expectation (based on past experience) or a hope that expenditures on R & D will bear fruit, but we cannot know ahead of time exactly in what way. If we did, the R&D expenditures would be unnecessary. While the “new growth economics” has done much to bring these important aspects once again within the scope of economics, the traditional equilibrium framework it has used must be judged inadequate to account for these important phenomena.⁴

The production function approach to growth amounts to noting that certain things (inputs) were (historically) present that might account for the growth experience and arguably could feature similarly in future growth. It is a black box approach to the extent that it does not fully “explain” the connection, the process in time. In particular it ignores any “extra-economic” factors like the political and institutional environment or, more accurately, these are, at best, implicit in the analyses. (However, for an attempt to account explicitly for these phenomena while remaining within the production function framework, see Scully 1992.)

The production function is a black box also to the extent that it subsumes individual decision-making. As Kirzner has noted:

A production function can be looked at “positively.” As such it represents simply a set of technological relationships. On the other hand, a production function can be looked at as representing opportunities, from among which a human being is able to make a choice. Clearly an economics in which market events are seen as the results of deliberately planned actions, ought to view production possibilities, in this second way, as alternatives from among which planned courses of action may be constructed.

(Kirzner 1966:45; see also Hayek 1941:147)

And in a footnote to this: “Current practice generally (and, as it seems to us, unfortunately) follows the ‘technological’ view. This is especially the case with respect to aggregate models . . . this practice is especially unfortunate in the capital theory context” (Kirzner 1966:45).

The Neo-Ricardian Challenge

The production function has proven to be a very resilient metaphor. Even prior to the emergence of the “new growth economics,” the production function approach was severely and, according to some, effectively, criticized by the neo-Ricardians. This debate between the Cambridges (England and United States) is well known and has been widely surveyed (Blaug 1974; Harcourt 1991; Harcourt and Laing 1971; Yeager 1976, just to mention a few). I will accordingly not attempt yet another comprehensive survey or evaluation here. Rather we will visit this approach only to take note of an episode in the history of capital that is instructive for what both sides of the debate took for granted; namely, equilibrium.

In many ways the challenge mounted by the neo-Ricardians revived familiar criticisms of the Böhm-Bawerkian attempt to measure capital. Growth theory is an implicit capital theory—it includes K as a factor of production, where K is some measure of the produced means of production. In addition, growth theory appears to address the related question of income distribution. Because capital, like any other input, is subject to diminishing returns, it will be accumulated up to the point where the value of its marginal product just repays the opportunity cost of its employment, conveniently expressed, for example, by the interest cost of the financing that facilitates it. In this way, the neoclassical (production function) approach supports the impression that thrift, by providing funds for investment, is a positive contributor to growth, in a measure directly related to the productivity of capital. This, incidentally, also provided a justification for the earnings of capital (owners of capital) which needed to be paid the value of its marginal product if it were to be wisely invested. The neo-Ricardians attacked these conclusions by attacking the very concept of capital employed. They marshaled new and varied examples to show (as we have seen in our examination of Böhm-Bawerk’s theory) that capital as a measurable quantity cannot be conceived of as being independent of income distribution and prices. They showed, for example, that a measure of the quantity of capital used in any technique of production varied with the interest rate at which the inputs are accumulated. Thus the same physical items will have a different measure at different interest rates. It is not possible to separate the value and the quantity of capital. Moreover, while one technique may prove optimal at one interest rate and give way to another at a lower interest rate, a paradoxical re-switching may occur at an even lower interest rate, where the first technique again may become preferred. Thus, no matter how one ranked the techniques in terms of “capital intensity” (a crucial notion for the production function that relied on variations in the capital-labor ratio), one could not, in general, say that lower interest rates would induce more “capital-intensive” techniques of production to be adopted, and one was thus left without a theory to explain the earnings of capital. (It is also possible to show that there are cases, even without re-switching, where a fall in the interest rate results in a “less capital-intensive” technique, a phenomenon of “capital reversing.”) The distribution of earnings between wages and profits appears to be arbitrarily exogenous to the economy.

This is a very quick overview of the neo-Ricardian approach. It does not capture the intricacies with which its proponents were able to fill many pages. At the end of the day they succeeded in convincing the neoclassicals with their technical virtuosity that, from a technical standpoint, truth was on their side. Indeed, a purely physical measure of capital is not to be had in a multicommodity world where incomes and prices may change. And, indeed, capital reversing and re-switching were theoretical possibilities. But, as Hicks put it, they looked like “being on the edge of the things that could happen” (Hicks 1973b:44). The neoclassicals retreated into the world of the practical and appealed to the use of the production function as a self-evidently useful metaphor (a parable; Samuelson 1962). In so far as the question of how one decides on the persuasiveness of this metaphor was left unanswered, the debate must be judged as having been inconclusive.

