The idea that capital theory might lead economists to discover forms of ‘paradoxical’ behaviour emerged in the economic literature of the 1960s largely as an outcome of developments in the field of production theory (linear production models leading to enquiries into discrete and discontinuous relations). What happened in capital theory is in fact a special instance of a more general phenomenon. Economists sometimes tend to examine a large domain of economic phenomena by adapting theoretical concepts that had originally been devised for a much narrower range of special issues. The discoveries of ‘paradoxical’ relations derive from the fact that their process of generalization often turns out to be ill-conceived and misleading, if not entirely unwarranted.

For a long time, in capital theory it had been taken for granted that there is a unique, unambiguous profitability ranking of production techniques in terms of capital intensity, along the scale of variation of the rate of interest. The discovery that this is not necessarily true has induced many economists to speak of ‘paradoxes’ in the theory of capital. But the roots of apparently paradoxical behaviour are to be found, not in the economic phenomena themselves, but in the economists’ tendency to rely on too simple ‘parables’ of economic behaviour.

Traditional beliefs about capital are deeply rooted in the history of economic analysis, and may be traced back to pre-classical literature. As will be shown in the next section, a long post-classical tradition was then developed on that basis. The length of ancestry might explain the survival of conventional beliefs.

The Emergence of the Conventional View

The notion of ‘capital’ was associated for a long time with investible wealth and its income generating power, and was largely independent of detailed consideration of the function of invested wealth in the production process. The earliest development of capital theory took place within the accounting framework of a pre-industrial economy (William Petty, John Locke, Richard Cantillon). Within this perspective, capital was often associated with purely financial transactions (lending and borrowing) and the relationship between capital and rate of interest came quite naturally to be conceived as the relationship between loanable funds and their price (see Cannan 1929, pp. 122–53). The origin of the belief in an inverse monotonic relation between the demand for capital and the rate of interest may be traced back to this phase of the literature. The distinction between capital as a fund of purchasing power and capital as a ‘sum of values’ embodied in physical assets remained in the background (see Hicks 1977, p. 152), but was bound, in time, to generate tension ‘between the physical and financial conceptions of capital’ (Cohen and Harcourt 2005, p. xli).

The association of capital with the process of production did not come to the fore until quite late, in spite of certain isolated anticipations. (John Hicks 1973, p. 12, even quotes Boccaccio’s Decameron on the issue.) The description of capital as a stock of means of production became common with the Physiocrats and the classical economists. In this period, Cesare Beccaria (1804, ms 1771–72) presented what Jean-Baptiste Say considered to be the first analysis of ‘the true functions of productive capitals’ (Say 1817, p. xliii). Soon after him, Adam Smith (1776) built upon the distinction between ‘productive’ capital and ‘unproductive’ consumption his theory of structural dynamics and economic growth. Finally, David Ricardo gave a definite shape to classical capital theory by examining the relationship between capital accumulation and diminishing returns and by considering in which way different proportions of capital in different industries might influence the relative exchange values of the corresponding commodities (Ricardo 1817, ch. 1, sections 4 and 5).

Classical capital theory is characterized by lack of interest in the purely financial dimension of investment. As a result, the relation between capital accumulation and the rate of interest recedes into the background and is substituted by the relation between real capital accumulation and the rate of profit. In this way, the foundations of capital theory shifted from the exchange to the production sphere, and the demand-and-supply mechanism was confined to the process by which the rate of interest is maintained equal to the rate of profit in the long run. However, a number of economists (starting with Johann Heinrich von Thünen, Mountifort Longfield and Nassau William Senior) continued to be interested in the income-generating function of capital at the level of the individual investor, and tried to combine this approach with the emphasis on the productive function of capital that had emerged in the classical literature. The marginal productivity theory of capital and interest was developed as an answer to this conceptual problem. The essential features of that theory may be clearly seen in Thünen, who suggested a relationship between the rate of interest (i) and the rate of profit (r) quite different from the one found in Ricardo. The reason for this is that Ricardo had taken r to be fixed for the individual entrepreneur, so that equality between i and r was brought about by adjustment between the supply and demand for loans in the financial markets. Thünen suggested a different adjustment mechanism by taking r to be variable for the individual entrepreneur, so that the attainment of the long-run equality between the rate of profit and the rate of interest came to depend on the change in the physical productivity of capital as much as on adjustment in the financial markets (see Thünen 1857).

