On Price Normalization and Choice of Techniques in Ricardo΄s Theory - Review of Economics and Economic Methodology

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Papers For More Mathematics and Theory in Economics How Mathematics Obscures Conceptual Errors ​in Arrow-Debreu General Equilibrium Model On Methodological Pluralism in Economics Control Theory Without Controls Controversies Regarding the TTIP Agreement in the Academic Literature A Complex but Interesting Relation: Keynes, Mathematics, and Statistics Social and Solidarity Economy: A Quest for Appropriate Quantitative Methods The Wealth of (Some) Nations: ​Imperialismand the Mechanics of Value Transfer Make economics relevant again Reframing the Labor Question: On Marginal Productivity Theory and the Labor Theory of Property In need for a drastic change: on the "evidence-based" debates ​in cultural economics and cultural policy research Why established macroeconomics is problematic and how this situation can be overcome Socialism and Marxian economics: An overview Choreography of Resolution: Conflict, Movement, and Neuroscience Should We Remember Ricardo? Surplus Approach Two Hundred Years After ‘On the Principles of Political Economy and Taxation’ Ricardo, Marx, Keynes: Distinct Economic Visions? Absolutne stroškovne razlike in trdovratna trgovinska neravnovesja: Harrodijanski proces usklajevanja Letters of Ricardo to Malthus (1810-1823) Body, Language and Meaning in Conflict Situations Economics and the 'Manchester system' From Paucity to Inefficiency: The Case ​of Democratic Economic Governance The lowest in the world and falling? Explaining the movements ​of income inequality ​in Slovenia during the financial crisis Institutions and Economic Development: ​A More Complete Viewto Understanding Economic Growth Financing obstacles to the realization of 21st century socialism On Price Normalization and Choice of Techniques in Ricardo΄s Theory Inequality in Marx and Piketty Theory and Policy Implication ​Inequity, Inequality, Insecurity: Introduction to the REEM's Special Issue Contemporary Factors of Economic Growth and Competitiveness: Theoretical Aspect OK, Capitalism Mainstream Discourse on Inequality: A Close Inspection Social Capital and Innovation: Evidence from the US (1997-2014) Tyranny of Merit: What’s Become of the Common Good? by Michael J. Sande Pluralism in economics versus a unified paradigm? Rethinking Corporations Marcora for Slovenia: What are the fiscal consequences ​of the much-needed legislative innovation? Enactive Economics: Interactionism, ​Social Cognition and Epistemic Horizons The Incommensurability, Incompatibility and Incomparability of Keynes’s and Walrasian Economics Unexpected Applications of Lund’s Early Retirement Extreme David Ricardo’s Theory of Foreign Trade vs Modern Comparative Advantage Theory

For More Mathematics and Theory in Economics

How Mathematics Obscures Conceptual Errors ​in Arrow-Debreu General Equilibrium Model

On Methodological Pluralism in Economics

Control Theory Without Controls

Controversies Regarding the TTIP Agreement in the Academic Literature

A Complex but Interesting Relation: Keynes, Mathematics, and Statistics

Social and Solidarity Economy: A Quest for Appropriate Quantitative Methods

The Wealth of (Some) Nations: ​Imperialismand the Mechanics of Value Transfer

Make economics relevant again

Reframing the Labor Question: On Marginal Productivity Theory and the Labor Theory of Property

In need for a drastic change: on the "evidence-based" debates ​in cultural economics and cultural policy research

Why established macroeconomics is problematic and how this situation can be overcome

Socialism and Marxian economics: An overview

Choreography of Resolution: Conflict, Movement, and Neuroscience

Should We Remember Ricardo? Surplus Approach Two Hundred Years After ‘On the Principles of Political Economy and Taxation’

Ricardo, Marx, Keynes: Distinct Economic Visions?

Absolutne stroškovne razlike in trdovratna trgovinska neravnovesja: Harrodijanski proces usklajevanja

Letters of Ricardo to Malthus (1810-1823)

Body, Language and Meaning in Conflict Situations

Economics and the 'Manchester system'

From Paucity to Inefficiency: The Case ​of Democratic Economic Governance

The lowest in the world and falling? Explaining the movements ​of income inequality ​in Slovenia during the financial crisis

Institutions and Economic Development: ​A More Complete Viewto Understanding Economic Growth

Financing obstacles to the realization of 21st century socialism

On Price Normalization and Choice of Techniques in Ricardo΄s Theory

Inequality in Marx and Piketty Theory and Policy Implication

​Inequity, Inequality, Insecurity: Introduction to the REEM's Special Issue

Contemporary Factors of Economic Growth and Competitiveness: Theoretical Aspect

OK, Capitalism

Mainstream Discourse on Inequality: A Close Inspection

Social Capital and Innovation: Evidence from the US (1997-2014)

Tyranny of Merit: What’s Become of the Common Good? by Michael J. Sande

Pluralism in economics versus a unified paradigm?