It is important to note, however, the rules of the game under which it proceeded. Neither side in the debate raises any questions relating to the availability or use of knowledge or expectations regarding production techniques. Both adopt the Ricardian assumption of a uniform rate of profit on capital invested equal to the rate of interest. This enables both sides to talk about capital earnings as interest or profits as though these were the same things. There is the implied presumption that all economic agents share knowledge about investment opportunities, so that capital markets are always at all times fully arbitraged. There is no room for differences and inconsistencies in plans and valuations. In fact the “re-switching” and “reversing” that occurs does not happen in time. It is a question of the comparison between alternative equilibria, between alternative steady states. As Yeager has remarked:

It is loose but convenient to speak of interest rate movements and of switches between techniques. Strictly speaking, the discussion concerns not changes or events but alternative states of affairs. They might best be thought of as prevailing in separate economies identical in all respects except those necessarily associated with different interest rate levels.

(Yeager 1976:313n.)

There is no analysis of transition from one equilibrium to another.

Thus, although the debate seemed to be about issues in real-world economies, the relevance of the models used is very questionable. The neoclassicals seemed to think that it was a question of the degree of substitutability between inputs, which the neo-Ricardians assumed to be low (their models involved discrete substitutability by “switching” from one fixed technique to another). Neither side wondered about the relevance of their framework to the market process as we know it.

There is a related point (see Kirzner 1996). The neo-Ricardian critique is dependent on the idea that interest is a return to capital. The neoclassical approach identifies interest as the surplus value generated by productive capital. This provides a justification for the incomes earned by capitalists who are merely enjoying the value of what their capital has created for consumers. The neoclassicals are then criticized because capital cannot be shown to be a factor of production whose price varies inversely with the quantity employed (owing to re-switching, reversing, etc.) Now there are two related problems with this critique, in addition to the question of relevance noted above. First, it should not be surprising that no measure of the capital stock that is independent of prices (the distribution of income) exists. Capital processes are composed of a variety of fundamentally incommensurable components applied over time. Changes in the rate at which values are capitalized and discounted are bound to yield ambiguous results. This, in itself, says nothing about the justification of the earnings of producers. But, second and more important, the neoclassicals are mistaken in their view that interest is the return to capital. We will show below that it is better understood as an intertemporal price ratio that expresses the phenomenon of time preference. If this is correct, then the earnings of producers are to be understood as something completely different. Producers as workers earn wages (or salaries) and as entrepreneurs, who add value by fulfilling consumers’ hitherto unperceived needs, they earn profits. We shall thus have to look more closely at the nature of interest, profits, and wages. I do this in Chapter 7.

Summary Conclusion

In this brief historical overview we have seen how some of the recurring questions in capital theory have been answered. We have become aware of the role of time. Capital describes a process in time. The passage of time has implications for knowledge and expectations. Different theorists have attempted to wrestle with this in different ways. In Smith’s corn economy, with a regular known cycle involving one homogenous product, it was hardly relevant. Ricardo tried to maintain the simplicity of the corn economy in a multicommodity world with durable capital (variable production cycles) by imagining the economy to settle down to some known state of affairs which is duplicated every period. Menger’s approach avoided any explicit consideration of equilibrium and highlighted clearly the role of time in any conception of the production process. Böhm-Bawerk tried to combine Menger and Ricardo, but since they offered essentially irreconcilable views of the world, he ended up with more than one theory of his own which, not surprisingly, gave rise to a varied progeny. The production function approach as well as the approach of its most severe critics, the neo-Ricardians, are both, in an important sense, Ricardian theories.

One capital theorist who defies categorization is John Hicks. While working in the formalistic idiom of neoclassical economics he was always sympathetic to those aspects of the subject that defied formalization. He was also a grand synthesizer and his work thus reflects aspects of many traditions. In his last extensive work he developed a framework that proves extremely useful in understanding a variety of issues. This is examined in the next chapter.

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¹“[T]he very idea of a ‘production function’ involves the astonishing analogy of the subject (the fabrication of things, about which it is appropriate to think in terms of ingenuity, discipline, and planning) with the modifier (a mathematical function, about which it is appropriate to think in terms of height, shape, and single valuedness)” (McCloskey 1985:79).

²s_i = , where F_i is the factor in question.

³The production function is used at both the “macro” and “micro” levels. It is the core concept in the neoclassical microeconomic theory of the firm. As such it has recently come under increasingly critical scrutiny. If the firm is portrayed as a complete CRS production function in a competitive market then one is at a loss to explain what limits its size and what makes it different from other firms. One is led naturally then to a discussion of scarce (inimitable) factor inputs that provide the firm with a (transient or permanent) competitive advantage. The nature of special knowledge, routines, capabilities, and the like feature heavily in this exciting literature. For a brief overview see Chapter 9 below.

⁴Solow has perceptively and provocatively noted: “The idea of endogenous growth so captures the imagination that growth theorists just insert favorable assumptions in an unearned way; and then when they put in their thumb and pull out a plum they have inserted, there is a tendency to think that something has been proved.” A theory of “easy endogenous growth” implies something like “[S]pend more resources on R & D, there will be more innovations per year, and the growth rate of A [in equation (5.1)] will be higher” (Solow 1994:53). It is Solow’s judgment that “there is probably an irreducibly exogenous element in the research and development process, at least exogenous to the economy” (ibid.:51). As we shall indicate at some length, innovation, although it may be fostered or inhibited by the existence of certain institutional environments, cannot be “explained” in the same way that physical laws and outcomes can. In this sense it is exogenous or, as I prefer to say, autonomous. This is not meant to undervalue in any way the importance of the shift of focus that the new growth economics has occasioned.