# Main Concepts and Principles of Political Economy - Production and Values, Distribution and Prices, Reproduction and Profits

Christian Flamant

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# Main Concepts and Principles of Political Economy

## Production and Values, Distribution and Prices, Reproduction and Profits

## Prelude to a Reconstruction of Economic Theory

by

Christian Flamant
**Main Concepts and Principles of Political Economy**

By Christian Flamant

Copyright © 2018 Christian Flamant

Deposit certificate: 3DVJ1J9-10 September 2018 (France)

Case # 1-6938394841 (US Copyright Office)

All rights reserved.

ISBN: to follow

<footnote>All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise without prior permission, in writing, from the publisher.</footnote>
To my wife Anne,

And my children Nicolas, Aurélie, Alexandre and Camille
# Preface

The intention to write this book arose from the double ascertainment that economic science is in deep trouble, to such a point that it can hardly be considered as a science, but that at the same time no prominent economist has so far published a book which could be widely considered as a true reconstruction of this science, based on solid foundations. When I use the terms "economic science", I refer to the theory that is socially considered as constitutive of this science, because it is widely taught in all universities over the world and developed in numerous articles published in the most prestigious economic reviews, i.e. the neo-classical theory, based on the paradigm of equilibrium on the market as the best way to allocate scarce resources.

This theory has nevertheless failed, on two different but related grounds. First, at a purely theoretical level the theory has been criticized and refuted over the years by a number of heterodox economists. The best synthesis of all these criticisms might well be the book of Steve Keen: "Debunking Economics", published in 2001 for the first edition, and in 2011 for a revised and expanded edition. Second, at a more concrete and empirical level, neo-classical theory has been the doctrinal basis of neo-liberal economic policies implemented in most developed economies by almost all governments, whatever their location on the political spectrum, since 1980. These policies have in the vast majority of rich countries encouraged changes in favor of liberalization, financialization and globalization, which have resulted in disastrous consequences worldwide, i.e. a global reduction in the rate of growth, a rise in unemployment and finally the world financial crisis of 2008, which the neo-classical theory had been unable to foresee, and is still unable to explain correctly.

To be sure a small number of economists had predicted the occurrence of the crisis, and Steve Keen was one of them. His own explanations can be found in a series of articles which have been regrouped in a new book, published in 2015: "Developing an economics for the post-crisis world". These explanations can also be found in chapter 14 of "Debunking Economics" (second edition), which develops a monetary model of capitalism. This model relies on a Keynesian theory of money, as interpreted in particular in the works of Minsky. On this basis Keen develops a theory of debt which is consistent and has great heuristic value. However none of his two books can be considered as a full-fledged economic theory, encompassing an integrated theory of production, distribution, values and prices.

Indeed, after examining various theoretical alternatives to neo-classical theory, in the last two chapters of Debunking Economics, Keen himself recognizes that "at present, however, the various non neo-classical schools of thought have no coherent theory of value as an alternative to the neoclassical school's flawed subjective theory of value" (page 443). This is undoubtedly a big problem, because it is a "central organizing concept" (ibid.). Economic theory still needs therefore to be reconstructed, and as for any construction, this has to start from the very foundations of economic analysis. These foundations are the essential characteristics which constitute the logical fundaments of a theory, and therefore its main and primary assumptions, allowing to define its basic concepts. This is the reason why they are
called principles. To try and define such fundamentals principles was therefore the main motivation for writing this book.

As for its epistemological bases, they rely on the idea that the world of experience exists independently of the theories that try to explain it, and that each theory is nothing else than a picture of the world. I refer here to what Wittgenstein wrote in his Tractatus Logico-Philosophicus: "a picture is a model of reality" (Wittgenstein, 1921, p. 9). Each theory has its own set of concepts, which constitute a paradigm, in the sense that Kuhn gave to this word, on second thoughts, i.e. " 'a disciplinary matrix' - 'disciplinary' because it is the common possession of the practitioners of a professional discipline and 'matrix' because it is composed of ordered elements of various sorts" (Kuhn, 1974, p. 3). These elements are symbolic generalizations, models, and exemplars, the latter being defined as "concrete problem solutions, accepted by the group as, in a quite usual sense, paradigmatic" (Kuhn, 1974, p. 4).

A paradigm can be analyzed as a system of concepts, and in such a system a special emphasis must be placed on these particular concepts which are the basic and simple assumptions underlying the other elements of the system. These assumptions constitute by themselves a set of concepts which are explained and defined on the basis of the experience that we can derive from our perception of the real world. A concept is always an element which can be defined only through its relations within the system of concepts to which it belongs. But what makes the specificity of these assumptions that can be considered as principles is that they must relate to a context which is outside the theory per se, because it is the world of experience as we perceive it.

The role of these principles is indeed a fundamental one, since they define the nature and give its signification to the object which is built by a theory. They constitute indeed the semantics of a theory. A problem can therefore arise when some of these assumptions are implicit, and when they happen to be contradictory to other implicit or explicit assumptions or to the process of construction of some concepts. The view on which this book is based is that it is indeed what has befallen to the neo-classical theory: despite the fact that the syntax of this theory, under the form of an impressive mathematical apparatus, is generally correct, the loose definition of its principles has resulted in a semantics which is highly flawed, and made it unable to give a coherent description of the world as it is. This inherent shift from reality will ultimately lead to the loss of its scientific statute.

It is to contribute to this process that this book has been written and it is why it will attempt to rebuild economic theory from its main concepts and principles. If it will do so by trying to stick to the common perception that ordinary human beings can have of the real world, it will also try to use the teachings of various theories, considered as scientific ones, which have been developed in other unrelated fields of knowledge, such as the mathematical theory of measurement or the physical theory of dimensional analysis.
"Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality."

Albert Einstein, *Ideas and Opinions* (1933)
# CONTENTS

<table><tr><td>LIST OF ABBREVIATIONS</td><td>13</td></tr><tr><td>LIST OF FIGURES AND TABLES</td><td>17</td></tr><tr><td>Chapter 1. Introduction: the basic concepts and principles. Commodities, Exchange and Money</td><td>19</td></tr><tr><td>PART I - PRODUCTION WITH A SURPLUS: A THEORETICAL DEAD-END</td><td>37</td></tr><tr><td>Chapter 2. The Ricardian theory of production, distribution and value</td><td>39</td></tr><tr><td>Chapter 3. Distribution and prices in Piero Sraffa's theory</td><td>61</td></tr><tr><td>Chapter 4. Commodities do not produce commodities: the biased concept of production in Sraffa's theory</td><td>83</td></tr><tr><td>PART II - PRODUCTION AND VALUES</td><td>103</td></tr><tr><td>Chapter 5. Production and values</td><td>105</td></tr><tr><td>Chapter 6. A first determination of values as physical values</td><td>117</td></tr><tr><td>Chapter 7. A few teachings about the concept of value</td><td>133</td></tr><tr><td>Chapter 8. The labor theory of value and the problem of joint production</td><td>151</td></tr><tr><td>Chapter 9. Conclusion on the concepts of production and value</td><td>181</td></tr><tr><td>PART III - DISTRIBUTION AND PRICES</td><td>187</td></tr><tr><td>Chapter 10. From physical values to social and monetary values</td><td>189</td></tr><tr><td>Chapter 11. Distribution, surplus-value and profit</td><td>209</td></tr><tr><td>Chapter 12. A more realistic model integrating capitalist consumption</td><td>243</td></tr><tr><td>Chapter 13. A numerical example of a model with capitalist consumption</td><td>273</td></tr><tr><td>Chapter 14. Distribution and prices: a close connection</td><td>285</td></tr><tr><td>Chapter 15. A coherent theory of rent</td><td>299</td></tr><tr><td>Chapter 16. Some theoretical lessons from the models</td><td>315</td></tr><tr><td>PART IV - REPRODUCTION AND THE RATE OF PROFIT</td><td>341</td></tr><tr><td>Chapter 17. Profits, rate of profit and reproduction</td><td>343</td></tr><tr><td>Chapter 18. The Marxist definition of the rate of profit in the simple model</td><td>367</td></tr><tr><td>Chapter 19. The evolution of the rate of profit as defined by Marx in the simple model</td><td>381</td></tr><tr><td>Chapter 20. The rate of profit as defined by Marx and its evolution in the model with capitalist consumption</td><td>393</td></tr></table>
<table><tr><td>Chapter 21. The evolution of the rate of profit - based on its current definition - and the share of profits</td><td>415</td></tr><tr><td>Chapter 22. Some reflections and first elements of discussion to introduce extended reproduction</td><td>445</td></tr><tr><td>Chapter 23 - Conclusion</td><td>465</td></tr><tr><td>APPENDIX 1 - FUNDAMENTAL PRINCIPLES OF POLITICAL ECONOMY</td><td>485</td></tr><tr><td>APPENDIX 2 - THE ERRONEOUS NATURE OF MARX'S REPRODUCTION SCHEMES IN BOOK II OF CAPITAL</td><td>493</td></tr><tr><td>APPENDIX 3 - SUMMARY TABLES OF VALUES AND PRICES OF THE MAIN VARIABLES OF THE MODELS</td><td>497</td></tr><tr><td>BIBLIOGRAPHY</td><td>503</td></tr><tr><td>DETAILED TABLE OF CONTENTS</td><td>511</td></tr></table>
## About the author

Christian R. Flamant is a retired Economist. He graduated in 1970 from Science Po - Paris, and obtained a PhD in Economics in 1977 from the University of Paris 1 - Panthéon-Sorbonne. He was a Research and Teaching Assistant in Economics at the University of Paris 1 from 1972 until 1979, and a lecturer in National Accounting and Economic Analysis, for the last two years (1977-1979). In 1979 he joined the French Development Agency (AFD), where he worked for 32 years, until his retirement at the beginning of 2012. He held various positions in several countries, in particular as a country or regional director in Vanuatu, Angola, Ghana, Horn of Africa (Djibouti, Ethiopia, etc.) and Yemen. He was seconded to the IMF from 1982 until 1985, and to the OECD as a Principal Administrator in the Development Cooperation Directorate, from 1998 until 2001.

After retiring he went back to what had been his first professional interest: pure economic theory. He wrote two articles published in the World Economics Association journals:

* the first one on Piketty: “Again on Piketty’s *Capital in the Twenty-First Century* or Why National Accounts and Three Simple Laws Should not be a Substitute for Economic Theory” in the *World Economic Review*, N°5 (July 2015);

* the second one on Sraffa: “Commodities do not produce commodities: A critical view of Sraffa’s theory of production and prices” in the *Real World Economics Review*, issue N° 72 (30 September 2015).

He devoted the last three years to the continuation of a reflection that was initiated with his thesis on “Hypotheses and concepts in the theories of value, capital and distribution”. More than forty years after his thesis had left some questions unanswered, he feels that he has at last succeeded in redefining the most basic concepts of production, values, distribution and prices, allowing to establish in this book a series of fundamental principles of political economy. His conviction is indeed that Economic Theory, even though it will always remain an incomplete science, needs to be rebuilt starting from its foundations, which these concepts and principles might now help to provide.
# LIST OF ABBREVIATIONS

*out of matrix equations

a : share in value of fixed capital obtained by section I, producing fixed capital, over total fixed capital produced during a defined period of time

$\alpha = \frac{a}{1-a}$ : ratio of the share in value of fixed capital obtained by section I, and the share in value of fixed capital obtained by section II, producing consumption goods

b : share in value of fixed capital obtained by section II, producing consumption goods, over total fixed capital ($b=1-a$)

c : share in value of capitalist consumption over total consumption

$c_I$ : share in value of consumption of capitalists of section I over total consumption

$c_{II}$ : share in value of consumption of capitalists of section II over total consumption

$$c_I + c_{II} = c$$

C : total value of all consumption goods produced during a period

$C_c$ : total value of consumption goods obtained by capitalists during a period

$C_w$ : total value of consumption goods obtained by workers during a period ($C = C_c + C_w$)

$\gamma = \frac{c}{1-c}$ : ratio of the share in value of capitalist consumption over the share in value of workers consumption

$c^*$: a parameter setting the consumption of all capitalists of section I which can be obtained through selling fixed capital to capitalists of section II at a price without capitalist consumption multiplied by $\frac{1}{1-c^*}$

$\gamma^* = \frac{c^*}{1-c^*}$ : the corresponding markup (because $\frac{1}{1-c^*} = 1 + \frac{c^*}{1-c^*}$)

I : price of total investment during a period of time, equal to the price of fixed capital produced during the same period ($I = I_I + I_{II}$)

$I_I$ : price of investment in section I producing fixed capital, during a period

$I_{II}$ : price of investment in section II producing consumption goods, during a period
$k$: primary rate of surplus-value in a model without capitalist consumption ($k = \frac{L_I}{L_{II}}$ i.e. the ratio of value produced in section 1 to value produced in section 2)

$k_c$: rate of surplus-value in the general case, with capitalist consumption ($k_c = \frac{k+c}{1-c}$)

$K$: total value of fixed capital previously accumulated and in operation during a given period of time

$L$: total value of all commodities produced in a given period of time ($L = L_I + L_{II}$)

$L_I$: value of all commodities produced in section I, producing fixed capital

$L_{II}$: value of all commodities produced in section II, producing consumption goods

$p_i$: price of commodity i

$P$: total price of the product of a given period of time ($P = P_I + P_{II}$), equivalent to Y

$P_I$: price of the product of section I, equivalent to $Y_I$

$P_{II}$: price of the product of section II, equivalent to $Y_{II}$

$\Pi$: overall amount of monetary profits

$\Pi_I$: amount of monetary profits in section I

$\Pi_{II}$: amount of monetary profits in section II

$\pi$: share of monetary profits in the price of total product ($\pi = \frac{\Pi}{Y}$)

$\pi_I$: share of profits of section I in the price of product of section I ($\pi_I = \frac{\Pi_I}{Y_I}$)

$\pi_{II}$: share of profits of section II in the price of product of section II ($\pi_{II} = \frac{\Pi_{II}}{Y_{II}}$)

$q$: organic composition of capital, originally a marxist concept equal to $\frac{K}{V}$, the ratio of the value of total accumulated fixed capital over the value of labor-power, in a simple model without capitalist consumption

$q_c$: organic composition of capital in a model with capitalist consumption
$r$: average rate of profit:

- can be calculated in terms of prices or values,

- can be calculated either the Marxist way $\left( r = \frac{S}{K+V} = \frac{k}{q+1} \right)$, or according to the current definition, i.e. $r = \frac{S}{K}$ (in terms of values) or $r = \frac{\Pi}{It}$ (in terms of prices)

$r_c$: average rate of profit in the general case with capitalist consumption

$r_{Ic}$: average rate of profit in section I, in the general case with capitalist consumption

$r_{IIc}$: average rate of profit in section II, in the general case with capitalist consumption

$S$: total amount of surplus-value (in value terms)

$t$: average lifespan of accumulated fixed capital

$u_i$: fraction of available fixed capital obtained by capitalists of section I at the $i^{th}$ stage of purchases of fixed capital within this section

$v_i$: value of commodity $i$

$V$: total value of labor-power ($V = V_I + V_{II}$)

$V_I$: value of labor-power in section I

$V_{II}$: value of labor-power in section II

$\bar{w}$: average monetary wage per unit of time, equivalent to total wages divided by the total number of (physical) hours of work

$W$: total price of labor-power, equivalent to total money wages

$W_I$: total price of labor-power in section I, equivalent to money wages paid in this section

$W_{II}$: total price of labor-power in section II, equivalent to money wages paid in this section

$Y$: total incomes distributed during a given period of time, equivalent to $P$, the price of the product ($Y = Y_I + Y_{II}$)

$Y_I$: amount of incomes distributed in section I during a given period, equivalent to $P_I$

$Y_{II}$: amount of incomes distributed in section II during a given period, equivalent to $P_{II}$

$\Psi$: a parameter which is by definition the sum $1 + \alpha + \gamma + \gamma^* + \alpha\gamma$
LIST OF FIGURES AND TABLES

LIST OF FIGURES

**Figure 19.1** - Representation of function $r (values) = f(q) = \frac{q}{(q+1)t}$, for different values of t
384

**Figure 19.2** - Representation of function $r (prices) = f(k) = \frac{k}{kt+(1-a)(1+k)}$ ($a = 0.25$)
387

**Figure 19.3** - Representation of function $r_I (prices) = f(a) = \frac{a}{at+1-a}$
388

**Figure 20.1** - Representation of function $r_c (values) = g(q_c) = \frac{q_c + \gamma t}{(q_c + 1)t}$
402

**Figure 20.2** - Representation of function $r_c (values) = f(k_c) = \frac{k_c}{(k_c - \gamma)t + 1}$ with $k_c > \gamma$
403

**Figure 20.3** - The invariance frontier of the average rate of profit $f(c) = \frac{1}{1+t}$
406

**Figure 21.1** - Evolution of the rate of profit $r_c$ with its current definition and in terms of values as a function of the primary rate of surplus-value $k$, for different values of $c$
426

**Figure 21.2** - Evolution of the rate of profit $r_c$ with its current definition and in terms of prices as a function of the primary rate of surplus-value $k$, for different values of $c$
428

**Figure 21.3** - Evolution of the rate of surplus-value $k_c$ with $k_c = f(c) = \frac{c+k}{-c+1}$
431

**Figure 21.4** - Comparison of curves representing the profit rate $r_c$ in terms of values and prices for two different values of $c$
433

**Figure 21.5** - Evolution of $\pi_I$ - the share of profits in section I - as a function of $a$ and for different values of parameter $c^*$
435

**Figure 21.6** - Evolution of $\pi_I$ - the share of profits in section I - as a function of $c^*$ and for different values of parameter $a$
436

**Figure 21.7** - Evolution of $\pi_{II}$ - the share of profits in section II - as a function of $k$ and for different values of parameter $c$
437

**Figure 21.8** - Evolution of $\pi_{II}$ - the share of profits in section II - as a function of $c$ and for different values of parameter $k$
438

**Figure 21.9** - Evolution of $\pi$ - the share of profits in the product - as a function of $k$
441
**Figure 22.1** - Curves representing function $r_c = \frac{q_c + \gamma t}{(q_c + 1)t}$ in year 0 and year 23 453

LIST OF TABLES

**Table 8.1** - Morishima's example of joint production presented in a make-use table. 161

**Table 8.2** - Schematic representation of a joint production system with by-products 168

**Table 8.3** - A Make-Use Input-Output table for joint production 170

**Table 11.1** - Prices, money values and profits in section I (first case) 220

**Table 11.2** - Prices, money values and profits in section I (second case) 223

**Table 11.3** - Prices, money values and profits in section I (third case) 224

**Table 12.1** - Prices, money values and profits in section I, with capitalist consumption 254

**Table 14.1** - Prices, money values and profits in section I, with capitalist consumption (generalization) 285

**Table 21.1** - Values of $\pi$ corresponding to different values of the other parameters 440

**Table 21.2** - Values of $r_c = f(\pi)$ corresponding to different values of the parameters 443

**Table 22.1** - A numerical example tracing the evolution of the rate of profit with extended reproduction and without capitalist consumption 448

**Table 22.2** - A numerical example tracing the evolution of the rate of profit with extended reproduction and capitalist consumption 451
# Chapter 1. Introduction: the basic concepts and principles.
## Commodities, Exchange and Money

This introductory chapter recalls what are the main steps of the approach pursued in this book, and how the theory that it develops is linked to its social background. It then shows why economic variables are dimensional variables, and explains what is dimensional analysis. At this preliminary stage, only the most basic variables are examined. These variables are commodities, exchange and money, and their dimensional nature has a bearing on their conceptual definition. This is in particular the case for the definition of money, which is closely related to the theory of measurement.

### 1. Main steps of the approach

In order to try and contribute to a reconstruction of economic theory, starting with its foundations, the book starts with this first introductory chapter devoted to the definition of the most basic economic variables: commodities, exchange and money. These are indeed the three most basic concepts corresponding to unavoidable realities with which most human beings have to deal with on a daily basis to survive in the real world. Defining these concepts will involve to state some basic principles, closely related to the world of experience.