Rethinking Corporations

Marcora for Slovenia: What are the fiscal consequences ​of the much-needed legislative innovation?

Enactive Economics: Interactionism, ​Social Cognition and Epistemic Horizons

The Incommensurability, Incompatibility and Incomparability of Keynes’s and Walrasian Economics

Unexpected Applications of Lund’s Early Retirement Extreme

David Ricardo’s Theory of Foreign Trade vs Modern Comparative Advantage Theory

On Price Normalization and Choice of Techniques in Ricardo ΄ s Theory Kosmas Manoudakis Panteion University of Social and Political Sciences kosmas.manoudakis@reemslovenia.com Abstract Ricardo’s corn model was the first model that made the unambiguous rank, comparison and choice of techniques, with respect to their profitability, possible. The problem of choosing techniques will be mathematically formulized and a solution based on the corn model will be provided. For those, who are familiar with the problem of price determination, it is well known that it is impossible to determine prices, for a given nominal wage, without determining profit rate first. From the price determination system, a relation between nominal wage and profit rate can be obtained, known as the w-r relation. To obtain the w-r relation, and fully determine prices, the price of a single commodity, or of a basket of commodities should be exogenously determined. The w-r relation as it will be shown in this paper is sensitive in the above arbitrary price determination. This vicious cycle was known to Ricardo. He concluded that these difficulties were just mathematical and technical natured and therefore ostensible. This vicious cycle broke, as it will be shown later in this paper, by determining profit rate before and independently of prices. Key words : Classical, Sraffian, Ricardian, Political Economy ​ JEL classifications : B12, B24, B51, E11, P16 The w-r relation and its dependence on price normalization We refer to the production system [A,l,X] , which produces the gross product X,X >0 by using the productive technique [A,l] , where A,A ≥0 is the nxn matrix of the technical coefficients and l,l ≥0 the 1xn row vector of labor inputs per unit of commodity produced. We further assume that matrix A,A ≥0 is decomposable and has the following canonical form: The canonical form of matrix A, shows that the production system using technique [A,l] produces both basic and non-basic commodities. Let the system produce m basic commodities and n-m non-basic commodities 1 . As a result, A 11 is a square indecomposable mxm matrix and A 22 is a square indecomposable (n-m)x(n-m) matrix. We further assume, that since rank(A)=n, rank (A 11 )=m and rank (A 22 )=n-m hold, then A 11 and A 22 are nonsingular. Let also matrix A 12 be indecomposable 2 . The previous assumption, that technique [A,l] is productive, implies that: ​ where is the maximum eigenvalue of matrix , corresponding to the maximum eigenvector ​ of of matrix . It also holds for and (the maximum eigenvalues of matrices, A 11 and A 22 ) respectively: ​Thus, it holds: ​We further assume that: The following definitions are implied: a) r is the given production system’s and its subsystems’ uniform profit rate. b) R represents the maximum profit rate of the given production system, which corresponds to a zero-nominal wage, w . c) R ̅ is the value of R, which assures positive or at least semi-positive prices for all commodities that [A,l,X] produces. It is obvious that R ̅ is the smallest positive value of R. Let wages be paid post factum. The following relationship describes commodity prices, p: Replacing w with zero in (1) it follows: ​From (2) it follows that: We furthermore denote: - R I (R II ) the maximum profit rate of subsystem I (II). - R ̅ I (R ̅ II ) the value of R I (R II ) , which ensures positive prices p I (p II ) for all commodities that subsystem I (II) produces as net product. We obtain for the prices of the two subsystems: And From (a) and (b) we can deduce that: What has been said about the given production system [A,l,X] , also holds for any other production system [A,l,V] , which uses production technique [A,l] (Stamatis, 1998). From equation (2) it follows that, to have a solution different than the trivial one ( p=0) it must hold: rank[I-A(1+R)] . When rank(A)=n (see Stamatis, 1991 p.67) it also holds: When (3) holds, then the inverse of [I-A(1+r)] also exists for r,0≤r≤R ̅ and consequently w,0 ≤w≤w max . From (1) we obtain: Equation (4) fully determines the price vector and up to a scalar for an exogenously given , or for an exogenously given w . It is concluded that prices cannot be fully determined for an exogenously given r , or w . To fully determine prices and obtain the w-r relation, it is mandatory to normalize prices first. The arbitrary determination of the price of a commodity or of a basket of commodities is called price normalization (Stamatis, 1991). The price normalization is accomplished by equating the price of a commodity or the price of a basket of commodities to a constant positive quantity of a homogeneous extensive thing. This equation is called normalization equation, the commodity mentioned above (basket of commodities) normalization commodity/ies and the positive quantity of a homogeneous extensive thing is called fictitious money. Let the normalization commodity be y, y ≥ 0. Let also the price of this commodity be equal to b units of fictitious money, or otherwise b units of that homogeneous extensive thing. Then the normalization equation is: ​When we multiply equation (4) with (5) we can obtain the w-r relation, which is: For r,0≤r≤R ̅ and w,0≤w≤w max . It also holds for w=0 : We can derive from production system [A,l,X] , its’ part [A*,l*,X*] , which is called the normalization subsystem, by eliminating from vectors x,y,l (and thus we obtain