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2021 | OriginalPaper | Buchkapitel

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

verfasst von : Albert E. Steenge

Erschienen in: A Reflection on Sraffa’s Revolution in Economic Theory

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel Sraffa’s Silenced Revival of the Classical Economists and of Marx

Nächstes Kapitel Definitions, Assumptions, Propositions and Proofs in Sraffa’s PCMC

Nur mit Berechtigung zugänglich

In this context the term ‘primary’ means that the production of this factor is not explained by the model.

Matrices and vectors are in bold type, scalars in standard. Vectors are column vectors, row vectors are indicated by an apostrophe.

See also the next section on additional properties of the system.

We assume that each industry employs labour, thus l′ > 0.

Each element of the vector f/L is equal to the corresponding element of the vector f divided by L.

See Kurz and Salvadori (1995) or Miller and Blair (2009) for comprehensive presentations of historical background and current research lines.

There are a few examples of work in which these commodity bundles are used. Although used in a different context, Quadrio Curzio (1967, pp. 43–44) provides an example. Seton (1977), in his analysis of Marxian price systems, provides another.

Note: if p′(f/L) = 1 and L = 1, w stands for the value of the economy’s net income.

These properties are basically self-evident. Indecomposability e.g. prevents that there are sub-systems capable of (re)production without inputs, directly or indirectly, from sectors not belonging to that sub-system. The full rank property means that there are no industries using the same technology. A dominant eigenvalue smaller than 1 guarantees that a surplus is possible. For further details, see e.g. Takayama (1974) or Seneta (1981).

See also footnote 4. With f > 0 and l′ > 0 also matrix (f/L)l′ > 0 and, consequently, matrix M.

That is, each element of A is multiplied by α in the new configuration.

For further background, see Steenge and Serrano (2012), or Steenge (2015).

However, we should point out again that the choices we have made in this section only are one example of possible extensions. Future work may be called for to explore further possibilities.

Matrix A in this model is assumed to have the same properties as matrix A in the Leontief model of Sect. 2.

For a derivation of this relation in the Sraffian system, see e.g. Pasinetti (1977, Sect. 12.2). For a recent discussion of Standard commodity and Standard system, see Sinha (2016, esp. ch. 7).

Symbols have the same interpretation as before.

We observe that vectors f and x are not proportional to each other, so there is no role for the Standard commodity, see the next section.

Sraffa is of course aware of the possibility that wages may at least in part consist ‘of the necessary subsistence of the workers’ and would thus enter ‘the system on the same footing as the fuel for the engines or the feed for the cattle’ (Sraffa 1960, p. 9). However, he decides to ‘follow the usual [post-classical] practice of treating the whole of the wage as variable’ (Sraffa 1960, p. 10).

Different types of closure in the two cases of stationary (or semi-stationary) conditions and of sustained growth are discussed in Quadrio Curzio and Scazzieri (1986), who introduce a distinction between the representations of production technology corresponding to the two above states of the economy (transformation apparatus and structural apparatus, respectively).

The concept of ‘Take Out’ has been introduced in John Hicks’s Capital and Time as ‘the difference between value of output and value of input’ (Hicks 1973, p. 30). Here it is used as a general expression to denote what is left after subtracting from the gross product the physical quantities needed to cover the self-replacement of means of production and workers’ consumption.

See Morishima (1966, p. 521) for an alternative proof of this proposition.

Abraham-Frois, G. and E. Berrebi (1997), Prices, Profits and Rhythms of Accumulation, Cambridge, Cambridge University Press.CrossRef

Kurz, H.D. and N. Salvadori (1995), Theory of Production: A Long Period Analysis, Cambridge, Cambridge University Press.CrossRef

Miller, R.E. and P.D. Blair (2009), Input-Output Analysis; Foundations and Extensions, Cambridge, Cambridge University Press.CrossRef

Pasinetti, L.L. (1977), Lectures on the Theory of Production, London and Basingstoke, Macmillan.

Quadrio Curzio, A. (1967) Rendita e distribuzione in un modello economico plurisettoriale, Milano, Giuffrè.

Seneta, E. (1981), Non-negative Matrices and Markov Chains, 2nd ed., New York, Springer-Verlag.CrossRef

Seton, F. (1977), ‘The Question of Ideological Obstacles to Rational Price Setting in Communist Countries’, in A. Abouchar (ed.), The Socialist Price Mechanism, Durham, NC, Duke University Press.

Steenge, A.E. (2015), ‘Limits, Resources and Distributional Trade-Offs: Structural Constraints and Opportunities’, in M. Baranzini, C. Rotondi and R. Scazzieri (eds.), Resources, Production and Structural Dynamics, Cambridge, Cambridge University Press, pp. 107–25.

Steenge, A.E. and M. Serrano (2012), Income Distributions in Input-Output Models, Economic Systems Research, 24: 4, pp. 391–412.CrossRef

Takayama, A. (1974), Mathematical Economics, The Dryden Press, Hinsdale, IL.

Hicks, J. (1973) Capital and Time. A Neo-Austrian Theory, Oxford, Clarendon Press.

Leontief, W. (1941) The Structure of American Economy, 1919–1929, Cambridge, Harvard University Press.

Leontief, W. 1991 [1928] ‘The Economy as a Circular Flow’, Structural Change and Economic Dynamics, 2 (1, June), pp. 181–212.

Morishima, M. (1966) ‘Refutation of the Nonswitching Theorem’, The Quarterly Journal of Economics, 80 (4, November), pp. 520–25, https://doi-org.ezp.lib.cam.ac.uk/10.2307/1882913 (accessed 25.2.2020).

Quadrio Curzio, A. and Scazzieri, R. (1986) ‘The Exchange-Production Duality and the Dynamics of Economic Knowledge, in M. Baranzini and R. Scazzieri (eds.), Foundations of Economics. Structures of Inquiry and Economic Theory, Oxford and New York, Basil Blackwell, pp. 377–407.

Sinha, A. (2016) A Revolution in Economic Theory: The Economics of Piero Sraffa, Basel, Springer.CrossRef

Sraffa, P. (1960) Production of Commodities by Means of Commodities, Cambridge University Press.

Steenge, A. (2015) ‘Limits, Resources and Distributional Trade-Offs: Structural Constraints and Opportunities’, in M. Baranzini, C. Rotondi and R. Scazzieri (eds.), Resources, Production and Structural Dynamics, Cambridge, Cambridge University Press, pp. 107–25.CrossRef

Steenge, A. (2020) ‘From Multipliers to the Distribution of Income: Connecting Leontief and Sraffa’, this volume.