From Multipliers to the Distribution of Income: Connecting Leontief and Sraffa

Abstract

The aim of this chapter is to contribute to our understanding of the relation between Leontief-based and Sraffa-based modelling. To this end we take a second look at the core properties of models belonging to either of these ‘schools’. We focus on the well-known open static Leontief model with one primary factor, and explore how this model behaves if we replace the traditional input coefficients matrix by a matrix of extended input coefficients that capture the real pay accruing to the wage earners. We show that capital can be straightforwardly introduced, and that this model generates a precise expression for the relation between the rate of profits and the wage rate. We finish by discussing the connections between this ‘extended’ Leontief model and Sraffa’s single product models.

Appendices

Comment

1.1 A Comment on Steenge’s ‘From Multipliers to the Distribution of Income: Connecting Leontief and Sraffa’

• Roberto Scazzieri 

In his contribution to this volume (Steenge 2020) Professor Albert Steenge outlines an ambitious framework aimed at bridging the approaches to the economy as a circular flow due, respectively, to Wassily Leontief (1941, 1991 [1928]) and Piero Sraffa (1960). In so doing, Steenge provides the coordinates of a general scheme for the investigation of a circular economy, of which the contributions of Leontief and Sraffa appear to be special cases. Steenge’s aim is ‘to understand the contributions of both these authors in terms of the analytical structure they have adopted’ by concentrating on Leontief’s open static model and on the model that Sraffa outlines in Part One of Production of Commodities by Means of Commodities. At the root of Steenge’s treatment is the consideration of the relationship x = Ax + f, ‘where A is the (n × n) matrix of direct [intermediate] input coefficients, f the final demand vector, also known as the final or net output vector, and x the gross output vector’ (Steenge 2020, this volume). In this model, labour requirements per unit of output are given by vector l′, which is ‘the (row) vector of direct labour input coefficients’ (Steenge 2020, this volume). An important consequence of Steenge’s approach is that, since labour is the only primary (non-produced) factor of production, it ‘receives the entire income or “surplus”, that is the commodity bundle f’ (Steenge 2020, this volume). If we solve equation x = Ax + f for x, we obtain x = (I − A)⁻¹f, in which matrix (I − A)⁻¹ is the mathematical operator known as the ‘Leontief inverse’. It is at this point that Steenge’s argument introduces an important twist concerning the role of labour in a circular production economy. For Steenge highlights that, if we assume that ‘workers are identical and receive the same wage irrespective of the sector in which they are employed’, then each unit of the labour input (say, each worker) may be assumed to consume ‘the same commodity bundle, to be called the real wage or real wage basket’ (Steenge 2020, this volume). This procedure entails introducing what Steenge calls ‘extended input coefficients ’, that is, input coefficients including both the quantity of each intermediate input that is needed to deliver one unit of any given product as well as the quantity of that input entering the ‘real wage basket’ needed to deliver each unit of that product. This approach leads Steenge, with L total employment, to the following expression for the wage basket: w = (p′f) / L = p′(f / L), and to the following expression for the price equation: p′ = p′A + wl′ = p′A + p′(f / L)l′ = p′[A + (f / L)l′]. In short, ‘the price of the i-th good consists of two parts. That is, first, a part consisting of the intermediate inputs used up in its production, represented by the i-th column of matrix A. The second part consists of the bundle of commodities consumed per unit of the labour input in each sector, here represented by the term (f / L)l′_i, where l′_i stands for the i-th element of the labour input coefficients vector l′’ (Steenge 2020, this volume). This approach to labour inputs entails that ‘the unit price of good i must cover the value of this basket of consumption goods which (again per unit) is equal to [p′(f / L)]l′_i’ (Steenge 2020, this volume). In turn, this means that the unit price of each produced commodity must cover not only the cost of intermediate inputs needed to deliver any unit of that commodity, but also the costs of what may in fact be considered as the ‘necessary consumption’ needed to produce each unit of that commodity via the unit labour input requirements for that commodity.