To do so, we will have to recall first that these concepts cannot be defined in a vacuum, but against the background of this real world. We will have also to acknowledge that the corresponding variables are used in quantitative reasoning, which implies that they must be measured. But such a measurement is possible only to the extent that these variables have a dimension, in the scientific sense of the term. This will lead us to recall briefly the main elements of dimensional analysis. Similarly, we will show that the definition of money implies to rely on the theory of measurement.

The main concepts being defined, and since commodities have to be produced before being obtained and used by people who need them, the first and second part of the book will deal with the production process and the different roles played by different kinds of commodities in this process. The first part will show that this process is still wrongly analyzed by economic theory, and why theories of production based on the idea of production as surplus are necessarily a theoretical stalemate.

This will allow to show in the second part that once production is correctly defined as a transformation process, it becomes possible to address the question of the value of commodities, basing their equivalence at this theoretical stage.

The third part will deal with the question of exchange, considered as an exchange between commodities and money. This will be done from a macroeconomic point of view, simultaneously shedding light on the question of distribution and prices, considered as monetary prices.
Finally the findings achieved in the preceding parts will give us some tools to deal in the fourth part with the controversial question of the rate of profit and its evolution, which is closely related to the reproduction of the economic system.

## 2. The background

At the outset let it be clear that the theory exposed here is not background independent. This notion is well known in physics, where one can refer for instance to a 2005 article by Smolin: "the case for background independence", and in particular in general relativity, which is essentially background independent. However it is generally left aside in economic theory, which is part of social sciences, and where there can be no such thing as background independence, which simply means that the defining equations of the theory cannot be independent from the social context, i.e. the social and political organization of the society in which individuals enter into economic relations.

This social and political background is to some extent the framework of the disciplinary matrix that Kuhn, as mentioned above, considers as constitutive of a paradigm. This means that it is impossible not to refer to this specific social and political background because it plays a role in the definition and determination of some important variables of the theory. But the reverse is also true: the solutions of the equations of a theory may have an influence on social relationships, if they become a constituent of the ideology of a society.

This background against which a theory must be built and principles established is therefore in this book the social structure of the capitalist mode of production, since it is the dominant mode of production in the real world. Apart from a country like North Korea, or from isolated tribes in the Amazon forest or in Papua New Guinea, this capitalist mode of production prevails indeed everywhere in the world, even to a great extent in a so-called Communist country like China.

It is not the objective of this book to enter into a full and detailed description of this mode of production. Suffice it to say that it is characterized by the separation of individuals into two broad categories: those who own the means of production, who for convenience will be called capitalists, and the other ones, with no other property than their capacity to work, who will be called workers.

To be sure, in the real world a number of workers own some means of production, through the ownership of stocks of various companies or enterprises, and could be considered at the same time as both workers and capitalists, but this is not relevant for the present analysis. This only shows that we are here not in the real world, but in the theoretical world of language and concepts, aiming at providing a picture of the world, distinct from the real world and giving only an approximate vision of it. Obviously the objective pursued here is that these concepts make a picture which is as close as possible of what is empirically perceived as the real world.
### 3. Economic variables have a dimension

At this stage, and without going as far as claiming that there is no science if there is no measurement, one has to recognize that economic science deals with quantities and must therefore be able to perform quantitative analyses. This makes the question of measurement a central one for economic theory, and brings to the fore a theory which is rarely heard of in economic circles, because of its origins in Physics, and which is dimensional analysis. This analysis has to be used because economists, like physicists, deal with quantities, and because it is this analysis which has to be applied when dealing with quantities in a scientific way. Dimensional analysis is well explained in several books of Physics, including a short but excellent course which was taught by Ain N. Sonin at the MIT, under the title: "The physical basis of dimensional analysis" (Sonin, 2001, pp.1-57). Although it does not seem necessary to expose here the full theory in all its details, it seems appropriate just to recall a few of its findings that can be useful also for economists, through a less than one page recapitulation which can be found in section 2.8 of Sonin's course:

1. A *base quantity* is a property that is defined in physical terms by two operations: a comparison operation, and an addition operation. The comparison operation is a physical procedure for establishing whether two samples of the quantity are equal or unequal; the addition operation defines what is meant by the sum of two samples of that property.

2. Base quantities are properties for which the following concepts are defined in terms of physical operations: equality, addition, subtraction, multiplication by a pure number, and division by a pure number. Not defined in terms of physical operations are: product, ratio, power, and logarithmic, exponential, trigonometric and other special functions of physical quantities.

3. A base quantity can be measured in terms of an arbitrarily chosen *unit* of its own kind and a *numerical value*.

4. A *derived quantity* of the first kind is a product of various powers of numerical values of base quantities. A derived quantity is defined in terms of numerical value (which depends on base unit size) and does not necessarily have a tangible physical representation.

5. The *dimension* of any physical quantity, whether base or derived, is a formula that defines how the numerical value of the quantity changes when the base unit sizes are changed. The dimension of a quantity does not by itself provide any information on the quantity's intrinsic nature. The same quantity (e.g. force) may have different dimensions in different systems of units, and quantities that are clearly physically different (e.g. work and torque) may have the same dimension.

6. Relationships between physical quantities may be represented by mathematical relationships between their numerical values. A mathematical equation that correctly describes a physical relationship between quantities is *dimensionally homogeneous*. Such equations remain valid when base unit sizes are changed arbitrarily.
7. The categorization of physical quantities as either base or derived is to some extent arbitrary. If a particular base quantity turns out to be uniquely related to some other base quantities via some universal law, then we can, if we so desire, use the law to redefine that quantity as a derived quantity of the second kind whose magnitude depends on the units chosen for the others. All base quantities that are transformed into derived quantities in this way retain their original physical identities (i.e. their comparison and addition operations), but their numerical values are measured in terms of the remaining base quantities, either directly via a defining equation or indirectly by using a unit that is derivable from the remaining base units.

8. A system of units is defined by (a) the base quantities, (b) their units, and (c) the derived quantities, each with either its defining equation or the form of the physical law that has been used to cast the quantity into the derived category. Both the type and the number of base quantities are open to choice (Sonin, 2001, pp. 32-33).

Recalling these main elements of dimensional analysis provides us with a useful tool to now address the definition of our main concepts, starting with the concept of commodities.

## 4. Commodities

The first sentence of Capital by Karl Marx has not lost its actuality. It states: “The wealth of societies in which the capitalist mode of production prevails appears as an ‘immense collection of commodities’; the individual commodity appears as its elementary form” (Marx, 1867, p. 27). In fact this starting point is not truly different from that of Walras, who names his main book, published in 1874: “Elements of Pure Economics or the Theory of *Social Wealth*”, and for whom commodities are likewise the components of this wealth.

Over the whole spectrum of economic schools, it seems therefore that a consensus exists to consider commodities as a central concept of economic theory. This provides us with a good reason similarly to start this analysis from commodities, the question being then to build up a concept of commodity which meets the criteria for being considered as a scientific one.

A first observation that can be made about commodities is that commodities appear in vast quantities of various types and dimensions: some commodities can have their quantities expressed in terms of mass (in tons or kilograms), some others in terms of length (in meters or kilometers, or in terms of hours (in hours, days, or years). This brings us back to the relevance of dimensional analysis to address the question of this diversity of dimensions. It is therefore useful to recall one important consequence of the main rules of dimensional analysis exposed in the introduction, again on the basis of Sonin's course:

> “Suppose that, in a specified physical event, the numerical value $Q_o$ of a physical quantity is determined by the numerical values of a set $Q_1...Q_n$ of other physical quantities, that is,

$$Q_o = f(Q_1, Q_2, ..., Q_n),$$
A physical equation must be dimensionally homogeneous. Dimensional homogeneity imposes the following constraints on any mathematical representation of a relationship like this equation:

(1) Both sides of the equation must have the same dimension;

(2) Wherever a sum of quantities appears in $f$, all the terms in the sum must have the same dimension;

(3) All arguments of any exponential, logarithmic, trigonometric or other special functions that appear in $f$ must be dimensionless” (Sonin, 2001, p. 21).

It is my deep conviction that not only physical equations, but also economical equations must be homogeneous, meaning that economical quantities must have the same dimension, in order for these economical equations to be as logically meaningful as physical ones, and thus for economic theory to be eligible to a scientific status.

For economists commodities must indeed have similar characteristics which make it possible to consider them as measurable quantities and as such to make them comparable. The problem which cannot be ignored is that there are millions of different commodities, with very different properties. This fact of observation is of paramount importance but at the same time so obvious that it cannot in itself be considered as a primary and overarching principal. It is more an element of the background underlying all others. It seems indeed more important to focus on the elementary form of commodity, which makes it indispensable to define, as a starting point, what we intend as a commodity, i.e. what is the signification of this word.

It is a fact of experience: commodities are objects which have numerous properties, some of them being measurable, which allows to treat them as quantities, with each type of quantity having by definition its own dimension. For instance some types of objects can be characterized inter alia by their length, a spatial dimension, which means that the corresponding numerical value has the dimension of a length. But for other types the mass of the objects can be the main element (in so far as it is indispensable in their case to set their price), and the corresponding numerical value has the dimension of a mass. These objects may also have derived quantities, like an area, a volume, a density, which have no physical meaning, but are also dimensions. Commodities have simultaneously a number of other unmeasurable properties, like their color, or their shape, etc...From this simple observation we can therefore derive a first principle:

**Principle 1:** An elementary commodity is a multidimensional object

As stated above by Marx there are vast numbers of commodities, and it is obvious that the dimensions of commodities differ widely from one commodity to another, which implies as a corollary to this first principle that:

**Corollary to Principle 1:** Commodities are heterogeneous
It is because of this heterogeneity from one commodity to another that economic theory has been obsessed by the quest for a particular property of commodities which could provide the basis for describing the economic world in quantitative terms. This implies to find a property allowing to define otherwise heterogeneous commodities in terms of a common quantity. This would make it possible to decide whether two quantities of commodities A and B having this property are equivalent (A = B), or non-equivalent (A ≠ B). Dimensional analysis tells us that there must also be an addition operation that defines what is meant by the sum C = A + B of two quantities of commodities sharing this property. If indeed it is the case, dimensional analysis ensures us that “base quantities with the same comparison and addition operations are of the same kind (that is, different examples of the same quantity). The addition operation A + B defines a physical quantity C of the same kind as the quantities being added” (Sonin, 2001, p. 10).

One of the reasons for this quest for a common property is that economic theory has founded the equivalence of commodities in the exchange of these commodities, by considering that exchange is an equivalence relation. But before addressing the question of exchange, one must first raise the preliminary question of the existence of commodities. Indeed, and this is an obvious truth, commodities cannot be created ex nihilo, by spontaneous generation. Whatever the way they come into existence, commodities must pre-exist to exchange. The recognition of this fact leads to a kind of overarching principle:

**Principle 2:** Commodities must have been produced before being exchanged

## 5. Exchange

Economists have always viewed the exchange of two commodities as an exchange between two equivalent objects, which has led some of them to write an equation such as for instance:

2 m of fabric = 5 kg of wheat

To be sure, these economists were aware that such an equation is a pure nonsense, because of the different dimensions of the two objects on each side of the equation. They tried therefore to find out what would be this hidden common dimension, shared by both commodities, which would make the equation homogeneous and give it a logical sense. In other words they intended to replace this equation by another one, where on each side of the equation there would be two common and comparable magnitudes having exactly the same measurement. Economists call value this magnitude, so that instead of the previous equation, we would get:

value of 2 m of fabric = value of 5 kg of wheat

Such an equation has now a logical sense, provided that this particular magnitude is properly defined. The question which has therefore to be answered is: what is the common dimension of this magnitude named value, which makes different values of various heterogeneous commodities comparable, and therefore measurable. This question was relevant for classical economists as well as neo-classical economists, despite the different answers that they gave to it.
## 5.1. The neo-classical approach

At the beginning of the neo-classical theory, a first attempt had been made by Jevons, in his “theory of political economy” published three years before Walras’s book, to identify the exchange relation, perceived as an exchange between quantities, to a ratio of utilities: “the ratio of exchange of any two commodities will be inversely as the final degrees of utilities of the quantities of commodities available for consumption after the exchange is effected” (Jevons, 1871, pp. 95-96). This is written in a symbolic form:

$$ \frac{\Phi(1)^a}{\Psi(1)^b} = \frac{y}{x} = m, \text{ with } a \text{ and } b \text{ two commodities, } x \text{ and } y \text{ their quantities and } \Phi \text{ and } \Psi \text{ their utilities.} $$

But the utility given to one commodity is a subjective feeling which has no material existence, and exists only in the minds of individuals. Therefore this homogeneity concerns only one individual. Even if we suppose that a single individual can establish a kind of “utility scale” between various commodities, which is quite debatable, such an homogeneity cannot be demonstrated when there are many of them, each one with his own subjectivity and his own scale of utility. Among the infinite space of all individual subjectivities, no homogenization is possible. Therefore homogeneity is not demonstrated, but only postulated, which does not allow correctly to produce the concept of value, as the common dimension of commodities associated to the relation of equivalence of exchange and to its measurement.

A more sophisticated approach in the neo-classical tradition started with Léon Walras (op. cit.) and was based on the well-known parable according to which, at the beginning of the story, all individuals have some endowments of diverse quantities of various commodities, and meet on a marketplace. There, after a number of bargaining rounds, and with the assistance of some kind of “market commissioner”, equilibrium of overall supplies and demands for all commodities is reached. This means that the exchange ratios between all commodities are determined in terms of one of them, which is taken as a standard of value and provides a unit of account. Then real exchanges can take place in accordance with these exchange ratios, and the market is cleared. These are the roots of the theory of general equilibrium.

This position has been proven wrong on purely logical grounds by a number of economists. A very good synthesis of all these criticisms is given in “Debunking Economics” the already mentioned book by Steve Keen. Let us just mention the circularity behind this kind of reasoning: prices are defined when equilibrium is reached, but in order to reach such an equilibrium prices must have been announced out of equilibrium at the beginning of the process.

## 5.2. The classical approach

The classical approach is somewhat different, because it does not seek in exchange as such the common characteristic which makes commodities equivalent, but goes upward to the
production process of commodities to emphasize the fact that all of them have been produced by human labor, which later on at the stage of the exchange process appears on both sides of the equivalence relation. This is so because for classical economists in the production process labor has been embodied, or for Marx “crystalized” into commodities.

Such an approach deserves nevertheless the same kind of criticism as the neo-classical one, on several grounds. First, labor in itself has no material content. It is not a substance but an activity, which is as such immaterial. There is no real thing in labor which can be transferred or embodied, or crystalized into a commodity. Second, there are many types of labor, from manual labor, implying physical activities of many kinds, to intellectual labor, which requires almost no physical activity. This is what Marx tried to conceptualize through the distinction he made between simple labor and complex labor. Moreover labor can differ greatly from one trade to another.

To be sure, Marx was fully conscious of this problem of heterogeneity of labor and tried to overcome it by developing the concept of abstract labor, supposed to be a homogeneous type of labor. But the pages that he devoted to that are much more philosophical and metaphysical than theoretically satisfactory. They do not provide the key which would allow to transform two commodities into comparable quantities having a common dimension of abstract labor. At this stage, abstract labor remains therefore an abstraction with no tangible meaning.

Moreover, both schools of thought remain in fact theoretically stuck by a conception of exchange which does not allow for a correct introduction of the concept of money.

### 5.3. Exchange and money

Both schools have a conception of exchange as an equivalence relation between two commodities. But such a conception is also factually wrong, because it has no empirical foundation: exchanges as they take place in the real world are not exchanges between commodities, since they have nothing to do with barter. In the world as it is commodities do not exchange against commodities, but against money.

Furthermore many empirical researches show that, despite the fact that money has long been represented through some kinds of commodities like gold or silver coins, and although it can still be represented through some objects of a material nature, like notes or coins, money in the world in which we live is essentially of an immaterial nature, and has lost most of its material representations. If money is not a commodity, an exchange between a commodity and money cannot be thought of as an exchange between two commodities.

If classical as well as neoclassical economists have overlooked this fact and made the same mistake on this question, although they were aware of the existence of money as a medium of exchange, it is simply because they could not abandon the view, which appeared to them well founded empirically, that money was a commodity. This was indeed the case for Marx as well as for all of the neoclassical economists. This means for instance that to transform the previous “equation” supposed to represent an exchange between two commodities into an equation reflecting an exchange between one commodity and money, they would write:
2 m of fabric = 3 g of gold (if gold is the commodity-money)

Or to write it more logically:

Value of 2 m of fabric = value of 3 g of gold

In such a formulation, it is clear that an exchange with money is still equivalent to an exchange between two commodities, with the only difference that one of these two commodities has a special status as a standard of value, allowing it to play the role of a universal medium of exchange. But this does not prevent this standard of value from keeping the dimension of a commodity.

One of the reasons behind the fact that money is not a commodity lies in the nature of money, which has more to do with the existence of debt than with the weight of a gold or silver coin. The recent and fascinating book by anthropologist David Graeber: “Debt, the first 5000 years”, provides us with a detailed historical account of this reality, showing in particular the very ancient linkages between debt and money. In his book Graeber sheds light on the way this relationship has emerged, much earlier than was previously thought, and has evolved over centuries in very different societies. He shows also that the nature of money as a general equivalent has always been disconnected from its material representation in various old or silver or bronze coins.

On a more theoretical ground, Keynes’s “Treatise on Money”, published six years before his “General Theory” and a little forgotten, has shown that a fundamental characteristic of money is liquidity. Finally, it is now widely recognized that in the real world money is created ex nihilo by commercial banks when they make new loans.

The question of the conceptual nature of money will be discussed later. Nevertheless all these observations, based on experience, are sufficient to enunciate two new principles:

**Principle 3:** Money is not a commodity

**Principle 4:** Commodities exchange against money and money exchanges against commodities: commodities do not exchange against commodities

At this juncture it is not enough to observe that the vast majority of commodities do not have their price set on a “market”, on which, through the intervention of an auctioneer, and at the end of a kind of iterative convergence process, relative prices as well as monetary prices would be established. It has moreover to be recognized that for an exchange relation to be considered as a relation of equivalence, both quantities on each side of the equality have to be of the same dimension.

Therefore, if exchange - as it is the case - is a monetary one, this means that both quantities have to be expressed in money. In other words, when exchanges take place, commodities exchanged against money must already be measured in money, i.e. expressed in terms of their money value. This logical necessity is indeed what can be observed in the real world: commodities which are on sale have already a monetary price which has been established
before commodities are made available to buyers, and buyers already possess money. This
deserves to be formulated in terms of a new principle:

**Principle 5:** Commodities have a monetary price before exchanges take place

Where these monetary prices and money come from will be dealt with later in the course of
this book, because the nature of money needs to be specified first in more details.

**6. Money**

The links between money and debt and money and liquidity have already been mentioned, but
money is more often defined in the economic literature through its three main functions:
money serves as a medium of exchange, as a store of value, and as a unit of account.

Of these three functions, it is the third one which is by far the most important at a conceptual
level, because it is the logical precondition for the existence of the two other ones. Indeed
money could not be used as a medium of exchange if it were not a unit of measurement, and
could neither be used as a store of value if one could not measure the quantity of money
which is stored.

The nature of a unit of measurement and of measurement is nevertheless a question which is
rarely dealt with by economic theory. It is indeed traditionally the mathematical theory of
measurement which addresses this issue, and it is more often associated to physics. Physicists,
more than economists, are familiar with this theory. But this question is of such an
importance, even for economic theory, that it seems quite useful to give now, if not an
exhaustive presentation of the mathematical theory of measurement, at least some rather
substantial explanations about its content.

**6.1. The mathematical theory of measurement**

On this technical question we rely on an old but rather simple presentation made by Doneddu,
in a classic book on the basics of modern mathematical analysis, where chapter 3 deals with
the “Measure of magnitudes” and is broadly used in the following pages (Doneddu, 1963, pp.
224-244). It is indeed a reliable presentation of this theory of measurement.