This view is founded upon a thorough transformation of the Ricardian theory of diminishing returns and provided the logical starting point for the later marginalist theory of diminishing returns from aggregate capital. The analytical and historical process leading to this outcome is a rather complex one, and it is best understood by distinguishing two separate stages. In the first stage, the law of diminishing returns, which Ricardo considered to hold for the economy as a whole in the long run, was applied to the short-run behaviour of the individual entrepreneur. As result, the change in input proportions within any given productive unit is associated with the change in the physical productivity of capital. Here the variation of the capital stock is unlikely to influence the system of prices, so that the decrease (or increase) in the return from the last ‘increment of capital’ could be unambiguously associated with an increase (or decrease) in the physical capital stock. The second stage consisted in extending the above result to the variations in the aggregate quantity of capital available in the economic system as a whole.

The process which we have described made it possible to transform the classical conception of diminishing returns from a macro-social law into a microeconomic relation derived from the law of variable proportions. This new type of diminishing returns was then extended to the ‘macro-social’ sphere once again. As a result, it became possible to think that the rate of interest and the rate of profit (tending to be equal to each other) are associated with the physical marginal productivity of aggregate capital: an increase in the relative quantity of capital with respect to the other inputs would be associated with lower marginal productivity of capital and thus with a lower equilibrium rate of interest and rate of profit. This inverse monotonic relation between the rate of interest (and the rate of profit) and the quantity of capital per head eventually became an established proposition of capital theory. The relevance of this relation can be seen from the attempts by William Stanley Jevons (1871), Eugen von Böhm-Bawerk (1889) and John Bates Clark (1899) to found on the theory of the marginal productivity of factors the explanation of the distribution of the social product among factors of production under competitive conditions.

Further light on the conceptual roots of the marginalist view of capital is shed by the contributions of Jevons and Böhm-Bawerk. In their theories, profit is considered as the remuneration due to the capitalist as a result of the higher productiveness of ‘indirect’ or ‘roundabout’ processes of production than of processes carried out by ‘direct’ labour only. The generalization of the marginal principles which they carried out is thus associated with the description of the production process as an essentially ‘financial’ phenomenon in which final output, like interest in financial transactions, could be considered as ‘some continuous function of the time elapsing between the expenditure of the labour and the enjoyment of the result’ (Jevons 1879, p. 266). The subsequent discovery of ‘anomalies’ in the field of capital accumulation was possible when economists started to question this extension of capital theory from the financial to the productive sphere, and when the technical structure of production was examined on its own grounds independently of the ‘financial’ aspect which might be considered to be characteristic of ‘the typical business man’s viewpoint’ (Hicks 1973, p. 12).

Anticipations of Debate

It has just been shown that microeconomic diminishing returns provided the foundations for a theory of the diminishing marginal productivity of social capital, which was extended from the microeconomic sphere by way of logical analogy.

The pitfalls of this approach did not take long to emerge, as economic analysis came to grips with the full complexity of the production process. Knut Wicksell, discovered that, in the case of an economic system using heterogeneous capital goods, it might be impossible to describe diminishing returns from aggregate capital. The reason for this is that a variation in the capital stock might be associated with a change in the price system that would make it impossible to compare the quantities of capital before and after the change (see Wicksell 1901–6, pp. 147 ff. and 180). Wicksell also recognized that this difficulty is characteristic of capital because ‘labour and land are measured each in terms of its own technical unit … capital, on the other hand, … is reckoned, in common parlance, as a sum of exchange value’ (1901–6, p. 149).

The special difficulty associated with heterogeneous capital goods is in fact an outcome of a particular procedure by which the fundamental theorems concerning capital and interest had been formulated with reference to the idealized setting of an isolated producer, and then extended by analogy to the case of the ‘social economy’. The drawbacks of this methodology were perspicaciously noted by Nicholas Kaldor in the late 1930s, when he complained that capital theory had been developed starting with ‘a … specialised set-up, with the picture of Robinson Crusoe engaged in net-making’ rather than with the ‘general case’ of ‘a society where all resources are produced and the services of all resources co-operate in producing further resources’ (Kaldor 1937, p. 228.) Kaldor also noted that, had the analysis started with the ‘general case’, ‘a great deal of the controversies concerning the theory of capital might not have arisen’ (Kaldor 1937, p. 228).

It is remarkable that so many ‘paradoxical’ results of modern capital theory were subsequently discovered precisely as an outcome of the procedure here described by Kaldor.