x*, y*, l* ), those components, related with the goods that only produce the given (and not the normal subsystem) system of production. Accordingly, we can derive from nxn matrix A , the kxk matrix A* , by eliminating rows and columns related to the above commodities. Correspondingly we can also derive p* from p . Let R* be the maximum profit rate of the normalization subsystem and R ̅* the respective profit rate, which guarantees positive process. Based on the above, we can derive two distinguished types of normalization: If the normalization commodity consists only of basic commodities, it follows that: A*=A 11 . Then it holds ​And moreover As R* we define the maximum profit rate of the normalization subsystem and as R¯* the profit rate which guarantees positive process. ​ On the other hand, if the normalization commodity consists of both basic and non-basic commodities, it follows that: A* = A . Then it holds (Stamatis G. 1998 p.12): Therefore, it is possible to have: ​for a normalization commodity consisting only of basic commodities and ​for a normalization commodity consisting of both basic and non-basic commodities. Based on the above, we can substitute relations (1), (2), (3), (4) , (5) , (6) , (7) with (1a), (2a), (3a) , (4a) , (5a) , (6a) , (7a) respectively: We have seen that the maximum profit rate of the normalization subsystem may be different from the maximum profit rate of the given subsystem. This implies that the position of the w-r curve may change with price normalization. ​ When normalization commodity changes, the normalization subsystem, that produces the normalization commodity, as its net product, also changes. With the normalization subsystem changing the obtained w-r relation also changes. The w-r relation does not belong to the technique [ A, I ] but to the normalization subsystem [ A*, I*, X* ] , instead. It is possible in other words to have as many w-r relations as the correspondent normalization subsystems. Consequently, when normalization commodity changes, the normalization subsystem and the correspondent w-r relation 3 also change. The place (the maximum profit rate, the labor productivity in price terms), the slope 4 and the shape (the second derivative of the w-r relation) of the w-r curve also change. Based on that, the phenomena of switching and reswitching may appear or disappear. Moreover, the choice, the order and the ranking of production techniques are generally impossible, since there are no production techniques to choose, order or rank but normalization subsystems instead. The only possible case for an unambiguous choice, order and ranking of techniques s is when the w-r relation is linear. The w-r relationship is linear, when prices were first normalized with Sraffa΄s (1960), Miayo΄s (1977) or Vassilakis’ (1983) standard commodity. Another example of a unambiguous choice is when the surplus product and the material inputs have the same composition. This is possible in Charassofian systems and of course in corn economies a la Ricardo (Manoudakis, 2010). Actually Ricardo was the first, who solved the above problem using the corn model. The corn system The Ricardian corn system consists of n production sectors 5 . The first sector produces corn (c) , using only corn as a material input. The n -1 other sectors (i) produce other commodities by using corn and the remaining other n -1 commodities. The system uses a linear production function and each production process produces a single commodity. The profit rate r,r >0 , is uniform and the real wages are represented by the vector d,d=(d c 0) . The form of real wages vector implies that real wage consists only of corn 6 . Therefore, the given production technique is a closed a la Leontief model. Obviously, it is assumed that production technique is productive 7 and the production system is viable. The Ricardian production system can be described by the triplet [ A R ,l R , X R ] , where b>l R ,l R >0 the labor inputs entering directly in the production process of a unit of commodity, and X R , X R ≥0 , the gross product of the Ricardian system. Let matrix A R ,A R ≥0 be the Ricardian material input matrix, with the following canonical form: ​ where is the quantity of corn entering in the production process of a unit of corn 8 and is the quantity of the other n -1 commodities (i) entering directly in the production process of a unit these commodities 9 (i) . The dl R matrix of real wages per unit of produced commodity is equal to: It is obvious that since real wages consist only of corn, material inputs and the real wages have the same composition. Accordingly we can define matrix which represents the input matrix of means of production and wage commodities per unit of produced commodity 10 . In the Ricardian corn model not only, real wages but profit rate as well can be expressed in corn terms. It follows, thus, that real wages (which are included in the material inputs) and profits have the same composition. In terms of modern economics, the Ricardian corn model is decomposed into two subsystems: the reproductive one, which produces corn, and the non-reproductive one, which produces all other n -1 non-reproductive commodities. Profit rate is therefore determined as the ratio of the net product of the corn system and the sum of material (in corn) and labor inputs. Competition among production process, leads to the dominance of profit rate in the production process of corn over the production processes of all other n -1 commodities and thus to be uniform. The abovementioned common composition of profits and real wages entails a price-independent profit rate determination. The corn economy therefore, has the properties of a one-good economy. ​Let the gross product, of the reproductive subsystem be equal to a unit of corn. Therefore, the net product