By definition a real measure (i.e. a measure by a real number) of a set of objects, named *E*,
comes down to the establishment of an isomorphism between all the equivalence classes that
can be defined on this set through a given equivalence relation and the additive semi-group of
*R*<sup>+</sup>, the sub-group of positive real numbers.

For such an isomorphism to exist, a certain number of assumptions have to be verified.

1. If one admits the existence of an operation of real measurement, it is necessary, within the set of all equivalence classes resulting from a given equivalence relation, to define an operation of addition, which to each couple of equivalence classes establishes a correspondence with a class named sum of the two first ones. It two elements belong to classes *p* and *q*, the sum of these two elements represents class *p* + *q*, under the condition
that we can define on set E an addition which is associative, commutative and has a neutral element (the zero class).

It is considered that the set of equivalence classes is moreover totally ordered by the following relation: if element $p$ is smaller than element $q$, then $p \leq q$.

Introducing an addition which is associative, commutative and has a neutral element makes the set E of equivalence classes a totally ordered commutative semi-group with a neutral element.

2. The existence of an isomorphism with $R^+$ also implies the introduction of the axiom of Archimedes.

Let us indeed give the following definition of measurement:

*Definition 1:* To measure the elements of a set E consists in establishing for any element $x$ of E a correspondence with a real number and only one, named “real measure of $x$”, written $\mu(x)$, and such that for a non-zero element $u$ of E, chosen once and for all and named “measurement unit”, the corresponding real number is 1, and also such that to the sum of two elements of E corresponds the sum of their measures:

$$ \mu(u) = 1 \\ \mu(x + y) = \mu(x) + \mu(y) $$

This means that function $x \to \mu(x)$ defined on E and with images in $R^+$ must be a homomorphy for addition.

The problem of measurement comes down to building this function $x \to \mu(x)$. It is easily solved for all the elements which are a multiple of the unit $u$. Solving the problem for all the elements of E which are not a multiple of unit $u$ needs the introduction of the axiom of Archimedes: “whatever the non-zero elements $x$ and $y$ of E, there is a natural number $n$ such that $n \cdot x > y$”.

If E verifies this axiom, its elements are classes of Archimedean magnitudes. The set E is then an Archimedean semi-group. If E is an Archimedean semi-group then it can be shown that to any couple of elements $(x, y)$ $(x \neq 0)$ of set E corresponds a natural number $q$, representing an equivalence class, and only one, such that :

$$ q x \leq y < (q + 1) x, \quad q \text{ being the Euclidean quotient of } y \text{ by } x $$

To find $q$ means to realize the Euclidean division of $y$ by $x$, with this division having a rest $r$ such that:

$$ [q x \leq y < (q + 1) x] \iff [y = q x + r \text{ and } r \leq x] $$

$q$ is only the approximated measurement of $x$ at the nearest whole.
The next step consists in going from the semi-Archimedean group isomorph to $Q^+$ (the positive rational numbers) to a set isomorph to $R^+$. Such an operation does not pose any particular problem, and therefore its demonstration will not be presented, in order not to unduly lengthen this section. It must just be recalled that it involves the introduction of the axiom of bisection and of the axiom of absence of gaps (this is Dedekind's method for the construction of real numbers).

It is now possible to draw from this construction of the operation of measurement, as it has been proposed, two important consequences:

*First consequence:* change of unit and stability domain of the equivalence relation of measurement.

The real measurement of a magnitude has been defined by assuming that the measurement of some particular magnitude $u$ was 1. Any other magnitude is then measured in relation to $u$, by comparing it to a multiple or sub-multiple of $u$. Let us take $v$ as a non-zero element of $E$. If we name $\mu_u(x)$ the measurement of $x$ when $u$ is taken as the unit, let us find the relation between $\mu_u(x)$ and $\mu_v(x)$.

We just have to consider the function $f(x) = \frac{\mu_u(x)}{\mu_u(v)}$ of $E$ into $R^+$ (with $\mu_u(v) \neq 0$)

By operating in $R^+$, we have:

$$1) f(v) = \frac{\mu_u(v)}{\mu_u(v)} = 1$$

$$2) f(x+y) = \frac{\mu_u(x+y)}{\mu_u(v)} = \frac{\mu_u(x)}{\mu_u(v)} + \frac{\mu_u(y)}{\mu_u(v)} = f(x) + f(y)$$

$f(x)$ is therefore the measurement of $x$ when $v$ is taken as the unit : $f(x) = \mu_v(x)$,

$$\text{Which implies: } \mu_v(x) = \frac{\mu_u(x)}{\mu_u(v)}$$

We have therefore the following property: whatever the non-zero elements $u$ and $v$ of an Archimedean semi-group: $\mu_u(x) = \mu_u(v)$. $\mu_v(x)$

This property is a characteristic of the measurement operation. It allows to define what is called the domain of stability of the equivalence relation. In the present case, i.e. the equivalence relation of exchange, the domain of stability is made of all of the systems where exchange takes place and which leave this relation invariant while maintaining the property which has just been defined : $\mu_u(x) = \mu_u(v)$. $\mu_v(x)$

Since a measurement relates to a domain of stability, the definition of the conditions according to which a system constitutes a domain of stability is an important one in the
context of a research focusing on the dimension associated to an equivalence relation. This definition will indeed imply to deal with the question of the standard of value corresponding to the measurement unit.

*Second consequence:* Influence of the construction of real measurement on the structure of the corresponding set of magnitudes.

Let us recall first that the construction of the set of magnitudes and the construction of the set of measurements of these magnitudes are simultaneous. The two sets have therefore the same mathematical structure, by construction. Let us recall also that to make the real measurement of a magnitude comes down to considering it as belonging to an additive semi-group isomorph to $R^+$. It derives from this that the set of magnitudes is complete, because it has been admitted, by definition, that every infinite suite of nested intervals included in E, and whose length tends to zero, never defines a gap.

If E is complete, then to each real number $\alpha$ corresponds one element y of E having $\alpha$ as its measurement when x is taken as unit.

$$x \neq 0, \exists y \mid \{y = \alpha. x\} \leq\Rightarrow \mu_x(y) = \alpha$$

It must be noted that this formula does not define the outer product of a number by an element of E, whatever the way it could be done. In particular the outer product of $\alpha$ by $x$ only exists if the measurement of y when $x$ is taken as unit is $\alpha$. This outer product is only another way to say that $\alpha$ measures y in terms of $x$. In particular such a product only exists within the domain of stability of the equivalence relation.

This remark having been made, we can introduce a second definition:

*Definition 2:* To any couple $\alpha, x$ of a number $\alpha \in R^+$ and an element $x \in E$ ($x \neq 0$) it is possible to associate one element of E, named product of $\alpha$ by $x$, noted $\alpha x$, and defined as the element of E having $\alpha$ as its measurement when $x$ is chosen as the unit.

$$\text{If } x = 0, \text{ then } \alpha.0 = 0 \quad \forall \alpha \in R^+$$

From this definition three properties can be deducted:

1st Property:

$\forall \alpha, \forall \beta, \forall x \in E \text{ Assuming that } x \neq 0 \text{ and taking } x \text{ as the measurement unit:}$

$$y = \alpha x \Leftrightarrow \alpha = \mu(y)$$

$$z = \beta x \Leftrightarrow \beta = \mu(z)$$

$$\text{Then: } \alpha + \beta = \mu(y) + \mu(z) = \mu(y + z)$$

$$\text{But } y + z = (\alpha + \beta)x$$

$$\text{Therefore: } (\alpha + \beta)x = \alpha x + \beta x$$

2nd Property:
$$ \forall \alpha, \forall \beta, \forall x \in E $$

Assuming that $x \neq 0$ and $\beta \neq 0$, and that $y = \beta \cdot x \neq 0$

Taking $x$ or $y$ as the unit and using the first property:

$$ \mu_x(z) = \mu_x(y) \cdot \mu_y(z) = \beta \cdot \mu_y(z) $$

With $z = \alpha \cdot y$,

We obtain $\mu_x(z) = \alpha \cdot \beta$

Therefore $z = (\alpha \cdot \beta)x$

But $z = \alpha \cdot y = \alpha(\beta \cdot x)$

Therefore: $\underline{\alpha \cdot (\beta \cdot x) = (\alpha \cdot \beta) \cdot x}$

### 3rd Property:

$$ \forall \alpha \in R^{+}, \forall x, \forall y \in E $$

Let us take any unit u:

$$ \begin{aligned} \mu[\alpha(x+y)] &= \alpha \cdot \mu(x+y) \\ &= \alpha[\mu(x)+\mu(y)] \\ &= \alpha \cdot \mu(x) + \alpha \cdot \mu(y) \\ &= \mu(\alpha \cdot x) + \mu(\alpha \cdot y) \end{aligned} $$

Therefore: $\underline{\alpha(x+y) = \alpha x + \alpha y}$

It must be emphasized that these theorems are valid only for the domain of stability of the equivalence relation. But they are then fully valid.

Definition 2 introduces only a quite specific outer operation and does not come down, in general, to accept that one can multiply any scalar number by any magnitude. By definition the operation $y = \alpha \cdot x$ is a notation for $\mu_x(y) = \alpha$, which is allowed because the objects to be measured are “infinitesimally adjustable”. Besides, the demonstrations of the three properties show well that you always have to go back to the definition to grasp the sense of outer product $\alpha \cdot y$. The application that defines real measurement: $x \to \mu(x)$ from $E$ into $R^+$, has the following properties:

$$ \mu(x+y) = \mu(x) + \mu(y) $$

$$ \mu(\alpha \cdot y) = \alpha \cdot \mu(y) $$

So we can say that real measurement is pseudo-linear.

It follows from this whole demonstration that all the objects belonging to set $E$ must have the same dimension, and that the theory of measurement is thus closely related to dimensional analysis. The mathematical theory of measurement having thus been fully exposed, it is also possible now to go further in the understanding of the nature of money.
## 6.2. The nature of money

From the definition of money as a unit of measurement most economists go one step further to define money as a measure of value, usually without providing any preliminary definition of the dimension of this value. However we know from principle 4 that money appears in exchange, and from principle 5 that exchange takes place when a quantity of money is equivalent, not to a value, but to the monetary price of a commodity. From this it follows that money is the unit of measurement of prices, and not of values. This also implies that the question of the nature of money must be coherent with the mathematical theory of measurement, which in turn necessarily constrains the answer that can be given to this question. Indeed, as a measurement, money cannot be anything else than a set of numbers belonging to the $R^+$ set. This provides us with the theoretical justification of the next principle:

**Principle 6:** As the measurement of a dimension of commodities which appears when it is exchanged against them and is called their price, money is a pure set of numbers belonging to the set of positive real numbers $R^+$. As such money has itself no dimension, and is composed of scalar numbers

Corollary to Principle 6: Monetary prices are therefore also dimensionless, scalar numbers

It results from this principle that money has no material content, which is perfectly coherent with the way it is created. In the world as it is, money is indeed created ex nihilo by banks. These economic agents are of such importance in present day capitalists economies, as the institutions responsible for money creation, that banks have to be added to the two categories of economic agents that were defined previously, those of workers and capitalists.

It is widely agreed as an empirical fact that money is created by banks, and this has a bearing on some other features of money which give more precisions about its nature, because money is created by banks when they make loans. This allows us to state that money is an “asset-liability”, to use a term coined by Bernard Schmitt, a French theorist who developed the circuit theory of money on the basis of Keynes’s theories, in his book “Money, wages and profits”, published in 1966. He shows that money is more precisely a double asset and a double debt.

It is a double asset because as soon as it is created through the registration of a loan in the books of a bank, the bank bears a claim on the borrower to whom it has extended a loan. It appears in the bank books on the assets side of its balance sheet. This claim is redeemable at the time which has been agreed for repayment, and it will stay in the books up to this time. Simultaneously the amount of money corresponding to the loan is credited to the account of the borrower, where it appears as a liquid monetary asset for this borrower. At the same time, it is also a double debt, because it is registered as a demand deposit on the liabilities side of the bank balance sheet, and it is a debt for the borrower, which will stay as such up to the time when it has to be repaid. When this repayment occurs, all these four accounts are cleared, and money disappears. This brings us to the next principle:
**Principle 7:** Money is created ex nihilo by banks through credits that they provide to other economic agents, taking the form of a double asset and a double debt: money is cancelled by the repayment of these credits

The fact that money is created ex nihilo does not imply that its creation happens by chance. Money is closely linked to production, because one of the essential features of the capitalist mode of production is that wages are paid in money. If we assume that there is no money, and situate ourselves at the beginning of a new production process or period, then there are capitalists owning the means of production, and workers which need to be hired in order to start a new production cycle. These workers have to be paid in money, and capitalists have to turn to banks and to borrow from them in order to get this amount of money.

This observation constitutes the starting point of what is usually considered as the “monetary theory of production”, which as recalled by Dillard in his 1980 article “A Monetary Theory of Production: Keynes and the Institutionalists”, goes back to Keynes, and to the title of his course of lectures at Cambridge University at the autumn of 1932. It is important enough to be considered as a principle, being understood also that wages paid to a worker are not a pure amount of money, but an amount of money paid for a defined period of time:

**Principle 8:** Wages are paid in money and have the dimension of dimensionless numbers per unit of time

It must be noted that because of the nature of money as a measurement, the corresponding quantities that money measures cannot be themselves dimensionless: there is logically no such thing as the measurement of a measurement. The only operations that can be performed on measures are changes of unit, which are of no help to answer the question of the dimension of the magnitudes measured by money, i.e. the values of commodities. So far this question is still left unanswered. This is the reason why we must now turn to production, which might bring about a solution to this problem.

Production will be dealt with in two parts. The first one will explore theories of production based upon the notion of surplus, and will show that they constitute a theoretical dead-end. The second one will try and shed some light on the nature of production as a transformation process.

## References

*Dillard, Dudley (1980). A Monetary Theory of Production: Keynes and the institutionalists. Journal of Economic Issues, vol. XIV, N°2.*

*Doneddu, Alfred (1963). Les bases de l'analyse mathématique moderne (« The bases of modern mathematical analysis »). Paris: Dunod.*

*Graeber, David (2011). Debt, the first 5000 years. New-York: Melville House Publishing.*

*Jevons, W. Stanley (1871). The theory of political economy. London and New York: Mac Millan.*
Keen, Steve (2001). *Debunking Economics: the Naked Emperor of the Social Sciences*. Australia: Pluto Press, and U.K: Zed Books. 335 pp.

Keen, Steve (2011). *Debunking Economics: the Naked Emperor Dethroned*. London and New-York: Zed Books. 478 pp.

Kuhn, Thomas (1974). *Second Thoughts on Paradigms*. In Suppe F. (Ed.), *The Structure of Scientific Theories* (pp. 459-482). USA - Urbana (Illinois): University of Illinois Press.

Marx, Karl (1867). *Capital: A Critique of Political Economy*. Volume I - Book One: The Process of Production of Capital. Source: First English Edition of 1887. Marx/Engels Archive [online]. Available at: https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-I.pdf. Accessed 22 Aug 2018.

Schmitt, Bernard (1966). *Monnaie, salaires et profits* (*Money, wages and profits*). Paris: PUF.

Smolin, Lee (2008). The case for background independence. *Perimeter Institute for Theoretical Physics*. arXiv:hep-th/0507235v1. 25 July 2005.

Sonin, Ain N. (1997). *The physical basis of dimensional analysis*. Cambridge (USA): MIT, Department of Mechanical Engineering (Second Edition: 2001).

Walras, Leon (1874-1877). *Eléments d’économie politique pure ou théorie de la richesse sociale*. Paris: Guillaumin et Cie. English edition: Walras, L. (1965). *Elements of Pure Economics or the Theory of Social Wealth*. (translation: W. Jaffé). London: G. Allen & Unwin.
# PART I - PRODUCTION WITH A SURPLUS: A THEORETICAL DEAD-END

"Nothing is created, nor in the operations of art, nor in those of nature, and we may assume as a principle that in any operation there is an equal amount of matter before and after operation ; that the quality and quantity of the ingredients is the same, and that there are only changes, modifications". Antoine Lavoisier (1789), in his Elementary Treatise of Chemistry.
# Chapter 2. The Ricardian theory of production, distribution and value

To understand conceptual problems still left unresolved today, it is often very useful to go back to the great economists of the past, who grappled with these same problems. This is the reason why the Ricardian theory of production and distribution, closely linked to Ricardo's theory of value and prices, will be exposed and criticized in this chapter.

## 1. The Ricardian problematics and its basic data

### 1.1. The Ricardian problematics

David Ricardo was concerned by the future of the capitalist mode of production, which was just starting its prodigious development during his lifetime. In the first words of the preface to his major book “Principles of Political Economy and Taxation” published in 1817, he considered that: “the produce of the earth, all that is derived from its surface by the united application of labor, machinery and capital, is divided among three classes of the community; namely, the proprietor of the land, the owner of the stock or capital necessary for its cultivation, and the laborer by whose industry it is cultivated” (Ricardo, 2001 ed., p. 5).

A little further, he continued: “But in different stages of societies, the proportion of the whole produce of the earth which will be allotted to each of these classes, under the names of rent, profit, and wages, will be essentially different; depending mainly on the actual fertility of the soil, on the accumulation of capital and population, and on the skill, ingenuity and instruments that will be employed in agriculture” (Ricardo, 2001, p.5).

Ricardo was interested in the reproduction of the system, and hence by the share going to the owners of capital, because he had well perceived that the accumulation of capital realized from the profits would play a major role in this reproduction. This is why he added: “To determine the laws which regulate this distribution, is the principal problem in Political Economy” (Ricardo, 2001, p. 5).

### 1.2. The exogenous data

The exogenous data of the Ricardian theory are first the quantities of the various commodities which are produced and the techniques available, meaning the quantities of labor and means of production used in the production of each commodity. But it is also the level of real wages, i.e. the quantities of goods consumed by workers, which are fixed in the long run exogenously at the minimum of subsistence, by the law of Malthus. This law was indeed accepted as perfectly valid by Ricardo, himself a friend of Malthus.

Ricardo also considers fixed capital as a wages fund: a machine is but the materialization of the past spending of the wages fund, which has gone one step further in the production process. Profit is thus the difference between what is produced, on the one hand, and the
current wages plus the wages fund necessary for this production, on the other hand: it is thus a
residue.

Finally Ricardo uses the important assumption that the share of production going to each
capitalist, divided by the amount of capital employed, must be equal for all of them, which
implies that the rate of profit must be uniform in all branches.

It is on the basis of this problematics and of these data that Ricardo has devised successive
solutions to the question of distribution that led him to view as the principal difficulty the
question of value. These solutions will now be examined.

**2. A first try to solve the distribution problem: “the Essay on Profits”**

**2.1. A mathematical presentation of the Essay on Profits**

It is a well-accepted opinion in the economic literature since the Introduction of Sraffa to the
Principles of Ricardo that within the framework of Ricardo’s simple model of the “Essay on
Profits” published in 1815, i.e. two years before the “Principles”, he had succeeded in solving
the problem of the determination of profits while avoiding the problem of valuation (Sraffa,
1951, pp. xiii- lxii).

In this model Ricardo assumes that the workers’ wages consist only of corn, which is a
generic term for wheat, since Ricardo makes all his calculations in terms of wheat. Therefore
the only wage-good is wheat, and the production of wheat is realized with only labor and
wheat, because the other means of production can also be represented by wheat, as a wage-
good.

In the branch which produces wheat, the share of profits in the net product and its rate on the
invested capital are consequently determined as a ratio of physical quantities as soon as the
level of wages is set and as the conditions of production on the marginal land are given, which
allows to get rid of the rent. With the assumption of the uniformity of the profit rate across the
branches, through the moves of capital from one branch to another, we get the principle put
forward by Ricardo that: “the general profits of stock being regulated by the profits made on
the least profitable employment of capital in agriculture” (Ricardo, 1815, p. 13)

On the basis of these rather simplifying assumptions, which will be discussed below, the
model allows for the determination, not only of the profit rate, but also of the exchange ratios
of all the other commodities in terms of wheat, since the means of production of the other
branches can always be expressed in terms of a quantity of wheat.