The stage of modern controversy was set by the consideration of two distinct problems: (a) the measurement of ‘aggregate capital’ in models with heterogeneous capital goods; and (b) the discovery that production techniques that had been excluded at lower levels of the rate of profit might ‘come back’ as the rate of profit is increased (this phenomenon is known as reswitching of technique).

Joan Robinson started the discussion by calling attention to the difficulties inherent in any physical measure of aggregate capital (Robinson 1953–4). She also pointed out the ‘curiosum’ that the degree of mechanization associated with a higher wage rate and a lower rate of profit might be lower than the degree of mechanization associated with a lower wage rate and a higher rate of profit. (She attributed this ‘curiosum’ to Miss Ruth Cohen, but later on she attributed it to her reading of Sraffa’s Introduction to Ricardo’s Principles.)

Immediately afterwards, David Champernowne discovered that, in general, we must admit ‘the possibility of two stationary states each using the same items of equipment and labour force yet being shown as using different quantities of capital, merely on account of having different rates of interest and of food-wages’ (Champernowne 1953–4, p. 119). Champernowne also admitted that the inverse monotonic relation between the rate of profit and the quantity of capital per head (as well as the inverse monotonic relation between the rate of profit and capital per unit of output) might not be generally true: ‘ it is logically possible that over certain ranges of the rate of interest, a fall in interest rates and rise in food-wages will be accompanied by a fall in output per head and a fall in the quantity of capital per head’ (Champernowne 1953–4, p. 118). Champernowne’s explanation of what appeared to be perverse behaviour from the point of view of traditional theory was that changes in the interest rate can be associated with changes in the cost of capital equipment even if the physical capital stock is unchanged. As a result, perverse behaviour was attributed to pure ‘financial’ variations and a physical measure of capital was still thought to be possible. This Champernowne tried to obtain by introducing a chain index method for measuring capital (Champernowne 1953–4, p. 125). A few years later, Joan Robinson again took up the same issue in her Accumulation of Capital (1956, pp. 109–10). The reason she gave for the ‘Ruth Cohen curiosum’ is quite different from the one proposed by Champernowne. She explicitly recognized that ‘financial’ factors such as a higher wage rate and a lower rate of interest would have ‘real’ consequences by influencing the actual choice of technique. (In the ‘perverse’ case a lower rate of interest would be associated with the choice of the less mechanized technique.)

When a few years later Michio Morishima attempted a multi-sectoral generalization of Joan Robinson’s simple model he confirmed the possibility of a positive relationship between the rate of interest and the degree of mechanization of a technique (Morishima 1964, p. 126). Finally John Hicks came up with the same problem when examining ‘the response of technique to price changes’ in the framework of a simple economy consisting of a consumption good ‘industry’ and a net investment good ‘industry’, and in which the same capital good is used in both industries (see Hicks 1965, pp. 148–56).

But, in spite of all these anticipations, it must be admitted that the issue of technical reswitching was not given an important place in economic theory before the publication of Piero Sraffa’s Production of Commodities by Means of Commodities (1960). It is with Sraffa’s work that the phenomenon took a prominent place. Sraffa was able to show that heterogeneity of capital goods and of ‘capital structures’ (different proportions between labour and intermediate inputs in the various processes of production) would normally give rise, with the variation of the rate of profit and of the unit wage, ‘to complicated patterns of price-movement with several ups and down’ (Sraffa 1960, p. 37). This phenomenon would in turn bring about changes in the ‘quantity of capital’ that are not generally related to the rate of profit in a monotonic way. Reswitching of technique and reverse capital deepening are thus derived from a general property of production models with heterogeneous capital goods. (See reswitching of technique and reverse capital deepening.)

Neoclassical Parables and the Capital Controversy

Following the publication of Sraffa’s book, a lively debate on capital theory suddenly flared up in the 1960s, and the way it did is itself an interesting event.

It has already been pointed out that, when propositions derived from individual behaviour are applied to the more complex case of the ‘social economy’, the extension is admittedly possible on condition that the social economy has a number of special features making it identical, from the analytical point of view, to the case of the isolated individual. To test these features, the social economy is often described in terms of a ‘parable’ in which those particular conditions are satisfied. This ‘parable’, though unrealistic, is taken to be useful, from an heuristic or a persuasive point of view.

