1.1 Introduction

In recent years, there have been a number of developments in what came to be known as “Cambridge Capital Controversies” (CCC). The debates sparked by Robinson’s (1953—1954) important article inspired, as she admitted, mainly by Sraffa’s (1951) introduction to Ricardo’s Principles (Works I), in which she laid bare the inconsistencies in the neoclassical theorization of capital as a “factor of production”. The controversy, however, essentially began after the publication of Sraffa’s (1960) book in which, besides the development of his own version of the classical theory of value and distribution, leveled a critique on the dominant neoclassical theory by raising questions about the conceptualization of capital and its measurement units. Sraffa’s major positive contribution was that he fundamentally changed the way of thinking of the notion of capital intensity and the movement in prices in the face of changes in income distribution. He argued that variations in income distribution elicit intricate price movements to all different directions, which become extremely difficult, if not impossible, to predict. Furthermore, Sraffa (1960) argued that the intricate movement of prices gives rise to wage rate profit (WRP) curves of various curvatures and not straight or monotonic lines as anticipated by the neoclassical theory. In so doing, Sraffa (1960) challenged the neoclassical principle that the prices of factors of production depend directly on their relative scarcities.

In the neoclassical perspective, straight-line WRP curves, as we will show in Chapter 3, are absolutely necessary for the derivation of well-behaved demand schedules for both capital and labor. The demand schedule for capital is hypothesized to be inversely related to interest rate while that for labor is inversely related to real wage. Based on these well-behaved demand schedules for capital and labor, the neoclassical economics derives a host of other schedules, which in their turn are indispensable in dealing with macroeconomic questions. Sraffa and economists inspired by him opined that the price trajectories are non-linear and the associated with these WRP curves, or in neoclassical economics, the price-factor frontier are also non-linear, thereby rendering unlike the construction of a well-behaved demand schedule for capital. The non-linearity in WRP curves indicates that substitution between capital and labor takes place in a way different from what the neoclassical principle of scarcity prices dictates. Samuelson (1962) assumed the defense of the fundamental propositions of the neoclassical theory in the context of a one-commodity world through his “surrogate production function” and claimed that his conclusions apply to economies with many commodities.

The assumption of a one-commodity world is no different from the assumption of a uniform capital-labor ratio across industries, and essentially from an economy without prices! Samuelson’s assumption was criticized for its lack of realism by Piereangelo Garegnani (1930—2012), initially as a participant in a seminar at MIT, where Samuelson presented for the first time his “surrogate production function”, and later in an article (1962). Garegnani (1970, 1976, 1990) criticized Samuelson’s efforts to rescue the neoclassical theory of value and distribution. Similarly, Pasinetti (1966) indicated that once we hypothesize different capital intensities across industries, the neoclassical principles no longer hold. The idea is that as relative prices change consequent upon changes in income distribution, the resulting revaluation of capital can go either way, and it is possible for an industry to start as capital-intensive in one income distribution and to become labor-intensive in another (see Pasinetti 1977, ch. 5). Consequently, we no longer derive Samuelson’s straight-line WRP curves (see Chapter 3), which are consistent with the cost-minimizing choice of technique and give rise to well-behaved demand schedules for capital and other related schedules. In the presence of many capital goods and various capital-intensities across industries, the WRP curves become non-linear and may cross over each other more than once. This is equivalent to saying that a capital-intensive technique may be selected, for both low and high rates of profit, a result that runs contrary to the neoclassical theory of value and income distribution, whose quintessence is the notion of scarcity prices.

Samuelson gave an intellectual struggle and, a few years later (Samuelson 1966), admitted the limitations of neoclassical theory, without pursuing further the implications of the findings. In other words, he conceded that in the more realistic case of the multi-commodity world, we could not rule out the issue of the reswitching of techniques; that is, the same technique might be the most profitable at several rates of profit (or real wages). Under these circumstances, there is no way to construct a consistent demand schedule for capital and the same is true for labor; as a result, it is not possible to determine long-run equilibrium prices and the associated with these equilibrium rate of interest (profit) or wage rate, as the rewards for the marginal productivities of capital and labor, respectively. Therefore, it becomes meaningless to discuss short-run deviations from the long-run equilibrium because such a long-run equilibrium does not exist in the first place.

Sraffa’s critique also casts doubt on the old classical economists’ view that prices change with income redistribution, according to their capital intensity vis-a-vis the economy’s average. In fact, Sraffa argued that the concept of capital intensity is frivolous because of the intricate price complexities developed from changes in income distribution. In other words, the capital intensity of an industry may be found higher than the economy-wide average in one income distribution, and in another, its ranking may be the other way around. It is important to stress that these conclusions were derived only at the theoretical level ranging from patently unrealistic numerical examples, starting from Samuelson’s (1962) parable of a one-commodity world, to a seemingly more realistic world of two commodities produced by different techniques as discussed by Samuelson (1966) and Pasinetti (1966), confirming reswitching scenarios. It is important to stress that in the utilized numerical examples no attention was paid to what extent, if any, they were representative of the operation of actual economies. The occurrence of reswitching along with the other problems related to the measurement of capital led, from the 1970s onward, to the development of an influential variant of the neoclassical theory known as the intertemporal equilibrium approach. The latter surpasses the problem of the theoretical inconsistencies in the measurement of capital goods, however, at the expense of spiriting away the long-run character of the analysis associated with the equalization of the rate of profit and, therefore, the determination of equilibrium prices (Garegnani 1970; Eatwell 1990, 2019; Petri 2017, 2020; Fratini 2019).