This can be easily demonstrated, taking the following notations:

$r$ is the profit rate

$L_i$ is the quantity of labor required for the production of the quantity $A_i$ of commodity $i$.
We designate by index $i = 1$ the branch which produces wheat, the wage-good consumed by workers. The amount of wages advanced in this branch is equal to $L_i$ multiplied by the quantity $w$ of wheat paid to workers per unit of labor time, so that $L_i w = A_{il}$.

We designate by $A_{ij}$ the quantity of commodity j used for the production of commodity i.

Then we note $a_{ij}$ the ratio $\frac{A_{ij}}{A_j}$, with $\sum_{i=1}^{n} a_{ij} \le 1 \quad \forall j$

If we resize to one the quantities of the various commodities, and if we name $p_1, p_2, \dots, p_n$, the price of the $n$ commodities, the system can be presented as follows :

$$ (1) \quad (a_{11}p_1)(1+r)=1 \\ (2) \quad \begin{cases} a_{21}p_1 + a_{22}p_2 + \dots + a_{2n}p_n(1+r) = p_2 \\ \dots \\ a_{n1}p_1 + a_{n2}p_2 + \dots + a_{nn}p_n(1+r) = p_n \end{cases} $$

In equation 1, the price of wheat is taken as the unit of measurement, equal to 1, corresponding to the overall quantity of wheat produced, which is the standard of value. Equation (1) of the system determines the profit rate and the system (2) of $n-1$ equations determines the $n-1$ prices of the commodities other than wheat in terms of the overall quantity of wheat produced, which corresponds to the measurement unit. In this system it is not possible for any commodity or basket of commodities other than wheat to play the role of standard of value, because otherwise the rate of profit could not be determined, and the system of equations (2) could not be solved. The prices, expressed in terms of the measurement unit (the number 1) are numbers which can only correspond to values which have the dimension of quantities of wheat, and nothing else. To demonstrate this assertion, let us transform the system into a matrix format.

Let $A$ be the matrix of technical coefficients and $P$ the vector of prices. The system can then be written:

$$ (3) \quad (1+r)AP=P $$

Which gives $[I - (1 + r) A] P = 0$

It is not possible to say that this system has a solution (positive prices and one unique positive profit rate) up to a scalar multiple. For the price vector obtained as a solution to be positive (i.e. $p_i \ge 0, \forall i$) it is indeed needed that matrix $A$ be irreducible, according to the Perron-Frobenius theorem, which is demonstrated by Gantmacher, in his 1959 book "Applications of the Theory of Matrices" (pp. 64-65).

But matrix $A$ is reducible, because one of its lines has only one non-zero element. One could then get negative prices, which would be devoid of economic significance.
The system cannot therefore be written under this format. It is necessary first to eliminate
from matrix A the first line, which gives the conditions of production of wheat. Let us call A'
the matrix obtained by doing so and P* the price vector (p₂, p₃, ...pₙ).

The system can then be written:

$$
(4) \quad \begin{cases} a_{11}p_1(1+r) = p_1 \\ (1+r)A'P = P^*. \end{cases}
$$

But such a system has no solutions, because matrix A' is not a square matrix and vector P has
not the same dimension as vector P*. It is therefore necessary to eliminate one column of
matrix A' by eliminating one price. The only price that can play this role is price p₁ because
otherwise the two price vectors on each side of the equation would not be identical and the
system would have no solution, since it would be written under the format:

$$
(1 + r) A'' P' + (1 + r) U = P^*
$$

Taking $p_1 = 1$, naming $C$ (for corn!) the column vector $(a_{21}, a_{31}, \dots, a_{n1})$, and $A^*$ the matrix resulting from the elimination of vector $C$, the system cannot but be written under the following format:

$$
(5) \quad \begin{cases} a_{11}(1+r) = 1 \\ (1+r)A^*P^* + (1+r)C = P^* \end{cases}
$$

$$
\text{Therefore } P^* = [I - (1+r)A^*]^{-1} (1+r)C \quad (6)
$$

$$
\text{Or: } P^* = \left[ I + (1+r)A^* + (1+r)^2 A^{*2} + \dots + (1+r)A^{*n} \right] (1+r)C \quad (7)
$$

If we develop this expression for any price, we get:

$$
p_i = (1+r)a_{i1} + \sum_{k=2}^{n} (1+r)^k A_i^{*k} C \quad (8)
$$

Where $A_i^{*k}$ is the $i^{th}$ line of matrix $A^{*k}$

From equation (8) we can observe that a price $p_i$, although being a scalar number, can be
analyzed as corresponding to the sum of the quantity of wheat employed directly in the
production of commodity $i$, to which is added the profit of branch $i$, on the one hand, and the
quantities of wheat employed indirectly in the production of intermediate goods used in the
production of commodity $i$, to which is added the profit corresponding to the different stages,
on the other hand. To choose the price of wheat as the measurement unit, since this
measurement corresponds to a system with the dimension of wheat, comes down to choosing
as a standard of value a pure quantity of wheat.
Although this simple model allows for the determination of the profit rate *r* without any need to determine the exchange ratios (the prices), it is clear that it allows nevertheless to calculate these exchange ratios in terms of wheat, the standard of value for the model. It is striking that the method used by Ricardo to determine the profit rate while avoiding the valuation problem allows however to determine the value of commodities, which has the dimension of a quantity of wheat! It is equally stunning to realize the correctness of Ricardo's findings, however within the theoretical boundaries corresponding to his peculiar and rather restrictive assumptions, without the assistance of the mathematical apparatus that we used.

A first conclusion is that the profit rate is determined exclusively in the sphere of production of the wage-good, wheat. And it is because the price of wheat is chosen as the measurement unit that a variation in the price of wheat cannot have any influence on the price of wheat, which is by definition invariable.

A variation in the profit rate can therefore come from only two causes: either a variation in the technical conditions of the production of wheat, i.e. in the number of workers necessary for the production of a given quantity of wheat on the marginal land; or a variation in the level of real-wage, i.e. the quantity of wheat advanced by worker. These two causes of variation in the profit rate can combine, even though it is always possible to determine precisely the exact influence of each of the two causes in this variation.

When the profit rate changes in the sector producing wheat, capital moves are supposed to ensure the transmission of this variation to the other sectors, where prices in terms of wheat adjust consequently.

When the only origin of this variation is a change in the technical conditions of wheat production, the only cause of the variation in the other commodities price is the change in the profit rate. When the modification of the profit rate comes from a change in the level of real wages, the variation in the other commodities price results not only from a change in the profit rate, but also from the change in the real wage of the workers employed in each of the corresponding branches (in other words, in a homothetic way if real wage is the same in all of the branches). But it is always possible to isolate within the variation in prices the shares resulting from the modification in the profit rate and from the change in real-wage, respectively.

Finally the price of commodities may change only because of a modification in the technical conditions of production of anyone of these commodities (if it enters directly or indirectly into the production of all the other ones). The influence on prices of such a modification can also be isolated.

But the most important conclusion that one can derive from this model is certainly that it is always possible to compare the price of commodities when they change, whatever the cause of this change: a change in either the profit rate, or the level of the real-wage, or the conditions of production. Values are indeed nothing else than quantities of wheat, and it is always possible to compare a quantity of wheat to another one. It is similarly always possible to measure the evolution of the amount and share of wages or the amount and share of profits
in terms of wheat in the product when the system itself changes, or/and to compare the whole product in terms of wheat during a period to the product in terms of wheat in another period. A given quantity of wheat is indeed in this model an invariable standard of value, corresponding to the measurement unit (the number 1) of the prices, which are scalar numbers.

One might nevertheless consider as superfluous to explore the properties of such an old and simplified model, dating back two hundred years ago. In fact it is not so, because this model gives us a good hint on what an economic system with money as a commodity would look like. It suffices indeed to replace wheat by gold, to say that gold is the money-commodity and that wages are paid in money (gold) to be transported into a world looking to some extent like the world of the eighteenth century!

## 2.2. The underlying conception of production and distribution

At the time of Ricardo, agriculture was still the most important sector, employing the majority of population, and for Ricardo production in agriculture depended on the fertility of the land. This fertility resulted itself from the laws of nature, giving more or less "productive power" to different types of land. Agricultural production resulted in the creation of a surplus, the size of which depended on the level of fertility of the different sorts of lands under cultivation.

In the "Essay on Profits" this fertility can be measured unequivocally, as a ratio of two quantities of wheat: wheat produced over wheat used in the process of production. This can be done independently of the price system, which is not needed to define fertility. Production in other branches is seen by analogy.

If we consider now that wages are part of the surplus, in each of the *j* branches there is at least the surplus needed to cover the cost of the wage-good (wheat) paid to their workers. This is the case when $\sum_{j=1}^{n} a_{1j} = 1$. There is also the possibility of an additional surplus for each of the commodities *i* for which $\sum_{j=1}^{n} a_{ij} < 1$.

In this model the distribution of the product among the three classes identified by Ricardo has nothing to do with the social or political background, or with the will of men. Indeed the rent of land comes from a purely natural phenomenon, the difference in the fertility of various types of land, and from the growth of population, which obliges to put under cultivation new lands of decreasing fertility, with the least fertile land paying no rent. As Ricardo writes: "Rent then is in all cases a portion of the profits previously obtained on the land. It is never a new creation of revenue, but always part of a revenue already created" (Ricardo, 1815, p.18).

The level of real wages is also depending on the growth of population, which keeps it at a level that is just sufficient to allow for the subsistence of laborers, in accordance with Malthus's law.
Finally the profit rate is determined in agriculture. Profit is the difference between the product of the least fertile land and the amount of wages in terms of wheat paid for the labor required, directly or indirectly, by the existing technique of production. The profit rate is the ratio of this last quantity and this same amount of wages. Once this rate is known, it is easy to compute the amount of profits in agriculture, given the fertility of the lands, first, and finally in all the other branches. Distribution is therefore a kind of natural phenomenon, which regulates the shares of each class in the output of the system, i.e. its net product.

## 2.3. A highly questionable model

A first and obvious remark can easily be made regarding the assumptions of this model. It is that they violate two of the principles that we already established in the previous chapter. First, they contradict principle 3, according to which money is not a commodity, because the standard of value here is a commodity, in this case wheat. And second, as a consequence, they also contradict principle 8, according to which wages are paid in money, as we defined it, because wages in this model are real wages, not paid in money, but in wheat.

This model relies indeed on very simplifying assumptions, according to which the only wage-good employed in the production of corn is corn itself, and that corn is produced only from itself and from labor. This is indeed a matter for criticism, which will now be discussed.

The fact is that Ricardo isolates the sector producing wheat, which is the only sector whose production enters in the production process not only of the wheat sector itself, but also of all the other sectors, without any exception, because wages in all of these other sectors are paid in wheat. Conversely the production of these other sectors does not enter at all into the production process of wheat.

It is this very peculiar feature which allows for the determination of the profit rate in the wheat sector, where the surplus is the difference between the amount of wheat produced and the amount of wheat employed in the production process of wheat, among others to pay for wages. It is the reason why the profit rate is determined independently of prices.

However, outside of the wheat sector and at the level of the system as a whole, there is absolutely no reason why there should necessarily be a surplus of wheat, which must be used as an intermediate good in some sectors, and not only for the payment of wages. There will be an overall surplus of wheat only when $\sum_{j=1}^{n} a_{1j} < 1$.

But then, if it is the case, such a surplus implies that wheat will be also a consumption good, which is quite a logical problem, because it is difficult to understand why the same good, wheat, can be an intermediate good when it is consumed by workers, and a consumption good when it is consumed by the other classes of society: the owners of the lands and the owners of capital.

This difficulty can be resolved only if we admit that, as a pure consumption good, wheat is not produced by the agricultural sector as such, in the raw form it has when it has just been
harvested, but rather by the agribusiness sector, where it is processed to be made edible for consumers. In this case there is no longer a surplus of raw wheat, because all the raw wheat produced is used either to pay wages or as an intermediate good transformed in other sectors.

Moreover edible wheat is then produced in a separate production process and is necessarily different from unprocessed wheat: it is another good, different from raw wheat. And since no one can imagine that workers should consume unprocessed, non-edible wheat, real wages paid in a genuine consumption good, i.e. processed wheat, imply that other goods than raw wheat must appear on the left side of the equation in the sector producing wheat. This would completely remove the possibility to determine the profit rate in the wheat sector alone, and in fact would invalidate the whole model!

But the same analysis can be made as soon as we consider that there are obviously other inputs than wheat (be it processed or unprocessed) in the sector producing wheat. Then it is no longer possible to determine the profit rate as a pure ratio of quantities. The price system must be determined otherwise, even though, as we shall see in the next section, it can still be possible to determine the profit rate before the price system. All this shows thus that this first Ricardian model is too simple to be considered as an appropriate image of the real world.

Another kind of criticism can be made as regards the conception of production as a technical or physical process, in particular in the wheat sector as such, where there is necessarily a big surplus, because a limited quantity of wheat is used to produce a certainly much bigger quantity, which will be used not only in the wheat sector itself, but also in all of the other sectors, if only to pay for wages. It is also this surplus, once realized through the sale of wheat, which in the wheat sector will take the form of profits alone, on the marginal land, and of profits and rents on the other, more fertile lands.

This conception of production as the creation of a surplus, inherited from the Physiocrats, does not seem to be a problem for Ricardo, because he perceives this phenomenon like a farmer of his time who sows one quintal of wheat seeds per hectare and harvests 8 quintals of wheat. The surplus of 7 quintals is attributed to the productive power of land, its fertility. Production can therefore be conceived, quite naively, as a kind of spontaneous generation process. However, in the physical world where cultivation takes place, there is no such thing as spontaneous generation. At the time when Ricardo wrote his Essay on Profits, Lavoisier had already discovered the principle of mass conservation, which implies that mass can neither be created nor destroyed, although it may be rearranged in time and space, or the elements associated with it may be changed in form.

In the case of wheat production Lavoisier's principle implies that the mass of the wheat harvested, and its difference with the mass of wheat sowed, has to come from physical elements other than seeds, brought into the system by other sources. Agronomists now know what these elements are and also from which sources they come from.

One of these elements is water, in the form of rain or irrigation, which provides hydrogen and oxygen. Another one is air, which provides carbon dioxide and through the role of chlorophyll allows the synthesis of sugars. Legumes directly absorb nitrogen from the air in the soil, with
the help of bacteria in their roots. For all of the other plants, like wheat, nitrogen is removed from the ground almost entirely in the form of nitrates. This brings us to this other source which is the ground itself, which acts as a substratum, and also provides multiple elements: at the height of its growth, one hectare of wheat absorbs 2 kg daily of nitrogen, 6 kg of potassium, and 1 kg of phosphorus, as well as sulfur, calcium, magnesium and trace-elements.

All this shows well that from a physical point of view there is therefore no such thing as the spontaneous generation of a surplus. One might think nevertheless that there can be an economical surplus, owing to the fact that a number of these elements are free and not appropriated. Therefore they have no price (or a price equal to zero), and do not have to enter in the economical production process. This is indeed true, but we will show later that for other theoretical reasons the notion of surplus has to be rejected.

# 3. The problem of value and prices for Ricardo

## 3.1. The problem in terms of labor values

For Ricardo, profit is determined as a ratio between production and the consumption necessary for such a production. In the model examined in the last section there was a branch where this ratio had the form $\frac{A - A_{11}}{A_{11}}$ and was thus perfectly determined.

Let us abandon now the assumption that workers' wages are constituted by only one commodity. Let us assume for instance that workers receive in exchange for their labor not only a given quantity of wheat (produced by branch 1) but also a given quantity of woolen cloth (produced by branch 2). Even if we imagine a branch *i* where production is realized only with labor, to the exclusion of any other means of production, the ratio between the surplus obtained in this branch and the consumption (under the form of real wages) needed for this production now takes the form $\frac{A_i - (A_{i1} + A_{i2})}{A_{i1} + A_{i2}}$. The quantities in the numerator and denominator are not homogeneous (they do not have the same dimension). The profit rate therefore cannot be determined in such a way.

The situation remains obviously the same if we examine the whole system with n branches,

$$ \text{where the profit rate is then: } r = \frac{\sum_{i=1}^{n} A_{ij} - \sum_{ij=1}^{n} A_{ij}}{\sum_{ij=1}^{n} A_{ij}}. $$

The determination of the profit rate implies that the numerator and denominator of the ratio be made homogeneous, i.e. that the quantities of commodities that appear here be expressed in a common measurement unit. Since profit is realized through the exchange of commodities, this measurement unit must be for Ricardo that of the exchange ratios of commodities, and must also be such that these ratios ensure the equalization of profit rates among the different
branches. The determination of the profit rate requires therefore a theory of measurement, which in turn implies a theory of value, as the dimension to be measured.

It is not simple to explain how Ricardo attempted to solve this problem, since he did not succeed in doing so in a satisfactory way. One can indeed find, inextricably mingled in his works, a theory of measurement of the exchange ratios of commodities through the labor time, direct or indirect, necessary to produce them, and a theory of measurement through the prices of production, defined as the prices that ensure the uniformity of profit rates.

Two texts, apart from the first chapter of Ricardo's "Principles" and from some letters in the correspondence, allow nevertheless to shed some light on his attitude when confronted to this question. These texts are the two successive versions of the study entitled "Absolute value and exchangeable value", written in the very last weeks preceding Ricardo's death, on September 11, 1823 (Ricardo, 1823, pp. 361-397 and pp. 397-412).

What appears fundamental to Ricardo from these texts is what he names the absolute value of commodities, i.e. the measurement of the values of commodities through a measurement unit which has to be invariable in space and even more in time, or whose variations can be referred to an invariable standard. The determination of such an absolute value seems to him necessary to understand the variation of the exchange ratio of a commodity with another one, which he names the exchange value. One has indeed to know in which one of these two commodities lies the cause of this variation.

It is on this theoretical basis that Ricardo considered since the "Principles" that the absolute value of a commodity had to reflect its production cost, which in turn had to be understood as the quantity of labor directly or indirectly necessary to produce it. This position seemed indeed to give him an invariable measurement of value:

> "It has been said that we are not without a standard in nature to which we may refer for the correction of errors and deviations in our measure of value, in the same way as in the other measures which I have noticed, and that such standard is to be found in the labor of men. The average strength of a thousand or of ten thousand men it is asserted is always nearly the same, why then not make the labor of man the unit or standard measure of value? If we are in possession of any commodity which requires always the same quantity of labor to produce it that commodity must be of uniform value, and is eminently well qualified to measure the value of all other things. And if we are not in possession of any such commodity, we are still not destitute of the means of accurately measuring the absolute value of other things, because by correcting our measure, and making allowance for the greater or less quantity of labor necessary to produce it we have always the means of referring every commodity whose value we wish to measure to an unerring and invariable standard" (Ricardo, later version, unfinished, 1823, pp. 401-402).

Let us forget, at least temporarily, what has been said in chapter one on the heterogeneity of labor, and suppose that this problem has been solved. The insoluble problem which Ricardo nevertheless had to face was to attempt to introduce the profit rate as such within the system of absolute values determined by quantities of incorporated labor.
To show it let us use again the notation of the preceding sections. Moreover let us call $v_i$ ($i=1,...,n$) the absolute value of commodities $i$ in terms of labor quantities. Commodities designated by $i=1,...,k$ are wage goods.