In this vein Paul Samuelson attempted to construct a ‘surrogate production function’ by analogy with microeconomic behaviour (Samuelson 1962). His work can be considered as the first explicit attempt to get rid of the complexities of an economic system with heterogeneous capital goods by constructing a model in which that system is described in terms of an ‘aggregate parable’ with physically homogeneous capital. After introducing the assumption that ‘the same proportion of inputs is used in the consumption-goods and [capital-] goods industries’ (Samuelson 1962, pp. 196–7), Samuelson was able to prove that ‘the Surrogate (Homogeneous) Capital … gives exactly the same result as does the shifting collection of diverse capital goods in our more realistic model’ (1962, p. 201). In particular, ‘the relations among w, r, and Q/L that prevail for [the] quasi-realistic complete system of heterogeneous capital goods’ could ‘be shown to have the same formal properties as does the parable system’ (1962, p. 203). This result was taken to be a justification for using the surrogate production function ‘as a useful summarizing device’ (1962, p. 203). In fact, Pierangelo Garegnani, who was present at a discussion of a draft of Samuelson’s paper, did point out that Samuelson’s result is crucially dependent on the assumption of equal proportions of inputs (see Garegnani 1970). Samuelson acknowledged Garegnani’s criticism in a footnote to his paper and admitted that it would be a ‘false conjecture’ to think that the ‘extreme assumption of equi-proportional inputs in the consumption and machine trades could be lightened and still leave one with many of the surrogate propositions’ (Samuelson 1962, p. 202n). But Samuelson and various other economists continued to look for conditions that would ensure a monotonic relation between the rate of profit and the choice of technique even in presence of a nonlinear relation between w and r.

The outcome appeared a few years later. David Levhari, a Ph.D. student of Samuelson’s, in his dissertation and then in a paper for the Quarterly Journal of Economics, claimed he had proved that reswitching of the whole production matrix would be impossible if this matrix is of the ‘irreducible’ or ‘indecomposable’ type (Levhari 1965). This property – Levhari claimed – would exclude reswitching and thus make it possible to extend the use of a ‘surrogate production function’ to the nonlinear case with production technologies for basic commodities.

However, Levhari’s theorem was disproved by Luigi Pasinetti in a paper at the Rome First World Congress of the Econometric Society in 1965. Pasinetti’s final draft of his paper was published in the November 1966 issue of the Quarterly Journal of Economics (Pasinetti 1966) together with papers written in the meantime by David Levhari and Paul Samuelson (1966), Paul Samuelson (1966), Michio Morishima (1966), Michael Bruno et al. (1966) and Pierangelo Garegnani (1966). This set of papers was called by the journal editor ‘Paradoxes in Capital Theory: A Symposium’, thereby originating the term. Paul Samuelson concluded the discussion with a ‘Summing up’ in which he admitted that ‘the simple tale told by Jevons, Böhm-Bawerk, Wicksell, and other neoclassical writers’, according to which a falling rate of interest is unambiguously associated with the choice of more capital-intensive techniques, ‘cannot be universally valid’ (Samuelson 1966, p. 568).

The various contributions to this discussion showed that reswitching might occur both with ‘decomposable’ and ‘indecomposable’ technology matrices. This result was proved in different ways by Pasinetti (1965, 1966), Morishima (1966), Bruno et al. (1966) and Garegnani (1966). Samuelson stated in his summing up that ‘reswitching is a logical possibility in any technology, indecomposable or decomposable’ (1966, p. 582). He then called attention to the associated phenomenon of reverse capital deepening and concluded that ‘there often turns out to be no unambiguous way of characterizing different processes as more “capital-intensive”, more “mechanized”, more “roundabout’” (1966, p. 582).

Although the logical possibility of reswitching was admitted by all participants in the discussion, Bruno, Burmeister and Sheshinski raised doubts as to its empirical relevance: ‘there is an open empirical question as to whether or not reswitching is likely to be observed in an actual economy for reasonable changes in the interest rate’ (Bruno et al. 1966, p. 545n). The same doubt was expressed in Samuelson’s summing up (Samuelson 1966, p. 582). Bruno, Burmeister and Sheshinski also mentioned a theorem, which they attributed to Martin Weitzman and Robert Solow, according to which reswitching of technique may be excluded, in a model with heterogeneous capital goods, provided at least one capital good is produced by ‘a smooth neoclassical production function’, if ‘labour and each good are inputs in one or more of the goods produced neoclassically’ (Bruno et al. 1966, p. 546). This theorem is based on the idea that ‘setting the various marginal productivity conditions and supposing that at two different rates of interest the same set of input–output coefficients holds, the proof follows by contradiction’ (Bruno et al. 1966, p. 546).