While this has been the state of knowledge up until recently, lately, we have witnessed many interesting developments in this crucial for the logical consistency of economic theories area. More specifically, already from the 1980s, there have been many studies showing that the price rate of profit (PRP) curves display monotonic trajectories in the face of changes in income distribution and the resulting WRP curves are nearly straight lines, thereby ruling out, in most cases, the reswitching scenarios of the Sraffian economists. These empirical findings are certainly less damaging for neoclassical economics, but by no means negate the remaining inconsistencies in its theory of value and distribution, which must be addressed and resolved. By contrast, in the classical approach, such findings become thought-provoking questions and enrich the approach adding additional qualifications and calling for further research efforts on the theoretical front but by no means cast doubt on the very core tenets of the classical theory of value and distribution. It should be stressed though that these explorations became possible thanks to Sraffa’s seminal contributions and especially with the aid of his conception of standard commodity, that is, the output of an industry that remains invariant to changes in income distribution (Sraffa 1960, ch. 5). The latter became instrumental in making meaningful comparisons and derive statements of general validity about the movement of prices and the related to these capital intensities of industries.

Recent studies have shown that technologies described by the input—output structure of the economies share similar properties persisting over time; more specifically, the particular distribution of eigenvalues is what gives rise to the observed quasi-linearities. In effect, the research has shown that in many countries and over the years, the distribution of eigenvalues follows an exponential pattern. Figuratively speaking, the eigenvalues form an “elbow” taking place at eigenvalues much lower than the dominant; that is, the second eigenvalue is by far lower than the dominant (or Perron-Frobenius) one. Our theoretical discussion and empirical Findings lend support to the view that the ratio of the second to the dominant eigenvalue, the so-called “spectral ratio”, is chiefly responsible for the observed quasi-linearities of relative price trajectories and of WRP curves and resulting frontiers.

1.2 Structure of the book

Chapter 2 of the book begins with the notion of capital as it has been conceptualized by the old classical economists (Smith, Ricardo and J.S. Mill) and Marx. The broad classical economists’ notion of capital is contrasted with that of the first neoclassical economists (Jevons, Menger, Walras, Bohm-Bawerk,

J. B. Clark and Wicksell). The two alternative views moved parallel to each other and, certainly, we cannot say that there has been any debate, since their differences were obvious and irreconcilable. Along the way, the reasons why the first neoclassical economists were dissatisfied with the classical definition of capital and thought of the need to theorize it in a way consistent with the requirements of their theory are explained. In so doing, we point out the differences in these two competing approaches and the challenges that they face in the presence of capital and changes in income distribution.

Chapter 3 deals with the capital theory controversies of the 1960s initiated by Robinson’s (1952—1953) article and Sraffa’s (1960) writings and teachings, whose very purpose was the critique of the economic orthodoxy of the time and the revelation of its inconsistencies in the measurement of capital. The chapter continues with Samuelson’s (1962) valiant efforts to provide satisfactory answers to the issues raised by Sraffa and his followers regarding the neoclassical theory of value and distribution and the treatment of capital goods.

Chapter 4 deals with the issues raised by Cambridge economists on both sides of the Atlantic. In fact, the empirical research, mainly conducted for the US economy, has shown that the actual PRP curves are not far from linearity and the case of reswitching of techniques, that so much ink was spilled on, is in effect derived from numerical examples that are not representative of the actual economies. The subsequent research for a number of diverse economies over time strengthened the view of the near linearities in price trajectories.

Chapter 5 is about the shape of the WRP curves in actual economies and for this reason presents estimates of such curves for the US economy using input—output data of various years and industry detail. In this chapter, we make an effort to fill the gap between the theoretical research and the compelling need for relevant empirical support of the actual shape of the WRP curves. The WRP curves from the USA and many economies show that the case of reswitching cannot be excluded, but at the same time by no means is the general case. The WRP curves are characterized by near linearity, a result that does not vindicate the neoclassical theory of value for reasons that we explain and does not dump the classical one as one may hasten to point out. On the contrary, these findings give credence to the explanatory content of the classical theory of value and distribution, and open new directions in grappling with issues relating to technological change.

The near linearities in both PRP trajectories and WRP curves, which by now have already become a stylized fact, compel an explanation of this lawlike regularity. Chapter 6 deals with the empirically repeatedly established near linearities in PRP paths and WRP curves through the spectral or ei- gendecomposition of the economic system matrices and their characteristic skew distribution of both eigenvalues and singular values that explain the low effective rank of the system matrices. Furthermore, we discuss the nature of technology as reflected in the technological coefficients matrices and, particularly, in their vertical integration form, which indicates the presence of

Chapter 7 reviews the estimation methods of Chapters 4—6 using an aggregated input—output table of five sectors of the US economy and the year 2014. In so doing, the interested reader can keep better track of the utilized estimating methods and techniques, and have a better understanding of the possible meaning of the characteristic distribution of eigenvalues and triangularization methods, by inspecting this simple but insightful representation of the economy.

Chapter 8 summarizes the findings and concludes with some general remarks about the current stage of the CCC and its future prospects and directions.

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