Ricardo then defines the profit rate as the ratio between the value of the surplus and the value of wage goods:

$$r = \frac{\sum_{i=1}^{n} v_i A_i - \sum_{i=1}^{k} v_i A_i}{\sum_{i=1}^{k} v_i A_i}$$

If we introduce the profit rate such as just defined in the calculation of values, the system becomes the following, using matrix notation:

$$\text{With } V = (V_i) \qquad i = 1, \dots, n \\ L = (L_i) \qquad i = 1, \dots, n$$

$$(1+r)Av + (1+r)L = V \\ \Rightarrow [I - (1+r)A]V = (1+r)L \\ \Rightarrow V = [I - (1+r)A]^{-1} (1+r)L$$

The value of a commodity $i$ is then:

$$v_i = (1+r)L_i + \sum_{k=2}^{n} (1+r)^k A_i^k L \quad \text{where } A_i^k = i^{\text{th}} \text{ line of matrix } A^k$$

If we develop this last expression, attributing the $k$ index to the quantity of labor used during the $k^{th}$ “period” ($L_{ik} = A_i^k L$), we get:

$$v_i = (1+r)L_{i1} + (1+r)^2 L_{i2} + \dots + (1+r)^n L_{in}$$

This expression allows us to reach the same conclusions as those that Ricardo himself drew from his assumptions. So defined, the value of commodities depends on the profit perceived on the means of productions employed, whose amount varies with the relative importance of these means of production: “every alteration in the permanent rate of profits would have some effect on the value of all these goods, independently of any alteration in the quantity of labor employed on their production” (Ricardo, 1817, p. 33).

Then, when the profit rate varies, the value of commodities varies whereas the quantity of labor has not varied. Consequently, the absolute value of commodities is equal to the quantity of labor required to produce them only when this rate is equal to zero. As Ricardo also writes: “If all commodities were produced by labor employed only for one day there could be no such thing as profits for there would be no capital employed, beyond that of which every laborer is in possession before he commences to work - there could as we have seen be no variation in
the value of labor, but commodities would vary as labor was more or less productive”
(Ricardo, 1823 – rough draft, p. 365).

Let us consider the exchange ratio between two commodities, *i* and *j*. It is expressed as follows:

$$ \frac{v_i}{v_j} = \frac{L_{i1}(1+r) + L_{i2}(1+r)^2 + \dots + L_{in}(1+r)^n}{L_{j1}(1+r) + L_{j2}(1+r)^2 + \dots + L_{jn}(1+r)^n} $$

This exchange ratio will not be equal to the ratio of the quantities of labor incorporated in each commodity, except in two cases:

- Firstly when the profit rate is zero, whatever the conditions of production of the two commodities,

- Secondly, when the conditions of production of the two commodities are identical, whatever the profit rate, which implies that we have:

$$ \frac{L_{i1}}{L_{j1}} = \frac{L_{i2}}{L_{j2}} = \dots = \frac{L_{in}}{L_{jn}} $$

When this condition is not met, the exchange ratio will be different from the ratio of the quantities of labor incorporated in the compared commodities. Ricardo was aware of that, since he wrote: "The difficulty then under which we labor in finding a measure of value applicable to all commodities proceeds from the variety of circumstances under which commodities are actually produced" (Ricardo, 1823 – rough draft, p. 368).

The exchange ratio might be equal to the ratio of quantities of incorporated labor only if profit rates in branches *i* and *j* were different and in a ratio that would authorize this equality, with $r_i$ and $r_j$ such as:

$$ \frac{L_{i1}(1+r_i) + L_{i2}(1+r_i)^2 + \dots + L_{in}(1+r_i)^n}{L_{j1}(1+r_j) + L_{j2}(1+r_j)^2 + \dots + L_{jn}(1+r_j)^n} = \frac{L_i}{L_j} $$

With $L_i = L_{i1} + L_{i2} + \dots + L_{in}$

And $L_j = L_{j1} + L_{j2} + \dots + L_{jn}$

But such a case obviously contradicts completely the basic assumption of the uniformity of profit rates, which is for Ricardo a fundamental feature of capitalist competition. The conclusion that Ricardo drew from his analysis was therefore that labor did not provide an invariable standard for the measurement of value:

"It must then be confessed that there is no such thing in nature as a perfect measure of value, and that all that is left to the Political Economist is to admit that the great cause of the variation of commodities is the greater or less quantity of labor that may be necessary to
produce them, but that there is also another though much less powerful cause of their variation which arises from the different proportions in which finished commodities may be distributed between master and workman in consequence of either the amended or deteriorated condition of the laborer, or of the greater difficulty or facility of producing the necessaries essential to his subsistence” (Ricardo, 1823 – later version, p. 404).

In fact, at the root of Ricardo’s problems there is a definition of the profit rate which is uncoherent, as soon as it is introduced in the calculation of values in terms of incorporated labor.

Indeed the formula $r = \frac{\sum_{i=1}^{n} V_i A_i}{\sum_{i=1}^{k} V_i A_i} - 1$ does not allow for the determination of the profit rate, as soon as these values $v_i$ are themselves depending on the profit rate. But in the opposite case where values are defined exclusively in terms of incorporated quantities of labor the calculation of the profit rate becomes contradictory with the hypothesis of uniformity of the profit rate and has no significance, because as we just shew there are as many different profit rates as there are commodities produced in different conditions.

## 3.2. The same problem in terms of production prices

What happens now if we reject the measurement of values in terms of incorporated labor? Is it possible to determine the profit rate in a coherent way, if we interpret the Ricardian analysis in terms of production prices, defined as the exchange ratios which ensure the equalization of the profit rates among branches? In other words, if we define now the profit rate by the following expression:

$$r = \frac{\sum_{i=1}^{n} p_i A_i}{\sum_{i=1}^{k} p_i A_i} - 1, \text{ with } p_i \text{ = production price of commodity } i$$

Will we not meet the same contradiction as when we interpret it in terms of labor values?

A whole school of economic theory and in particular Von Thunen, Bohm-Bawerk and Walras, answered yes to this question. For more detailed comments on this point, one can refer to Dobb’s article “The SRAFFA system and critique of the neo-classical theory of distribution” (Dobb, 1980, pp. 347-362).

Among others, Walras thus accused Ricardo of having one equation to determine two unknowns in affirming that the price is determined by the sum of profits and wages and that profits are determined by the difference between the price of commodities produced and the amount of wages.
However, as soon as 1898, an early Russian mathematician economist named Dmitriev, in his “Economic essays on value, competition and utility” did justice to these criticisms and provided the demonstration that the profit rate can be determined within the framework of the Ricardian model interpreted in terms of production prices (Dmitriev, 1974).

To show it let us take the preceding notation. Moreover, since we abandon the measurement through the quantities of incorporated labor, and given the conception of wages adopted by Ricardo, labor must be replaced by the quantities of wage-goods paid to the workers.

The equations which determine the production prices are therefore the following:

$$ (a_{11}p_{1} + a_{12}p_{2} + \dots + a_{1n}p_{n})(1+r) = p_{1} $$

$$ (a_{21}p_{1} + a_{22}p_{2} + \dots + a_{2n}p_{n})(1+r) = p_{2} $$

$$ (a_{n1}p_{1} + a_{n2}p_{2} + \dots + a_{nn}p_{n})(1+r) = p_{n} $$

This system, once transcribed in matrix form, is represented by the equation:

$$ (1+r)AP = P $$

For mathematical reasons already exposed this system only provides significant solutions, from an economical point of view, when matrix A is irreducible. This requirement translates at an economical level in the fact that each commodity has to enter directly or indirectly in the production of all of the other ones. But for Ricardo only wage-goods meet this condition, since we can always resolve the production of a good to an expense of labor during successive periods. It must be noted however that in fact the expression “wage-goods” has in this model a broader meaning than the one admitted by Ricardo, because it encompasses not only the goods consumed by workers, but also the goods that enter directly or indirectly in their production.

Let us continue to consider that commodities designated by $i=1,...,k$ are wage goods (in this broader sense). From a mathematical point of view, this means that matrix A can always be reduced in the following way:

$$ (1+r) \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1k} & 0 & 0 & \dots & 0 \\ a_{21} & a_{22} & \dots & a_{2k} & 0 & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & 0 \\ a_{k1} & a_{k2} & \dots & a_{kk} & 0 & 0 & \dots & 0 \\ a_{k+1,1} & a_{k+1,2} & \dots & a_{k+1,k} & a_{k+1,k+1} & a_{k+1,k+2} & \dots & a_{k+1,n} \\ a_{k+2,1} & a_{k+2,2} & \dots & a_{k+2,k} & a_{k+2,k+1} & a_{k+2,k+2} & \dots & a_{k+2,n} \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ a_{n,1} & a_{n,2} & \dots & a_{n,k} & a_{n,k+1} & a_{n,k+2} & \dots & a_{n,n} \end{bmatrix} \begin{bmatrix} p_1 \\ p_2 \\ \dots \\ p_k \\ p_{k+1} \\ p_{k+2} \\ \dots \\ p_n \end{bmatrix} = \begin{bmatrix} p_1 \\ p_2 \\ \dots \\ p_k \\ p_{k+1} \\ p_{k+2} \\ \dots \\ p_n \end{bmatrix} $$

In matrix notation, this translates into the following system:
$$ (1+r) \begin{bmatrix} A_1 & 0 \\ W & A_2 \end{bmatrix} \begin{bmatrix} P_1 \\ P_2 \end{bmatrix} = \begin{bmatrix} P_1 \\ P_2 \end{bmatrix} $$

We have called $0, A_1, W$ and $A_2$ each of the different submatrices. Submatrix $A_1$ means that the $k$ wage-goods enter in their own production. Since wage-goods constitute some kind of basket of goods which are all needed for the subsistence of workers, no element of this submatrix should be equal to zero. The existence of submatrix $0$ reflects the fact that the production of the $k$ wage-goods requires the use of wage-goods, but do not require any input from the $n-k$ sectors producing other goods. These other goods are not consumed by the workers, and are therefore considered by Ricardo as “luxury goods”. Submatrix $W$ means that the production of luxury goods obviously requires wage-goods. Submatrix $A_2$ means that luxury goods can enter in their own production. $P_1$ is the price vector of wage goods and $P_2$ the price vector of “luxury goods”.

Under this format, such a system has no economically significant solutions, because matrix $A$ is reducible, and it must therefore be reduced into two sub-systems:

$$ (1) (1+r)A_1P_1 = P_1 $$

$$ (2) (1+r)WP_1 + (1+r)A_2P_2 = P_2 $$

Sub-matrix $A_1$ is an irreducible square matrix (of order $k$). Matrix $A_2$ is also a square matrix of dimension $(n-k, n-k)$. The two other matrices have no reason to be square matrices, and are rectangular ones: matrix $W$ is of dimension $(n-k, k)$, and matrix $0$ is of dimension $(k, n-k)$.

For this reason only the first matrix equation (1), which contains the conditions of production of the wage-goods and is equivalent to $A_1P_1 = \frac{1}{1+r}P_1$, allows the calculation of the profit rate.

The Perron-Frobenius theorem indeed ensures us that the only eigenvalue of an irreducible matrix with non-negative terms to which corresponds a positive eigenvector is the dominant eigenvalue of this matrix, i.e. the highest one.

If we call $\alpha$ the dominant eigenvalue of matrix $A_1$, equation (1) is equivalent to:

$$ A_1 P_1 = \alpha P_1, \text{ with } \alpha = \frac{1}{1+r}, \text{ and this implies that } r = \frac{1-\alpha}{\alpha} $$

The profit rate is thus directly deducted from the dominant eigenvalue $\alpha$ of matrix $A_1$, and vector $P_1$ is itself the eigenvector associated to this eigenvalue. Moreover we know that the eigenvalue of a matrix is determined independently of the eigenvector, as a solution to the equation $(A_1 - \alpha I) = 0^1$.

<footnote n="1">¹ Indeed $A_1P_1 = \alpha P_1 \Rightarrow A_1P_1 = \alpha IP_1 \Rightarrow (A_1 - \alpha I)P_1 = 0$</footnote>
We arrive again to the same conclusion as that which Ricardo never ceased to uphold since the “Essay on Profits”, according to which the profit rate is determined exclusively by the conditions of production of wage-goods, independently of their price. This means that in such a system the production prices and the profit rate are not determined simultaneously: the profit rate is determined first.

Once the profit rate is determined the eigenvector $P_1$ of matrix $A_1$ can be determined, up to a scalar multiple, and the price of wage-goods is therefore also determined, which comes down to adopting as a measurement unit the price of anyone of these goods.

Once we know the profit rate and the price of wage-goods, we can also determine the price of the other goods, as a solution to equation (2), such as:

$$P_2 = [I - (1+r)A_2]^{-1} (1+r)WP_1$$

For a good i, this price is expressed as:

$$P_i = (1+r)W_i P_1 + \sum_{k=2}^{n} (1+r)^k A_{2,i}^k W_i P_1 \quad \text{with } A_{2i}^k = i^{\text{th}} \text{ line of matrix } A_2^k$$

It should be no surprise that we obtain again the result of the “Essay on Profits” according to which the price of “luxury goods” can be expressed solely in terms of the price of wage-goods and the profit rate. When the whole system of production prices is known it is then possible to determine the amount and the share of wages, as well as the amount and the share of profits.

Moreover, we can see that a production price can be identified to a pure production cost only for luxury goods. This cannot be the case for wage-goods, because the price of anyone of these wage-goods depends not only on the proportion in which the other ones enter into its own production, but also on the proportion in which it enters itself in the production of the other goods. In passing we also see that, contrary to what we could think at first glance, the level of wages as such is not exogenous: these are indeed the quantities of wage-goods which are exogenous. Their price is endogenous.

It has therefore be demonstrated that the Ricardian theory, at least from a mathematical point of view, and when interpreted in terms of production prices, allows to determine in a coherent way the distribution of the net product between the workers and the capitalists.

### 3.3. The contradictions and limitations of the Ricardian analysis

First we must ask ourselves whether this second interpretation in terms of production prices is able to answer the fundamental concern of Ricardo: to understand the evolution of distribution over time. This comes down to asking if it is possible to compare the share of wages and profits from one period to another, when these shares change, and therefore to measure the variation of income distribution between wages and profits.

Let us begin by noting that for given conditions of production the profit rate is determined independently of the price which is chosen as a measurement unit. On the other hand, the
price of all the commodities, and consequently the level of wages and the amount (and not the rate) of profits, depend on the price chosen as a measurement unit. As Ricardo indicates: "We are possessed then of plenty of measures of value and either might be arbitrarily selected for the purpose of ascertaining the relative value of commodities at the time they are measured" (Ricardo, 1823 – rough draft, p.381).

Let us note however, that the share of wages and profits in the income is always the same, whatever the measurement unit. Indeed the price vectors are collinear.

The problem is that in the Ricardian system such as the one that we just exposed every variation in wages has to be analyzed as a variation in the conditions of production, because wages are made of goods. When such wages change, matrix A of technical coefficients is modified and especially the conditions of production of the commodity whose price is adopted as the measurement unit. Obviously, it is still possible to calculate the profit rate, the price system and the distribution resulting from a variation in the wage level. But it is not possible to compare this new system to the previous one, and consequently the new distribution to the previous one, because the measurement unit has been modified and is no longer the same.

We cannot but conclude that the evolution as such of prices and distribution is undetermined, because of the impossibility to find out an invariable measurement unit as soon as we are at the level of exchange ratios implying the introduction of a uniform profit rate into the analysis. We can nevertheless affirm that if the conditions of production of wage-goods deteriorate, or if the quantities of wage-goods increase, these circumstances will translate into a decrease in the profit rate. The dominant eigenvalue α of matrix A₁ is indeed an increasing function of the elements of this matrix, and the profit rate is consequently a decreasing function, since $r = \frac{1-\alpha}{\alpha}$.

This is exactly the definitive conclusion which Ricardo had reached: "We are without any (measurement unit) by which to ascertain the variations in the values of commodities for one year, for two years or for any distant portions of time" (Ricardo, 1823, rough draft, p. 381). And: "there can be no perfect measure of the variations in the value of commodities arising from an alteration in these proportions (into which commodities are divided for wages and profits) as the proportions will themselves differ according as the commodity employed for the measure may be produced in a shorter or longer time" (Ricardo, 1823, later version, p. 404).

At a first level, we can conclude from this analysis of Ricardo's Political Economy that if he did not succeed in producing a correct concept of value, his approach, as it appears particularly in his last text, "Absolute value and exchangeable value", written shortly before his death, provides nevertheless some interesting elements.

<footnote n="2">² On this point, see again Gantmacher. Op.cit.</footnote>
One of his problems and in fact his main mistake was that Ricardo wanted to integrate
production and distribution, and therefore absolute values in terms of incorporated labor with
exchangeable values as exchange ratios resulting in a uniform rate of profits, i.e. production
prices. He could not succeed, because values and prices do not have the same dimension,
which he almost understood, but not to the point of finding out a way to pass from one to the
other. In any case his treatment of wages as real wages, or wage-goods, would have prevented
him from progressing towards a possible solution.

This leads us to a second level, which is the conception of production and distribution. In fact
it had not changed much since the Essay on Profits, since the essential change was the
replacement of a single wage-good, wheat, by a number of different wage-goods, which can
come from any sector, and not only from agriculture. To be sure, it is a little more realistic to
consider that workers survive not only by consuming wheat, but also a variety of other goods.

But this does not allow to get rid of the criticism that his system continues to violate two
principles already established. First, it contradicts principle 3, according to which money is
not a commodity, because the standard of value here is still a commodity, even if it is no more
wheat, but can be any one of the various wage-goods. And second, as a consequence, it also
contradicts principle 8, according to which wages are not established as real wages, but in
money, in which they are paid.

Instead of wheat alone, these are now all of the wage-goods, in the broad sense that have been
defined, which are singled out as the only goods that enter, directly or indirectly, in the
production of all the other ones. This derives from the assumptions made by Ricardo, because
the payment of wages in wage-goods makes them the perfect substitute to labor, i.e. the only
element in the system to enter in the production of every good.

It is the reason why the determination of the profit rate takes place in the wage-goods sector,
where there is a surplus, as the difference between the amount of wage-goods produced and
the amount of wage-goods employed in the same production process. It has to be so, because
on top of the payment of wages in the wage-goods sector there must remain an amount of
wage-goods available to pay for wages in all of the other sectors, those producing “luxury
goods” (in fact goods that, once produced, do not re-enter into the production process of any
wage-goods). It is also the reason why the profit rate is determined independently of prices,
through the eigenvalue $\alpha$ of matrix $A_1$.

However, outside of the wage-goods sector and at the level of the system as a whole, there is
absolutely no reason why there should necessarily be a surplus of wage-goods. It is all the
more the case that, due to the broad definition of wage-goods, they also include all the goods
entering in their production (otherwise they could not be produced). Therefore wage-goods
must be used not only to pay for wages, but also as intermediate goods in some sectors, and
not only for the payment of wages. There might be an overall surplus of a given wage-good $j$
only when for this particular wage-good $\sum_{i=1}^{n} a_{ij} < 1$.
But then, if it were the case, such a surplus would imply that this wage-good, for the quantity not entering into the production process, would be also a "luxury good", i.e. in fact a consumption good, which is quite a logical problem, because it is impossible to understand why the same good, whatever it is, could be an intermediate good when it is consumed by workers, and a consumption good when it is consumed by the other classes of society: the owners of the lands and the owners of capital. In fact, even though it is certainly true that these other classes consume some specific goods that are not necessary for their subsistence (the reason why they are called "luxury goods") they also have to consume for their subsistence a number of goods, like food etc., that are similar to the goods consumed by workers.

Such a contradiction can be resolved only if we admit that wage-goods are true consumption goods, which have to be treated as such, meaning as goods that do not re-enter as intermediate goods into the production process. In passing it is of great importance to be very clear about what that means to enter in the production process, since it has to do with the very definition of the concept of production. Let us precise therefore that to enter (or re-enter) in the production process implies first to be bought by a producer from another producer, and second and furthermore to be transformed one way or another into another good through the production process. This is the definition of intermediate goods, which are part of what is often called circulating capital. In this sense wage-goods, which are not bought on the market by producers to be given in kind to the workers whom they employ, for payment of their labor, and are not transformed in the production process, do not meet the criteria to be considered as intermediate goods. They are pure consumption goods, which once produced are bought, not by producers, but by individuals, be they workers or capitalists, and never go back to re-enter for transformation into the production process.