It is worth noting that Weitzman–Solow’s theorem is simply a consequence of the idea that, in the case of a commodity produced by a neoclassical production function, each set of input–output coefficients ought to be associated in equilibrium with a one-to-one correspondence between marginal productivity ratios and input price ratios. No ratio between marginal productivities would be associated with more than one set of input prices, and this is taken to exclude the possibility that the same technique be chosen at alternative rates of interest, and thus at different price systems. The Weitzman–Solow theorem is at the origin of a line of arguments that has been followed up by a number of other authors, such as David Starrett (1969) and Joseph Stiglitz (1973). These authors have pursued the idea that ‘enough’ substitutability, by ensuring the smoothness of the production function, is sufficient to exclude reswitching of technique. However, non-reswitching theorems of this type involve that, for each technique of production, the capital stock may be measured either in physical terms or at given prices. For in a model with heterogeneous capital goods, if we allow prices to vary when the rate of interest or the unit wage are changed, there is no reason why the same physical set of input–output coefficients might not be associated with different price systems: even in the case of a continuously differentiable production function, the marginal product of ‘social’ capital cannot be a purely real magnitude independent of prices. Once it is admitted that ‘in general marginal products are in terms of net value at constant prices, and hence may well depend upon what those prices happen to be’ (Bliss 1975, p. 195), it is natural to allow for different marginal productivities of the same capital stock at different price systems. It would thus appear that reswitching of technique does not carry with it any logical contradiction even in the case of a smoothly differentiable production function.

But Pasinetti also pointed out that the concept of neoclassical substitutability is itself a very restrictive concept indeed, as it requires the possibility of infinitesimal variations of each input at a time. In fact, Pasinetti noted that it is possible to have a continuous variation of techniques (that is, continuous substitutability) along the w–r relation and yet wide discontinuities in the variation of many inputs between one technique and another, thus making reswitching a quite normal phenomenon (see Pasinetti 1969). Moreover, and even more significantly, a non-monotonic relation between the rate of profit and capital per man may well be obtained even in the absence of reswitching (Pasinetti 1966; Bruno et al. 1966). This last possibility calls attention to the phenomenon that lies at the root of the various ‘paradoxes’ in the theory of capital: the fact that, unless special assumptions are made, a change in the rate of profit and in the unit wage at given technical coefficients is associated with a change of relative prices.

This debate continued for a few years in the late 1960s and early 1970s, with a series of journal articles (see for example Robinson and Naqvi 1967) and books (see for example Harcourt 1972). In particular, John Hicks presented a ‘Neo-Austrian’ model in Capital and Time (1973), concluding that reswitching of technique can be excluded only in the special case in which all the techniques have the same ‘duration parameters’, which means the same ‘construction period’ and ‘utilization period’ (1973, pp. 41–4).

In the end, numerous details were added. Yet the basic essential results remained those that had come out of Sraffa’s book and of the symposium on ‘Paradoxes in Capital Theory’. It is instructive to see that, in a recent exchange of views that has appeared in the Journal of Economic Perspectives (2003, Spring and Winter issues), Franklin Fisher (2003), Geoff Harcourt in Cohen and Harcourt (2003) and Luigi Pasinetti (2003), when asked to succinctly summarize the issues at stake, have essentially restated their original positions.

Aftermath and Ways Ahead

The discovery of paradoxes in capital theory has had a number of important repercussions, mostly beyond its original context. For it stimulated a large amount of analytical and empirical research on some of the issues that had been discussed in the controversy, without pressing the attention towards the fundamentals, as had been the case with the original debates. In many instances, the recent developments have been motivated by the need to face the problem of measuring the stock of capital goods in economic systems subject to advances of technical knowledge and structural change, or some of the associated issues in the theory of economic dynamics. In this section we shall refer to some of these developments without pretending to give a complete picture, but with the purpose of identifying the main lines of inquiry.