As Steenge points out, this treatment of labour inputs has already been followed by a few previous authors, such as Alberto Quadrio Curzio and Francis Seton. Quadrio Curzio introduced extended coefficients in a discussion of a 2-commodity Leontief closed system in which ‘consumption only takes place by way of the labour employed in production’ (Quadrio Curzio 1967, p. 43), so that the following system is obtained:

$$ {\left[\begin{array}{l}{a}_{11}+{l}_1{c}_1\kern1.125em {a}_{12}+{l}_2{c}_1\\ {}{a}_{21}+{l}_1{c}_2\kern0.875em {a}_{22}+{l}_2{c}_2\end{array}\right]}^{\prime }\ \left[\begin{array}{l}{p}_1\\ {}{p}_2\end{array}\right]=\left[\begin{array}{l}{p}_1\\ {}{p}_2\end{array}\right] $$

Quadrio Curzio argued that solving the above system would lead to determining a ‘subsistence wage’ c₁p₁ + c₂p₂, which would be the value of a wage basket consisting of the necessary consumption of commodities 1 and 2 entering each unit of labour input (Quadrio Curzio 1967, p. 44). In a similar vein, Francis Seton called attention to the possibility of representing the production system by means of an ‘augmented technology’ including both intermediate input coefficients as well as the coefficients of necessary consumption (Seton 1977). As Steenge noted in a previous contribution, this approach ‘opens up the possibility to formulate also this model [the Leontief open static model] in terms of ‘commodities being produced by commodities’, since ‘labour inputs are still present […] but expressed in terms of the remuneration labour receives in exchange for its productive inputs’ (Steenge 2015, p. 113).

The consequences of switching from a ‘mixed’ representation of production technology separately considering intermediate inputs and labour inputs to an ‘augmented’ representation of technology embedding labour inputs (via real wage baskets) into the physical representation of commodity flows, are far reaching. First of all, Steenge’s extended coefficients allow the introduction of an augmented technology matrix with the remarkable property that a value of 1 of its dominant eigenvalue expresses a condition of uninterrupted circular flow. This means that the viability (self-replicability) condition of the circular economy is now expressed by explicitly considering ‘final consumption’ (defined as the consumption of final goods entering the workers’ real wage basket) as a constitutive component of the circular flow, and therefore as a condition for the economy to be in a self-replacing state (i.e. as a condition for the whole production process to be repeated from one period to another). In addition, by attributing ‘weights’ α and β to matrices A and (f / L)l′, under the condition that the dominant eigenvalue of matrix αA + β(f / L)l′ must be equal to 1, consequences of varying the distribution of income can be followed straightforwardly in a situation of uninterrupted circular flow.

Steenge’s analysis suggests several remarks and several questions. The consideration of extended technical coefficients makes it possible to generalize the point of view of the circular flow beyond the domain of conventional intermediate goods (means of production needed for the formation of means of production, including themselves) and to include within the circular flow itself the final consumption goods needed for the economy to be in a self-replacing state. This is clearly a return to the classical concept of ‘necessary consumption’, which is not in the foreground when considering Leontief’s open static input-output model or Sraffa’s treatment of ‘the whole of the wage as a variable’ (Sraffa 1960, p. 10).¹⁸ Steenge’s emphasis on extended technical coefficients leads to a more comprehensive theory of the circular flow that might be of considerable heuristic value. For example, an augmented technology matrix with dominant eigenvalue greater or less than 1 would point to an ‘interrupted’ circular flow, which could in turn signal that some necessary product flows have been disregarded, or that the economy is following a technical change trajectory associated with bottlenecks and emergent scarcities. This point of view suggests the formulation of a ‘flexible’ circular model in which multiple different closures of the circular flow may be possible depending on different representations of the circular flow and on different policy questions to be addressed. For example, a stationary or semi-stationary economy may require a type of closure through necessary consumption (‘necessary’ final demand) that might not be required when considering a dynamic economy growing at a sustained rate triggered by intermediate flows internal to the production system.¹⁹ Or, the closure of a circular flow governed by internal demand may be different from the closure of a circular flow driven by external demand (export-led closure). Steenge’s approach to system closure leads to a remarkable twist in the analysis of distribution between wages and profits, which Steenge investigates by focusing on the physical circular system. Here too the starting point of Steenge’s contribution is the consideration of extended technical coefficients including workers’ necessary consumption. This representation of technology highlights the (aggregate) physical commodity basket going to workers and the (aggregate) physical commodity basket entering the ‘take out’ (what is left after replacing the means of production and remunerating workers).²⁰ In this case, wages and the aggregate ‘take out’ (profits in Sraffa’s terminology) are determined independently of prices, and an inverse relationship may be envisaged between the aggregate wage basket and the aggregate ‘take out’ (aggregate profits), expressed in terms of the physical commodity shares assigned to the workers’ wage basket and to the ‘take out’ respectively. This result should be assessed against Sraffa’s own formulation of the relation r = R(1 − w) between wages (as a proportion of the Standard net product) and the rate of profits (Sraffa 1960; Sinha 2016). Steenge does not provide a detailed discussion of this issue but highlights that his analytical formulation leads to an inverse (and generally non-linear) relationship between the workers’ share of the national income, here denoted by β, and the rate of profits r.²¹ He also argues that if the final output or net output vector f ‘has the proportions of the Perron-Frobenius eigenvector of matrix A, the relation between r and β is a linear one’ (Steenge 2020, this volume). This latter result replicates, in a different setting, Sraffa’s own formulation within the Standard system. Steenge’s argument suggests a possible generalization, which however raises new questions concerning the interpretation of net output under an ‘augmented’ production technology (extended technical coefficients) and, correspondingly, the interpretation of the rate of profits under that technology.