If therefore wage-goods are considered as what they truly are, i.e. consumption goods that do not enter in the production process, this implies that it is no longer possible to make them appear on the left side of the equations which were used to represent this process. They will only appear on the right side or the equations. This would make them quite similar to luxury goods, which would be quite inappropriate to designate usual consumption goods. But this should also be the case for all the goods so far considered as "luxury" goods, which - once produced, have no reason to appear on the left side of the equations, as if they went back into their own production process. In other words, submatrix $A_2$ should also be a zero submatrix. As for the most appropriate common term for both kinds of consumption goods, it should be "final goods".

Does it mean that the presentation used so far must be completely abandoned? No, because there still remain true intermediate goods, which are finished products, but not final products. Indeed all of these goods enter directly or indirectly in the production of all other goods, but are never used, neither as consumption goods, nor as capital goods.

This leads us to state first a corollary to the last principle, stated at the end of chapter one:

**Corollary to Principle 8:** Wages are not paid to workers in wage-goods.
We can also state two new principles, to be added to the first eight ones:

- First, the following principle:

**Principle 9:** Workers buy consumption goods out of their wages previously paid in money.

- With the following corollary:

**Corollary to Principle 9:** there are no specific consumption goods bought by the workers that could be isolated as pure wage-goods.

- And second, as an additional principle:

**Principle 10:** Intermediate goods are bought, not by individuals, but by producers, to enter the production process, where they are physically transformed into other intermediate goods or final goods.

- With this last corollary:

**Corollary to Principle 10:** Goods bought by workers as consumption goods, as well as all the other consumption goods, are not intermediate goods, but final goods.

Moreover, one must observe that the conception of distribution underlying the main and the last works of Ricardo (the “Principles” and the text on “Absolute and Exchangeable Value”, respectively) has not really changed from that of the “Essay on Profits”. Indeed, if the model that we analyzed in this chapter is actually more complex than the previous one, distribution is still completely disconnected from its social background, let alone from the balance of power between the various classes of society. The level of wages is given and considered as a natural phenomenon, because it is still Malthus’s law of population which maintains it at the minimum level of subsistence. Since the technical conditions of production, and in particular the number of workers in each branch, are also given, this determines the overall amount of wages. Then the amount of profits is obtained by difference, as a residue. In this sense, distribution still depends on the conditions of production.

In any case and finally, the problem that also stays unresolved is that prices in this last Ricardian model still have the dimension of a commodity, whose price is chosen as the measurement unit, whatever it may be (as long as it is an intermediate good). It follows, as Ricardo himself recognized it, that no comparison in time between two price systems is possible as soon as there is the slightest variation in the conditions of production, which would cause necessarily a variation in the conditions of production of the good chosen as the standard of value.

The Ricardian system remains however quite a rich one, owing to all the questions it raises, and despite or maybe because of its flaws. It provides us with a number of insights on the problems that need to be overcome to answer actual economic questions. But however instructive it may be, it has to be seriously modified to make some progress toward the resolution of these questions.
Is it still possible nevertheless, on a basis where there would be no wage-goods, and therefore where distribution would have to be defined differently, to design a coherent system to determine both distribution and prices, as well as their variation over time? This is what is going to be examined in the next chapter, devoted to the theory of Sraffa, such as exposed in his well-known book "Production of commodities by means of commodities".

## References

Dmitriev, Wladimir Karpowitsch (1898). *Economic Essays on Value, Competition and Utility*. In D. M. Nuti (Ed.) and D. Fry (translator). New-York and London: Cambridge University Press (1974).

Gantmacher, Feliks Ruvimovich (1959). *Applications of the Theory of Matrices*. Translated from the Russian by J. L. Brenner, with the assistance of D. W. Bushaw and S. Evanus. New-York: Interscience Publishers.

Ricardo, David (1815) An Essay on the Influence of a Low Price of Corn on the Profits of Stock. In Piero Sraffa (Ed.), *The Works and Correspondence of David Ricardo*. Vol. IV. Cambridge (UK): Cambridge University Press.

Ricardo, David (1817) *On the Principles of Political Economy and Taxation*. 1st Edition: London: John Murray.
Online edition: third edition 1821. Kitchener (Canada): Batoche Books (2001).
https://socialsciences.mcmaster.ca/econ/ugcm/3113/ricardo/Principles.pdf
Accessed 23 Aug 2018.

Ricardo, David (1823) Absolute value and exchangeable value. In Piero Sraffa (Ed.), *The Works and Correspondence of David Ricardo*. Vol IV. Cambridge (UK): Cambridge University Press (pp. 361-412). Rough draft: pp. 361-397. Later version - unfinished: pp. 397-412.

Sraffa, Piero (1951). "Introduction" to "Principles of Political Economy" in Sraffa, P. (Ed.) *The Works and Correspondence of David Ricardo*. Vol. I. Cambridge (UK): Cambridge University Press.
**Chapter 3. Distribution and prices in Piero Sraffa's theory**

This chapter will expose first Sraffa's theory of production as production of a surplus, and of distribution as a sharing of this surplus between workers and capitalists. It will show also that Sraffa's treatment of the problem of reproduction is quite different from Ricardo's. As for its construction of an invariable standard of prices, it is closer to Ricardo's "Essay on Profits" than to Ricardo's "Principles". Moreover, the way according to which this standard is built makes it impossible to analyze changes in the system of production. The main utility of Sraffa's system has been in fact to destroy theoretically the neo-classical theory of capital and its view of the rate of profit as the marginal productivity of capital.

To begin with, let us recall that the neo-Ricardian or neo-Cambridgeian theory, which has developed after the publication in 1960 of Sraffa's only book, "Production of commodities by means of commodities", has been widely perceived as a return to the classical and more particularly Ricardian political economy.

Such a conception seems however somewhat questionable, to the extent that it relies on a superficial interpretation of Ricardian analyses. It starts from the idea, which is right, that the problem of finding an invariable measurement of values has been an essential concern for Ricardo. Since Sraffa manages to build up such an invariable standard of value, this conception then deducts ipso facto that he solves the problem which had been left unsolved by Ricardo, and is therefore following his steps in the same footprints, with more success.

However, the analysis of the preceding chapter has shown that Ricardo, far from leaving the problem unsolved, has given it an unequivocal answer, but a negative one, by showing that there cannot be such an invariable measurement of values as soon as conditions of production change. Therefore the solution proposed by Sraffa can only be developed in another type of problematic, which is not exactly the Ricardian one.

This will be shown at several levels: first the level of reproduction, second the level corresponding to the link between the conception of wages and the construction of a standard of measurement. Finally it will be shown that the nature of Sraffa's assumptions have a bearing on the meaning of production prices.

# 1. Sraffa's problematic of reproduction: quite different from Ricardo's

## 1.1. Reproduction and accumulation of capital

The divergence in problematics between Sraffa and Ricardo appears as soon as the first paragraph in the preface of "Production of commodities", where Sraffa underlines that: "the investigation is concerned exclusively with such properties of an economic system as do not depend on changes in the scale of production or in the proportions of 'factors'." (Sraffa, 1960, Preface, p. 5)
If we could consider that Ricardo's problem consists in determining the distribution of the product at a given point in time, Sraffa would be perfectly right in adding that: "this standpoint (...) is that of old classical economists from Adam Smith to Ricardo" (Sraffa, 1960, p. 5). It is true indeed that for Ricardo the problem is not simultaneously to determine equilibrium prices and quantities, but to determine prices and distribution for produced quantities of commodities and production techniques which are both given.

But for Ricardo this constitutes only a logical necessity, and as such only a first step for an analysis which is embedded in a larger framework, i.e. the reproduction of the whole economic system. The determination of profits is of such an importance for Ricardo because it represents a logical precondition of a second question, which is the accumulation of capital. This will allow in turn the reproduction of the economic system on an enlarged scale, when the additional means of production newly accumulated start producing.

When Sraffa warns us at the very beginning of his book that the change of the economic system stays outside of his analysis, the implicit message is that he does not intend to address the reproduction of the system, unlike the Ricardian analysis. The reason must be sought after in the nature of the invariable standard that Sraffa wants to build up. This standard, as we shall see, is indeed invariable only to changes in distribution, whereas for Ricardo the problem which is unsolvable is that of the invariability of the standard in case of changes in production. Indeed for Ricardo any variation in distribution must be analyzed as a variation in the production system, as it was shown in the preceding chapter.

## 1.2. The question of returns

These previous remarks allow us to better understand a second precision provided by Sraffa in his preface, which is that "no changes in output and (at any rate in Parts I and II) no changes in the proportions in which different means of production are used by an industry are considered, so that no question arises as to the variation or constancy of returns" (Sraffa, 1960, p. 5). It is because the concept of return implies a comparison between the price systems of two distinct production systems, and because such comparison would need a standard that would be invariable to changes in the system, which is not the case. Sraffa's theory leads therefore to the absolute rejection of any assumption about returns.

This impossibility of formulating an assumption about returns is quite significant of the difference in problematics between Sraffa's and Ricardo's analyses. For the last one, an analysis of reproduction cannot indeed be separated from an assumption on returns. This does not mean that the Ricardian concept of returns is identical to the neo-classical one. On the contrary Ricardo does not intend to make the quantity of one factor vary while the other ones remain constant, in order to determine simultaneously distribution, quantities and prices, since for Ricardo the quantities which are produced and the technique are given, as well as the quantities of wage-goods.

The problem is to determine the evolution in the value of wage-goods in order to determine the amount available for accumulation, when all the data are changing. And the evolution in
the rent of land is not the only change to be concerned, as the whole chapter on wages in the “Principles” clearly shows. The Ricardian concept of returns does not refer to “factors of production”, but to the productivity of labor, the measurement of which requires a theory of value based on a measurement unit invariable with respect to changes in the system of production. This explains the use by Ricardo of the labor theory of value.

It is indeed by assuming a decreasing productivity of labor in the sector producing wage-goods that Ricardo concludes to the possibility of a deadlock in the accumulation of capital, because the profits rate will fall below the level needed for capitalists to continue investing. To be sure, this analysis is not fully rigorous to the extent that the link between the fall in returns in terms of labor value and the fall in profits rate is not correct, at least in the “Principles” (if not in the “Essay on Profits”). It remains that the concept of returns with the meaning that has just been defined plays an essential role in the Ricardian analysis of reproduction, whereas it is completely absent from Sraffa’s analysis.

This particular point concerning returns seems thus to constitute a good test of the change in problematics, since it shows that the accumulation of capital, which is central to Ricardo, is outside of the ambit of the theory of production prices. One can certainly agree that a system that does not register any change, neither in the quantities that are produced, nor in the technique of production, is reproducing identically. But such a reproduction has nothing to do with the conception of reproduction for Ricardo, because it belongs to a logic which is exterior to the accumulation of capital.

However, it is by examining the way through which Sraffa’s constructs his invariable standard that one can best perceive the effects of this difference in problematics.

## 2. Wages and the construction of an invariable standard

### 2.1. Two Ricardian models with wages as wage-goods

One could think that Sraffa deals with the problem of the invariable standard only in chapter 4, entitled “the Standard commodity” of “Production of commodities”, thus starting to address it from paragraph 23 of his book. In fact the reasoning which will lead him to the construction of the standard commodity is touched upon as soon as paragraph 8, in which Sraffa abandons the Ricardian conception of wages.

Up to this point Sraffa develops two models. The first one, that Sraffa names “production for subsistence” can be represented in matrix format by the following equation:

$$AP = P \text{ with } \sum_{i} a_{ij} = 1, \forall j$$

This system has a solution other than $P = 0$, as soon as one of the prices is taken as a measurement unit, since the determinant of matrix $(I - A)$ is zero, because of the condition $\sum_{i} a_{ij} = 1$, which means the absence of a surplus. In this model no problem of distribution
arises, although one can only interpret it as a situation in which the totality of output is used, either as means of production or for the consumption of workers.

The second model, entitled “production with a surplus” is based on the implicit assumption that quantities produced are higher than quantities used in the production process or as means of subsistence for the workers. It is the condition for a surplus to appear, and this condition is not a uniquely technical one, since it implies that workers receive only the quantities of goods necessary for their subsistence. As Sraffa writes: “wages (are) consisting of the necessary subsistence of the workers and thus entering the system on the same footing as the fuel for the engines or the feed for the cattle” (Sraffa, 1960, § 8, p. 10).

This model is therefore nothing else than the Ricardian model exposed in the previous chapter, and which takes the form $(1+r)A.P = P$. It is important to note that the apparition of a profit rate depends on the existence of a surplus.

In this model and for reasons already exposed, Sraffa shows that only the commodities which enter directly or indirectly in the production of all the others play a role in the determination of the system, and he decides for this reason to name them “basic products”. The profit rate is determined only by the conditions of production of these basic products. As for the equations of production of non-basic products or “luxury goods”, they can determine only the price of these goods themselves, once the price of basic products is known (anyone of these prices being chosen as a measurement unit).

In order to clarify in advance some ideas that will be exposed later, it is important to note that in this “Ricardian” system the only data relating to wages are the quantities of wage-goods by labor unit, and not the price of these quantities of wage-goods. In particular the fact to add to the price system an additional equation like: $S_1p_1 + S_2p_2 + ... + S_kp_k = w$ (where $S_1, S_2, ... , S_k$ correspond to given quantities of the commodities that are part of the wage-unit), does not help in solving the system. Indeed the information conveyed by this equation is already embodied in the system itself through the production coefficients of wage-goods.

As a matter of fact, one could think to use such an equation to present the system of production prices under the format:

$(1+r)(A^1P + wL) = P$, where $A^1$ is the column block of the production coefficients of basic goods other than wage-goods.

But this would have no sense, since $A^1$ is not a square matrix, and therefore has no inverse matrix.

It follows than in a “Ricardian” system of production prices, one has to determine first the profit rate, then the production prices, including the price of the wage-goods, which allows finally, and only then, to determine the level of $w$, which is not given, but is an unknown variable of the system, in terms of any price that has been chosen as a measurement unit.
As we showed in the previous chapter, the conception of wages which is at the basis of this model does not allow to take into account a change in distribution, because of the impossibility to define an invariable measurement unit within this framework. It is therefore no surprise that the first step made by Sraffa in the construction of his invariable standard consists in abandoning this conception of wages. But then, what conception does he adopt?

## 2.2. A new conception of wages as part of a surplus

Sraffa starts by expressing the idea that: “besides the ever-present element of subsistence, they (wages) may include a share of the surplus product. In view of this double character of the wage it would be appropriate, when we come to consider the division of the surplus between capitalists and workers, to separate the two component parts of the wage and regard only the “surplus” part as variable; whereas the goods necessary for the subsistence of the workers would continue to appear, with the fuel, etc., among the means of production” (Sraffa, 1960, § 8, p. 10).

Sraffa nevertheless abandons this conception to adopt that which consists in considering the totality of wages as variable.

We can however note that this distinction between wages partially or totally taken out of the surplus has no consequence from a formal point of view. This was demonstrated in 1974 by Maurisson in his PhD thesis, devoted to the works of Piero Sraffa (pp. 153-154). Indeed, in the case where only part of the wages are a part of the surplus, the matrix of production methods A contains two column blocks: the first one is $A^W$ reflecting the conditions of production of subsistence goods, and the second one is $A^N$, for the other goods entering the production process.

Thus $A = (A^N, A^W)$.

Let us call $L$ the vector of direct quantities of labor, and $P$ the column vector of production prices, which subdivides into a vector $P_W$ for the price of subsistence goods, and a second one $P_N$ for the price of the other goods:

$$P = \begin{pmatrix} P_N \\ P_W \end{pmatrix}$$

The system of production prices then can be written under a matrix format:

$(1+r)AP + w'L = P$, with $w'$ the wage corresponding only to the part payed out of the surplus.

This is equivalent to: $(1+r)(A^N P_N + A^W P_W) + w'L = P$

And gives us: $(1+r)A^N P_N + rA^W P_W + A^W P_W + w'L = P$
This last equation allows us to express the total amount of wages in nominal terms (meaning in terms of the measurement unit) w. We can indeed write:

$$wL = A^W P_W + w'L$$

By pre-multiplying by vector $L^{-1}$, inverse to the left of column vector $L$, we get:

$$w = L^{-1}A^W P_W + w'$$

We define $W = L^{-1}A_W$ as the line vector of quantities of subsistence goods by labor unit, which gives:

$$w = WP_W + w'$$

The whole system of production prices can therefore be written:

$$(1+r)A^N P_N + rA^W P_W + wL = P$$

When all of the wages are considered as a part of the surplus, it means that there are no wage-goods and that all the elements of vector W are zero, which implies that the column block $A^W$ is zero. In such a case $w = w'$ and the system of production prices can be written:

$$(1+r)A^N P_N + wL = P$$

It is therefore under a mathematical format which is identical to the previous system [which was $(1+r)AP + w'L = P$], of which it constitutes therefore a particular case. The essential difference between both systems is that wage-goods appear no more in the second system as basic goods and therefore do not participate any more in solving this system. Solving it will now be considered within the framework of this last model, which is the one kept by Sraffa.

## 2.3. The consequences of wages not being advanced

Before exploring the functioning of this model, it seems useful to open a parenthesis in order to understand that when wages are no longer considered as advanced (whether it is a fraction or the totality of these wages), which means that no profit is perceived on them, their expression in terms of wage-goods becomes ipso facto undetermined. It means that wages cannot in any case be identified to a quantity of wage-goods. To show it the simplest thing to do is to argue by contradiction, precisely by imagining that wages are identified to a quantity of wage-goods, and that this quantity is not advanced, but taken from the surplus.

If therefore we imagine a hypothetical system of production prices where wage-goods exist as such but are not advanced, this system will be of the form:
$$ (a_{11}p_{1} + \dots + a_{1k}p_{k})(1+r) + a_{k,k+1}p_{k+1} + \dots + a_{1n}p_{n} = p_{1} $$

$$ (a_{k1}p_{1} + \dots + a_{kk}p_{k})(1+r) + a_{k,k+1}p_{k+1} + \dots + a_{kn}p_{n} = p_{k} $$

$$ (a_{k+1,1}p_{1} + \dots + a_{k+1,k}p_{k})(1+r) + a_{k+1,k+1}p_{k+1} + \dots + a_{k+1,n}p_{n} = p_{k+1} $$

$$ (a_{n,1}p_{1} + \dots + a_{n,k}p_{k})(1+r) + a_{n,k+1}p_{k+1} + \dots + a_{n,n}p_{n} = p_{n} $$

With $P_1 = (p_1, p_2, \dots, p_k)$ = the price vector of basic goods other than wage goods

And $P_2 = (p_{k+1}, p_{k+2}, \dots, p_n)$ = the price vector of wage-goods

Is it possible to solve this system? This comes down to asking whether it is possible to
calculate the unknown variables which are the profit rate and the production prices.

This can be done by presenting the system under the following matrix format:

$$ A'P = P $$

$$ \text{With } A' = [(1+r)A^1 : A^2] $$

$A^1$ and $A^2$ are the column blocks corresponding respectively to the technical coefficients of the basic goods other than wage-goods, on the one hand, and the wage-goods, on the other hand.

$$ \text{We get then: } (I - A')P = 0 $$

For the system to have a solution other than the trivial one $P = 0$, it is necessary and sufficient that we have: determinant of $(I - A') = 0$.

If it were the case, solving the equation canceling the determinant would allow to determine
the profit rate and the prices.

But it cannot be the case. Let us show it, naming $A'$ the transposed matrix of matrix $A'$.

$$ \begin{gather*} \text{Indeed: } \det (I - A') = 0 \Rightarrow \det (I - A'^t) = 0 \Rightarrow \exists X \neq 0 \mid {}^t A'X = X \\ \Rightarrow X {}^t (A'^t) = X \Rightarrow X A' = X \end{gather*} $$
$$ \Rightarrow \exists \sigma \neq 0 | \sigma A' = \sigma, \text{ avec } \sigma = X^t $$

The fact that the determinant is zero entails therefore that we must have $\sigma A' = \sigma$, which means that the n equations are non-independent, and implies that anyone of them can be deducted from all the others.