A first area of research has been the analysis of the necessary conditions for the empirical measurement of aggregate capital. Franklin Fisher elaborated a research line he had himself started in an earlier contribution (Fisher 1969) and called attention to the fact that the aggregation of outputs, as well as that of productive factors, ‘requires separability in each firm’s production function’ (Fisher 1987, p. 55). He also noted that, under constant returns, the two highly restrictive assumptions of no specialization and generalized capital augmentation are necessary, whereas, in most cases of non-constant returns, aggregation would not be allowed even when assuming the same production function for all firms (Fisher 1987, p. 55). Robert Gordon proposed to measure collections of heterogeneous capital goods, under condition of embodied technical change, by considering the associated ‘net revenue at a given set of prices (w) of variable inputs’ (Gordon 1993, p. 106; see also Gordon 1990). Edward Denison did find Gordon’s proposal objectionable and proposed instead to ‘equate’ new capital goods with the old ones by ‘what their relative costs would be if both were produced at a common date’ (Denison 1993, pp. 89–90). An interesting link between this literature and the capital controversy debate has been suggested by Charles Hulton, who has called attention to the advantages of a ‘recursive description of the production possibility set’, in which the assumption of capital as an original input is dropped, and ‘capital and labour are assumed to produce gross output and capital which is one period older’ (Hulten 1992, p. S15). Hulten’s formulation highlights the central role of knowledge advances embodied in new capital goods and suggests the relevance, for distinct purposes, of gross outputs and net outputs ‘as indicators of capacity and economic welfare’ (Hulten 1992, p. S11). Alexandra Cas and Thomas Rymes have specifically addressed the issue of whether ‘knowledge of the constant-price aggregate stock of capital would, for the comparison of economies, permit one to “predict” certain variables’ (Cas and Rymes 1991, p. 7; emphasis added). In particular, they investigated capital measurement issues brought about by embodied technical change, and proposed a set of ‘new measures’ aimed at taking the fact into account that ‘the net capital stocks of each industry and at the aggregate are themselves being produced with increased efficiency when the capital goods industries are experiencing advances in technical knowledge’ (Cas and Rymes 1991, p. 67). The same authors relate their measures of changing capital stocks under conditions of structural change to ‘Pasinetti’s concepts of vertically integrated sectors and productivity aggregated by end use’ (Cas and Rymes 1991, pp. 90–1). This point of view highlights the common ground behind recent attempts to measure stocks of heterogeneous capital goods in terms of an aggregate concept of productive capacity, be it Pasinetti’s ‘unit of vertically integrated productive capacity’ (Pasinetti 1973, 1981), Cas and Rymes’ ‘new measures of multifactor productivity’ (Cas and Rymes 1991), or Hulten’s ‘accounting for capacity’ (Hulten 1992). In all these cases, the producibility of capital goods is emphasized, as is the close connection between advances of technical knowledge and the reshuffling of inter-industry relationships (particularly those affecting intermediate goods). Philippe Aghion and Peter Howitt have commented on recent discussions about capital measurement for an economy subject to advances of knowledge by recalling Joan Robinson’s view that the real issue is not so much about the measurement of capital as rather about the meaning one wishes to assign to any given collection of capital goods (Aghion and Howitt 1998, p. 435).

Another line of investigation has concerned the attempt to assess the empirical (or computational) relevance of capital paradoxes, as distinct from their theoretical possibility. In this connection, Stefano Zambelli has used computer simulations in order to investigate the ‘realism’ of capital paradoxes in artificial economies (Zambelli 2004). This author has found a significantly higher likelihood that the capital–labour ratio be positively related to the rate of profit, contrary to the conventional belief of a negative relationship between these two variables. This result is consistent with the empirical investigation carried out by Zonghie Han and Bertram Schefold (2006). These authors have compared pairs of techniques from the OECD input–output database, and have found that ‘observed cases of reswitching and reverse capital deepening are more than flukes’ (Han and Schefold 2006, p. 22), even if we are far from observing what has been called an ‘avalanche of switchpoints’ (Schefold 1997, pp. 278–80).

A third line of research has carried the discussion of capital paradoxes into the field of dynamic economic theory. The literature relevant in this connection is itself quite differentiated. For example, Frank Hahn (1966) called attention to his earlier discovery of zones of instability in economies with heterogeneous capital goods, and pointed out that reswitching should be considered as one amongst the multiple causes of instability in capital markets (Hahn 1982). It is interesting that this line of argument, while maintaining that reswitching is a special case of a larger class of phenomena, at the same time and rather surprisingly also makes reswitching to be more general than was the case with earlier treatments of the same phenomenon. For capital paradoxes are no longer mainly associated with an economy with heterogeneous capital goods and a uniform rate of profit, but are ‘ extended’ to the case of multi-sectoral economies with many different capital goods and a multiplicity of rates of interest (and rates of profit). Luigi Pasinetti followed a different approach, and examined the analytical features of a dynamic economy in which market interactions are not explicitly examined (Pasinetti 1981). In this case, too, there are reasons to think that reswitching and reverse capital deepening would not represent exceptional cases, and would not be limited to the institutional framework of a perfectly competitive economy. Other authors have examined the relationship between capital paradoxes and dynamic stability, and have argued that reswitching of technique and reverse capital deepening are neither necessary nor sufficient conditions for the economic system to show lack of stability and irregular behaviour (Mandler 2005). It has also been emphasized that ‘reswitching’ adds an important element of instability, the importance of which depends on the process of adaptation, but also on the utility function’ (Schefold 2005, p. 467).