is equal to As it has already been assumed ​consists also of labor inputs. In that case net product is also the surplus product of this economy. Equation (9) represents the price system of the corn economy From (9) is implied: From equation (9) it is obvious that profit rate is independent from price normalization, while from equation (10) is implied that profit rate is the ratio of two magnitudes, sharing the same composition, namely the surplus product and the material inputs. Given equations (9) and (10), prices of the non-reproductive products can also be determined 11 . We have seen, how ingeniously had Ricardo solved the profit determination problem, independently of prices and how he had made the unambiguous choice of techniques possible. Concluding remarksWe have seen how price normalization changes the w-r relation in decomposable production systems. The reason is quite simple: we cannot fully determine the prices without first normalizing the prices. Price normalization can change the dimensions of the production system, converting the production technique used by the given system, into a normalization subsystem. The existence of multiple subsystems can change the maximum profit rate of the normalization subsystem. Therefore, there is no need for the profit rate of the given production technique, and the profit rate of the normalization subsystem to coincide . Price normalization also changes the slope of the w-r curve (in other words the capital intensity of the normalization subsystem in price terms). The latter also changes the shape of the w-r curve. It is well known that w-r curve serves as the basis, for applying the profit maximization criterion. This criterion has been for decades a popular tool for economists to choose among techniques although, as has been shown above, they do not actually choose between techniques, but among the normalization subsystems. The fact that prices depend on the rate of profit and the rate of profit depends on the normalization of prices makes it impossible to choose techniques in the general case. Ricardo approximately 150 years before the so-called Cambridge Controversy, has found a way to overcome the above difficulties, although even then he was not aware of it. Having identified the vicious circle of prices, profit rate and wage determination, he had ingeniously introduced the framework of "common composition". He knew, in determining prices, an income distribution variable (he choose real wages) should first be determined exogenously. Real wages and moreover labor itself consisted as one commodity (re)produced in the Ricardian production system. Considered that, real wage is determined based technological attributes only, independently of price normalization and profit rate. As a result, the material inputs, the net product (at the same time the surplus product) and the real wages had the same composition. Given all the above assumptions and conclusions, he managed to determine profit rate independently of price normalization. ​ In the corn model price normalization was not mandatory for determining the profit rate. Profit rate could easily be determined for each positive vector of prices. The above determination made it possible to choose unambiguously among techniques, since in corn models all the normalization subsystems coincide with the given production technique. So 150 years before Sraffa, Ricardo introduced a well-defined method of determining the profit rate. 1. It is obvious that 1≤ m≤n 2. In other words, the given production system is decomposed into two subsystems: subsystem I which produces only basic commodities and subsystem II which produces basic and non-basic commodities. 3. The w-r relation can be re-written as: where π ε ≡(p[I-A])/lq is the productivity of labor in the normalization subsystem and K q ≡pAq/lq is the capital intensity, and the slope of the w-r curve, of the normalization subsystem. 4. The slope of the w-r curve is equal to It holds for ​ In the special case that the w-r is linear then it holds otherwise it holds 5. In the Ricardian framework there are three solid social classes: i) landowners own the land, which they rent, ii) capitalists who possess the capital, by which they buy labor power and organize the production and iii) workers who sell their labor force to the capitalists. In our model we have assumed that land can be extended indefinitely, and therefore, we have included it in the material input matrix. We further assume that all land pieces are equally productive. Consequently, all landowners receive the same rent. The above assumptions do not affect the results our analysis, but simplify it instead, since in Ricardo all the income variables are expressed in corn terms. In Pasinetti (1980) the analysis is more complex since the production of corn is referred to decreasing returns to scale and the luxurious product (gold) is subjected to constant returns to scale. 6. It also holds that where Y R , represents the net product of the Ricardian system. 7. Therefore it holds 8. For the production process of corn holds 9. Obviously is a square n - 1 xn - 1 matrix 10. It is obvious that corn is a reproductive commodity and the other commodities are non-reproductive commodities. Thus Ricardo’s corn system can be represented in a canonical form with a decomposable non-singular matrix. 11. It holds for the price of the non-reproductive system:

​ where

is the maximum eigenvalue of matrix , corresponding to the maximum eigenvector

is the quantity of corn entering in the production process of a unit of corn 8 and

As it has already been assumed

6. It also holds that

where Y R , represents the net product of the Ricardian system.

7. Therefore it holds

8. For the production process of corn holds

9. Obviously

is a square n - 1 xn - 1 matrix