However the system does not verify the condition $\sigma A' = \sigma$, although it is necessary for solving it, because it would imply:

$$ \sigma(1+r)A^1 = \sigma_1, \text{ with } \sigma_1 \text{ the vector made of the k first elements of } \sigma, $$

$$ \sigma A^2 = \sigma_2, \text{ with } \sigma_2 \text{ the vector made of the n-k last elements of } \sigma. $$

But only the condition $\sigma(1+r)A^1 = \sigma_1$ can be verified, whereas we have $\sigma A_2 \neq \sigma_2$ in the general case. There is indeed no reason why we should have $\sigma A_2 = \sigma_2$ because this condition, translated in economic terms, would mean that the calculation of the profit rate and prices would need the assumption that there is no surplus of wage-goods. Such an assumption is not acceptable, because the consumption of capitalists must at least partly be made of the same goods as the consumption of the workers.

It has thus been demonstrated that the representation of the system of production prices with wage-goods as part of a surplus (and consequently non-advanced), and such as $A'P = P$, cannot be accepted. But is the representation under the form $(1+r)A^1P_1 + A^2P_2 = P$ acceptable? The answer is again a negative one, because matrices $A^1$ and $A^2$ do not have an inverse (they are not square matrices).

The only remaining possibility for a representation of the production system consists in distinguishing two sub-systems, the first one which gives the price of basic goods other than wage-goods, the second one which gives the price of wage-goods. These sub-systems are:

$$ (1+r)A_1^1P_1 + A_1^2P_2 = P_1 $$

$$ (1+r)A_2^1P_1 + A_2^2P_2 = P_2 $$

We can see that for solving one sub-system only, one has to consider as given either $P_2$, in order to determine $r$ and $P_1$, or $P_1$, in order to determine $r$ and $P_2$. But then one of the two matrix equations has no more reason to exist, because either $P_2$ or $P_1$ are already determined. We can note that if it is necessary to give oneself vector $P_1$ in order to determine vector $P_2$, the converse is not true. It is indeed sufficient to give oneself the level of wages: $w = WP_2$, with $W$ the vector of the quantities of wage-goods by unit of labor, because we have:

$$ A_1^2P_2 = WP_2L_1 $$
It remains that when we give ourselves w to determine r and $P_1$, the sub-system giving the price vector $P_2$ of wage-goods cannot be calculated. The system is indeed over-determined because it contains $n-k+1$ equations, which are:

$$ (1+r)A_2^1P_1 + A_2^2P_2 = P_2 \quad (n-k \text{ equations}) $$

$$ w = WP_2 \quad (1 \text{ equation}) $$

This for $n-k$ unknowns (the $n-k$ prices of wage-goods).

We arrive thus to this conclusion, at first sight rather curious, that as soon as we give ourselves $w$, i.e. the level of wages as a share of the surplus, while wanting at the same time to continue to consider wages as a quantity of wage-goods, the nature of these wage-goods is not classifiable, their quantity is undetermined, and the price of wage-goods stops having any significance. In fact they cannot even be calculated as long as we try to define wages simultaneously as the price of labor and as a quantity of wage-goods.

One other important conclusion is that the conception of wages as quantities of wage-goods is compatible only with the classical conception of wages as advanced wages. As soon as wages are not advanced their economic expression in terms of wage-goods is undetermined. The classical notion of real wages disappears with the assumption that wages are a part of the surplus. This does not mean that we cannot in this case calculate the price of “wage-goods”, but that this calculation is made ex-post, with the level of wages $w$ being given as the price of labor, which cannot be identified beforehand as a quantity of wage-goods.

The converse of the proposition that we just demonstrated that any wages considered as quantities of wage-goods are necessarily advanced is however not true. It is possible to consider wages as advanced without necessarily considering them as quantities of wage-goods.

It must finally be noted that this demonstration has a corollary, which is that it is not sufficient, for a commodity to be a basic commodity, that it enters directly or indirectly in the production of all the other ones, but it is also necessary that its utilization gives rise to the perception of a profit rate. In fact a commodity cannot be perceived as entering directly or indirectly in the production of all the other ones unless it is designed beforehand to give rise to this perception.

# 3. The construction of the invariable standard

## 3.1. A new system with $n+2$ unknown variables

After the explanations provided in the last section on the theoretical status of wages, we can now understand that since the search for an invariable standard requires from Sraffa to renounce to the classical conception of wages as a quantity of wage-goods, an essential difference appears in the solving of the system, with regard to the second model, i.e. the “Ricardian” one, analyzed by Sraffa.
This difference lies in the fact that the system of production prices now presents itself with $n+2$ unknowns, i.e. the $n$ prices, the profit rate $r$ and the wage $w$, and this for $n$ equations. To give oneself one of the prices, chosen as a unit of measurement, no longer suffices for solving the system, which has still one unknown, and remains undetermined. The only means to remove this indetermination, in order to be able to solve the system, is to give oneself also one of the distribution variables, chosen as an independent variable.

If we focus on the equations of production for basic goods only, in which the wage-goods corresponding to the wage $w$ obviously have no longer their place, the system of production prices presents itself under the following format:

$$ (1+r)AP + wL = P $$

Matrix $A$ is by assumption irreducible and supposed to be “productive”, which implies that $\sum_{i} a_{ij} \le 1$ for every $j$, and $\sum_{i} a_{ij} < 1$ for at least one $j$. Such a system has therefore always a solution in the form of a positive price vector, provided that one of the prices is taken as a measurement unit and one of the two distribution variables is given.

However it must be acknowledged that solving the system cannot be done directly, through the simultaneous determination of $n$ unknowns, i.e. $n-1$ prices and the dependent distribution variable. The calculation of the second distribution variable is indeed logically prior to calculation of prices.

The system can indeed be represented in the following matrix format:

$$ P = w[I - (1+r)A]^{-1}L $$

Such a matrix equation can be solved only after $w$ and $r$ are known. On this basis, how is a standard supposed to be invariable with respect to the variations of distribution going to be built? We will try to understand that in the next sub-section.

## 3.2. The determination of the second distribution variable

What is now at stake is to define the conditions that have to be met for a price chosen as a measurement unit to become an invariable standard. To understand it, the simplest solution consists in examining the way in which the system of production prices is solved when the price of any one of the $n$ commodities is chosen as a measurement unit.

Let us therefore take the price $p_i$ of commodity $i$ as this unit, such as $p_i = 1$. The calculation of the second distribution variable, the first one being supposed known, can be made from the equation giving the price of this commodity, by developing this equation. We get then:

$$ p_i = w[l_{i0} + (1+r)A_i L + (1+r)^2 A_i^2 L + \dots + (1+r)^n A_i^n L] $$
$$1 = w \left[ l_{i0} + (1+r)l_{i1} + (1+r)^2 l_{i2} + \dots + (1+r)^n l_{in} \right]$$

This equation defines a relation between the wage and the profit rate. This means that for a given level of the profit rate $r$, between 0 and a maximum reached when $w = 0$, this equation allows to calculate the corresponding level of wage. Reciprocally, for a given level of wage, between 0 and a maximum reached when $r = 0$, this equation allows to calculate the corresponding profit rate. For reasons of positivity of prices, it must be noted that the profit rate $r$ calculated this way is the highest of the roots of this equation of degree $n$ in $r$. It must also be noted that if the profit rate is a pure number, as regards the wage it is necessarily expressed in terms of the price chosen as the measurement unit.

Moreover any price, of a commodity or an aggregate of commodities – provided that they are basic commodities, can be chosen as a measurement unit. It entails that there can be an infinite number of relations between $w$ and $r$, as many as the number of prices that can be chosen as measurement units.

Any price having been chosen as the measurement unit, and once the distribution variables have been determined (one or the other being given) by the equation of this price, the totality of prices can then be determined by the matrix equation:

$$P = [I - (1+r)A]^{-1} wL$$

### 3.3. Changes in distribution and the measurement unit

What happens now if distribution changes, with production techniques remaining the same?

It can be directly deducted from the preceding equation that the whole price system is going to be modified. But contrary to what one might think at first sight, the new price system corresponding to the new state of distribution is not comparable to the previous one, although it is still the same commodity whose price is taken as the measurement unit. Indeed the fact that this price stays equal to 1 does not prevent the change in distribution from influencing it. This appears clearly if we develop the equation giving the price $p_j$ of commodity $j$ in terms of the price $p_i$ of commodity $i$, which is equal to 1:

$$\frac{p_j}{1} = \frac{l_{j0} + (1+r)l_{j1} + (1+r)^2 l_{j2} + \dots + (1+r)^n l_{jn}}{l_{i0} + (1+r)l_{i1} + (1+r)^2 l_{i2} + \dots + (1+r)^n l_{in}}$$

It is clear from this equation that the modification in prices following a change in distribution is the result of two separate effects. First an obvious effect which results from the modification of the equations of these prices themselves (i.e. a change in the numerator of the preceding ratio). Let us call intrinsic effect the effect on the prices. Second a less obvious effect at first sight which is due to the repercussion of the variation in distribution on the price taken as measurement unit itself (i.e. a change in the denominator of this same ratio). We will call extrinsic effect this effect produced on the prices by this variation in the “numéraire”,
which results from the intrinsic effect produced by a change in distribution on the “numéraire” itself.

Thus the new price system resulting from a variation in distribution is not comparable to the old one. It follows necessarily that a comparison is no more possible for the net product of the system $\sum (I - A)P$ as well as for the amount of profits $\sum rAP$ and the amount of wages $\sum [I - (1+r)A]P$. It is of course the same for the share of profits $\frac{\sum rAP}{\sum (I - A)P}$ and the share of wages $\frac{\sum [I - (1+r)A]P}{\sum (I - A)P}$.

The evolution of distribution is therefore undetermined.

### 3.4. The existence of an invariable measurement unit

To suppress this indetermination comes down to building up an invariable measurement unit. It is now possible to affirm that for a commodity to be able to play this role, it is sufficient for its price not to be submitted to the intrinsic effect produced by a change in distribution.

$$ \text{Since the price of such a commodity is } p_i = (1+r)A_i P + w l_i \quad (1) $$

It means that we must have:

$$ \forall (w,r), \quad dp_i = (1+r)a_i dP + drA_i P + dwl_i = 0 \quad (2) $$

But the absence of any intrinsic effect when distribution changes implies that an increase in profits must be exactly compensated by a decrease in wages, and reciprocally, a necessary condition which writes as:

$$ drA_i P + dwl_i = 0 \quad (3) $$

The conjunction of equations (2) and (3) entails necessarily that we have also:

$$ (1+r)A_i dP = 0, \quad \text{and therefore} \quad dP = 0 \quad (4) $$

As a consequence a necessary condition for the price of a commodity to be free from any intrinsic effect is that the price of its means of production must be equally invariable as regards the effects of changes in distribution. The same condition applies to the means of production of these means of production and so on. We can deduct immediately from this demonstration that the standard commodity cannot consist in a unique good and that it necessarily includes all the commodities entering directly or indirectly in the production of all the others, i.e. all the basic goods.
The price of each of these commodities must necessarily verify equation (2), which can be translated in synthetic terms by summing up all these equations. It must be noted that, since the invariable standard must necessarily include all the basic goods, the wage rate $w$ is necessarily expressed in terms of the price of all these commodities. And since the maximum amount that $w$ can reach when $r$ is zero is the price of the net product or $\sum (I - A)P$, the simple requirement of simplifying the calculation leads to express $w$ in terms of this price, which makes $w$ vary from zero to 1. If moreover we assume that $\sum l_i = 1$, the sum of equations (2) can be written under the form:

$$ \sum (I - A)dP = r \sum AdP + w \sum (I - A)dP + dr \sum AP + dw \sum (I - A)P = 0, \forall (w, r) \quad (5) $$

From the definition of the standard we have $dP = 0$, and therefore equation (5) is verified only if we have:

$$ dr \sum AP + dw \sum (I - A)P = 0 \quad (6) $$

$$ \text{Or: } -\frac{dr}{dw} = \frac{\sum (I - A)P}{\sum AP} = \text{constant} \quad (7) $$

Indeed, because $dP$ equals zero by definition, then $P = \text{constant}$.

In this case, we have also:

$$ \sum (I - A)P = r \sum AP + w \sum (I - A)P \quad (8) $$

$$ \text{Or: } \sum (I - A)P = r \sum Ap + w \sum (I - A)P \quad (9) $$

Therefore by combining equations (7) and (9), we get:

$$ -\frac{dr}{dw} = \frac{\sum (I - A)P}{\sum AP} = \frac{r}{1 - w} = \text{constant} \quad (10) $$

And since this relation must be verified for every value of the couple $(w, r)$, it is necessarily verified for $w = 0$. In this case we get $\frac{\sum (I - A)P}{\sum AP} = R$, i.e. the maximum profit rate of the system under review. And therefore we have also: $-\frac{dr}{dw} = \frac{r}{1 - w} = R \quad \forall (w, r)$

It results from this demonstration that a necessary condition for the existence of an invariable measurement unit is the existence of a linear relationship between $w$ and $r$, such as
$r = R(1-w)$. It is worth remarking that this whole demonstration has been conducted without even asking the question of the exact nature of this measurement unit!

Indeed it was sufficient to make the assumption that such a unit existed. However, since the demonstration was done under the definitional constraint that $dP = 0$, it follows that the wage $w$ is necessarily expressed in this invariable measurement unit, without ours knowing what it is made of. We thus find back, but in a much quicker way, Sraffa's ascertainment according to which: "It is curious that we should thus be enabled to use a standard without knowing what it consists of" (Sraffa, 1960, p.37).

We must note in passing that the necessity to have a linear relationship between $w$ and $r$ for a standard commodity to exist implies conversely that it is impossible to define a standard commodity when wages are advanced, even though no reference is made to quantities of wage-goods, and wages are expressed in terms of net product. In this case indeed the system of production prices should be written as: $(I - AP) = rAP + (1+r)wL$

And it is obvious that the relationship between $w$ and $r$ would be no more linear.

### 3.5. The nature of the standard commodity and its consequences

The uniqueness of the standard commodity has so far not been demonstrated, but this can be proven quite simply. Indeed we just need to observe that in order to build this commodity it is necessary to find a set of multipliers $Q = (q_1, q_2, ..., q_n)$ such as:

$$ \frac{Q(I-A)P}{QAP} = R \quad \forall P \qquad [\text{see equation (5) above}] $$

$$ \text{Or } QAP = \frac{1}{1+R}QP $$

Vector $Q$ is the left eigenvector of matrix $A$, corresponding to the eigenvalue $\frac{1}{1+R}$, which is the dominant eigenvalue of matrix $A$. This matrix being irreducible, the elements of vector $Q$ are positive, and this vector is unique within a homothety. For it to be perfectly defined it is sufficient to add to the system the relation $\sum QL = 1$, which has the advantage of identifying the total quantity of labor of the initial system of production prices to the total quantity of labor of the Standard system. The standard commodity, whose essential function is to express wages in the standard system, is thus the net product of the standard system, such as $Q(I-A)P = 1$.

We can remark that the invariable nature of the price of the standard commodity comes from the invariability of the ratio between its price and that of its means of production, always equal to $R$, the standard ratio of the system, whatever the situation of distribution. This property results from the fact that this ratio can be assimilated to a simple ratio of quantities, with the same commodities that appear in the same proportions in the net product and in the
means of production. The standard commodity is thus an homothetic commodity, which is produced with only itself. This constitutes an analogy to the Ricardian system where wheat is equally a homothetic commodity. It is both the only basic commodity and an invariable measurement unit. This was emphasized by Benetti and Cartelier in an article on Production prices and Standard of prices, published in 1975.

This analogy is nevertheless not total, because wheat in this last system remains an invariable standard even when its standard ratio is modified (on this point we refer to the previous chapter), which is not the case here, because any modification of R leads to a modification of Q and of the standard commodity (as well as of the linear relation between R and w).

We have seen so far that the existence of a linear relation between r and w, with w being expressed in terms of the standard commodity, is a necessary condition for the existence of this standard. The converse of this proposition can now be demonstrated, which will make this necessary condition appear as a sufficient condition. For this purpose all that we need is to show that the linear relation between r and w is not confined to the standard system, but remains verified in the initial system of production prices, provided that wages are paid in terms of the standard commodity. This point was highlighted as soon as 1962 by Newman, in an article on "Production of commodities by means of commodities", published in the Swiss Journal of Economics and Statistics (Newman, 1962).

Let us name r* the profit rate in the initial system. This profit rate r* is the ratio between the amount of the initial net product remaining available after payment to the workers of a fraction w of the net standard product, on the one hand, and the price of the means of production of the initial system; on the other hand.

Thus we have:

$$r^* = \frac{\sum_j p_j \left( 1 - \sum_i a_{ij} \right) - w \sum_j p_j \left( q_j - \sum_i q_i a_{ij} \right)}{\sum_j p_j \sum_i a_{ij}} \quad (1)$$

Since r* is the profit rate in each branch, we have also:

$$r^* = \frac{p_i - \sum_j a_{ij} p_j - l_i w \left( \sum_j p_j \left( q_j - \sum_i q_i a_{ij} \right) \right)}{\sum_j a_{ij} p_j} \quad (2)$$

From equation (2) and the fact that $q_i > 0$ for every i, we obtain:
$$r^* = \frac{q_i p_i - q_i \sum_j a_{ij} p_j - q_i l_i w \left[ \sum_j p_j \left( q_j - \sum_i q_i a_{ij} \right) \right]}{q_i \sum_j a_{ij} p_j} \quad (3)$$

And by summing up for each good j:

$$r^* = \frac{\sum_i q_i p_i - \sum_j p_j \sum_i q_i a_{ij} - w \sum_i q_i l_i \left[ \sum_j p_j \left( q_j \sum_i a_{ij} \right) \right]}{\sum_j p_j \sum_i q_i a_{ij}} \quad (4)$$

Since by construction of the $q_i$ multipliers, we have $\sum_i q_i a_{ij} = \frac{1}{1+R} q_j$, and that, again by construction, vector $q$ being defined within a homothety, we have $\sum_i q_i l_i = 1$; since finally $R$ has the same value in the initial system and in the standard system, equation (4) can be put under the form:

$$r^* = \frac{\sum_i q_i p_i - \frac{1}{1+R} \sum_j p_j q_j - w \left( \sum_i p_i q_i - \frac{1}{1+R} \sum_j p_j q_j \right)}{\frac{1}{1+R} \sum_j p_j q_j} \quad (5)$$

Since each $p_i$ and each $q_i$ are positive, equation (5) reduces itself to:

$$r^* = \frac{\left( \sum_i q_i p_i - \frac{1}{1+R} \sum_j p_j q_j \right) (1-w)}{\frac{1}{1+r} \sum_j p_j q_j} \quad (6)$$

And since $\sum_i q_i p_i = \sum_j p_j q_j$, we finally get:

$$r^* = \frac{\left(1 - \frac{1}{1+R}\right)(1-w)}{\frac{1}{1+R}} = \frac{\frac{R}{1+R}(1-w)}{\frac{1}{1+R}} \quad (7)$$

$$\Rightarrow r^* = R(1-w) \text{ in the initial system} \quad (8)$$

We showed thus that the rate of profit $r^*$ of the initial system is identical to the rate of profit of the standard system, when $w$ is expressed in terms of the standard commodity. On the other
hand, the actual share of wages in the initial system $w^*$ is not identical to the share of wages $w$ in the standard system.

$$ \text{Indeed } w^* = \frac{(I - A)P - rAP}{I - AP}, \qquad (9) $$

$$ \text{Whereas } w = \frac{Q(I - A)P - rQAP}{Q(I - A)P} \qquad (10) $$

It is obvious that the two ratios are identical only when $w = 0, r = R$.

From this property we can draw an important conclusion: it is that wages can no more be considered as an independent variable, so that only the profit rate is able to play this role.