More generally, the discovery of capital paradoxes has stimulated a deeper understanding of the features of continuity and discontinuity in the dynamics of economic systems. This line of research has its point of departure in a phenomenon detected by Luigi Pasinetti shortly after the climax of the controversy (Pasinetti 1969). In Pasinetti’s more recent words, ‘the vicinity, even the infinitesimal vicinity, of any two techniques on the scale of variation of the rate of profits does not entail at all vicinity of such techniques … discontinuities in input use.’ (Pasinetti 2000, p. 409). John Barkley Rosser Jr. has picked up such suggestions and has investigated the discontinuities in order to identify the implications of capital paradoxes for the analysis of the optimal dynamic path followed by an economy characterized by ‘an infinite, differentiable technology’ (Rosser 1983, p. 182). This author acknowledges that it may sometimes be impossible to directly observe reswitching along optimal adjustment path (as maintained, for example in Burmeister and Hammond 1977), but he notes that this would only happen ‘at the price of dynamic discontinuities’, that is, on the condition that the economic system be able to ‘jump over’ the zone associated with intermediate techniques. The above result has been interpreted as showing that ‘in a world of infinite and smooth technologies, reswitching is to be “observed” by observing discontinuities in optimal dynamic paths’ (Rosser 1983, p. 183; see also Rosser 2000, pp. 213–20). This point of view emphasizes the analytical importance of capital paradoxes as characteristic instances of the discontinuities that may be generated by the nonlinearity of certain structural relationships. In this way, the propositions discovered during the capital controversies of the mid-20th century are found to be consilient with much later developments in the economic analysis of nonlinear dynamic systems.

Synthesis

The source of most of the difficulties that have emerged in capital theory may be traced back to the fact that ‘capital’ may be conceived in two fundamentally different ways: (a) as a ‘free’ fund of resources, which can be switched from one use to another, without any significant difficulty: this is what may be called the ‘financial’ conception of capital; (b) as a set of productive factors that are embodied in the production process as it is carried out in a particular productive establishment: this is what may be called the ‘technical’ conception of capital.

The idea that there exists an inverse monotonic relation between the rate of interest and the demand for capital was born in the financial sphere. The parallel idea of an inverse monotonic relation between the rate of profit and the ‘quantity of capital’ employed in the production process is the outcome of a long intellectual process of extensions and generalizations reviewed earlier in this essay. But the recent debate on capital theory has conclusively proved that such extensions and generalizations are devoid of any foundation. It is logically impossible to make the ‘financial’ and the ‘technical’ conceptions of capital coincide, except under very restrictive conditions indeed. More precisely, there is no unambiguous way in which a decreasing rate of profit may be related to the choice of alternative techniques, in terms of monotonically increasing capital intensity, be this considered in terms of capital per unit of output or of capital per unit of labour.

These analytical results are hardly in dispute by now. But their ultimate significance and relevance for economic theory have been, and remain, controversial.

A group of economists have been so impressed by the new discoveries in capital theory, concerning the relations between rate of profit, capital per head, capital per output, and technical progress, as to become convinced that these discoveries are calling for a reconstruction of economic theory from its very foundations. It is stressed that the traditional beliefs are due to mistaken generalizations from the theory of short-run microeconomic behaviour, and it is argued that the economic theory (‘marginal economic theory’) that led to mistakes and inconsistencies should be abandoned. It is also pointed out that the obvious alternative is a resumption and development of the more comprehensive approach to value, distribution and growth of the classical economists (see Garegnani 1970, 2005, and, in a different context, Pasinetti 1981).

A second line of interpretation maintains that economic theorists should be prepared to give up the analytical tools of equilibrium analysis and concentrate much more on the actual historical dynamics of economic systems. In this vein, reswitching of technique is acknowledged as a logical possibility but doubts are expressed on its importance in actual economic history (see Robinson 1975, pp. 38–9; Hicks 1979, p. 57).