Reply

1.1 A Response to Scazzieri’s Comment

• Albert E. Steenge 

I have very little to add to Roberto Scazzieri’s remarks on my contribution to this volume. The chapter, as pointed out excellently by Scazzieri, tries to find a bridge between the works of the two great contributors to multi-sectoral analysis, Wassily Leontief and Piero Sraffa. Such efforts have been made before, naturally, but a gap has remained, in the sense that none of these efforts has been fully successful in providing an overall conceptual framework. The present contribution should be seen as belonging to that same category. However, the premise from which we start is less familiar, since we proceed by endowing the standard model of one of the two, Leontief, with a less well-known attribute, that is the concept of extended input-output (I-O) coefficients. We then proceed to explore the potential of this concept.

First of all, obviously these two directions or ‘schools’ (if you like) have quite a bit in common. Here we may think of the strict distinction between produced and non-produced goods, and the distinction between economies with a surplus and without a surplus. Nonetheless, the focus of both these schools of thought has been quite different. In Leontief-based research and applied work, the focus has largely been on policy and policy analysis based on multipliers. However, in terms of theoretical underpinnings, quite a few structural aspects still require further study; for instance in cases of multiple primary resources or multiple final destinations for the produced products.

In comparison, the Sraffa-based research direction has a different focus. In addition to structure, much attention is given to the distribution of income between the owners of capital and labour, and the corresponding price movements. Furthermore, Sraffian analysis includes a significant role for standards of value, a concept that is absent in the Leontief world.

A promising research direction here is offered, in my view, by the concept of extended input coefficients . Standard input coefficients stand for the amount of produced goods used as intermediate inputs in unit production of produced goods. To these intermediate inputs, extended input coefficients (partly or wholly) add the standardized elements of the consumption bundle that are consumed by the primary non-produced factor—labour—and paid for out of the wage (there are no savings). A shift in this consumption bundle, given some additional conditions, is then interpreted as a shift in income.

As we have shown, these extended input coefficients can straightforwardly be introduced in a Leontief framework. If there is equilibrium in the sense of each industry being in equilibrium, that is being in a state where it sells its goods for the same total amount as it paid for its inputs (including the labour input), all coefficients can be assembled into a coefficients matrix with a number of special properties. In our contribution, this matrix consists of two non-negative matrices to which ‘weights’ can be attributed. In terms of this structure, shifts in these weights can then be interpreted as shifts in the distribution of income, per industry.

Here we are building on the work of authors such as Quadrio Curzio (1967) or Seton (1977), who utilized the concept of the so-called extended I-O coefficients. How to proceed precisely in special cases is basically a matter of choice, and of selecting a required context. In fact, quite a few questions then can be formulated while still remaining in a Leontief-based methodology. The present contribution just provides the outlines of an early exploration in this area. Formal ‘bridges’ can be constructed along the way.

A number of challenges definitely remain. Roberto Scazzieri correctly suggests a closer look at the possible role of ‘flexible’ circular models, thereby partly returning to classical thought. In fact, as he points out, economic theory may suggest several ways of ‘closing’ the model, each of which is motivated by economic or political necessity. Proceeding along these lines may also generate intriguing further links between economic theory and linear algebra, in particular Perron-Frobenius theory, which are now found mainly in multiplier and growth theory.