Indeed, like equation (9) shows well, the share of wages $w^*$ in the net product of the initial system depends on the price system and the profit rate: $w^*$ can therefore be calculated only when the profit rate and the prices are known. As a consequence, it is this share $w^*$ which constitutes a residue. If we took thus the share of wages in the standard system $w$ as the independent variable, we would arrive to the contradictory phenomenon that the same profit rate $r$ would be a dependent variable and the share of profit a residue in the standard system, but an independent variable in the initial system, because – being as such different from $w^*$, it does not depend on prices.

The unique means to get rid of this contradiction consists in considering that only the profit rate can be an independent distribution variable in the production price system, as soon as we define an invariable measurement unit. We must note indeed that this proposition is verified only when wages are expressed as a proportion of the net product, which implies first the construction of the standard commodity.

Sraffa himself was well aware of that, because he notes: “the choice of the wage as the independent variable in the preliminary stages was due to its being regarded as consisting of specified necessaries determined by physiological or social conditions which are independent of prices or the rate of profits. But as soon as the possibility of variation in the division of the product is admitted, this consideration loses much of its force. And when the wage is to be regarded as “given” in terms of a more or less abstract standard, and does not acquire a definite meaning until the prices of commodities are determined, the position is reversed. The rate of profits, as a ratio, has a significance which is independent of any prices, and can well be given, before the prices are fixed. It is accordingly susceptible of being determined from outside the system of production, in particular by the level of the money rates of interest” (Sraffa, 1960, § 44, p.39).
# 4. Nature of the assumptions and significance of production prices

## 4.1. The invariable measurement unit and the conception of distribution

As we demonstrated in the previous section Sraffa succeeded in constructing a measurement unit invariable as regards changes in distribution, something that Ricardo precisely considered impossible. But the theoretical change to achieve such a result has been important, first because it involves a radical change in the problematic of production prices, linked to the abandonment of the Ricardian assumption concerning the nature of wages. This cuts the links, which to Ricardo was essential, between production and distribution. This rupture between production and distribution is certainly an essential ingredient of the post-Ricardian economics.

In Ricardo's paradigm, this is indeed the conception of wages as a quantity of wage-goods that provides the link between production and distribution. Wage-goods are produced and enter directly in the production of all commodities: in Sraffa's system, they would be basic goods. Therefore any variation in wages implies a variation of the whole production system. The reproduction of the capitalist mode of production, which is the main problem for Ricardo, does not take place in a homothetic way, but through a deformation in the structure of production and distribution. Moreover, once the quantity of wage-goods by unit of labor is given (through Malthus' law of population), the rate of profits and the level of wages in terms of prices are endogenous variables of the Ricardian system.

These features disappear in Sraffa's theory. The process to construct the standard commodity indeed forbids to consider the wage, not only as a quantity of wage-goods<fnref n="3" />, but also as advanced, and this even if the wage is nothing else than a fraction of the net product. Because the construction of the standard implies an invariant structure of the production system, it is clear that the reproduction of this system completely defies intelligibility, and on this question Sraffa, despite the fact that he has been able to construct an invariable standard, albeit from another theoretical ground, joins Ricardo.

More important still is the fact that distribution itself cannot be explained within the theoretical boundaries of Sraffa's system. Indeed, in the Ricardian system of production prices, distribution is determined within the system once the quantity of wage-goods (which is not a distribution variable) is known, and it is the evolution of distribution which is left undetermined, because of the lack of a numéraire which would be invariable to the changes in distribution. In other words, if the concept of real wage is meaningful in the Ricardian theory, the notion of a change in real wage is meaningless.

In Sraffa's theory, and precisely because of the conception of the wage which is implemented for purely logical reasons - that have been analyzed, distribution has to be completely imported from outside in the production price system, through the need to give oneself the

<footnote n="3">³ The measurement of the wage in terms of the price of a given good is however formally equivalent to the conception of the wage as a quantity of wage-goods with only one wage-good, i.e. the good whose price is chosen as the measurement unit.</footnote>
profit rate. Distribution thus goes out of the ambit of this analysis, it is undetermined and
therefore unexplained. In this connection, Sraffa's remark, previously cited, and stating that
the rate of profits "is accordingly susceptible of being determined from outside the system of
production, in particular by the level of the money rates of interest" cannot be taken seriously.
As Kregel clearly showed in a paper presented to the Sraffa symposium of 1973 in Amiens
Kregel, 1973), there is no conception of money that would be compatible with Sraffa's system
of production prices.

For these various reasons, Sraffa's theory cannot be fully interpreted as constituting a return
to the Ricardian theory, from which it clearly departs because some of its basic assumptions,
and therefore its problematic, are different. If we want nevertheless to reconcile Sraffa's
model and Ricardo's analysis it is more with Ricardo's first model, that of the "Essay on
Profits", that an analogy exists, because Sraffa's standard commodity plays the same role, as
wheat in the "Essay on Profits", and for the same reasons: it is an homothetic commodity. It
seems therefore important not to confuse both theories. However we will not go as far as
saying that Sraffa's system is a regression from this last system, to the extent that wheat
offered at least the advantage of being an invariable numéraire with regard to changes in the
production system.

We must nevertheless be conscious of the very peculiar character of the underlying
assumptions that allow for the construction of the standard commodity. These assumptions are
always implicit in "Production of commodities", because, as Joan Robinson rightly pointed
out in the review of Sraffa's book that she wrote in 1961, shortly after its publication, there is
no preliminary discussion of assumptions in this book.

Going back to the assumptions regarding wages, Sraffa first adopts, for just a while, the
assumption that part of wages consist in subsistence goods, the remaining part being taken
from the surplus. We already showed that it is equivalent, on a formal level, to the conception
adopted later, of wages totally paid out from the surplus. However, in case of a variation of
the part of wages coming from the surplus, it requires a supplementary assumption according
to which the share of wages made of subsistence goods is not modified, because otherwise it
would become impossible to build up the standard commodity. But this in turn implies that
any variation in w must necessarily translate into a change in the budget coefficients of
subsistence goods such as the quantity of these goods consumed by workers remains
unchanged, in spite of the change in the wage level. In other words the income elasticity of
subsistence goods must be zero, whatever the variation in w. One must admit the eminently
restrictive character of such assumptions and can understand therefore why Sraffa, although
he did not discuss this kind of assumptions, did not pursue this "hybrid" case.

What to think now of the assumption according to which the wage is defined in terms of the
standard commodity, this one being made of basic goods (means of production), owing to the
conception of wages finally adopted by Sraffa ? If we took it literally, it might mean that
workers are paid with the means of production corresponding to their share of the surplus, and
hence they would become capitalists, which is absurd. In order not to face such a nonsense,
one must get back in the analysis the specificity of the goods consumed by workers, although
they are rejected by Sraffa into the limbo of non-basic goods. We arrive therefore to this paradoxical observation, even though it is an implicit assumption of the system: although labor enters directly into the production of every commodity, those commodities that are bought by workers are not basic goods, otherwise they would become capitalists.

We are thus facing the following alternative:

- Either we take into consideration only the sub-system of basic commodities, which are in fact the means of production of this system. Then wages are nothing else than a fraction of the means of production and workers become ipso facto capitalists, which is absurd;

- Or we admit this last assumption, according to which commodities consumed by workers are non-basic ones, but we need to re-introduce in the analysis the sub-system of non-basic commodities. What happens then when the wage is modified?

We could think first that the quantities of non-basic commodities consumed by workers will be modified, but such a modification can be defined only if the quantities of these goods are determined. However, as we saw previously, this determination implies to have an additional equation identifying $w$ and the price of these goods, which makes the system over-determined. As a consequence, the variation in the quantities of consumption goods remains undetermined. In any case this variation (undetermined) must necessarily cause a variation in the use of basic commodities needed for the production of these non-basic goods, and then a variation (undetermined) of the standard system. In fact all this should not be a surprise, since Sraffa warned us that changes in the economic system were outside the scope of his analysis. We must therefore admit that a change in distribution leaves the produced quantities of commodities and the economic system unchanged, a very peculiar and restrictive hypothesis indeed.

If the produced quantities are thus given, how can we imagine a variation in distribution? Some authors have considered that it would be sufficient to add the assumption that workers can save and that capitalists can consume the same goods as workers. But this would imply that any change in distribution would be equivalent to a change in workers' savings and capitalists' consumption, in identical and opposite proportions, and furthermore such an hypothesis does not hold, since it comes down to adding to the production price system additional equations: these equations would identify total savings (from workers and capitalists) to the global price of basic commodities, and total consumption to the global price of non-basic commodities. As we already saw, such a system, and in particular the distribution variables, would be overdetermined. The production price system is indeed a close system<fnref n="4" />, in which it is irrelevant to inject assumptions regarding the realization of the net product, realization being external to the logic of the system. Therefore any variation in distribution, especially concerning the wage, is illusory in such a system. And the invariable

<footnote n="4">⁴ As soon as the profit rate has been given - and it alone can be given, the system has as many equations as unknowns and is completely determined.</footnote>
standard of production prices, invariable to illusory distribution changes, is itself but a dummy
construction.

4.2. The blurred significance of production prices

First, we must now emphasize something which should not have escaped the reader familiar
with Sraffa's theory, which is that this whole chapter has addressed only part I, entitled
"Single product and circulating capital", of Sraffa's book. On this somewhat limited basis,
what can be the significance of production prices?

On the negative side, they are characterized mainly by a series of impossibilities:

- Impossibility to provide an instantaneous determination of distribution, precisely because the link between production and distribution that existed in the Ricardian system is severed, albeit rightly so: indeed the Ricardian conception of wages does not constitute an economic theory of distribution, even though it provides a measurement for a given state of distribution;

- Impossibility to reflect otherwise than in an illusory way the variations in distribution,
the origins of which remain in any case unexplained;

- Impossibility to analyze the accumulation of capital, i.e. the reproduction of the
capitalist mode of production and therefore to provide any analysis of growth, as well
as technical progress;

- Impossibility to reflect, if not the existence of a stable structure of differentiated profit
rates, at least the process of differentiation of profits rates.

The only object for which Sraffa’s theory of production prices is adequate is the measurement
of production prices, but it refers back to a field which is external to the analysis, with the
indetermination of the given variable that is the rate of profits.

On the positive side, the sub-title of Sraffa’s book: “Prelude to a Critique of Economic
Theory” seems nevertheless to be fully justified. Maybe not so much because the book has
allowed on certain points to demolish the dominant neo-classical theory, but because it
demonstrates, by reduction to the absurd, the vanity of attempting to build up an economic
theory based on exchange ratios between commodities taking into account the existence of
profit, without asking the question of its nature. It is thus clearly a prelude.

Moreover this prelude is not without bearing some fruits, because it allows to get rid of the
neo-classical misconception of capital as a factor of production paid for at its marginal
productivity. The system of production prices again clearly demonstrates the vanity of this
attempt, as soon as we are outside of an economic system with only one good, because the
measurement of capital made of heterogeneous goods depends on the profit rate, and is
therefore unable to define this variable. But this analysis, which forbids us to consider capital
as a factor of production, does not allow in itself to define clearly the concept of capital. If we
know indeed that capital is made of means of production fetching a profit, i.e. of basic commodities, we do not understand what is this profit which is indispensable to provide a definition of capital as such.

These problems come obviously from the initial assumptions of Sraffa's analysis, and from the fact that they do not respect some of the basic principles that were highlighted at the end of the previous chapter. To be sure, and this is some progress compared to Ricardo's paradigm, workers are no longer paid in wage-goods. But they are still not paid in money, because there is no money in this theory. Workers are paid by sharing a part of the standard net product, made itself of the standard commodity, and this standard commodity cannot play the role of money because it is not invariable to changes in the system. In other words the standard commodity is not a scalar, but has the dimension of a basket of basic commodities. Therefore it cannot play the role of a bridge between past, present and future. If there is progress compared to Ricardo, it is therefore small progress.

And this all the more so that the rate of profits is no more a residue, but on the contrary has to be given before prices are determined, which implies that exchange, i.e. the realization of the commodities at such prices, has taken place. One cannot see indeed how any producer could make any profit without having sold first his commodities on the market. It follows that production and exchange are not clearly distinguished and therefore "collide" theoretically: production prices are supposed to be known before exchange takes place, but can be calculated only when the rate of profits is known, i.e. after exchange has taken place.

If we are brought to such problems and contradictions, it might be the sign that there are in fact deeper flaws in Sraffa's conceptual foundations. That it is indeed the case, in particular as regards the very conception of production, will be shown in the next chapter.

## References

Kregel, Jan Allen (1973). Sraffa and Keynes: the rate of interest and susceptible rate of profit. Colloque Sraffa. Amiens. Translated in *Cahiers d'Economie Politique (Papers in Political Economy)*, 1976, N° 3, pp.155-163.

Newman, Peter (1962). Production of commodities by means of commodities. *Swiss Journal of Economics and Statistics*, Swiss Society of Economics and Statistics (SSES), Vol. 98 (issue 1, March 1962), pp. 58-75.

Sraffa, Piero (1960). *Production of commodities by means of commodities. Prelude to a critique of economic theory*. Cambridge (UK): Cambridge University Press.
**Chapter 4. Commodities do not produce commodities: the biased concept of production in Sraffa's theory**

In this chapter we will go further in exploring Sraffa's theory of production as a circular process and as production of a surplus, and show why it entails a number of theoretical contradictions, in particular with the introduction of fixed capital. The introduction of land and non-produced resources also creates an intractable logical problem, which ends up by invalidating Sraffa's whole theoretical system.

The previous chapter discussed the links between distribution, prices, and the invariable standard in Sraffa's theory, and highlighted a few problems concerning these questions. But it confined itself to the first part of *Production of commodities by means of commodities*, where there is only circulating capital, without questioning the underlying conception of production. It did so because Sraffa himself does not clarify his conception of production at any time neither within this first part, not in the two other parts of the book.

Our interpretation of this strange attitude (not defining production at first while dealing with *production prices*) is that Sraffa's theoretical target was the neo-classical theory of distribution, seen as a mere extension of the neo-classical theory of prices to this particular kind of goods that are “factors of production”. In fact in 1960, when Sraffa published *Production of commodities by means of commodities*, it was perceived by a number of economists as a kind of theoretical bomb. Indeed it hit a heavy blow to the neo-classical theory of distribution, by showing that in a system with multiple heterogeneous goods it was impossible to define capital as a factor of production of which the marginal productivity was determining the rate of profit. However, doing so implied keeping some assumptions corresponding to the neo-classical paradigm, as regards the very nature of production and of the goods involved in this process, in particular fixed capital. To what extent this was done unknowingly is difficult to say, but it is this preservation that brought about a number of flaws.

In the present chapter we will continue to revisit Sraffa's theory, trying now to go further by digging up to its very foundations, i.e. its most basic assumptions regarding the nature of production. This will lead to analyzing the definitions of the different categories of goods appearing in Sraffa's book, and playing a role in the production process as described by him: basic goods, subsistence goods, luxury goods, consumption goods and fixed capital goods. We will show why this classification is wrong, explaining in particular why subsistence goods should not be considered as intermediate goods, that consumption goods are final goods which are not part of a surplus, and that there is no surplus of intermediate goods.

We will demonstrate then that the method chosen by Sraffa to introduce fixed capital goods through joint production does not allow for the existence of a surplus for this kind of goods, and leads to several contradictions. These contradictions entail some serious consequences: neither the definition of basic goods nor the notions of surplus and of Standard commodity can be upheld at the level of the whole economic system. It comes therefore as no surprise that the problems created by the introduction of land or other non-produced means of
production cannot be resolved. We will end up by showing how all these limitations explain why Sraffa's system has remained over the years such a puzzle, on which nothing has been built: in particular the system is incompatible with a Keynesian theory of money.

**1. Sraffa's approach: production as a circular process**

It is only in Appendix D of *Production of commodities by means of commodities*, i.e. in the very last pages of the book, that Sraffa finally discusses the question of production! He connects his conception of production to the old classical economists Quesnay and Ricardo who regarded the system of production and consumption as a circular process. He even adds that this conception, which he develops in his book, is “in striking contrast to the view presented by modern theory, of a one-way avenue that leads from factors of production to consumption goods” (Sraffa, 1960, Appendix D, p. 111-113).

As we indicated in the previous chapter, Sraffa says that his system is a generalization of that of Ricardo wherein wheat is both a factor of production and consumer good, which allows one to define a surplus, irrespective of the values or prices, and to determine the rate of profit regardless of them. In his system, what he calls the basic goods are generally playing the role of wheat, when, through the construction of the standard commodity, they appear in the same proportions in the means of production and the net product, which thus seems to validate his reasoning by generalizing the Ricardo's case to a system with multiple heterogeneous goods.

It was important to start by recalling this, because, as far as science is concerned, the field of validity of a scientific theory is generally constituted by the field delimited by its own assumptions, the first of them consisting in the definition of its concepts. It is also generally admitted that, for a theory to be considered as scientific, there is a need for consistency, both internally and externally. By this we mean that the theory's assumptions must not contradict themselves, which is the internal consistency, and that the assumptions must have a coherent link with the reality which the theory wants to describe, which is the external consistency.

Everybody will certainly acknowledge that the definition of production is a very important assumption, a fundamental one indeed, and in this regard it should normally have appeared, not in an appendix at the very end of *Production of commodities...*, but on the contrary in an introduction, at the very beginning of the book. However, and quite strangely, this is not what Sraffa does. Before starting to analyze his theses regarding production, let us state again that for Sraffa production means production of a surplus, and that to be able to compare this surplus to the means of production, there is an absolute logical need for the commodities to appear both in the means of production and in the surplus, which is therefore a forced corollary to his circular conception of production.

**2. Production for subsistence**

It is difficult to understand why Sraffa put back to the end of his book something as fundamental as his definition of production, but maybe the reason was that he did not want to put it too close to the first chapter of the book, entitled “Production for subsistence”, because
in this three-page chapter, there is no surplus, which seems completely contradictory to this
definition of production. Indeed what Sraffa wants to show is that there can be a price system,
even when there is no surplus. But can there be any production? Certainly not by Sraffa's own
definition, unless we realize that the title of the chapter means that there is some kind of
"individuals" or "producers" behind the scene, which barely survive (subsist), by the
consumption of some of the commodities that are produced.

Saying that, however, which Sraffa implies when he writes that the system includes “the
necessaries for the workers”, means that on the left-hand side of the equations (like also on
the right-hand side) some of the means of production are in fact consumption goods, or more
precisely subsistence consumption goods, i.e. some kind of goods that are just sufficient to
allow the “producers” to survive.

However this raises another difficulty, which has to do with the definition of consumption,
something which, quite strangely also, does not appear in Sraffa’s book. Let us therefore give
a definition, and then discuss it. This definition is borrowed from Wikipedia (French version),
and indicates that:

> "...consumption characterizes the act of an economic agent (consumer) that uses (final consumption) or transforms (intermediate consumption) goods and services. This use or transformation causes the immediate (non-durable goods) or progressive (durable goods) destruction of the items consumed. From a general point of view, consumption (value-destroying) opposes production (value-creating)."

This definition is both interesting and misleading. Interesting, because it establishes a
distinction between the production process as an intermediate process, involving intermediate
goods and services, and final consumption as a final process. But this definition is also
misleading, because it tends to consider nevertheless both processes as consumption, since
both would cause the destruction (whether productive or final) of the goods involved. The
definition introduces however another distinction between final consumption, which "uses",
and intermediate consumption, which "transformes".

What we would like to argue is that this distinction is not a subtle one, but a fundamental one,
which should lead economists to reserve the use of the word "destruction" for only the goods
which are used for final consumption. A good reason for that is in fact given by the very last
sentence of this definition, pointing out that consumption, which is value-destroying, opposes
production, which is value-creating. It is indeed logical that a process, like final consumption,
which causes the immediate or progressive destruction, and in fact the disappearance of the
consumption goods involved, be value-destroying, whereas conversely it is not at all logical
that a process which is value-creating, like production, would cause the destruction of the
intermediate goods involved. On the contrary, and if we want to stay connected to reality, it is
easy to recognize that these goods used in the production process are not really destroyed in