A third line of interpretation is taken by more traditionally minded theoretical economists. It is argued that the discovery of ‘anomalies’ in the field of capital theory does point to an important deficiency in ‘marginal’ economic theory, which leads to the inevitable abandonment of the concept of ‘aggregate capital’. However, it is also argued that there is a way of overcoming this deficiency without giving up the basic premises of traditional theory, and in particular without rejecting the application of the demand-and-supply framework to the study of production. This way induces to concentrating the analysis either on the study of ‘short-run’ (‘temporary’) equilibria, in which the physical stocks of capital are given, or on the equilibrium of an intertemporal economy, in which goods are described by taking their dates of delivery into account. In either case, the logical possibility (or ‘existence’) of an equilibrium price vector is studied without explicitly considering the movement of ‘free’ capital from one use to another. In this approach, the importance of ‘capital paradoxes’ is explicitly recognized, but the associated difficulties are transferred either to the field of stability analysis or to the theory of the long-period supply of saving as financial capital (see, respectively, Hahn 1982; Bliss 2005).

A fourth line of interpretation has been pursued by many empirically oriented economists. It is acknowledged that the notion of ‘aggregate’ technical capital is untenable in terms of theory, but it is also argued that the utilization of aggregate production functions may be justified on pragmatic terms, due to supposedly satisfactory econometric fit (see, for example, Fisher 1971; Fisher et al. 1977). This view however, is by no means widely accepted. It has in fact been vigorously challenged by Paolo Sylos Labini (1995), who has reviewed the estimates that have emerged from using the Cobb–Douglas production function and has shown that such a ‘production function, when estimated econometrically, tends to yield, in general, poor results’ (Felipe and Fisher 2003, p. 251; see also McCombie 1998; and Felipe and Adams 2005). In a recent evaluative essay on aggregation in production functions, Jesus Felipe and Franklin Fisher have sharply criticized the continued use of aggregate parables. In particular, they maintain that ‘the revival of growth theory during the last two decades no doubt has produced important discussions, and seemingly interesting empirical results’ but ‘ authors do not realize that they are using a tool whose lack of legitimacy was demonstrated decades ago’ (Felipe and Fisher 2003, pp. 250–1). The same economists emphasize that ‘the impossibility of testing empirically the aggregate production function’ is ‘substantially more serious than a mere anomaly’, and that ‘macroeconomists should pause before continuing to do applied work with no sound foundation and dedicate some time to studying other approaches to value, distribution, employment, growth, technical progress etc., in order to understand which questions can legitimately be posed to the empirical aggregate data’ (Felipe and Fisher 2003, pp. 256–7). It is interesting that the theoretical and empirical researches that have taken up this challenge have devoted attention to the construction of a ‘capacity measure’ of the stock of technical capital that would allow comparisons across different states of technology without having recourse to the traditional ‘parables’ (see, for example, Pasinetti 1973, 1981; Cas and Rymes 1991; Hulten 1992).

Finally, let us note how the discovery of ‘paradoxes’ in capital theory has contributed to stimulating research into the dynamic properties of economic systems outside the world of steady state comparisons. In particular, some economists have attempted the theoretical investigation of regularities in the long-run dynamics of economic systems by suggesting a reformulation of the classical theory of structural change in a disaggregated framework (see Pasinetti 1981, 1993; Hagemann et al. 2003). Others have investigated the complex interaction of behavioural patterns along a dynamic trajectory, and have called attention to increasing returns and other nonlinear phenomena in structurally adaptive economic systems (see Anderson et al. 1988; Arthur et al. 1997).

Whatever the view that is taken, the major victim of the debate has been the Böhm-Bawerk–Clark–Wicksell theory of capital that was so patiently constructed towards the end of the 19th century. This theory relied on a conception of ‘aggregate capital’ that was taken as measurable independently of the rate of profit and of income distribution. Such a conception of ‘capital’ has had to be jettisoned, which has stimulated reformulations of the pure theory of capital. There has been on the one hand a return to the Walrasian general equilibrium theory in its intertemporal formulation, and on the other hand a remarkable revival of classical political economy. The controversy had also a number of less striking but perhaps longer-term consequences. The consideration of paradoxes has alerted economists to the richness and complexity of economic relationships, and to the need to avoid a process of generalization from the consideration of special cases. In any case the debate seems to have compelled theoretical economists to be more rigorous about the nature and limits of their assumptions. In many important cases, it has also brought about a change in the main focus of their analysis.

All this leads one reasonably to expect as unlikely that the next generation of economists will leave the issue of capital theory at rest.

See Also